Liquid Silicate Equation of State: Using Shock Waves to Understand the Properties of the Deep Earth Citation Thomas, Claire Waller (2013) Liquid Silicate Equation of State: Using Shock Waves to Understand the Properties of the Deep Earth. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8JTY-MQ03. https://resolver.caltech.edu/CaltechTHESIS:04162013-132730413 Abstract The equations of state (EOS) of several geologically important silicate liquids have been constrained via preheated shock wave techniques. Results on molten Fe 2 SiO 4 (fayalite), Mg 2 SiO 4 (forsterite), CaFeSi 2 O 6 (hedenbergite), an equimolar mixture of CaAl 2 Si 2 O 8 -CaFeSi 2 O 6 (anorthite-hedenbergite), and an equimolar mixture of CaAl 2 Si 2 O 8 -CaFeSi 2 O 6 -CaMgSi 2 O 6 (anorthite-hedenbergite-diopside) are presented. This work represents the first ever direct EOS measurements of an iron-bearing liquid or of a forsterite liquid at pressures relevant to the deep Earth (> 135 GPa). Additionally, revised EOS for molten CaMgSi 2 O 6 (diopside), CaAl 2 Si 2 O 8 (anorthite), and MgSiO 3 (enstatite), which were previously determined by shock wave methods, are also presented. The liquid EOS are incorporated into a model, which employs linear mixing of volumes to determine the density of compositionally intermediate liquids in the CaO-MgO-Al 2 O 3 -SiO 2 -FeO major element space. Liquid volumes are calculated for temperature and pressure conditions that are currently present at the core-mantle boundary or that may have occurred during differentiation of a fully molten mantle magma ocean. The most significant implications of our results include: (1) a magma ocean of either chondrite or peridotite composition is less dense than its first crystallizing solid, which is not conducive to the formation of a basal mantle magma ocean, (2) the ambient mantle cannot produce a partial melt and an equilibrium residue sufficiently dense to form an ultralow velocity zone mush, and (3) due to the compositional dependence of Fe 2+ coordination, there is a threshold of Fe concentration (molar X Fe ≤ 0.06) permitted in a liquid for which its density can still be approximated by linear mixing of end-member volumes. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: equation of state; liquid silicate; shock wave Degree Grantor: California Institute of Technology Division: Geological and Planetary Sciences Major Option: Geology Thesis Availability: Public (worldwide access) Research Advisor(s): Asimow, Paul David Thesis Committee: Rossman, George Robert (chair) Jackson, Jennifer M. Stolper, Edward M. Asimow, Paul David Defense Date: 11 April 2013 Funders: Funding Agency Grant Number NSF EAR-0855774 NSF EAR-1119522 Record Number: CaltechTHESIS:04162013-132730413 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04162013-132730413 DOI: 10.7907/8JTY-MQ03 Related URLs: URL URL Type Description http://dx.doi.org/10.1029/2012JB009403 DOI UNSPECIFIED Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 7616 Collection: CaltechTHESIS Deposited By: Claire Thomas Deposited On: 22 Apr 2013 17:14 Last Modified: 04 Oct 2019 00:00 Thesis Files Preview PDF (full thesis) - Final Version See Usage Policy. 5MB Preview PDF (Introduction) - Final Version See Usage Policy. 76kB Preview PDF (Ch. 1) - Final Version See Usage Policy. 34kB Preview PDF (Ch. 2) - Final Version See Usage Policy. 2MB Preview PDF (Ch. 3) - Final Version See Usage Policy. 2MB Preview PDF (Ch. 4) - Final Version See Usage Policy. 1MB Preview PDF (Data sheets) - Supplemental Material See Usage Policy. 221MB Repository Staff Only: item control page

Liquid silicate equation of state: Using shock waves to understand the properties of the deep Earth Thesis by Claire Waller Thomas In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2013 (Defended April 11, 2013) ii  2013 Claire Waller Thomas All Rights Reserved iii ACKNOWLEDGEMENTS There are many to whom I owe thanks in the creation of this thesis. It takes a lot of people to blow up so many things—first thanks to Paul Asimow for being a pretty fantastic advisor, teacher, and mentor. He is likely the smartest person I know, but also manages to be the most patient and good-humored as well. That is a rare combination for which I am abundantly grateful. Thanks to the rest my committee: George Rossman, for academic and general life guidance; Jennifer Jackson, for an incredibly friendly and all-around pleasant demeanor, which makes me regret not throwing in a DAC chapter just for fun; and Ed Stolper, who despite having an incredibly full schedule makes time to serve on my committee and also provide both insightful and entertaining contributions to PRG each week. Thanks to the shock wave staff: Mike Long, Papo Gelle, Russ Oliver, and Oleg Fat’yanov for advice, patience, and insight as I learn from the masters of making things explode. My officemates Kristin Bergmann, Ajay Limaye, and Katie Stack are amazing people, but I can’t really thank them for helping me with my thesis as we would rarely get work done when we were all in the office at the same time. Thanks to the delightful Dodd family and the Caltech swim team for making me have a life outside of lab. Thanks to my roommates/neighbors of the past 5 years, Fay Bi and Clara O’Farrell, for being such great friends despite being only thrown together by lovely chance of the Caltech housing lottery. Thanks to my parents as I am sure raising me was unpleasant as was having to deal the with the confusion of what it is I have been doing the past 5 years,“ I thought you were a geologist.” And finally thanks to my husband Daniel. He is the best person I know. iv ABSTRACT The equations of state (EOS) of several geologically important silicate liquids have been constrained via preheated shock wave techniques. Results on molten Fe2SiO4 (fayalite), Mg2SiO4 (forsterite), CaFeSi2O6 (hedenbergite), an equimolar mixture of CaAl2Si2O8-CaFeSi2O6 (anorthite-hedenbergite), and an equimolar mixture of CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 (anorthite-hedenbergite-diopside) are presented. This work represents the first ever direct EOS measurements of an iron-bearing liquid or of a forsterite liquid at pressures relevant to the deep Earth (> 135 GPa). Additionally, revised EOS for molten CaMgSi2O6 (diopside), CaAl2Si2O8 (anorthite), and MgSiO3 (enstatite), which were previously determined by shock wave methods, are also presented. The liquid EOS are incorporated into a model, which employs linear mixing of volumes to determine the density of compositionally intermediate liquids in the CaOMgO-Al2O3-SiO2-FeO major element space. Liquid volumes are calculated for temperature and pressure conditions that are currently present at the core-mantle boundary or that may have occurred during differentiation of a fully molten mantle magma ocean. The most significant implications of our results include: (1) a magma ocean of either chondrite or peridotite composition is less dense than its first crystallizing solid, which is not conducive to the formation of a basal mantle magma ocean, (2) the ambient mantle cannot produce a partial melt and an equilibrium residue sufficiently dense to form an ultralow velocity zone mush, and (3) due to the compositional dependence of Fe2+ coordination, there is a threshold of Fe concentration (molar X Fe ≤ 0.06) permitted v in a liquid for which its density can still be approximated by linear mixing of endmember volumes. vi TABLE OF CONTENTS Acknowledgements ............................................................................................ iii Abstract ............................................................................................................... iv Table of Contents ................................................................................................ vi List of Tables and Figures ................................................................................ viii Chapter I Introduction................................................................................................. I-1 References................................................................................................... I-4 Chapter II Multi-technique equation of state for Fe2SiO4 melt and the density of Fe-bearing silicate melts from 0-161 GPa ............................... II-1 Abstract ...................................................................................................... II-2 Introduction................................................................................................ II-3 Methods ..................................................................................................... II-6 1-atm double-bob Archimedean methods .......................................... II-6 Sink/float methods............................................................................... II-7 Shock wave methods ........................................................................... II-8 Sample synthesis and preparation of preheated (1573 K) shots .. II-8 Preparation of room temperature (300 K) shots ......................... II-11 Experimental setup ....................................................................... II-11 Results ...................................................................................................... II-13 Density and ultrasonic results at ambient pressure .......................... II-14 Sink/float results ................................................................................ II-14 Shock wave results ............................................................................ II-14 Comparison to previous data ....................................................... II-14 Solid Hugoniot shocked to liquid / χ2 test of weighted values .... II-17 Grüneisen parameter .................................................................... II-17 Thermal equation of state fitting .................................................. II-20 Correction and re-analysis of silicate liquids.............................. II-26 Implications ............................................................................................. II-26 Mixing of oxide volumes at high pressure and temperature; application to a magma ocean liquidus ............................................ II-26 Buoyancy in the current temperature regime; application to the solidus and ULVZ .................................................................. II-32 Conclusions.............................................................................................. II-35 Acknowledgements ................................................................................. II-36 References................................................................................................ II-37 Chapter III Direct shock compression experiments on pre-molten Mg2SiO4 and progress towards a consistent high-pressure equation of state for CaO-MgO-Al2O3-SiO2-FeO liquids .................................................. III-1 vii Abstract ..................................................................................................... III-2 Introduction............................................................................................... III-3 Methods .................................................................................................... III-5 Sample preparation of preheated (2273 K) shots .............................. III-5 Experimental setup ............................................................................. III-6 Results ....................................................................................................... III-8 Linear Hugoniot ................................................................................. III-8 Grüneisen parameter ........................................................................ III-10 Thermal equation of state fitting ...................................................... III-12 Revised Equations of State for MgSiO3 and CaAl2Si2O8 liquids .. III-14 Enstatite, MgSiO3 ........................................................................ III-14 Anorthite, CaAl2Si2O8 .................................................................. III-15 Discussion ............................................................................................... III-18 Sound speed of Mg2SiO4 liquid ...................................................... III-18 Isentropes for a full mantle magma ocean ...................................... III-20 Evaluating the partial melt hypothesis for Ultralow velocity zones ................................................................... III-23 Conclusions............................................................................................. III-30 Acknowledgements ................................................................................ III-32 References............................................................................................... III-33 Chapter IV Preheated shock liquids Hd, An-Hd-Di, An-Di: a test for linear mixing at high pressure and temperature ................................................ IV-1 Abstract .................................................................................................... IV-2 Introduction.............................................................................................. IV-3 Methods ................................................................................................... IV-4 Sample preparation............................................................................ IV-4 Experimental setup ............................................................................ IV-6 Results ...................................................................................................... IV-7 Hedenbergite (CaFeSi2O6) liquid...................................................... IV-8 An-Hd liquid................................................................................... IV-10 An-Hd-Di liquid .............................................................................. IV-10 Discussion .............................................................................................. IV-11 Linear mixing assumptions ............................................................. IV-11 Glass and 1-bar melt data ................................................................ IV-12 Tests of linear mixing ..................................................................... IV-16 Conclusions............................................................................................ IV-19 Acknowledgements ............................................................................... IV-21 References.............................................................................................. IV-22 viii LIST OF TABLES AND FIGURES Chapter II Page Table 1: Double-bob density measurements........................................... II-41 Table 2: Shock compression data ........................................................... II-42 Table 3: Parameters used......................................................................... II-43 Table 4: Equation of state fits for molten Fe2SiO4 ................................. II-43 Table 5: BM/MG fits for silicate liquid .................................................. II-44 Table 6: Major oxide proportions for model mantle .............................. II-44 Figure Captions........................................................................................ II-45 Figure 1: Comparison of previous sink/float and shock wave results ... II-49 Figure 2: Schematic of shock wave arrival selection ............................. II-50 Figure 3: Double-bob density linear fit ................................................... II-51 Figure 4: Density-pressure diagram of sink/float results ....................... II-52 Figure 5: US-up preheated fayalite Hugoniot .......................................... II-53 Figure 6: Grüneisen parameter of liquid and solid silicates .................. II-54 Figure 7: P-ρ preheated fayalite Hugoniot.............................................. II-55 Figure 8: Model mantle isentropic temperature profiles ........................ II-56 Figure 9: Silicate liquid densities along a modern geotherm ................. II-57 Figure 10: Density of liquid mixtures for ULVZ ................................... II-58 Chapter III Page Table 1: Shock compression data........................................................... III-38 Table 2: Parameters used........................................................................ III-38 Table 3: Equation of state fits for molten Mg2SiO4............................... III-39 Table 4: Equation of state fits for molten silicate liquids ...................... III-39 Figure Captions....................................................................................... III-40 Figure 1:Modeled sound speed parameter space ................................... III-43 Figure 2: US-up preheated forsterite Hugoniot....................................... III-44 Figure 3: Grüneisen parameter of liquid silicates .................................. III-45 Figure 4: P-ρ preheated fayalite Hugoniot............................................. III-46 Figure 5: MgSiO3 Hugoniot dataset ....................................................... III-47 Figure 6: Estimates of Mg2SiO4 liquidus .............................................. III-48 Figure 7: Silicate liquid and solid densities along a geotherm ............. III-49 Figure 8: Re-modeled mantle isentropic temperature profiles .............. III-50 Figure 9: Partial melting of ULVZ composition .................................. III-51 Chapter IV Page Table 1: Shock compression data.......................................................... IV-25 Table 2: Parameters used....................................................................... IV-26 ix Table 3: Equation of state fits for molten CaFeSi2O6........................... IV-26 Table 4: Equation of state fits for molten An-Hd ................................. IV-27 Table 5: Equation of state fits for molten An-Hd-Di ........................... IV-27 Figure Captions...................................................................................... IV-28 Figure 1: US-up preheated CaFeSi2O6 Hugoniot................................... IV-31 Figure 2: P-ρ preheated CaFeSi2O6 Hugoniot ...................................... IV-32 Figure 3: US-up preheated An-Hd Hugoniot ......................................... IV-33 Figure 4: P-ρ preheated An-Hd Hugoniot ............................................ IV-34 Figure 5: US-up preheated An-Hd-Di Hugoniot ................................... IV-35 Figure 6: P-ρ preheated An-Hd-Di Hugoniot ....................................... IV-36 Figure 7: Hd mixing model .................................................................. IV-37 Figure 8: Comparison of An-Hd mixing models .................................. IV-38 Figure 9: Comparison of An-Hd-Di mixing models ............................ IV-39 Figure 10: Comparison of mantle mixing models ................................ IV-40 I-1 Chapter I Introduction Claire W. Thomas I-2 An equation of state (EOS) (i.e., density as a function of pressure and temperature) for silicate melts at high pressure is of primary importance for understanding processes occurring within terrestrial interiors. There has previously existed only limited knowledge of melt density and compressibility determined at low pressures (< 40 GPa) on a restricted range of compositions. The goal of my graduate work documented by this thesis was to constrain the EOS for a number of important end-member liquids using shock wave studies and to incorporate them into an empirical, predictive model of melt density as a function of temperature, pressure and composition. Chapter II describes the multi-technique collaboration of the University of Michigan, the University of New Mexico, and Caltech to constrain the EOS of molten Fe2SiO4 (fayalite). Shock wave experiments further extended the fayalite Hugoniot of Chen et al. [2002], which aided in resolving previously discrepant results for that study and the sink/float data set of Agee [1992]. The majority of the shock wave methods used for pre-heated experiments are described in this chapter. Subsequent chapters cover only changes made during each new set of experiments. The equations and derivation of the model used for creating isentropes of liquid mixtures is presented in detail in this chapter and is employed throughout the rest of the thesis. The determination of the molten Mg2SiO4 (forsterite) EOS is presented in Chapter III, along with revised EOS parameters for CaMgSi2O6 (diopside), CaAl2Si2O8 (anorthite) [Asimow and Ahrens, 2010], and MgSiO3 (enstatite) melts [Mosenfelder et al., 2009] . This new set of end member EOS, including fayalite, is used to determine the isentropic temperature profile of a fully molten magma ocean of two hypothetical bulk mantle compositions, chondrite [Andrault et al., 2011] and peridotite[Fiquet et al., 2010], by way of the isentrope mixture model presented in Chapter II. The temperature and pressure of first crystallization was determined from the I-3 intersection of the isentrope and the experimental liquidus. This chapter also describes a melting model which employs linear mixing of liquid volumes to determine whether a partial melt and its equilibrium residue could produce an aggregate density that is comparable to that estimated by Rost et al.[2006] for a ultralow velocity zone and still maintain a low melt-residue density contrast (~1%) employed by the seismic modeling of Williams and Garnero [1996]. The EOS for CaFeSi2O6 (hedenbergite), a 50-50 mixture of CaAl2Si2O8-CaFeSi2O6 (anorthite-hedenbergite), and an equimolar mixture of CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 (anorthite-hedenbergite-diopside) are determined and described in Chapter IV. A series of tests on the validity of using linear mixing of volumes to predict the densities of multicomponent liquids at high temperature and pressure are presented. Results for these tests indicate that Febearing silicate liquid densities can only be approximated as ideal under certain compositional restrictions. I-4 REFERENCES Agee, C. B. (1992), Isothermal compression of molten Fe2SiO4, Geophys. Res. Lett., 19(11), 1169–1172. Andrault, D., N. Bolfan-Casanova, G. L. Nigro, M. A. Bouhifd, G. Garbarino, and M. Mezouar (2011), Solidus and liquidus profiles of chondritic mantle: Implication for melting of the Earth across its history, Earth and planetary science letters, 304(1 – 2 ), 251 – 259. Asimow, P. D., and T. J. Ahrens (2010), Shock compression of liquid silicates to 125 GPa: The anorthite-diopside join, J. Geophys. Res., 115(B10), B10209. Chen, G. Q., T. J. Ahrens, and E. M. Stolper (2002), Shock-wave equation of state of molten and solid fayalite, Physics of The Earth and Planetary Interiors, 134(1 – 2), 35 – 52. Fiquet, G., A. L. Auzende, J. Siebert, A. Corgne, H. Bureau, H. Ozawa, and G. Garbarino (2010), Melting of Peridotite to 140 Gigapascals, Science, 329(5998), 1516-1518. Mosenfelder, J. L., P. D. Asimow, D. J. Frost, D. C. Rubie, and T. J. Ahrens (2009), The MgSiO3 system at high pressure: Thermodynamic properties of perovskite, postperovskite, and melt from global inversion of shock and static compression data, J. Geophys. Res., 114. Rost, S., E. J. Garnero, and Q. Williams (2006), Fine-scale ultralow-velocity zone structure from high-frequency seismic array data, J. Geophys. Res., 111(B9), B09310. Williams, Q., and E. J. Garnero (1996), Seismic Evidence for Partial Melt at the Base of Earth's Mantle, Science, 273(5281), 1528 – 1530. II-1 Chapter II Multi-technique equation of state for Fe2SiO4 melt and the density of Fe-bearing silicate melts from 0 – 161 GPa Claire W. Thomas Qiong Liu Carl B. Agee Paul D. Asimow Rebecca A. Lange Accepted to Journal of Geophysical Research August 27, 2012 II-2 ABSTRACT New equation of state measurements on liquid Fe2SiO4 have been conducted in a collaborative, multi-technique study. The liquid density (ρ), the bulk modulus (K), and its pressure derivative (K’) were measured from 1 atm to 161 GPa using 1-atm double-bob Archimedean, multi-anvil sink/float, and shock wave techniques. Shock compression results on initially molten Fe2SiO4 (1573 K) fitted with previous work and the ultrasonically- measured bulk sound speed (Co) in shock velocity (US)-particle velocity (up) space yields the Hugoniot: US =1.58±0.03 up+ 2.438±0.005 km/s. Sink/float results are in agreement with shock wave and ultrasonic data, consistent with an isothermal KT=19.4 GPa and K’=5.33 at 1500°C. Shock melting of initially solid Fe2SiO4 (300 K) confirms that the Grüneisen parameter (γ) of this liquid increases upon compression where γ= γo(ρo/ρ)q yields a q value of –1.45. Constraints on the liquid fayalite EOS permit the calculation of isentropes for silicate liquids of general composition in the multi-component system CaO-MgO-Al2O3-SiO2-FeO at elevated temperatures and pressures. In our model a whole-mantle magma ocean would first crystallize in the mid-lower mantle or at the base of the mantle were it composed of either peridotite or simplified “chondrite” liquid, respectively. In regards to the partial melt hypothesis to explain the occurrence and characteristics of ultra-low velocity zones, neither of these candidate liquids would be dense enough to remain at the core mantle boundary on geologic timescales, but our model defines a compositional range of liquids that would be gravitationally stable. II-3 INTRODUCTION The formulation of an accurate equation of state (EOS) for silicate melts at high pressure and temperature is necessary to understand fully the important role of liquids in the differentiation and formation of terrestrial planets. The early history of the Earth may well have included one or more deep magma ocean events, perhaps extending to the core-mantle boundary (CMB) [Labrosse et al., 2007]. During crystallization of magma oceans and also during partial melting events in predominantly solid mantle environments, the formation of chemically distinct reservoirs is controlled by gravitational differentiation and therefore intimately linked to the solid and liquid EOS. In particular, the possibility of crystal-liquid density cross-overs — that is, for a given liquid and a given mineral or assemblage of minerals, a horizon above which the liquid rises but below which it sinks [Agee, 1988; Stolper et al., 1981] — implies dramatic bifurcation points in the possible paths of differentiation, leading to formation of either rising or sinking enriched reservoirs. These rising or sinking liquids will either potentially form crust or else potentially be sequestered at depth. Furthermore, changes in chemical equilibria with increasing pressure are defined by molar volume, the pressure derivative of the Gibbs free energy [Asimow and Ahrens, 2010]. Thus, accurate models of both the thermodynamic and fluid dynamic evolution of the Earth demand a fundamental knowledge of silicate liquid volumes at extreme pressure and temperature conditions. In addition, detection of ultra-low velocity zones (ULVZ) [Garnero and Helmberger, 1995] at the base of the mantle may indicate the presence of silicate partial melts [Lay et al., 2004; Mosenfelder et al., 2009; Williams et al., 1998] or more specifically iron-bearing melts [Labrosse et al., 2007]. Seismic observations such as S- and P-wave speeds and inferred density have given clues as to the nature of the materials in ULVZs, but there are a number of candidate II-4 iron-bearing solid phases that can produce the correct drops in wave speed and increased density without partial melting. Proposed phases include ferropericlase (Fe, Mg)O [Wicks et al., 2010] and iron‐enriched post‐perovskite [Mao et al., 2006]. Thus, improved constraints on iron-bearing silicate liquid densities at elevated pressure and temperature will aid in answering whether a liquid or liquid-solid mush could be dynamically stable at the CMB and similarly what the relative buoyancy of melt and its co-existing solids would be. There are presently no published experimental constraints on the density of iron-bearing silicate liquids beyond 47 GPa [Chen et al., 2002; Miller et al., 1991a]. Silicate liquids occupy a large, multi-dimensional continuum of compositions and therefore understanding their range of behavior requires either data on a very large number of compositions or data on a minimum set of end-member compositions and a reliable means of interpolating in composition space. At low pressure, ideal mixing of volumes has proven to be a reliable tool [Lange and Carmichael, 1987], but it remains poorly tested at high pressure [Asimow and Ahrens, 2010]. Here we present results on a pure iron-bearing end-member, fayalite (Fe2SiO4). In the future, this will allow assessment of intermediate compositions with lower iron contents without the need for extrapolation and provide an essential tie-point for the assessment of linear mixing along iron-magnesium mixing joins. Previous shock experiments on fayalite (Fe2SiO4) liquid achieved pressures up to 47 GPa [Chen et al., 2002], which corresponds to about mid-mantle range [Dziewonski and Anderson, 1981]. For this work, we extended this pressure range to 161 GPa, beyond CMB pressures, so we may apply our results to understanding the effects of iron on deep mantle systems. Additionally, this collaborative work sought to resolve inconsistencies in previous reports on Fe2SiO4 liquid compressibility and its change with pressure. Published isothermal static compression data from II-5 sink/float experiments [Agee, 1992] and shock wave results [Chen et al., 2002] are in close agreement for bulk modulus (Ko) values but differ greatly on the value of the pressure derivative (K’), 10.1 (isothermal) and 5.36 (isentropic), respectively, which give very divergent predictions for density at higher pressure. Isothermal and isentropic equations of state need not be the same, but a larger K’ for the isotherm implies a negative coefficient of thermal expansion, which is contrary to expectations. The anomaly is even more apparent when the Hugoniot data at estimated shock temperatures are compared to the extrapolated isotherm (Figure 1). For instance, at 15 GPa, the estimated shock state is at temperature (TS) 1783K and density ~4.75 g cm–3, somewhat denser than the extrapolated isotherm at the same P and T, ~4.5 g cm–3. At 50 GPa however, the estimated Hugoniot state is at temperature 2943 K and density ~6.0 g cm–3, much denser than the extrapolated 1773 K isotherm at 50 GPa, 4.75 g cm–3, whereas one expects density to decrease with increasing temperature. This discrepancy was also noted by Ghiorso (2004), whose equation of state fitting exercise produced an internally consistent prediction of all fusion curve data and Hugoniot volume data, but underestimated the density of neutral buoyancy experiments from Agee [1992] by 1%. We have conducted new measurements of the equation of state of molten fayalite (Fe2SiO4) using the following techniques: (i) double-bob Archimedean method for melt density and thermal expansion at ambient pressure (University of Michigan), (ii) multi-anvil sink/float technique to measure melt density to 4.5 GPa (University of New Mexico), and (iii) shock wave measurements of P-V-E equation of state to 161 GPa (Caltech). The aim of this multi-technique study has been to measure the density of molten Fe2SiO4, to resolve the discrepancy in published EOS for this material, and to use the measured equation of state to address processes of early earth differentiation and stability of a liquid at the CMB. II-6 METHODS 1- atm double-bob Archimedean methods The shared starting material for the double-bob density measurements and the shockwave experiments was synthesized at the University of California, Berkeley, by Mark Rivers and contains 64.48 mol% Fe as FeO, 34.97 mol% SiO2, and 0.56 mol% Al2O3 (wet chemical analysis is reported in Rivers and Carmichael [1987]), slightly off fayalitic composition (66.66 mol% FeO, 33.33 mol% SiO2). This is the same material used for ambient ultrasonic sound speeds reported in Ai and Lange [2004]. The starting material was a brick-red powder with iron predominantly in the form of ferric iron (Fe3+). To produce a starting composition with iron as Fe2+, approximately 75 g was loaded step-wise in four batches (~18-20 g each) into a straight-wall molybdenum crucible (7.8 cm deep and 3.6 cm inner diameter) placed in a Deltech furnace at 1300°C in a reducing stream of 1%CO-99%Ar. Each batch of melt was held at 1300°C for 12 hours until all 75 grams were loaded into the crucible and reduced. The Mo crucible was then placed on a platform in the Deltech furnace such that the top 3 cm of the liquid corresponded with the isothermal hotspot in the furnace. A thermal profile from the top of the crucible down into the sample had been previously documented for a Mo crucible filled with alumina powder for each temperature of measurement. Temperature was measured at the top of the crucible with an S-type thermocouple, calibrated against the melting temperature of gold. Liquid density measurements were made on the Fe2SiO4 liquid using the double-bob Archimedean method, described in detail in Lange and Carmichael [1987]. The method was adapted for a reducing atmosphere (1%CO-99%Ar) in this study by using molybdenum instead of platinum for the two bobs and the crucible. An electronic balance with a precision of ±0.0001 II-7 g was mounted on an aluminum platform above the furnace to measure the weights of the Mo bobs before and after immersion into the Fe2SiO4 liquid. By using two Mo bobs of different mass (~20 and ~7 g), but identical stem diameters (1 mm), the effect of surface tension on the stem was eliminated. The density of the liquid is calculated from the following equation: ρ (T ) = B L (T ) − B S (T ) V L (T ) − V S (T ) (1) where BL(T) and BS(T) are the buoyancy of the large and small bob, respectively, and VL(T) and VS(T) are the immersed volume of the large and small bob, respectively. The buoyancy is defined as the mass of the bob in the gas stream (corrected to vacuum) minus the mass of the bob during the immersion in the Fe2SiO4 melt. Four density measurements were made at each temperature with two different large bobs and two different small bobs. The accuracy of using Mo bobs instead of Pt bobs was tested by measuring the density of NaCl liquid at 1286 K and the result (1.445 g/cm3) closely matches that recommended by NIST [Janz, 1980] at this temperature (1.441 g/cm3). Sink/float methods Fayalite used for sink-float experiments was the same sample used in the original Agee [1992] study. This is a different starting material than used in the double-bob density and shockwave measurements of this study. The fayalite was synthesized in a Deltech gas-mixing furnace for 24 hours at controlled fO2 (IW-1). All sink-float experiments were carried out in a Walker-style multi-anvil device at the University of New Mexico, as opposed to a pistoncylinder device at the Bayerisches Geoinstitut used in Agee [1992]. Two types of gem-quality marker spheres were used: a spessartine garnet from the Malagasy Republic with composition Alm10.6Sps89.4 (also used in Agee [1992]) and a San Bernardino almandine-rich garnet II-8 Alm80.0Grs8.9Sps6.4Pyr4.7. Experiments were run in high purity molybdenum capsules, with run times of approximately 30 seconds at 1500°C. For further details of the experimental method, see Circone and Agee [1996]. Shock wave methods Sample synthesis and preparation of preheated (1573 K) shots The shared starting material used for the shock wave and double-bob experiments was reduced in a similar manner as described above in the double-bob methods section to produce a starting composition with the correct oxidation state (Fe2+). The powder was melted in a deep (3.5 cm deep, 1.7 cm inner diameter) molybdenum crucible hung on molybdenum wires in a Deltech furnace at 1300°C in a reducing gas stream of 1%CO-99%Ar. The powder was melted in short 30-45 minute runs; quenched and re-filled several times to reach the crucible’s capacity; and then left to equilibrate for 18 hours at temperature. The quenching process used was unfortunately slow to avoid CO gas release or rapid oxidation on hot exposure to air. An extralong ceramic hanging rod was used to push the crucible out of the hot spot to the bottom of the furnace tube to cool within the reducing atmosphere. Once the crucible was no longer glowing red (approximately 10 minutes), nitrogen was streamed for 10 minutes followed by a finishing quench in water. A combination of volume reduction on crystallization and surface tension at the crucible walls typically left a 5mm diameter triangular pit in the center, which reached nearly half way to the bottom of the crucible. Mössbauer spectra measured by Bjorn Mysen at the Carnegie Institution of Washington confirmed that Fe in the run products was reduced and dominantly in the form of fayalite and wüstite (FeO) with minor amounts of an iron-bearing accessory mineral, most likely magnetite. This aggregate was too fragile to be lapped into disks to fit into shock wave experiment sample II-9 holders, so it was re-ground and hot-pressed in a piston-cylinder device at 10 kbar and 725°C using a graphite sleeve and CaF2 base plug. The maximum fO2 in our graphite sample assembly is controlled by the CCO buffer[Holloway et al., 1992; Médard et al., 2008] with a calculated log fO2 of -16 at our P-T conditions [Jakobsson and Oskarsson, 1994; Ulmer and Luth, 1991].Experimental results from Médard et al. indicate that the fO2 within graphite capsules is likely more reducing than the CCO buffer (average value: CCO-0.07; IW+1.5), therefore the amount of Fe3+ introduced during this process is assumed to be negligible. The sample underwent synchronized slow decompression (1 MPa/minute) and cooling (1 K/minute) to reduce cracking during recovery, but usually the cylinder of fayalite naturally separated into rough 2–3 mm-thick disks. The hot-pressing increased the bulk density and sintered the aggregate into a machinable form; the sample disks were cored to 8 mm outer diameter. The composition was checked for carbon contamination from the graphite hot-pressing sleeve on a scanning electron microscope using energy-dispersive x-ray spectroscopy, but none was found. There were minor amounts of molybdenum contamination in the form of Fe-Mo metal oxides, but total bulk Mo content was constrained to less than 2 wt% with electron microprobe analyses. The disks were then lapped down to under-fill the volume of the molybdenum sample holder (see below) by 7-13%, such that — accounting for thermal expansion of the Mo holder [Miller et al., 1988] and the Fe2SiO4 liquid density at ambient pressure and 1300°C — a void space 1–1.5 mm-deep would remain at the top of the heated capsule. This small void space volume prevents blow out, excessive warping, or failure of the welded cap and yet remains far enough above the imaged area of the shock wave to preserve one-dimensional wave propagation. The molybdenum sample holders are comprised of a driver plate, sample well, and cap. The outer and inner portions of the rear-surface of the driver plate and both sides of the cap were II-10 polished to a mirror-finish with 1-micron alumina grit. For the two completed preheated shots, both caps had a thickness of 1mm, and all sample wells were 8mm in diameter and approximately 2.5mm deep. Shot 409 had a driver plate 1mm thick, and shot 396 had a 2 mmthick driver plate. After polishing, a line profile of the driver plate topography inside the sample well was measured with a depth gauge micrometer using a 0.7 mm flat-end tip in 0.2 mm increments along the centerline, which is aligned to the streak camera slit position. An outside contractor (Electron Beam Engineering Inc., Anaheim, CA) electron-beam welded the caps to the sample holder wells enclosing the sample to prevent leaking during experimental heating. After bake-out at 100±3 °C, an initial circular pass of a defocused, lowpower beam under 10–4 torr vacuum helps residual gases to escape; trapped volatiles could lead to bowing of the sample cap during experimental heating. Sample holders are leak-checked at room temperature in a helium bomb and additionally in a heat test chamber at 1573 K by visual inspection for escaping fayalite liquid. We note that fayalite is an especially troublesome liquid to weld into capsules due to its low melting point, high surface tension, and poor glass-forming; ten capsules were lost to bad welds in obtaining the two successful capsules that were used for our experiments. The final driver and top hat shape were measured with a depth gauge micrometer on a digital mill table in 0.1 mm increments along the bright vertical line in Figure 2a. The sample capsules were carbon coated on both sides to ~80-200 nm thickness using a Cressington model 208C coater except for a thin rectangular masked area where the streak camera slit images the shock arrival. The layer of carbon provides an oxygen sink to reduce oxidation the molybdenum during heating, which can degrade the reflection into the streak camera [Asimow and Ahrens, 2010]. II-11 Preparation of room temperature (300 K) shots One solid room-temperature (300 K) shot was performed, extending the pressure range of previous room-temperature shocks on fayalite [Chen et al., 2002] from 212 GPa to 229 GPa. This was done to verify that the highest-pressure 300 K shots (107, 161, 126, 412; See Table 2) melted during impact, which is necessary for calculation of the liquid Grüneisen parameter (γ). Details of this calculation are in the Results section under Grüneisen parameter. The sample used was a single crystal of fayalite synthesized by H. Takei [Takei, 1978]. The boule was cut perpendicular to the growth axis into an elliptical disk with a minimum diameter of 7.8 mm and a maximum diameter of 8.8 mm. The disk was doubly polished to 2 mm thickness (within ±0.002 mm of uniform thickness). The crystal contained some glassy inclusions, but their estimated volume from electron backscatter images was less than 4%. The sample was directly mounted on a 0.5 mm tantalum driver plate and was not encapsulated. A series of fused quartz mirrors were glued to the down-range side of the driver plate and sample [cf. Mosenfelder et al., 2007] for recording of the shock arrival. Experimental setup The experimental methods used in our study for molten materials were originally developed by Rigden et al. [1984] and refined by Rigden et al. [1988; 1989], Miller et al. [1988; 1991a], Chen et al. [1998; 2002] and Asimow et al. (2008). The experimental configuration is summarized here; for additional description, see Asimow and Ahrens [2010]. The experiments for this study were performed in a 90/25- mm two-stage light-gas gun; previous shots from Chen et al. (2002) were performed using a 40- mm propellant gun. Shots 396 and 409 were preheated with a H2O-cooled copper induction coil powered by a 10 kW Lepel radio frequency generator [Chen and Ahrens, 1998] to 1300°C (1573 K), 95°C above the melting II-12 point of pure Fe2SiO4. For preheated experiments, the sample holders are mounted through a hole drilled in a Zircar™ alumina board and are held in place by a “guy-wire” set up in which W-Re alloy wire is threaded through two Mo posts screwed into the sample side of the target. Tantalum flyer plates were pressed into Lexan projectiles and launched at velocities (ufp) of 4.997-7.085 km/s, measured by a double-flash x-ray system and a two-magnet induction detector [Asimow et al., 2008]. For all three shots reported here, the two measurements agreed to within the stated uncertainty, and the average of the two measurements was used for the ufp. For room-temperature experiments (shot 412), the streak camera is electrically triggered by signals from shorting pins. For preheated shots, the camera and xenon spark lamp (which illuminates the target) are triggered using a “non-contact” system from the flyer-magnet signals and an up-down counter [Asimow and Ahrens, 2010]. The light is filmed by a Hadland Imacon 790 streak camera equipped with a 25 μm slit focused horizontally across the center of the sample holder (Figure 2a). As the shock wave reaches the free surface of the driver and later the sample cover, it changes the reflectance of the polished surface, creating a visible cutoff on the film. The shock wave velocity (Us) through the sample is calculated from the time difference of these two cutoffs, corrected for the known shape of the driver face and cap and thermal expansion of the assembly, and subtracting the shock transit time through the Mo cap. The final shock state is calculated iteratively using the technique developed in Ridgen et al. [1988], which takes into account the thickness and EOS of the hot molybdenum cap. The camera writing rate was calibrated using a test streak modulated by a radio-frequency tuner at 147.9993 MHz, which provides a time signal every 6.757 ns. The streak photo was scanned at 2400 dpi in 8-bit gray scale and matched to the calibration streak scanned at the same resolution to assign each column of pixels a time value. Driver and sample cutoffs were picked by selecting the inflection point of II-13 an arctangent function fit by cross-correlation to the gray scale values of each row of pixels parallel to the time axis (Figure 2c, shows only 3 pixel rows). If the correlation coefficient for the arctangent fit for a given pixel line was less than 0.95, then that pick was discarded. Discarding of pixels usually occurred in dark portions of the photo that were either poorly exposed or dark in the pre-experiment static photo due to imperfections in the polish. A few hundred points on either side usually constrain the driver cutoff (except in shot 396, see below); these picks are coregistered with the previously measured driver topography profiles and corrected for deviations from flat. Although tilt and bowing of the impactor and hence the shock front are often approximated as parabolic [Mitchell and Nellis, 1981a], we found that fourth order polynomial fits were suitable for extrapolating the shape of the shock front within the sample (Figure 2b). Only the center 3mm portion of the sample arrival was used for the calculation of shock transit time to reduce edge effects. The iterative solutions [Rigden et al., 1988] to particle velocity (up), pressure (PH), and density (ρH) of the shock state were determined from impedance matching, the Rankine-Hugoniot equations, and the following input parameters: sample geometry; standard Hugoniot data (ρo, Co, s) for the metal flyer, driver and cover plates [Asimow et al., 2008; Mitchell and Nellis, 1981b]; and the initial density (ρo) and best-initial guess values for 1 bar bulk sound speed (Co) and Hugoniot slope s for fayalite liquid at 1300 °C. It should be noted that regardless of the initial guess for the Co and s the same resulting shock state converges after three iterations). Uncertainties in all derived parameters (US, up, PH, ρH) were obtained from the uncertainties in the measured and standard parameters by analytical error propagation [cf. Jackson and Ahrens, 1979]. RESULTS II-14 Density and ultrasonic results at ambient pressure The double-bob density measurements between 1551 and 1831 K are reported in Table 1 and plotted against temperature in Figure 3. A linear equation: ρ(T) = 3699±1 – 0.30±0.01*(T1573 K) was fit to the data. Sound speed (C) data on the same sample between 1516 and 1879 K are reported in Ai and Lange [2004] using a frequency-sweep acoustic interferometer. Those data can also be described as a linear function of temperature: C(T) = 2438±5 – 0.33±0.03*(T-1573 K) m/s. Sink/ float results The results from the sink/float experiments are shown in Figure 4. We observed floating of the almandine garnets at 3.5 and 4.5 GPa and floating of spessartine garnets at 4.0 GPa. We were unsuccessful in observing sinks or floats at 3.0 GPa because of marker sphere dissolution or breakdown during the run. Therefore, our new data do not bracket the fayalite liquid density. However the observed floats do give a minimum density of the liquid within the experimental pressure range 3.5-4.5 GPa, and thus the new data do not agree with the sink/float measurements of Agee [1992] requiring that the compression curve for fayalite liquid be at higher density values in the range 3.5-4.5 GPa. On the other hand, the measurements of Agee [1992] at lower pressure using ruby spheres are still consistent with the new multi-anvil measurements. Reasons for possible inconsistencies in the Agee [1992] are discussed below. Shockwave results Comparison to previous data The shock wave data are reported in Table 2, including shot number, flyer/driver material, temperature prior to firing, ufp, up, US, PH, and peak density (ρH). Figure 5 shows the Hugoniot of this data in US-up space. A Hugoniot is a family of shock states achieved in a II-15 material from the same starting conditions (in this case, liquid fayalite at 1573 K and 1 bar). Empirically the Hugoniot of a well-behaved material forms a line in US-up space, given to third order in strain by US = Co + s up [Jeanloz, 1989] where the intercept Co = (KoS / ρo)1/2 is the bulk sound speed of fayalite liquid at room pressure and the slope s = (K’S + 1)/4 is related to K’S, the isentropic pressure derivative of the isentropic bulk modulus (KoS) [Ruoff, 1967]. Previous work from Chen et al. [2002] consists of 7 experiments completed on the Caltech 40-mm propellant gun. All Chen et al. data are taken from the original publication and were not re-analyzed. These data have widely varying reported uncertainties; Chen et al.’s linear regression was weighted by errors on Us only: Us = 2.63±0.02 + 1.59±0.01 up km/s. To these data, we added two preheated shocks, extending the Hugoniot to 161 GPa. Shot 409 from this study had clear cutoffs and well-defined shock arrivals. Conversely, shot 396 had a poorly exposed driver arrival, but showed a simple, un-tilted planar shock arrival at the top hat. The driver arrival for this shot was approximated as linear and un-tilted, which introduced error in the travel time measurement but is a suitable assumption due to the flatness of the top hat. An un-weighted linear fit to all nine preheated fayalite liquid data points from Chen et al. [2002] and this study yields Us = 2.67±0.08 + 1.50±0.03 up km/s (r2= 0.996), whereas a York regression (a regression which accounts for errors in both Us and up) gives Us = 2.80±0.18 + 1.488±0.071 up km/s (MSWD = 8.9). The unconstrained intercept of the York fit to the full data set is consistent within the 99% confidence interval with the bulk sound speed determined to much higher precision by ultrasonic techniques, Co= 2.438±0.005 km/s [Ai and Lange, 2004]. This suggests we cannot confidently reject the null hypothesis that this material, like other silicate liquids measured in the same way, displays relaxed (liquid-like), as opposed to un-relaxed (glass-like), behavior upon II-16 shock compression [Rigden et al., 1988]. Therefore, to reduce the error on the linear fit, we fixed the intercept at the ultrasonic value measured in this work. The un-weighted, constrained linear fit in US-up space yields US = 2.438±0.005 + 1.58±0.03 up km/s (r2=0.998). A weighted, constrained fit yields poor results because the uncertainties of US reported for some of Chen et al.’s data are anomalously small. The selected slope s corresponds to a K’S value of 5.33±0.10, and overall this is in reasonable agreement with the previous work by Chen et al. [2002], which estimated K’S of 5.39±0.04. Figure 5 includes both the unconstrained and constrained linear fits to the entire data set, as well as the previous fit by Chen et al. [2002]. It is worth noting that the slopes of the new and old linear fits are very similar despite having slightly different though nominally fayalitic compositions. Thus within the errors of the experiments, our results seem to be insensitive to minor compositional differences. Our newest shockwave and sink/float results also resolve previously discrepant measurements for K’S. Isothermal static compression data from sink/float [Agee, 1992] and shock wave [Chen et al., 2002] experiments gave K’ values of 10.1 and 5.36, respectively. The newest sink/float measurements are shown in Figure 4 alongside two 3rd-order Birch-Murnaghan isotherms (dashed and dotted lines).The isotherms are fit with ρo from this study, the bulk modulus from ultrasonic experiments (KT=19.4) [Ai and Lange, 2004], and K’ from the slope of the newest constrained linear Hugoniot (K’=5.33)(dashed in Fig. 4) or K’ from the 3rd-order Birch-Murnaghan/Mie-Grüneisen (3BM/MG) (K’=7.28) from this study (dotted in Fig. 4). The details of the 3BM/MG fit are given below in the Thermal equation of state fitting section. The solid line is calculated using the previous data from Agee [1992]. The three experiments showing flotation are consistent with the dashed and dotted lines and discount the previous measurements (>3 GPa) on identical material used in the 1992 study. It is conceivable that piston “bottoming- II-17 out” or other unknown friction effects at high compressions and high temperatures produced lower pressures than expected from calibration runs. In any case, the new multi-anvil sink/float data are in good agreement with the ultrasonic and shockwave results. Solid Hugoniot shocked to liquid / χ2 test of weighted values Shot 412 was carried out on single crystal fayalite and is believed to have undergone a phase transition to liquid during the shock. A phase transition to liquid along the Hugoniot can often be detected by an offset in pressure-volume [Akins et al., 2004] or US-up space. To establish where in US-up space the phase transition occurs, we used a χ2 test of weighted values on linear fits to all room-temperature shock data (Table 1). The goal was to minimize the χ2 value of fitting two separate linear Hugoniots to solid and shock-melted data. Excluding shot 116, which has an anomalously low Us, the best fit (χ2= 0.0014867) occurred when shots 107, 161, 126, and 412 were calculated as liquids. The pressure limit of 190 GPa for shock melting implied by placing the boundary between shots 99 and 107 is also consistent with the melting thresholds along the Hugoniots of many iron-free silicate minerals shocked from room temperature: quartz at 120 GPa [J. A. Akins, 2002; Lyzenga, 1983], diopside at ≤144 GPa [Svendsen and Ahrens, 1990], and diopside-anorthite (Di64An36) at 133 GPa [Asimow and Ahrens, 2010] all of which have higher ambient melting temperature than that of fayalite (previous descriptions of shock melting of forsterite at 142 GPa [Mosenfelder et al., 2007] enstatite at 174 GPa [Mosenfelder et al., 2009] may not be accurate and are discussed in Chapter III). We have therefore used the four highest-pressure data points, presumed to have melted on shock, to calculate a value for the liquid Grüneisen parameter. Grüneisen parameter The thermodynamic Grüneisen parameter (γ) is a macroscopic parameter that relates II-18 thermal pressure to the thermal energy per unit volume, where thermal pressure is defined as the increase in pressure due to heating at a constant volume. Because a solid shocked to a liquid and an initially molten sample obtain different pressures and internal energy states at equal density (ρH) upon compression, γ can be directly determined by comparison of the two Hugoniots (e.g., [Luo et al., 2002; Mosenfelder et al., 2007, Asimow & Ahrens, 2010]). We calculated the Grüneisen parameter from the following expressions; the parameter definitions and the values used are given in Table 3. By definition, γ= 1  ∂P    . ρ  ∂E  ρ (2) The Mie-Grüneisen approximation defines γ as a function of ρ only, which simplifies the above expression to a finite difference: γ= 1 ∆Pth 1 ( PH −liquid − PH − solid ) = , ∆E ρ ∆E ρ (3) where ρ is the density of interest and ΔPth and ΔE are the offsets in pressure and internal energy between the two Hugoniot states at that density. The subscripts H-liquid and H-solid correspond to states on the Hugoniots that are both liquid in the shock state but that were initially a liquid at 1573 K and a solid at 300 K, respectively. The energy offset is calculated from ∆E = ∆E H − solid − ∆E H −liquid + ∆E tr , (4) where ∆Etr , the transition energy, is the internal energy difference at ambient pressure conditions and is estimated by the enthalpy difference ( ∆H tr ) because the transition takes place at nearly zero pressure [Asimow and Ahrens, 2010]. ∆H tr is composed of two parts, the integral II-19 of the isobaric heat capacity (CP) and the enthalpy of fusion, and was calculated with MELTS [Ghiorso and Sack, 1995]. The internal energy change across each shock front (ΔEH) was calculated using the 3rd Rankine-Hugoniot relation, ∆E H = PH  1 1    − 2  ρ o ρ H  . (5) There are several empirical forms for describing the density dependence of γ(ρ) consistent with the Mie-Grüneisen assumption [Stixrude and Lithgow-Bertelloni, 2005]. We will look only at the power law form, which is both the simplest and most prevalent in the literature: γ (ρ ) = γ o (ρ o / ρ ) q . The Grüneisen parameter at initial conditions is derived from the expressions (6) αK T ρCV (7) CP = 1 + αγT , CV (8) γ= and where α is the isobaric coefficient of thermal expansion, CV is isochoric heat capacity, and CP is isobaric heat capacity. Fitting the offset between the Hugoniots of fayalite liquid shocked from liquid and solid initial states to equation (5) yields a q value of –1.45 based on data in the vicinity of ρo/ρ =0.49. In solid materials γ generally decreases upon compression (Figure 6), but a negative q value demonstrates that γ for this liquid increases upon compression. Moreover, this q value is similar II-20 to values obtained on other silicate liquid compositions using Hugoniot offsets, such as forsterite (q = –1.47) (corrected from the Mosenfelder et al. value, see Table 5; Chapter III includes a new EOS from preheated forsterite), enstatite (q = –1.71) [Mosenfelder et al., 2009] (revised in Chapter III), and anorthite-diopside eutectic (q = –1.63) [Asimow and Ahrens, 2010]. We take this result as strong evidence that iron-bearing silicate liquids follow the same general behavior as iron-free silicate liquids, which are all described by –2.0 ≤ q ≤ –1.45 for the compression range 1 ≥ ρo/ρ ≥ 0.49. Figure 6 shows the similarities in volume dependence of γ for liquids of composition MgSiO3 (En), Mg2SiO4 (Fo), and Fe2SiO4 (Fa) as well as the opposite dependence displayed by mantle solids. Negative q values for liquids have been interpreted qualitatively as a result of increasing cation coordination by analogy to structural phase changes in related solids [Stixrude and Karki, 2005] and, more recently, quantitatively in terms of liquid configurational entropy in the context of a hard-sphere model of atomic packing in liquids [Jing and Karato, 2011a]. The consistent behavior of fayalite with other liquids so far studied does not obviously resolve the question of the microscopic origin of the behavior, but it shows that the behavior does not depend very much on the major cations in the liquid. Thermal equation of state fitting A Hugoniot is a one-parameter curve through a P-V-E thermodynamic space of two independent variables, in this case V(P, E). However, we are generally interested in defining the behavior of liquids in regions that lie off the Hugoniot curve. In particular, at high compressions, the Hugoniot reaches temperatures and energies much higher than mantle conditions of geophysical interest, even for the early earth. In order to investigate material properties that lie off the Hugoniot, we attempt to define the entire P-V-E surface for a given composition by fitting II-21 a thermal equation of state. There are several formalisms for thermal EOS; we have chosen to investigate three of them (all of which are plotted in Figure 7): (1) the shock wave equation of state (SWEOS, derived from both the constrained and unconstrained linear fit above), (2) 3rd and 4th order Birch-Murnaghan Mie-Grüneisen equation of state (3BM/MG and 4BM/MG), and (3) de Koker-Stixrude fundamental relation (dKSFR) [de Koker and Stixrude, 2009]. The SWEOS is defined by a linear Hugoniot in US – up space, transformed into pressure (P)- density (ρ) space using the first and second Rankine-Hugoniot equations [e.g., Ahrens, 1987]: ρH = ρ oU S US − up PH = ρ oU S u p (9) . (10) States that lie off the Hugoniot are found using the Mie-Grüneisen thermal pressure approximation with temperature-independent γ(ρ), as calculated above (6). The 3rd- and 4th-order Birch-Murnaghan/Mie-Grüneisen equations of state are defined by 3rd or 4th order Birch-Murnaghan isentropes centered at 1 bar and 1573K plus a Mie-Grüneisen thermal pressure approximation. Thus, total pressure P at any volume and internal energy is given by P = PS + Pth . (11) Ps is the Birch-Murnaghan isentropic pressure given to 4th order in strain by 5 2 PS = 3K oS f (1 + 2 f ) (1 + 3 3 143 2 ( K ' S −4) f + ( K " S K oS + K ' S ( K ' S −7) + ) f ) (12) 2 2 9 II-22 where f is the Eulerian finite strain parameter 2    1  ρ  3 f =    − 1 . 2  ρo     (13) [Birch, 1947]. The thermal pressure term (Pth) is given by Pth = γρ ( E − E S ) (14) where E is the energy of interest. The internal energy along the isentrope (ES) is found by integration of dES = –PSdV: ES = 9 K oS 2 3 143 4 ( f + ( K ' S −4) f 3 + ( K " S K oS + K ' S ( K ' S −7) + )f ) 2 ρo 4 9 (15) For 3rd-order BM isentropic pressure and energy, K” in the previous expressions is fixed to the value that makes the 4th-order term vanish: K " S = −( K ' S ( K ' S −7) + 143 ) / K oS 9 (16) In order to infer a temperature, we need to define an isochoric heat capacity (CV). Currently no shock temperature measurements exist to constrain an expression for CV(ρ) for an iron-bearing silicate liquid, so CV was fixed at the constant value determined from constant CP at ambient pressure and 1573 K [Lange and Navrotsky, 1992]. The temperature of interest is then calculated with the equation ρ T = To E γ (ρ ) 1 + d ρ ∫ρ ρ ∫E CV dE o o S , (17) II-23 which is composed of two parts: (1) integration along the reference isentrope to the density of interest with To = 1573 K as the foot temperature and (2) addition of the energy offset proportional to CV. A method for fitting shock wave data to the 3BM/MG and 4BM/MG forms is found in Asimow and Ahrens [2010]. To account for uncertainties in both pressure and volume, the function  P i ( ρ i ) − P i  2  ρ i − ρ i  2  H  χ 2 = ∑i =1  * * i + * i H     σ    σρ ρ      n (18) is minimized, where pressure is evaluated along the model Hugoniot at the trial density P*i = PS − γρ *i ( E S − E tri )  γ  ρi 1 −  *i − 1 2  ρo  . (19) The subscript * denotes a trial pressure or density, and σ denotes the respective uncertainties for each pressure-density experimental pair, (PiH, ρiH). Uncertainties in the fitted parameters are estimated by bootstrap iteration [Efron, 1982]. The input parameters that remain fixed are KoS, CV, ρo, and γo; the best-fit parameters (K’S and q, plus K’’ for 4BM) are given in Table 4. The 3BM/MG fit result is KS’ = 7.28±0.35, q = –0.95±0.51, and a reduced χ2 = 1.526. The 4BM/MG fit result is KS’ = 6.64±0.58, KS” = –0.47±0.27 GPa-1, q = –0.54±1.49 and reduced χ2 = 0.941. The 4BM/MG chi-squared value indicates probable over-fitting, and the fairly large error bars on all output parameters seem to indicate a very unstable fitting routine. On the other hand, the 3BM/MG seems relatively stable. The unconstrained-SWEOS appears to have similar form to the 3BM/MG and matches the highest pressure data better than the constrained SWEOS. The II-24 stiffness in the constrained SWEOS is likely due to being tied to the intercept and the most precise of Chen et al.’s data points—the precision of which is likely overestimated. The q value for the SWEOS is just within the stated errors of the 3BM/MG value, and the 3BM/MG reduced chi-squared value is reasonable. Additionally with little justification to implement the 4th-order fit, we recommend the 3rd-order fit for fayalite liquid. The initially solid data are included in the fits to the 3BM/MG and 4BM/MG thermal EOS for the liquid (figure 7). The 4th-order fit recovers these data very well, but only the 3rdorder fit appears justified by the fitting statistics. The Eulerian finite strain Birch-Murnaghan EOS has been used widely in the literature for describing the volumetric behavior of liquids [Agee, 1998; Y Ai and Lange, 2008; Asimow and Ahrens, 2010; Bottinga, 1985; Lange, 2007; Rigden et al., 1989; Suzuki et al., 1995; Tenner et al., 2007]; it has thus far proven to be adequate for fitting although it lacks any rigorous basis in theory as an EOS for liquids. The BM formalism has been criticized for its theoretical weakness in giving unphysical interatomic potentials for high K’ materials (gases and liquids) [Hofmeister, 1993] and for singularities that occur at high temperature and low pressure [Ghiorso, 2004]. Consequently, although the BM EOS can fit our data and allows interpolation between experimental points, it should be used only with caution for extrapolation beyond experimental constraints. It is a difficult task to capture the nature of liquids, which have short-range order but lack long-range order, in a single EOS. Two recent EOS formalisms have been developed specifically for silicate liquids that are both thermodynamically and theoretically self-consistent in their approach to describing the volume behavior at elevated temperatures and pressure: (1) the deKoker and Stixrude fundamental relation (dKSFR) [de Koker and Stixrude, 2009] and (2) the II-25 hard sphere equation of state (HSEOS) [Jing and Karato, 2011a]. The dKSFR is an extension of the finite strain description given in Birch [1952; 1978] but additionally takes into account the thermal free energy contribution and temperature dependence of liquid properties. The thermodynamic variables are self-consistent in that they are derived from a single equation for the Helmholtz energy, and the contributions to the Helmholtz energy are separable [McQuarrie, 1984] into an ideal gas term, an excess term, and an electronic term. Recent work by Muñoz Ramo and Stixrude [in review] using first principles molecular dynamics (FPMD, GGA+U) of Fe2SiO4 liquid obtained a fit with a fourth-order expansion in finite strain and first-order expansion in the reduced temperature using the dKSFR. Their Hugoniot, shown in Figure 7, is in good agreement with our shockwave data. However, contrary to the monotonic increase in γ with increasing pressure inferred from shock wave studies, their results show γ increasing up to a compression of ρo/ρ ≈ 0.49 followed by a decrease in γ coinciding with a change from the high-spin state of Fe. The HSEOS assigns a hard sphere for each cation species, which move freely in the liquid. The geometrical arrangement of these spheres gives the contribution of entropy to the compression of the liquid [Jing and Karato, 2011a]. This formalism does well in explaining the complex behaviors of melts, such as the dependence of γ with pressure. However currently, the calibration of the Fe sphere diameter is based on two experiments within a close range of temperature. Therefore, the Fe sphere diameter dependence on temperature remains somewhat poorly constrained for extreme temperature conditions along the Hugoniot. Calibration would be improved with shock temperature measurements or any other independent measurements of Febearing silicate liquids within a wider temperature range. II-26 Corrections and re-analysis of silicate liquids Also in this study, the anorthite and diopside liquid data of Asimow and Ahrens [2010] were re-analyzed, only sampling the center 3 mm of the arrival top hat. This was motivated by a hydro-code model of shock propagation through our capsule geometry carried out at Lawrence Livermore National Laboratory by J. Nguyen, which indicated that the outer part of the top hat arrival is influenced by edge effects. The corrected BM/MG fits are given in Table 5, and the reanalyzed data points are given in Table 1 of Chapter III. The 4th-order fit for anorthite liquid excludes 3 shots: 382, the highest pressure shot from Asimow and Ahrens [2010], due to an abnormally shaped shock arrival; and shots 663 and 672, the highest pressure shots from Rigden [1988] due to possible un-relaxed behavior upon compression. For further discussion of glasslike behavior in anorthite under some shock conditions, see Asimow and Ahrens [2010], section 4.4. The re-analysis of diopside liquid excludes 378, the lowest pressure shot from Asimow and Ahrens [2010] due to irregular and over-exposed top hat arrival. Also presented in Table 5 is the corrected forsterite liquid γo (which previously contained a sign error), and its subsequent re-fitted 4BM parameters. The entire EOS of forsterite is revised in Chapter III. IMPLICATIONS Mixing of oxide volumes at high pressure and temperature; application to a magma ocean liquidus Our extension of the EOS of fayalite liquid completes the basis set needed for estimation, subject to the assumption of linear mixing, of liquid densities in the full CMASF (CaO-MgOAl2O3-SiO2-FeO) major element space for mafic to ultramafic liquids at elevated temperatures and pressures. Shockwave experiments completed thus far that span this component space are II-27 forsterite (Fo, Mg2SiO4) [Mosenfelder et al., 2007], enstatite (En, MgSiO3) [Mosenfelder et al., 2009], anorthite (An, CaAl2Si2O8), diopside (Di, CaMgSi2O6) [Asimow and Ahrens, 2010], and now also fayalite (Fa, Fe2SiO4). All of the EOS parameters for each of these compositions, corrected (see above) or recommended by the original authors, are given in Table 5.The revised EOS are discussed in detail in the subsequent chapter (Tables 3 and 4) In this study we examine only one simple model of many possible scenarios for the crystallization of an early terrestrial magma ocean. Our initial condition is a well-mixed, convecting liquid composed of one of two plausible bulk silicate Earth compositions — “chondrite” or peridotite — that reaches the base of the mantle. To learn at what depth and critical potential temperature such a whole-mantle magma ocean would begin to crystallize, we used mixtures of silicate liquids in the CMASF system and calculated the intersections or tangency points between isentropes of these two liquid compositions and their respective liquidus curves. The basis of our model is derived from an alternative definition of the Grüneisen parameter, γ = (∂ ln T ) /(∂ ln ρ ) S , (20) which expresses the adiabatic temperature gradient with increasing density [Miller et al., 1991b]. Figure 8 shows the two mantle compositions that we examined. The first (Fig. 8a) is a synthetic “chondrite” mix [Andrault et al., 2011] with relative proportions of the five major oxides akin to the primitive mantle but lacking any minor components (TiO2, Na2O, K2O… etc.); its major element ratios are similar to pyrolite composition [Ringwood, 1975]. The second composition (Fig. 8b) approximates the major oxide composition of KLB-1 peridotite [Fiquet et al., 2010], a fertile spinel lherzolite. We also considered the KLB-1 composition given by Davis et al. [2009], but found that the isentropes were nearly indistinguishable from those calculated from the II-28 published composition in Fiquet et al. [2010]. Table 6 gives the major oxide proportions for each composition as well as the normalized percent of each end-member liquid used in the model. We also were concerned with what the density contrast of our model liquid would be with its first forming solid—whether it would sink or float. As a conservative measure of where solid flotation is possible, we assumed the liquidus solid to be pure Pv (MgSiO3, magnesium perovskite) using the EOS from Mosenfelder et al. [2009]. Mg-rich Pv was the liquidus solid in both Andrault et al. [2011] and Fiquet et al.[2010] at the pressures of interest. We determined each CMASF model isentrope for a given potential temperature (TP) in a stepwise fashion, solving for temperature (T) at each fixed increment of pressure (P) from 0 to 140 GPa. In this calculation, each CMASF isentrope is a combination of the pure end-member isentropes of An, Di, Fo, En, and Fa; and each of the end-member isentropes is defined by its own potential temperature TiP (superscript i denotes an end-member mineral), where TiP is a separately defined value from the TP of the CMASF model isentrope. The separate TP and TiP values permit the mixture of the end-member isentropes to define a point on the final CMASF model isentrope in which the total entropy (Stot) remains a constant, such that S tot = ∑i f i S i + S mix = constant, (21) where f is the mole fraction (Table 6) and Si is the molar entropy for each end-member in the model liquid. By assuming the entropy of mixing (Smix) is constant for a given liquid composition, the potential temperature of each end-member liquid isentrope (TiP) was solved by setting the derivative of the total entropy equal to zero: Ti  d ∑ f i S i = ∑ f i C Pi ln P  = 0 . i i  TP  The potential temperature for each end-member liquid isentrope is further defined as (22) II-29  γ i V i q  i q  V   i TP = T exp oi  Pi  −  i   .  q  Vo   Vo    i i (23) This expression is derived from (20) by integrating q ρ  T ∂T = γ o  o  ∂ρ .  ρ  ρ (24) and is dependent on the following three variables: (1) the temperature at each iteration along the model isentrope (T), (2) the volume of each end member at T and P (Vi), and (3) the volume of each end member at TiP and 0 GPa (ViP). These three variables are solved iteratively to satisfy equation (22) and to minimize the expression ∑ ( P − P) + ( P ) , i S i P 2 (25) i where PiP is the BM isentropic pressure at TiP and equals 0 GPa at the target volume ViP and PiS is the BM isentropic pressure for each end member at T. Minimization of this expression ensures that the isentropic pressure for each end member converges to the defined pressure in question and that PiP is in fact equal to 0 GPa. For Fa, we used the BM3 parameters given in Table 4; for all other end members, we used the BM parameters in Table 5. In total, 11 parameters (T and two volumes, Vi and ViP, for each of the 5 end members) were solved iteratively for each step increment in pressure along a model isentrope. The densities of the mixtures were calculated using linear mixing of volumes. This assumption has been previously tested for mixing of anorthite and diopside liquids [Asimow and Ahrens, 2010] and was found to be well-behaved at high temperatures and pressures but began to break down when applied at low temperatures requiring very large extrapolation downwards from the Hugoniot temperature. Furthermore, anorthite liquid may be a special case due to its highly polymerized structure at low pressure [Mysen and Richet, 2005], although we examine II-30 this further in Chapter III. Hence, despite examination of three compositions along the anorthitediopside join, it remains unclear whether or not linear mixing of volume at high pressure is valid for most mafic to ultramafic silicate liquids. Nevertheless, given five measured compositions in a five-oxide system, the assumption of linear mixing is necessary to make progress at this time and remains consistent with data presently available. We examine this assumption in detail in Chapter IV. The resulting model for the Andrault et al. [2011] “chondrite” mix shows that crystallization would begin at the base of present-day mantle (~135 GPa) with a critical adiabat of TP = 2600K (Fig 8a). Andrault et al. found Pv to be the liquidus phase at all pressures above 60 GPa; therefore we calculated the density for a first forming crystal and found it to be denser than the surrounding liquid (ρPv, 5.38 g cm–3 > ρliquid, 5.28 g cm–3). Thus a basal magma ocean similar to that described in Labrosse et al. [2007] in which a large portion of the lowermost mantle is zone-refined to concentrate incompatible elements would be unlikely as crystallization of the mantle would occur from the bottom up. A chondritic basal magma ocean is likely only able to exist within the thermal boundary layer (highest estimated outer core of 4150K) [Hernlund et al., 2005] or at the elevated basal temperatures that occurred during early Earth accretion [Canup, 2004]. The critical adiabat for the peridotite composition of Fiquet et al. (2010), on the other hand, occurs at a higher TP of 2800 K and, notably, is tangent to the liquidus at 85 GPa (Fig 8b). Consequently crystallization would begin in the middle of the lower mantle. A Pv liquidus phase would, like the model above, sink upon formation (ρPv, 4.94 g cm–3 > ρliquid, 4.63 g cm–3). This model is more akin to the picture described in Labrosse et al. [2007] as the basal magma ocean would include a portion of the lower mantle which could be used to distill incompatible elements II-31 into a potential present-day ULVZ, yet the sinking liquidus phase if not conducive to the dynamic formation of a basal ocean. It is worth mentioning that the surface temperatures in both of these models could be too hot for simple, realistic magma ocean models; Andrault et al. [2011] suggest that such high foot temperatures would lead to large portions of the mantle vaporizing and rapid cooling if thermal blanketing is inefficient. Within this exercise, the location of first crystallization is likely dominantly controlled by two factors: (1) the topology of the liquidus curve and (2) the amount of Fo end-member in the model liquid. It is unclear whether different shapes of liquidus curves stem from differing experimental techniques in the two synchrotron-based laser-heated diamond anvil studies considered or instead if they reflect thermodynamic differences within that range of composition space. However, Fig. 8 makes clear that without a notable decrease in slope in the liquidus curve compared to the isentrope at high pressure, crystallization would always be predicted at the base of the mantle (Fig. 8b). Secondly, it has been observed that the forsterite component has a large control on the spacing and steepness of the isentropes. This behavior can be seen to be true in peridotite, which has a greater fraction of Fo component (Table 6) and has a much steeper isentrope series than the En-rich chondrite liquid (Fig. 8). The effect that Fo has on the isentrope mixing can be predicted from its 1-bar thermodynamic parameters (Table 5) with larger γo , α, and KS. These larger parameters are inversely proportional to the amount of SiO2 in Fo, since the dV/dT value of SiO2 at ambient pressure (from which these parameters are derived) is zero [Lange and Carmichael, 1990]. The bulk modulus of SiO2 is notably low compared to other oxides likely because of topological degrees of freedom in tetrahedral framework liquids. Thus the component with the II-32 least SiO2 gives the largest γo, α, and KS and so the steepest isentrope, which contributes directly to the steepness of the isentrope of the mixture. In Chapter III, we used the revised EOS of Fo, An, and En to re-model these liquid isentropes (Figure 7). The results are fairly similar, but it is important to note that the γo , α, and KS of forsterite liquids are not as large as modeled by the 1-bar data, and thus forsterite does not play such a dominant role on the steepness of the isentrope. A full discussion is given in Chapter III, section Isentropes for a full mantle magma ocean. We can conclude from this exercise in modeling the Earth’s accretion, acknowledging the simplifying assumptions and presumed bulk compositions, that the first crystallizing solid will likely occur within the lower mantle or at the base of the mantle. This is made more likely if early earth core temperatures are assumed to be elevated above current estimates [Canup, 2004]. Although largely controlled by the chosen bulk composition, within our exercise the liquidus phase would be Pv and would sink upon formation. Whether pure Mg-Pv is a correct approximation to the liquidus phase, as opposed to a more Fe-rich (Mg,Fe)-Pv, is unclear based on the spread of high T-P Fe partitioning values in the current literature (DFe = 0.6 to > 0.1) [Andrault et al., 2012; Nomura et al., 2011]. Regardless, since pure Pv is the least dense endmember of the (Mg, Fe)SiO3 solid-solution, greater partitioning of Fe into the solid would only further increase the density and magnitude of the already observed negative buoyancy. Buoyancy in the current temperature regime; application to the solidus and ULVZ For long-term gravitational stability, a presumed molten silicate liquid of the ULVZ must be denser than or at least neutrally buoyant compared to the ambient lower-most mantle bulk composition that exists today. Shown in Figure 9 are the densities of fayalite, enstatite and forsterite liquids along a plausible modern mantle adiabat including a thermal boundary II-33 correction, with a potential temperature of 1673 K and a gradient of 10 K/GPa. A pure molten fayalite composition with density of 7 g cm–3 would pond at the core mantle boundary if present, as it is much denser than any likely solid assemblage at lowermost mantle conditions. However the model of a pure fayalite liquid ULVZ is inconsistent both with the density of 6.12 g cm–3 in the ULVZ inferred from seismic reflection coefficients [Rost et al., 2006] (Figure 9) and with the non-zero shear velocity of this layer. Instead, a less-dense multicomponent partial melt needs to be considered as a plausible state for the ULVZ. Assuming linear mixing of the oxide volumes calculated from the BM EOS given in Tables 4 and 5, Figure 10a displays lines of constant density within a Fa-Fo-En-Fs space (where ferrosilite (Fs) = En+ .5Fo -.5Fa). Figure 10b is the same plot, but uses the revised EOS in Chapter III Tables 3 and 4. The y-axis is Mg# (Mg# = MgO/FeO+MgO), and the x-axis is (MgO+FeO)/SiO2. The solid curves delineate compositions that are neutrally buoyant with respect to a pure Pv solid at ~135GPa (roughly the pressure at the CMB) at the given temperature conditions (3900 K–4500 K). The dashed lines indicate a constant density of 6.12 g/cm3, which represents the estimated upper limit of a 10% density jump of an ULVZ [Rost et al., 2006] with respect to PREM [Dziewonski and Anderson, 1981]. Any liquid mixture that lies above the solid lines where Pv sinks (white in Fig. 10) is inferred to be gravitationally unstable with respect to upwards percolation into the solid mantle, and any mixture below the dashed lines is denser than has been inferred for ULVZs. In this exercise, potential liquid compositions for a high-melt fraction ULVZ must span a narrow range in composition space (striped blue in Figure 10), eliminating pure enstatite, and fayalite end-members as candidate compositions. Previously (Fig. 10a), pure forsterite was not denser than coexisting Pv crystal, but in the revised EOS (Fig. 10b), this is no longer shown to be true. For reference, the simplified chondrite and peridotite liquids modeled in the section above II-34 are also plotted on Figure 10. These results indicate that these liquid compositions are unlikely to be gravitationally stable at the CMB today (although peridotite liquid is denser than previously modeled Fig 10b); if residual liquids were to be derived from magma oceans of either composition, they would need to be enriched in Fe, depleted in Si, or both during some process of re-crystallization and distillation from the original bulk mantle. The gray arrows from each composition indicate the direction of compositional evolution if Pv (MgSiO3, En) were to crystallize and be removed. It also gives a rough estimate of what the final liquid composition would need to be for Pv to be buoyant. When evaluating whether a liquid would be gravitationally stable, Figure 10 assumes that a pure Pv end member is an acceptable estimation for bulk lower mantle density. Adding Fe to the pertinent solid phase in this scenario (modeled as Pv) would qualitatively narrow the region of gravitationally stable liquids to only low Mg# (<0.5) compositions in Fig. 10 (blue-lined area). Also the densities shown here are for fixed compositions that are unlikely to be in chemical equilibrium. For a complete examination of stability, we would need a better grasp on the actual bulk composition of the solid and an understanding of Fe/Mg partitioning at lower mantle conditions. Moreover, it is likely that the bulk lower mantle contains some amount of ironbearing Pv and calcium-silicate Pv, which would be denser than pure Mg-Pv and also effectively narrow the range of acceptable, stable compositions for a ULVZ liquid shown in Figure 10. Similarly, incorporation of Al2O3 and CaO into the model partial melt using the EOS of anorthite and diopside would narrow this same region of Fig. 10 by decreasing the density of the liquid. Therefore, our simplified model in Fig. 10 indicates the minimum amount of Fe enrichment that the liquid must have to be a high-fraction, gravitationally stable silicate melt at the CMB, but the amount of enrichment is likely to be much greater. II-35 CONCLUSIONS The density for Fe2SiO4 liquid as well as values for K and K’ have been constrained. Additionally, the Grüneisen parameter for Fe-bearing silicate liquids is shown to change with compression in a manner similar to non-Fe bearing liquids and the opposite to that of solids. Although the Grüneisen parameter has been measured for formulation of a thermal equation state, as with any thermal EOS formalism, there are trade-offs especially when examining states far from the Hugoniot including extrapolations in pressure beyond experimental pressures (161 GPa). Temperature estimates could be greatly improved by either shock temperature measurements or any other off-Hugoniot temperature measurements to constrain Cv. Constraints on the end-member liquid fayalite EOS permit interpolation between CMASF silicate liquids densities, assuming linear mixing of volumes. The calculation of isentropes at elevated temperatures and pressures for model whole-mantle magma oceans composed of peridotite and simplified “chondrite” liquid indicate crystallization initiating at the mid-lower mantle (85 GPa) and the base of the mantle, respectively. Finally, a compositional range of gravitationally stable liquids was determined to examine the likelihood of a gravitationally stable ULVZ composed of a large degree of partial melt. There exists a region of melt composition that is denser than Pv at the CMB, but neither chondrite nor peridotite is sufficiently enriched in Fe to be stable. Fractional crystallization of Pv would need to take place before either liquid would be dense enough to remain at the CMB. To elucidate fully the proper stable liquid composition, formulation of multicomponent phase equilibria or partition coefficients at high pressure and temperature are necessary for accurate description of the CMB. II-36 ACKNOWLEDGEMENTS The authors would like to thank the following: the shock wave lab technical staff—Michael Long, Eprapodito Gelle, and Russel Oliver; additionally, Bjorn Mysen for conducting Mössbauer measurements, and Jeff Nguyen for running simulations of our shock wave experiments. 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Stolper, and T. J. Ahrens (1991b), The Equation of State of a Molten Komatiite 2. Application to Komatiite Petrogenesis and the Hadean Mantle, J. Geophys. Res., 96. Mitchell, A. C., and W. J. Nellis (1981a), Diagnostic system of the Lawrence Livermore National Laboratory two-stage light-gas gun, Review of scientific instruments, 52(3), 347. Mitchell, A. C., and W. J. Nellis (1981b), Shock compression of aluminum, copper, and tantalum, Journal of Applied Physics, 52(5), 3363-3374. Mosenfelder, J. L., P. D. Asimow, and T. J. Ahrens (2007), Thermodynamic properties of Mg2SiO4 liquid at ultra-high pressures from shock measurements to 200 GPa on forsterite and wadsleyite, J. Geophys. Res., 112. Mosenfelder, J. L., P. D. Asimow, D. J. Frost, D. C. Rubie, and T. J. Ahrens (2009), The MgSiO3 system at high pressure: Thermodynamic properties of perovskite, postperovskite, and melt from global inversion of shock and static compression data, J. Geophys. Res., 114. Muñoz Ramo, D., and L. Stixrude (in review), Spin crossover in Fe2SiO4 liquid at high pressure, Physical Review Letters. Mysen, B. O., and P. Richet (2005), Silicate Glasses and Melts: Properties and Structure, 544., Elsevier, Amsterdam. Nomura, R., H. Ozawa, S. Tateno, K. Hirose, J. Hernlund, S. Muto, H. Ishii, and N. Hiraoka II-40 (2011), Spin crossover and iron-rich silicate melt in the Earth's deep mantle, Nature, 473(7346), 199-202. Rigden, S. M., T. J. Ahrens, and E. M. Stolper (1984), Densities of liquid silicates at high pressures, Science, 226(4678), 1071-1074. Rigden, S. M., T. J. Ahrens, and E. M. Stolper (1988), Shock compression of molten silicate: results for a model basaltic composition, J. Geophys. Res., 93(B1), 367-382. Rigden, S. M., T. J. Ahrens, and E. M. Stolper (1989), High-Pressure Equation of State of Molten Anorthite and Diopside, Journal of Geophysical Research, 94(B7), 9508-9522. Ringwood, A. E. (1975), Composition and Petrology of the Earth’s Mantle, edited, p. 618, McGraw-Hill, New York. Rost, S., E. J. Garnero, and Q. Williams (2006), Fine-scale ultralow-velocity zone structure from high-frequency seismic array data, J. Geophys. Res., 111(B9), B09310. Ruoff, A. (1967), Linear Shock-Velocity-Particle-Velocity Relationship, Journal of Applied Physics, 38(13), 4976. Stixrude, L., and B. Karki (2005), Structure and Freezing of MgSiO3 Liquid in Earth's Lower Mantle, Science, 310(5746), 297-299. Stixrude, L., and C. Lithgow-Bertelloni (2005), Thermodynamics of mantle minerals – I. Physical properties, Geophysical Journal International, 162(2), 610-632. Stolper, E., D. Walker, B. H. Hager, and J. F. Hays (1981), Melt Segregation From Partially Molten Source Regions: The Importance of Melt Density and Source Region Size, J. Geophys. Res., 86. Suzuki, A., E. Ohtani, and T. Kato (1995), Floatation of diamond in mantle melt at high pressure, Science, 269(5221), 216-218. Svendsen, B., and T. J. Ahrens (1990), Shock-Induced Temperatures of CaMgSi2O6, J. Geophys. Res., 95(B5), 6943-6953. Takei, H. (1978), Growth of fayalite (Fe2SiO4) single crystals by the floating-zone method, Journal of Crystal Growth, 43(4), 463-468. Tenner, T. J., R. A. Lange, and R. T. Downs (2007), The albite fusion curve re-examined: New experiments and the high-pressure density and compressibility of high albite and NaAlSi3O8 liquid, American Mineralogist, 92(10), 1573-1585. Ulmer, P., and R. Luth (1991), The graphite-COH fluid equilibrium in P, T, & fO2 space, Contributions to Mineralogy and Petrology, 106(3), 265-272. Wicks, J. K., J. M. Jackson, and W. Sturhahn (2010), Very low sound velocities in iron-rich (Mg, Fe) O: Implications for the core-mantle boundary region, Geophysical research letters, 37(15), 15304. Williams, Q., J. Revenaugh, and E. J. Garnero (1998), A Correlation Between Ultra-Low Basal Velocities in the Mantle and Hot Spots, Science, 281(5376), 546-549. II-41 TABLES Table 1. Double-bob density measurements T (K) 1551 1632 1723 1818 ρ g cm–3 3.710 3.679 3.652 3.630 ± 0.002 0.002 0.002 0.001 Table 2. Shock compression data Shot # Chen et al. (2002) 999a a This work 1000 110 120 116 113 099 107 161 126 412 Chen et al. (2002) 995 996 This work 981 993 990 994 998 396 409 Flyer/ ufp T (°C) Driver km s–1 Al2024/ Al2024 W/Al2024 Cu/Cu Cu/Cu Ta/Ta Ta/Ta Ta/Ta Ta/Ta Ta/Ta Ta/Ta Ta/Ta Al2024/ Mo Al2024/ Mo Ta/Mo Ta/Mo Ta/Mo Ta/Mo W/Mo Ta/Mo Ta/Mo ± up km s–1 ± 25 1.96 0.001 0.754 0.26 25 25 25 25 25 25 25 25 25 25 2.03 4.63 5.30 5.171 5.38 5.77 6.25 6.494 6.51 7.085 0.01 0.01 0.002 0.006 0.005 0.01 0.001 0.001 0.005 0.012 Us km s–1 ρH g cm–3 ± PH GPa ± 7.05 0.26 4.90 0.03 23.3 0.4 1.51 0.18 2.87 0.08 3.29 0.08 3.58 0.11 3.69 0.17 3.95 0.05 4.27 0.1 4.44 0.11 4.57 0.1 4.864 0.019 6.95 8.2 8.71 9.02 9.44 9.79 10.15 10.33 10.57 10.90 5.61 6.76 7.00 7.26 7.21 7.34 7.59 7.65 7.73 7.81 46.2 103.3 124.7 141.4 153.0 169.4 190.5 200.3 211.9 229.2 0.9 0.8 0.8 1.7 2.1 0.7 1.5 2.1 1.6 2.1 ± 0.18 0.08 0.08 0.11 0.17 0.05 0.1 0.11 0.1 0.04 0.05 0.05 0.06 0.06 0.1 0.03 0.07 0.06 0.08 0.07 1300 1.05 0.004 0.43 0.02 3.23 0.02 4.33 0.12 5.25 0.03 1300 1.95 0.81 0.01 3.91 0.31 4.73 0.05 11.9 0.7 1300 1300 1300 1300 1300 1299 1300 1.52 0.02 1.890 0.002 2.00 0.02 2.52 0.01 2.50 0.01 4.997 0.003 6.050 0.002 4.72 4.94 4.92 5.82 5.85 8.45 9.50 5.09 5.43 5.61 5.70 5.96 6.76 7.15 0.03 21.9 0.14 28.3 0.34 30.1 0.02 43.4 0.1 47.6 0.02 119.5 0.04 161.0 0.3 1.5 2.5 0.2 0.9 0.3 0.6 0.01 1.24 0.02 1.53 0.08 1.63 0.04 1.99 0.01 2.17 0.02 3.82 0.003 4.58 0.01 a 0.01 0.01 0.54 0.01 0.14 0.03 0.05 40mm shots on Caltech polycrystalline fayalite II-42 II-43 Table 3. Parameters used Units ΔEtr ≈ ΔHtr α Cp Cv ρo Co s kJ kg-1 K-1 –1 –1 J kg K J kg–1 K–1 g cm-3 km s-1 Fe2SiO4 liquid Molybdenumd (1573 K) (1573 K) -1584.2194a 8.1949e-05b 1182.35c 1123.13c - Tantalume (300 K) 9.96 4.927 1.288 16.65 3.293 1.307 a Ghiorso and Sack [1995] bderived from this work, (see text and Table 1 for dV/dT) cLange and Navrotsky [1992] dAsimow et al. [2008] eMitchell and Nellis [1981] Table 4. Equation of state fits for molten Fe2SiO4 To ρo Co s γo q KoS KS' KS'' 𝜒2 Units SWEOS SWEOS* 3BM/MG 4BM/MG K g cm–3 m s–1 1573 3.699 2438±5 1.58 ± 0.03 0.412 -1.45 21.99 5.33 1573 3.699 2670±8 1.50±0.03 0.412 -1.04 26.38 4.99 1573 3.699 1573 3.699 0.412 -0.95 21.99 7.28 0.412 -0.55 21.99 6.64 -0.47 0.941 GPa GPa–1 1.526 Source this work AL04 fitted derived fitted derived fitted fitted SWEOS= shock wave equation of state. SWEOS*= shockwave EOS derived from the unconstrained linear fit (see text). 3BM/MG = 3rd order Birch-Murnaghan isentrope. 4BM/MG = 4th order BirchMurnaghan isentrope. Sources: Fitted indicates an adjustable parameter, AL04 is Ai and Lange [2004] II-44 Table 5. BM/MG fits for silicate liquid To ρo γo q KS KS' KS' CP CV Units Fo 4BMa En 4BMb An 4BMc Di 3BMc K g cm–3 1673 2.825 0.816 -1.47 42.19 4.73 -0.11 1926.18 1684.30 1673 2.618 0.365 -1.71 24.66 9.18 -1.87 1761.15 1690.54 1673 2.584 0.18 -1.71d 20.39 2.31 0.60 1528.72 1512.56 1673 2.643 0.49 -1.28 24.57 6.98 GPa GPa–1 J K–1 kg–1 J K–1 kg–1 1612.56 1506.21 Source LC90; L97 LC90 fitted derived derived fitted LN92 derived corrected γo and refit to 4BM b from Mosenfelder et al. [2009] c resampled shock arrivals and refit to BM and excluding some shots, see text. d fixed q value LC90 is Lange and Carmichael [1990]; L97 is Lange [1997]; LN92 is Lange and Navrotsky [1992]. a Table 6. Major oxide proportions for model mantle SiO2 FeO “chondrite” Andrault et al. (2011) mol % 2.6 45.9 1.8 43.5 6.2 KLB-1 Fiquet et al. (2010) mol % 1.5 53.7 1.2 38.3 5.3 KLB-1 Davis et al. (2010) mol % 2.8 50.8 1.8 38.7 5.9 Di An Fa Fo En mol % 1.95 4.40 7.57 24.2 61.9 mol % 0.70 3.32 7.16 56.1 32.7 mol % 2.93 4.98 8.25 55.0 28.8 CaO MgO Al2O II-45 FIGURE CAPTIONS Figure 1. A comparison of previous static sink/float and shock wave compression methods on fayalite liquid. The purple dashed line is an extrapolation of the 1773K isotherm from Agee [1992] with KT’=10.1. The blue solid line is the fayalite liquid Hugoniot from Chen et al. [2002] with KS’=5.36. Estimated shock temperatures are from Chen et al. [2002]. Figure 2. (a) Static image of the back of the molybdenum sample holder inverted 90 degrees clockwise such that the top is to the left. The centermost circle is the sample chamber or “top hat”; the guy wire posts are the smaller circles to the left and right. The vertical yellow lines correspond to the portion of the drivers (two outside lines) and the top hat (center line) that are sampled for shock wave arrival cutoffs. The bright vertical line is a superposed static image of the 25 micron slit in position. (b) On the streak image, the y-coordinate corresponds to position along the slit image in (a) where the x-coordinate is time increasing to the right. Yellow lines denote observed shock arrivals. Blue dashed lines are the 4th-order polynomial extrapolation of the shape of shock front from the drivers. The calibration streak shown at bottom converts an xcoordinate on the photo to a time during the experiment, giving the time between the shock wave passing from driver into sample (red line) and reaching the free surface of the top hat (rightmost yellow line). (c) An enlarged view of the top hat arrival with examples of pixel gray scale value vs. distance along the time axis profiles on the right. The arrival point is picked as the inflection point of the arc tangent function fit (black lines). The amplitude of the arctangent functions is arbitrary. II-46 Figure 3. The double-bob density measurements between 1551 and 1831 K, plotted against temperature, are fit by the line ρ(T) = 3699±1 – 0.30±0.01*(T-1573 K). Figure 4. Density versus pressure diagram showing the new multi-anvil sink/float results for fayalite liquid at 1500°C. “Up” triangles represent observed floats, in which the fayalite liquid was denser than the mineral marker buoy. No sinks were accomplished in the new study. The green curve is the calculated compression for the spessartine garnet marker; the blue curve is the calculated compression for the almandine-rich garnet marker. Fayalite liquid density at 1-bar is from this study (purple diamond) determined by the Archimedean technique. The dashed and dotted curves are the compression curves calculated for fayalite liquid at 1500°C using the K from ultrasonic data [Ai and Lange, 2004] and the K’ derived from the shock wave EOS and the 3rd-order Birch-Murnaghan EOS in this study, respectively (see Table 4). The thin purple solid curve is the compression curve for liquid fayalite from Agee [1992]. Float observations should plot below the correct liquid compression curve. Figure 5. Preheated (1573K) fayalite Hugoniot in shock velocity (Us)-particle velocity (up) space. Blue circles at the highest velocities were performed in this study. The red filled squares are taken from Chen et al. [2002] and were fit with the dotted blue line given in that study. The red solid line is the unconstrained un-weighted linear fit for all data points, and the light green solid line is the constrained un-weighted linear fit for all data points where the intercept is fixed at the bulk sound speed of liquid fayalite measured at 1573K. The hollow symbols are the initially solid experiments (300K) that were shocked to liquid from Chen et al. [2002] (hollow red II-47 squares) and this work (hollow blue circle). The dashed green line represents a cold Hugoniot offset from the fitted hot liquid Hugoniot using the derived model for the liquid Grüneisen parameter. Figure 6. The behavior of Grüneisen parameter for liquid and solid silicates with compression. The reference densities for the liquids (solid lines) are at 1673K for MgSiO3 [Mosenfelder et al., 2009] and Mg2SiO4 [Mosenfelder et al. 2009, corrected in this chapter] and at 1573 K for Fe2SiO4 (this study). The gamma functions for the solid materials are shown by dashed lines: magnesium perovskite (Pv) [Mosenfelder et al. 2009]-“BM3S model”; majorite (Mj) and enstatite (En) [Stixrude and Lithgow-Bertelloni, 2005]. Figure 7. Hugoniot data plotted in pressure-density space with thermal EOS fits for fayalite liquid and shock-melted single crystal all fixed at the Archimedean density at zero pressure. Data symbols are the same as Figure 5. Abbreviations: SWEOS- shock wave equation of state; BM/MG – Birch-Murnaghan/Mie-Grüneisen EOS; dKSFR- de Koker –Stixrude fundamental relation Figure 8. Models for the early mantle upon cooling showing the comparison of the liquidus curves (black solid line) with the isentropic temperature profile (dotted line) of the same composition. (a) “Chondritic” composition and liquidus from Andrault et al. [2011]. (b) Peridotite KLB-1 composition and liquidus from Fiquet et al. [2010]. The critical isentrope with II-48 its mantle potential temperature is shown in red. The EOS parameters used for En, Fo, An, and Di are given in Table 5 and Fa 3BM/MG parameters in Table 4. This figure is redone with the revised EOS in Chapter III Figure 8. Figure 9. Calculated densities as function of pressure along a 10K/GPa mantle geotherm. The density at base of mantle (135 GPa) for each composition is thermally corrected from 3073K to 4400K (shaded cones). Also shown is the density profile from PREM [Dziewonski and Anderson, 1981] and the inferred density for a ULVZ from seismic reflection coefficients [Rost et al. 2006]. This figure is redone with the revised EOS in Chapter III Figure 9. Figure 10. The density of liquid mixtures within En-Fs-Fo-Fa space at 135 GPa. The solids lines represent liquid mixtures that are neutrally buoyant with respect to solid magnesium perovskite (Pv) at different core-mantle-boundary temperatures from 3700-4500K. Any liquid compositions above these lines are less dense than Pv (white area) and any below are denser (blue striped area). Constant density lines for liquid compositions at 6.12 g/cc (dashed lines) represent the upper density limit for ULVZ’s inferred from Rost et al. [2006]. The blue C represents the “chondrite” composition from used in Andrault et al. [2011], and the yellow P is the peridotite composition from Fiquet et al. [2010]. Both compositions are given in Table 6. The gray arrows indicate the direction of liquid composition change with fractional crystallization of Pv (disregarding partitioning of Fe). a) The EOS parameters used for En and Fo are given in Table 5 and Fa 3BM parameters in Table 4. b) The EOS parameters used for En and Fa are given in Table 4 and Fo 3BM parameters in Table 3 of Chapter III. II-49 Figure 1 II-50 Figure 2 II-51 Figure 3 II-52 Figure 4 II-53 Figure 5 II-54 Figure 6 II-55 Figure 7 II-56 Figure 8 II-57 Figure 9 II-58 Figure 10 III-1 Chapter III Direct shock compression experiments on pre-molten Mg2SiO4 and progress towards a consistent high-pressure equation of state for CaO-MgO-Al2O3-SiO2-FeO liquids Claire W. Thomas Paul D. Asimow For submission to Journal of Geophysical Research April 4, 2013 III-2 ABSTRACT We performed shock compression experiments on preheated forsterite liquid (Mg2SiO4) at an initial temperature of 2273 K and have revised the equation of state (EOS) that was previously determined by shock melting of initially solid Mg2SiO4 (300 K). The linear Hugoniot, US = 2.674±0.188 + 1.64±0.06 up km/s, constrains the bulk sound speed within a temperature and composition space as yet unexplored by 1-bar ultrasonic experiments. We have also revised the EOS for enstatite liquid (MgSiO3) to exclude experiments that may have been only partially melted upon shock compression and also the EOS for anorthite liquid, which now excludes potentially unrelaxed experiments at low pressure. The revised fits and the previously determined EOS of fayalite and diopside were used to produce isentropes in the multicomponent CaO-MgO-Al2O3-SiO2-FeO system at elevated temperatures and pressures. Our results are similar to those previously presented for peridotite and simplified “chondrite” liquids such that regardless of where crystallization first occurs, the liquidus solid sinks upon formation. This process is not conducive to the formation of a basal magma ocean. We also examined the chemical and physical plausibility of the partial melt hypothesis to explain the occurrence and characteristics of ultralow velocity zones. We determined that the ambient mantle cannot produce an equilibrium partial melt and residue that is sufficiently dense to be a ULVZ mush. The partial melt would need to be segregated from its equilibrium residue and combined with a denser solid component to achieve a sufficiently large aggregate density. III-3 INTRODUCTION The volumetric behavior of silicate melts at high pressures and temperatures is not well constrained despite its importance for accurately modeling the formation and early evolution of the Earth. Our ability to predict the chemical and physical consequences of the Earth’s differentiation, for example, after the putative moon-forming impact, depends on a formulation of a liquid equation of state (EOS) accurate enough to determine the buoyancy of liquids relative to coexisting solids. Furthermore, the liquid EOS is also needed for rigorous interpretation of present-day deep Earth seismic observations: detection of ultra-low velocity zones (ULVZ) [Garnero and Helmberger, 1995] at the base of the mantle have been interpreted to possibly indicate the presence of liquid—potentially a partial melt [Lay et al., 2004; Williams et al., 1998] or a dense liquid that has remained un-sampled at the core-mantle boundary since differentiation of the planet [Labrosse et al., 2007]. This latter idea is based in part on the simplified modeling of the mantle as an isochemical MgSiO3 system [Mosenfelder et al., 2009], which would begin crystallization in the mid-mantle for a full mantle magma ocean. This concept provides the potential for creating two separately evolving chemical reservoirs in the upper and lower mantle that would allow distillation of a presumed dense melt. Yet applying this same idea to slightly more complex multicomponent mantle compositions in the CaO-MgO-Al2O3-SiO2-FeO (CMASF) system demonstrates that mid-mantle crystallization is not very likely or at least unlikely to lead to dynamically separate reservoirs [Thomas et al., 2012]. Similarly, our current knowledge of the density differences between complex multicomponent silicate liquids and the present-day ambient mantle are not adequate to make precise predictions of the gravitational stability of a liquid-mush ULVZ. III-4 Most of our understanding on the volume and elastic behavior of silicate liquids has been limited to pressures and temperatures within the upper mantle. Data are derived by density and sound speed measurements at 1 bar [Ai and Lange, 2008; Ghiorso and Kress, 2004; Lange and Carmichael, 1990] or by sink-float experiments at higher pressures up to a few GPa [Agee, 1992; 1993; Ohtani et al., 1995; Suzuki et al., 1998]. Diamond anvil cell (DAC) experiments are conducted at temperature and pressure regimes comparable to the lower mantle, but even distinguishing whether melting has occurred remains a non-trivial exercise (techniques for detecting melt vary among laboratories and results can often differ [Andrault et al., 2011; Fiquet et al., 2010]) and measuring the density of silicate melts in situ in the DAC remains an elusive goal. Shock experiments that start at ambient conditions and produce melt upon dynamic compression have achieved pressures equivalent to those in the deep mantle but can often be difficult to interpret due to solid-solid phase transitions along the Hugoniot or challenges in deciphering how far the melt reaction has been over-stepped [Akins, 2003; Akins et al., 2004; Luo et al., 2002; Mosenfelder et al., 2007; Mosenfelder et al., 2009]. Recently, preheated shock techniques have achieved pressures equivalent to those in the lower mantle for fully molten anorthite (An; CaAl2Si2O8), diopside (Di; CaMgSi2O6), An36-Di64 eutectic [Asimow and Ahrens, 2010], and fayalite (Fa; Fe2SiO4) [Thomas et al., 2012] compositions. To this shock dataset of liquids we have added a series of novel experiments, which preheat Mg2SiO4 (forsterite, Fo) beyond its melting point at ambient conditions to 2273 K prior to dynamic compression. These experiments compliment the shock experiments performed on initially solid forsterite and wadsleyite (300K) [Mosenfelder et al., 2007], and permit reevaluation of the previous EOS without the ambiguities of distinguishing solid-melt transformations along the Hugoniot. III-5 These experiments are also are the first to measure thermodynamic properties of Mg2SiO4 melt directly. The high melting temperature of forsterite (2163±20 K; [Bowen and Andersen, 1914]) has made in situ experimental measurements of melt properties difficult for this composition. They are instead typically calculated using linear mixing of partial molar properties extrapolated from lower temperatures and higher SiO2 compositions [Lange, 1997; Lange and Carmichael, 1990; Lange and Navrotsky, 1992; Stebbins et al., 1984]. Our newest measurements constrain the sound speed of silicate liquid within temperature and composition space yet unexplored by ambient-pressure ultrasonic experiments [Ai and Lange, 2008; Ghiorso and Kress, 2004]. METHODS Sample Preparation of Preheated (2273 K) Shots The Mg2SiO4 used for shock wave experiments was un-doped, single crystal forsterite (Morion Co. Gems) cored and lapped into disks. The sample disks under-filled the volume of the molybdenum sample holders by 7-13% at room temperature [cf. Thomas et al., 2012]. The room temperature under-filling—accounting for thermal expansion of Mo and the thermal expansion and volume of fusion of forsterite—results in a 1 – 1.5 mm-deep meniscus at the top of the heated capsule at 2273 K. This bubble is above the area imaged by the streak camera and therefore still allows observation of simple one-dimensional wave propagation. A void space is preferred to over-filling of the capsule, which can result in warping or failure of the welded cap. The surfaces within and outside the sample well including both sides of the cap were polished to a mirror finish with 1-micron alumina grit. Smoothing all the surfaces prevents bubbles from clinging to the walls of the sample holder during heating and potentially interfering with the imaged shock wave. The caps were electron-beam welded (Electron Beam Engineering III-6 Inc., Anaheim, CA) to the sample holder wells enclosing the sample. The details of this process are described in Thomas et al. [2012]; we note that forsterite, unlike fayalite, welded easily with only occasional and minor oxidation of the cap. Before and after welding, a profile of the topography aligned to the streak camera slit was taken with a depth gauge micrometer using a 0.7 mm flat-end tip. The inner portion of the sample well was measured in 0.2 mm increments, and the final driver and welded cap (or “top hat”) were measured in 0.1 mm increments. Typically the topography varied no more than ±0.008mm from flat for the inside of the sample well and ±0.015mm for the outside driver and top hat. Sample capsules were carbon coated on both sides to reduce oxidation of the reflecting molybdenum surface during heating [cf. Asimow and Ahrens, 2010; Asimow et al., 2008] except for shots completed on the 40-mm propellant gun. Oxidation of the target was less of a problem in the 40-mm gun possibly due to better vacuum for the smaller target chamber and catch tank volume compared to that of the 90mm/25mm two-stage light-gas gun (LGG). Experimental Setup The pioneering work for shock studies on molten materials is Rigden et al. [1984]. The description of our methods below builds on this and the work of Rigden et al. [1988; 1989], Miller et al. [1988; 1991a], Chen and Ahrens [1998], and Chen et al. [2002]. For more recent changes in experimental techniques and data analysis, the reader is directed to Asimow et al. [2008], Asimow and Ahrens [2010], and Thomas et al. [2012]. For this study, six total experiments were performed — four in the Caltech 90/25-mm twostage LGG and two using the Caltech 40-mm propellant gun. All shots were preheated to 2000°C (2273 K) using a H2O-cooled copper induction coil powered by a 10 kW Lepel radio frequency heater [Chen and Ahrens, 1998]. The targets were held in place by a “guy-wire” set up and III-7 mounted through a hole drilled in a high-temperature Zircar™ zirconia board as opposed to the alumina board used for lower-temperature experiments [cf. Asimow and Ahrens, 2010].The boards were also cut with a slit from the center hole through to the bottom including two separate inch-deep slits at the top (a Y shape) in order to relieve mechanical stresses that arose due to thermal expansion of the board while at high temperatures. Lexan projectiles with molybdenum flyer plates (see Table 1) were used in this study; flyer plate velocities (ufp) ranged from 1.04 5.963 km s-1. For the LGG shots, ufp was measured redundantly by a double-flash x-ray system and a two-magnet induction detector as described in Asimow et al. [2008]. The ufp for the 40-mm gun shot was also measured by two methods: double-exposure x-ray image and laser cutoffs. The 40-mm x-ray system utilizes a double-flash x-ray, which doubly exposes the flyer image on a single piece of film. Velocity is determined by the separation of the two flyer images — taking into account the magnification on film and the previously calibrated parallax distance — divided by the time between the two x-ray discharges [Rigden et al., 1988]. The 40-mm gun was also outfitted with a new laser timing system using 3 lasers aimed across the projectile path onto small active-area photodiode detectors. The distance between the lasers was measured to within ±0.05 cm, and the time between cutoffs is recorded on a 1 GHz oscilloscope. The velocities measured by the two techniques for each gun agreed to within the stated uncertainty for all the shots, and the average of the two values was used for the ufp given in Table 1. During each experiment, the rear face of the target was illuminated by a Specialised Imaging xenon spark lamp and filmed by a Hadland Imacon 790 streak camera through a narrow (25 μm) slit focused horizontally across the center of the driver and sample cap. The image of the shockwave transit was recorded using a new Specialised Imaging 2DR digital readout system. The system collects the streak image using a 4008 x 2688 pixel CCD, which was specifically III-8 designed to affix to the streak camera tube. The image is captured digitally, in contrast to the previous method, which relied on digital scanning of analog images captured on Polaroid film [cf. Asimow and Ahrens, 2010; Thomas et al., 2012]. The precise measurement of the shock transit time through the sample and molybdenum cap was measured from the extinguished (or sharp intensity change in) reflected light as the shock wave reached the free surface of the driver plate followed by the sample cap. The procedure for picking cutoffs and the calibration of the streak rate is described in Thomas et al. [2012]. The selected cutoffs were co-registered with the previously measured driver topography profiles and corrected for deviations from flat. The shape of the shock front as it enters the sample is determined by interpolating the visible driver cutoffs using a 4th-order polynomial. The 95% confidence interval for the fit was used as the upper and lower bounds for the location of the shock front and was the greatest source of error in the final calculated shock state — i.e., the error bars for shock pressure (PH), shock density (ρH), particle velocity (up), and shock wave velocity (Us). The time that the shock wave spends in the Mo cap is estimated using the iterative method of Rigden et al. [1988] and subtracted from the offset in time between the two cutoffs to calculate the shock wave velocity Us in the sample. Then up, PH, and ρH follow from impedance matching, the Rankine-Hugoniot equations, and the following input parameters: standard Hugoniot data (ρo, Co, s) for the cold metal flyer and hot Mo driver plates [Asimow et al., 2008; Chase, 1998] (see Table 2), the initial sample density (ρo) [Lange, 1997], and an initial guess for Co and the s parameter of Mg2SiO4 liquid to seed the iteration (the converged result is independent of this guess). RESULTS Linear Hugoniot The shock wave data for Mg2SiO4 liquid are reported in Table 1, including shot number, flyer/driver material, temperature prior to firing, ufp, up, US, ρH, and PH. III-9 A Hugoniot is a family of peak shock states achieved in a material by progressively stronger shocks from the same starting conditions [Ahrens, 1987] (in this case, liquid Mg2SiO4 at 2273 K and 1 bar). Empirically, the Hugoniot of a well-behaved material forms a line in US-up space, given to third order in strain by US = Co + s up [Jeanloz, 1989]. The slope (s) is related to K’S, the pressure derivative of the isentropic bulk modulus (KoS) by s = (K’S + 1)/4; the intercept, Co, is the bulk sound speed of the material at room pressure (Co2 = KoS / ρo) [Ruoff, 1967]. Typically, the intercept of the un-weighted linear Hugoniot in US-up space falls within error of the 1-bar value of the sound speed measured by ultrasonic techniques indicating relaxed (as opposed to “glass-like”) behavior upon compression. Currently there are no 1-bar sound speed measurements on CaO-MgO-Al2O3-SiO2 (CMAS) liquids with temperatures in excess of 1727K and mole fractions of MgO greater than 0.46 (Figure 1). Sound speed values can be calculated from the models of Ghiorso and Kress [2004] or Ai and Lange [2008] assuming linear addition of oxide component Co and dCo/dT but must be extrapolated well beyond the composition and temperature space used in either calibration (which is not recommended by the authors). The sound speed for Mg2SiO4 liquid at 2773K from Ghiorso and Kress [2004] is 3.195 km/s and from Ai and Lange [2008] is 3.126 km/s. Figure 2 shows linear fits to shock data in US-up space fixed at these 1-bar model values as well as the unconstrained fit. The unconstrained, un-weighted linear fit to all preheated Mg2SiO4 liquid data points yields US = 2.674±0.188 + 1.64±0.06 up km/s (r2 = 0.995). The unconstrained Co falls below the derived model values potentially indicating that the dCo/dT value is negative for Mg2SiO4 liquid, contrary to the positive [Ghiorso and Kress, 2004] and zero [Ai and Lange, 2008] values given in the previous studies. This is not evidence for unrelaxed behavior in the shock compression experiments; this would be expected to yield a higher bulk III-10 modulus than expected from relaxed ultrasonic data. Instead it is evidence that the temperature or composition dependence of those models cannot be extrapolated to the current conditions; see discussion below. Consequently, the KoS of 18.56±2.61 GPa (derived from the expression KoS = ρo Co2 and the unconstrained Co) is lower than the bulk moduli derived from either the Ghiorso and Kress [2004] or the Ai and Lange [2008] values, 26.55 and 25.38 GPa, respectively. This KoS is also much lower than the previously reported value given in shock wave studies, 41 GPa at To=1673K [Mosenfelder et al., 2009], which was derived from the Lange and Carmichael [1990] data set. If corrected to an initial temperature of 2273K using dV/dT and d2V/dPdT [Lange and Carmichael, 1990], KoS is 27.87 GPa and still much stiffer than our newest value of 18.56 GPa. The slope of our unconstrained Hugoniot fit corresponds to a K’ of 5.58±0.24, compared to that of the previous estimate of 4.73 [Thomas et al., 2012]. Grüneisen parameter The thermodynamic Grüneisen parameter (γ) is a macroscopic parameter that relates thermal pressure to the thermal energy per unit volume. The Mie-Grüneisen approximation this can be expressed as γ= 1 ∂Pth ρ ∂E (1) where thermal pressure (Pth) is defined as the increase in pressure due to heating at a constant volume [Poirier, 2000]. The Grüneisen parameter can be directly determined by the comparison of two Hugoniots with different initial densities [e.g. Asimow and Ahrens, 2010; Luo et al., 2002]. When compared at an equal density, a liquid obtained by shock compression of an initially solid material will have a distinct pressure and internal energy state than that obtained by direct shock compression of the liquid of the same composition. This method also applies for III-11 other changes in the initial state of shocked samples, including differences in initial solid phase and variations in initial porosity [Mosenfelder et al., 2009]. We determined the Mg2SiO4 liquid γ by comparison of our 2273K liquid Hugoniot with the initially solid 300K forsterite Hugoniot from Mosenfelder et al. [2007]. Mosenfelder et al. used two Mg2SiO4 starting compositions, polycrystalline forsterite and wadsleyite, and they asserted that the two highest pressure shots from each dataset were shock melted (shots #350 and #349 forsterite and #350 and #349 wadsleyite). However, the calculations of de Koker et al. [2008] indicate that shock compression of crystalline forsterite along the 300K principal Hugoniot produces incongruent melting to periclase (MgO) and a more silica rich liquid in the pressure range of 150-170 GPa; only above 170 GPa is pure Mg2SiO4 liquid present on the calculated Hugoniot. Correspondingly, their calculated wadsleyite Hugoniot was found to be 1000-1400 K cooler than the forsterite Hugoniot. Therefore even the highest-pressure wadsleyite datum (shot #350) is unlikely to be fully molten and does not provide a secure point for evaluation of liquid properties. We therefore have used only the highest forsterite point (#350, 188.5 GPa) from the Mosenfelder et al. [2007] dataset for determination of the liquid Grüneisen parameter. For the equations and details of this calculation, the reader is directed to Asimow and Ahrens [2010] and Thomas et al. [2012]. Using the power law form for γ ρ  γ ( ρ ) =  o   ρ  q (2) and γo = αK oS , ρoCP (3) III-12 yields a q value of –1.51 and γo = 0.448. This corroborates the general trend that has been observed by both calculation [Adjaoud et al., 2008; Boehler and Kennedy, 1977; de Koker et al., 2008; Stixrude and Karki, 2005] and experiment [Asimow and Ahrens, 2010; Mosenfelder et al., 2009; Thomas et al., 2012] that liquid γ increases upon compression, opposite to the behavior of solids. This q value is also fairly similar to the Mg2SiO4 fit from Thomas et al. [2012] (which corrected a sign error in the re-fit from Mosenfelder et al. [2009]), but differs greatly in the value for γo. The difference in these fits can been seen most clearly in Figure 3, which displays the previous γ curve (dotted) and the newest fit tied at a much lower γo. This offset is due to the very discrepant values for KoS used in (3), 41 GPa and 18.56 GPa for the previous and current study, respectively. Our newest fit for γ is also in much closer agreement to the linear fit given in de Koker et al. [2008] (Figure 3) and other silicate liquid γ functions derived from shock experiments. Thermal Equation of State Fitting The Hugoniot reaches temperatures and energies much higher than those of geophysical interest at lower mantle pressures (even for early Earth processes). It is therefore pertinent to select and apply a thermal equation of state formalism to investigate material properties that lie off the Hugoniot. We attempt to define the entire P-V-E surface of Mg2SiO4 liquid using the shock wave equation of state (SWEOS) and the 3rd- and 4th-order Birch-Murnaghan/MieGrüneisen equations of state (3BM/MG and 4BM/MG). The results and uncertainties for each fit are given in Table 3, and the Hugoniots are plotted in Figure 4. The SWEOS is defined by a linear Hugoniot in US – up space, converted to P - ρ space using the first and second Rankine-Hugoniot equations [e.g., Ahrens, 1987]. The slope of our linear Hugoniot, US = 2.674±0.188 + 1.64±0.06 up km/s, corresponds to a K’S value of III-13 5.58±0.24. States that lie off the Hugoniot are found using the Mie-Grüneisen thermal pressure approximation with a temperature-independent power law expression for the thermodynamic Grüneisen parameter (2). A q value of –1.51 was independently determined in the above Section 3.2 using selected experimental results from Mosenfelder et al. [2007]. The 3rd- or 4th-order BM/MG EOS is defined by a 3rd- or 4th-order Birch-Murnaghan isentrope centered at 1 bar and 2273K and a Mie-Grüneisen thermal pressure approximation. The 3BM/MG and 4BM/MG EOS are global fits, which include the six newest initially molten data points (Table 1) and the initially solid shot #350 forsterite from Mosenfelder et al. [2007]. Including the initially solid point (#350) reduces the error bars slightly for both fits; however, excluding the point returns nearly identical EOS parameters. We estimated uncertainties in these fits using bootstrap resampling of the data set; however, with only seven data points each at a distinct pressure, rather than many random samples from a population of Hugoniot constraints, this exercise yields rather artificial results. We applied two sets of 3rd- and 4th-order BM/MG fits at To= 2273 K, one where KoS was fixed to 18.56 GPa (derived from the unconstrained Co) and a second where KoS was permitted to vary within the stated error of the unconstrained Co. For the first set of fits, the 3BM/MG EOS result is KoS= 18.56 GPa, K’S = 6.37±0.04, q = –2.16±0.14, and reduced χ2 = 2.13. The 4BM/MG fit result is KoS= 18.56 GPa, K’S = 3.12±2.39, KS” = 1.26±1.47 GPa-1, q = 1.31±6.74 and reduced χ 2 = 1.18. The 4BM/MG evidently has very large error bars and strong correlations among the output parameters indicating a very unstable fitting routine. The 3BM/MG fit by contrast has less severe correlation between the parameters and appears justified by the fitting statistics. For the second set of fits in which we allowed KoS to vary, the best KoS value was 16.41 GPa. This KoS is the limit at which the derived Co still yields a linear Hugoniot in US-up space. III-14 The 3BM/MG fit with this KoS value is K’S = 7.37±0.81, q = –2.02±1.03, and reduced χ2 = 1.65. The 4BM/MG fit is KoS = 16.41 GPa, K’S = 4.27±2.02, KS” = 1.31±1.83 GPa-1, q = 0.47±9.86, and reduced χ 2= 1.11. This 4BM/MG fit is also unstable with large error bars and strong correlations among the parameters, but the 3BM/MG fit is much more reasonable. This 3BM/MG fit has a smaller reduced χ2 than the 3BM/MG fit above; hence for this dataset, we prefer the 3rd-order fit with KoS = 16.41 GPa. Revised Equations of State for MgSiO3 and CaAl2Si2O8 liquids Enstatite, MgSiO3 The isentrope of MgSiO3 derived from the recommended global EOS fit to shock wave data from all initial polymorphs and porosities of MgSiO3 [Mosenfelder et al., 2009] displays an inflection point (concave down to concave up) in T-P space. Although this topology is not necessarily unphysical, it conflicts with the simple concave down behavior of isentropes derived from shocked liquids only (Fa, Di, Di64An36 and Fo). It is possible that the volume decreases along the Hugoniot documented for the MgSiO3 shock wave data [Mosenfelder et al., 2009] may indicate that the sample is only partially molten as opposed to fully molten upon shock. Such an explanation would be consistent with the interpretation of discrepancies between shock data and MD simulations noted above for some Mg2SiO4 experiments [de Koker et al., 2008]. However, this idea is contrary to the previous assumption that silicate minerals require such large overstepping of the liquidus that partial melting does not occur [Akins et al., 2004]. To test whether the enstatite global fit may include non-liquid data, we exclude all but six of the highest pressure points from the glass [Mosenfelder et al., 2009], enstatite crystal [Akins et al., 2004], oxide mix [Marsh, 1980], and porous [Simakov and Trunin, 1973] data sets. These III-15 points are the most likely to be unambiguously molten during passage of the shock wave. The hollow points in Figure 5 shows the data used in this fit. The resulting 4BM/MG fit yields KoS= 24.66 GPa, K’S = 10.06±0.95, KS” = 2.34±0.82 GPa-1, q= –0.88±0.68, and reduced χ2= 1.72 at To = 1673.15 K. Due to the low number of points being fit, the bootstrap trial values for the output parameters for K’ and q were highly correlated and were in two very distinct parameter populations — one with reasonable q ≤ 1, and another with very high K’ (~14) and unreasonably high q values (40-90). This higher set is disregarded when determining the error bars on q. This fit has a similar reduced χ2 value to the fit given in Mosenfelder et al. [2009] (χ 2 = 1.89) but overall displays a simple concave down isentrope in TP space. Although not conclusive, this result may indicate that the inflection point on the model isentrope in dT/dP space was an artifact of fitting both molten and non-molten data as opposed to showing a unique physical behavior of enstatite liquid. Therefore this result also demonstrates that the technique of detecting shock melting via significant (~3%) volume [Akins et al., 2004] and temperature drops [Luo et al., 2004] may not be as straightforward as previously assumed and may only indicate partial melting of silicate samples. Anorthite, CaAl2Si2O8 In Thomas et al. [2012], the anorthite and diopside liquid data of Asimow and Ahrens [2010] were re-analyzed only sampling the center 3 mm of the arrival top hat (see Table 1). This re-sampling was motivated by a hydro-code model of shock propagation through the capsule geometry that indicated that the outer part of the top hat arrival was influenced by edge effects. The Di EOS and the An EOS parameters were both adjusted, but only the An fit displayed an inflection point for the isentrope in T-P space. As discussed in the MgSiO3 section above, it is III-16 believed this topology is more likely an artifact of fitting non-similar experimental states as opposed to representing the actual isentropic behavior of the liquid. The anorthite liquid Hugoniot is composed of two separate experimental data sets—one at low pressure (< 36 GPa) [Rigden et al., 1989] and one at high pressure (< 126 GPa) [Asimow and Ahrens, 2010]. The Rigden et al. [1989] experiments were re-analyzed in Asimow and Ahrens [2010], but the Co derived from this Hugoniot in US-up space still did not fall within error of the Co determined in 1-bar ultrasonic measurements [Lange and Carmichael, 1990]. In fact, anorthite liquid is the only composition studied thus far by preheated shock wave experiments (for which an ultrasonic velocity is available at similar temperature) that displays un-relaxed behavior. In the original Rigden et al. study, the data was fit with two separate Hugoniots due to a sharp offset observed in US-up space for the two highest pressure points. These same data points within the 25-36 GPa range have since been interpreted to be un-relaxed during compression [Asimow and Ahrens, 2010; de Koker, 2010; Ghiorso et al., 2009] despite being in the middle of the full pressure range now studied. Yet excluding these points alone does not resolve the discrepant behavior obtained by simultaneously fitting the lowest pressure points of Ridgen et al., the high pressure data of Asimow and Ahrens, and the 1 bar bulk sound speed [Lange and Carmichael, 1990]. One potential reason for this discrepancy is that the two shock data sets should not actually lie on the same Hugoniot, meaning the difference in the initial temperatures for two sets of experiments may be greater than stated. The Asimow and Ahrens experiments were heated to initially 1932 K and measured using a pyrometer; the Ridgen et al. experiments ranged from 1902 K to 1919 K and were measured using a thermocouple (actual experimental temperatures of Ridgen et al. are published in Asimow and Ahrens [2010]). Typically a 30 K temperature III-17 difference should not be significant for fitting the Hugoniot, but perhaps temperature errors using the thermocouple are larger than expected with consequently some of the experiments being conducted at significantly cooler conditions. Other possibilities include the experiments not being held at a high enough temperature above the melting point (1826 K) or held long enough above the liquidus before firing. Finally, it is also possible that there are no experimental discrepancies but that the volumetric and elastic behavior of anorthite at low pressure may be difficult to probe due to its highly polymerized structure. The rise time for the shock may be comparable to the relaxation time at least until much stronger shocks and higher temperature conditions along the Hugoniot are achieved. Although several plausible explanations can be put forward for the seemingly unusual character of the Ridgen et al. anorthite data, preliminary shock results on mixed liquids with high mole fractions of anorthite component (An50Hd50 and An33Hd33Di33) (see Chapter IV), in contrast, display predictable, relaxed behavior upon compression. Although it does not provide conclusive evidence, this newest data does support the likelihood of experimental differences between the two studies as opposed to unique low-pressure behavior for anorthite liquid. In summary, although we lack a definite understanding of the source of the low-pressure anorthite anomaly, we proceed here to derive a new fit for anorthite that can be used in conjunction with other compositions to synthesize the overall behavior of silicate liquids over a broad composition range. This is obtained from Ridgen et al. shot 665, the re-analyzed Asimow and Ahrens [2010] points [Thomas et al., 2012] and the experimental 1-bar sound speed at 1932 K [Ai and Lange, 2008]. The parameters are for the 4BM/MG fit are KoS = 19.77 GPa, K’S = 3.72±2.14, KS” = 0 .37±2.21 GPa-1, q = –1.86±1.14, and reduced χ2= 3.09 at To = 1932 K. This fit displays the same two highly correlated output parameter populations due to the low number III-18 of points included in the fitting as seen in the MgSiO3 fitting in the section above. The errors are determined using only the population of bootstrap trials with q ≤ 1. The errors for this fit are still unavoidably large, but the fit recovers the data well and also displays a concave down isentrope in T-P space. DISCUSSION Sound Speed of Mg2SiO4 liquid Although our derived Co is not within the error of the 1-bar models, it is unlikely that our liquid is un-relaxed or crystallized upon compression especially at the lowest experimental pressure (9.1 GPa, see Table 1). First, it would be typical for glass-like behavior to display bulk moduli higher than the ultrasonic values [Rivers and Carmichael, 1987]. Although the model values are extrapolated beyond their parameter space, it is at least encouraging to observe that the liquid does not have stiffer behavior than expected from these models. Second, it has been demonstrated that crystallization during the time of the experiment is improbable since the rise time of the shock wave in the sample is much less than the time necessary for crystallization to occur [Rigden et al., 1988]. Rise time (τr) can be estimated from τr = uPη U S ∆Pmax (4) [Jeanloz and Ahrens, 1979], where η is the effective viscosity and ΔPmax is the maximum pressure offset between the equilibrium Hugoniot and the Rayleigh line at PH(Us, up). Although η is not well constrained, the experimental 1-bar viscosity at ~2000°C for near Mg2SiO4 liquid composition (65.1 mol% MgO) is less than 0.1 Pa s [Urbain et al., 1982]. Using conservative estimates of 0.1-3 Pa s for the change in viscosity with elevated temperature and pressure [Adjaoud et al., 2008], τr calculated at our lowest pressure point (9.21 GPa) is on order of 10-10 III-19 seconds. This rise time is more than five orders of magnitude less than the time needed for crystallization for basaltic liquids derived from the time-temperature-transformation curves given in Uhlmann et al. [1982] (~10-5 s). Fang et al. [1983] further show that crystallization times increase with decreased polymerization; therefore crystallization is most likely kinetically impeded on the timescales of the experimental shock in Mg2SiO4 liquid. Finally, the most compelling evidence that crystallization did not occur during shock compression is that the temperature-pressure path of the Hugoniot is unlikely to have crossed into the solid forsterite stability field. The Mie-Grüneisen model to estimate shock temperature (TH) is given by TH = PH − PS + TS γCV ρ (5) where TS and PS are temperature and pressure along the BM isentrope, CV is the specific heat at constant volume, and γ is the Grüneisen parameter defined as a function of volume only (1). Using the 3BM/MG EOS fit (calculated in Section 3.3), Figure 6 shows that for even the highest temperature estimates of the Mg2SiO4 liquidus curve [de Koker et al., 2008; Ohtani and Kumazawa, 1981] the 2273 K Hugoniot lies within the liquid field. Consequently, we are confident that our experiments are probing relaxed liquids and that the unconstrained intercept of the linear Hugoniot of 2.674±0.188 km/s is the most accurate description of the bulk sound speed at 1 bar and 2273 K. This result implies that the dCo/dT for Mg2SiO4 is negative, i.e., sound speed decreases as temperature increases. This result is actually more in line with intuition than the previous estimates of zero [Ai and Lange, 2008] and positive dCo/dT values [Ghiorso and Kress, 2004]. Unless there are special structural rearrangements in liquid that overcome the general tendency of materials to become more anharmonic with increasing temperature and vibrational energy, one would expect liquids to act like solids and display decreasing sounds speed as temperature III-20 increases. Results from de Koker et al. [2008] also indicate a decrease in sound speed with increasing temperatures from 3000 K to 6000 K; yet extrapolation of these results to 2273 K gives a value of 3124 m/s, which does not however agree with our measured value. Preliminary modeling of this apparent complex behavior for sound speed suggests it may be due to cation coordination change and the greater degrees of structural freedom of liquids compared to solids [Wolf et al., 2012]. Isentropes for full mantle magma oceans The examination of liquidus topologies has been previously employed to describe dynamics of a solidifying magma ocean [Andrault et al., 2011; Miller et al., 1991b; Mosenfelder et al., 2009; Thomas et al., 2012]. Crystal/liquid density crossover points and isentrope-liquidus intersections are both key phenomena, and the relationship between these points is important to understand. Originally treated by Miller et al. [1991b], the crystallization of a chondritic (irondepleted CI composition—see Agee and Walker [1988]) magma ocean was depicted as a sequence of isentropes with decreasing specific entropy (or potential temperature, TP). The tangency of the liquid isentrope with the liquidus (maximum in S-P space) gives the pressure of first crystallization. The liquidus solid is the composition of the first forming crystal, and its density contrast with respect to the ambient liquid (whether is sinks, floats, or is neutrally buoyant) has important ramifications for understanding the first steps in the dynamic modeling of the Earth’s differentiation. The one-component liquid systems enstatite and forsterite [Mosenfelder et al., 2009] have been used as compositional analogues for the mantle and exhibited deep (>80 GPa) maxima in SP space. The liquidus-isentrope intersections for two multicomponent systems, KLB-1 peridotite III-21 [Fiquet et al., 2010] and synthetic chondrite [Andrault et al., 2011], have also been previously examined [Thomas et al., 2012]. The liquidi were taken from the above referenced studies; the details for calculating the isentropes of liquid mixtures in CaO-MgO-Al2O3-SiO2-FeO composition space are given in Thomas et al. [2012]. Isentropes derived using the newest Mg2SiO4 EOS (Table 3) and other re-analyzed end-member EOS of MgSiO3 and CaAl2Si2O8 (Table 4) yield slightly different results yet similar consequences to those previously presented. The critical isentrope for simplified chondrite composition shown in Figure 7 (Ch=.62En+.24Fo+.08Fa+.04An+.02Di) has a potential temperature (Tp) of 2960 K (compared to the previous estimate of Tp =2600 K), and the intersection occurs similarly at the base of the mantle (135 GPa). The peridotite (P=.33En+.56Fo+.07Fa+.03An+.007Di) critical isentrope is Tp = 3050 K and intersects the liquidus at 105 GPa, a slightly higher temperature (Tp =2900 K) and higher pressure than previously reported (~85 GPa). The first crystallizing phase observed on the liquidus in both experimental studies [Andrault et al., 2011; Fiquet et al., 2010] is Mg-perovskite (MgSiO3, Pv). Using the BM3S model given in Mosenfelder et al. [2009] (which includes the static data of Saxena et al. [1999]), the density of Pv at each intersection point can be calculated to determine whether the first forming crystals would sink or float. For chondrite, the density of Pv at 135 GPa and 4690 K is 5815 kg m-3, whereas chondrite liquid is 5150 kg m-3. Peridotite liquid is 4874 kg m-3 at 105 GPa and 5035 K, but Pv is 5401 kg m-3. In both cases, the liquidus solid is ~15% more dense than the liquid from which it formed and would therefore sink upon formation. This calculation can be redone to include partitioning of Fe, but current published values (DFe = 0.6 to 0.07) [Andrault et al., 2012; Nomura et al., 2011], which favor Fe partitioning into the liquid over high-pressure III-22 phases, would only further increase the density difference by incorporating at least some Fe into the solid Pv to form (Mg,Fe)SiO3 perovskite (MgPv). Crystallization from the middle or base of a magma ocean has implications for potentially creating a chemically stratified early mantle [Labrosse et al., 2007; Mosenfelder et al., 2009] provided that the lowermost mantle were to remain molten and mechanically separated from the upper mantle [Abe, 1997; Tonks and Melosh, 1993]. It has been postulated that a dense liquid at the core-mantle boundary (CMB), isolated from convection, could serve as a location for an un-degassed, primordial reservoir, which could hold a missing budget of incompatible elements [Labrosse et al., 2007; Lee et al., 2010]. Yet, similar to the previous conclusion given in Thomas et al. [2012], because the first crystals sink upon forming, this does not support the dynamic creation of a chemically stratified mantle. This conclusion is highly dependent on the choice of composition for the early bulk mantle and the liquidus topology of that composition. As shown here, despite the simplified chondrite and peridotite liquids having very similar bulk composition, the liquidus of each composition has a different maximum in T-P space. This discrepancy in liquidus shape in turn gives rise to very different predictions of where crystallization would begin. It is unclear whether these very different liquidus topologies are reflective of nature or are potentially derived from the different procedures used in identifying melt in the DAC [Andrault et al., 2011; Fiquet et al., 2010]. Additionally, choosing a bulk earth mineralogy that was more Fe-rich than either of these compositions would result in a smaller density difference with respect to the liquid and liquidus solid, but such a composition would need to be justified theoretically with respect to the timing and sequestration of the core. III-23 Evaluating the Partial Melt Hypothesis for Ultralow Velocity Zones Ultra low velocity zones (ULVZs) occur in thin (5-40 km) patches above the core-mantle boundary (CMB) [Garnero and Helmberger, 1995]. They are distinguished by several key features, the first being that they are best fit with a 3:1 ratio of S-wave to P-wave velocity anomaly, which has been inferred to indicate of the presence of liquid [Williams and Garnero, 1996]. They are also discontinuous along the CMB and can be often associated with the edges of large low shear velocity provinces and the source areas of plumes and large igneous provinces [McNamara et al., 2010; Rost et al., 2005; Williams et al., 1998]. They exhibit a non-zero shear wave velocity, so the assumed liquid cannot be present as a pure melt but must be within a liquid-solid “mush.” Modeling of different melt structures for this mush permits melt fractions of 0.01 % to 30% [Williams and Garnero, 1996], although more than 20% could lead to the loss of structural integrity for some melt configurations [Hernlund and Jellinek, 2010]. At select locations where reflection coefficients have been constrained, the ULVZ is inferred to be 6-14% denser than the overlying mantle, and there is an increase in S-wave velocity with depth within the ULVZ itself [Rost et al., 2006]. In order for a molten silicate liquid to comprise some portion of a ULVZ, it must be denser than or at least neutrally buoyant compared to the ambient lower-most mantle; otherwise the liquid would percolate upwards away from the CMB over geologic time. The question is whether such a melt can exist. The presence of liquid in ULVZ’s as a partial melt was first inferred by Williams and Garnero [1996] with the assumption that the density contrast between the liquid and solid would be relatively small (~1%) so that the variations in inferred seismic velocities would be small. This idea of low density contrast is based on shock-loading experiments [Brown et al., 1987; Rigden et al., 1984], which showed liquid densities overtaking III-24 their equivalent solids at moderate pressures (6-10 GPa for basaltic liquid). Figure 8 shows the densities of silicate liquids Fa, En, and Fo with major mantle solids, including periclase (Pe) and Pv, along a modern mantle adiabat (10 K/ GPa) with a TP of 1673K. This figure indicates that the assumption of density crossovers is highly dependent on chemistry as MgSiO3 liquid remains less dense than MgPv at all pressures. The density curve for each liquid in Figure 8 is shown at the base of the mantle with a range of uncertainty accounting for thermal expansion across a boundary layer for CMB temperatures of 3073-4400 K. Notably, the liquid compositions have a greater sensitivity to temperature than the solids, which is in accord with their large γ and CV values (α = γρCV/KT). It is easy to imagine that a simple mixture of any of the liquids could produce a density comparable to that of a ULVZ. In fact Mg2SiO4 liquid without the additional Fe at 3073K is 10% denser than PREM [Dziewonski and Anderson, 1981], but the important questions are whether a chosen liquid mixture is probable both 1) chemically based on our current knowledge of lower mantle chemistry and temperatures and 2) physically based on a plausible process by which it could be produced. There are two proposed processes for creating the putative ULVZ melt. The first is a melt left over from the differentiation of the Earth, perhaps distilled from the chemistry of the lower mantle, having been segregated from the surface by mid-mantle crystallization [Labrosse et al., 2007]. A second method of producing the ULVZ melt is by partial melting of the ambient mantle. We discuss these each in turn. As shown above, our current calculations appear to oppose segregation of a magma ocean into distinct reservoirs, but in fact our ability to constrain this problem so far applies only to the very initial stage of crystallization at the liquidus and is strongly dependent on the chosen composition. Development of a solid septum at later stages of magma ocean crystallization III-25 cannot be entirely ruled out. In detail it remains difficult test the distillation hypothesis for chemical and physical plausibility. The liquid chemistry, initially or during progressive differentiation, could occupy a wide range of compositions within CMASF space. Furthermore, the model of Hernlund and Jellinek [2010] suggests that stirring of the ULVZ piles by viscous coupling to the overlying mantle could create a stable mush layer for all geologic time without ongoing melt production. This model is also rather insensitive to density contrast, requiring only that the liquid be at least slightly denser than coexisting solids and disregards any chemical interaction with the mantle. Hence it is difficult to bring much insight to bear on this model using new constraints on the composition-dependent equation of state of silicate melts. Turning to the partial melting hypothesis, its plausibility has been supported by experiments showing that magnesiowüstite and perovskite assemblages melt at temperatures comparable to estimates for the CMB — e.g., (Mg0.9,Fe0.1)2SiO4 has a solidus temperature of 4300±270 K at 130±3 GPa [Holland and Ahrens, 1997]. Yet presently the melting temperatures for both Mg-Pv [Sweeney and Heinz, 1998; Zerr and Boehler, 1993] and MgO [Asimow and Fat'yanov, 2011] remain controversial, and the effect of Fe on the melting curve remains poorly constrained. Here, we limit our discussion to the plausibility of generating a stable ULVZ via partial melting of the ambient mantle. We will construct a model of phase and bulk densities under conditions of chemical equilibrium, which searches for an initial bulk mantle composition that would yield an equilibrium assemblage of 30% volume fraction of melt and 70% volume fraction residue that could form a gravitationally stable liquid mush (i.e., melt ≤ 1% denser than residue) with bulk density 10±4% denser than PREM (~ 6123±223 kg/m3). To calculate the densities of the mantle solids, (Mg, Fe)SiO3 perovskite (MgPv) and (Mg,Fe)O ferropericlase (Fp), we used the EOS provided in Mosenfelder et al.[2009] for pure III-26 Mg-Pv and pure Mg-Pe and calculated the addition of small amount of Fe2+ by ignoring changes in molar volume upon substitution. This is certainly an oversimplification but has already been shown to be a good approximation for estimating densities of (Mg, Fe)O solid solutions (for Fe contents up to ~60 mol%) at pressures greater than 70 GPa where Fe2+ is in the low-spin state and has an effective ionic radius similar to Mg2+ (~0.72 Å) [Fei et al., 2007]. Given the low Fe contents expected in Pv solid solutions in equilibrium with melt and the substantial uncertainty that remains in the liquid EOS, this is an adequate approximation for assessment of relative melt/solid buoyancy. For calculation of liquid densities, we assume linear mixing of the oxide volumes derived from the BM/MG EOS given in Tables 3 and 4. To define the distribution of Fe and Mg at chemical equilibrium among melt, Fp, and MgPv we need to define two partition coefficients, for example a solid-solid partition coefficient DFeMgPv/Fp = XFeMgPv/XFeFp (where XFe is molar iron content of the phase) and one solid-melt partition coefficient DFeMgPv/melt = XFeMgPv/XFemelt. The solid-solid DFeMgPv/Fp can be as low 0.09 [Auzende et al., 2008] in Al-free systems but approaches unity for Al-bearing systems at high pressure [Wood and Rubie, 1996]. A similar range of experimental values exists for the solidmelt DFeMgPv/melt for non-Al bearing [Nomura et al., 2011] (~0.07) and Al-bearing systems [Andrault et al., 2012] (~0.47-0.6). We have chosen to use the values derived from high pressure Al-bearing experiments as they are likely a closer approximation to natural systems. Hence for our calculations, we assumed DFeMgPv/Fp= 1and DFeMgPv/melt = 0.47 (we also explore the sensitivity of our result to this choice of solid-melt partition coefficient for Fe). In our calculations, we also fix the SiO2 concentration of the MgPv-Fp eutectic liquid. We chose two compositions: 1) a eutectic liquid with an SiO2 concentration of 41 mol % such that 70% of the melt is MgPv composition (fpvE = 0.7) and 30% is Fp composition and 2) a liquid III-27 with 47 mol% SiO2 concentration (fpvE = 0.9). These two estimates bracket the eutectic composition determined by Liebske and Frost [2012] for the pure MgO-MgSiO3 system such that SiO2 is 44 mol % (fpvE = 0.8). Ambient mantle is often approximated as 80% (Mg, Fe)SiO3 and 20% (Mg, Fe)O with some proportion of Fe [Boehler, 2000]. For our model, we evaluate whether this composition as well as other more exotic compositions would be able to produce a partial melt and an equilibrium residue with a low density contrast (a melt 1% denser than the residue, equivalent to the assumption in Williams and Garnero [1996]) and also produce an aggregate density comparable to a ULVZ allowing 30% volume fraction of melt. Results for this calculation at 3700 K and 135 GPa for fpvE =0.7 are given in Figure 9a. The solid lines represent curves of constant density contrast for the melt and residue, where positive percentages are a partial melt denser than the residue and negative percentages are a residue denser than the melt .The ULVZ density region is delineated by dotted lines representing a density 6-14% denser than PREM (~6123±223 kg/m3). An estimate for ambient mantle composition of 80% MgPv +20% Fp (fpv =0.8) with a Mg# of 0.85 (Mg# = XMg/(XFe+XMg)) would produce a partial melt 5% denser than its equilibrium residue, but the aggregate density would be 5503 kg/m3, which is actually 1% less dense than PREM. To produce an aggregate density comparable to PREM, the ULVZ bulk composition would need a very high Fe concentration (Mg# ≤ 0.6), but this would yield a very large (at least 14%) melt-residue density contrast. In fact for a eutectic composition of fpvE =0.7 and the selected partition coefficients, there is no starting composition that can fulfill both the aggregate density and density-contrast constraints. The unphysical region on the graph represents aggregate densities that are unachievable with the given amount of Fe in the system (Mg# for the partial melt is negative III-28 within this space). If we allow DFeMgPv/melt to approach unity (higher than the current experimentally determined values), thereby decreasing the extent to which Fe favors the melt and also thereby decreasing the density contrast between melt and residue, the unphysical region expands and does not permit aggregate densities within the ULVZ region. This result is also insensitive to percent of partial melting, as decreased volume fraction of melt also allows for a diminished melt-residue density contrast, but likewise further limits the highest aggregate densities that are achievable. It is an important conclusion therefore that if the eutectic composition is fpvE=0.7, then there is no composition in the ternary MgO-FeO-SiO2 system (that is, any combination of Mg# or fraction of MgPv or Fp), regardless of DFeMgPv/melt or melt fraction, that will produce an equilibrium assemblage equivalent to what is believed to be present in a ULVZ. In fact, this is also true for eutectic compositions fpvE = 0.8 [Liebske and Frost, 2012] and fpvE =0.9 with any experimentally determined DFeMgPv/melt value [Andrault et al., 2011]. Only at DFeMgPv/melt = 0.8 and a eutectic liquid of fpvE = 0.9 is the minimum estimate of the aggregate density achievable with a melt 1% denser than its residue (Figure 9b). The initial composition would also need to be exotic compared to typical ambient mantle compositions: richer in Fp (fpv = 0.2 -0.5) and Fe (Mg# = 0.6-0.7). Even disregarding the lack of experimental evidence for such high DFe or MgPv rich eutectic liquids, such high Fe contents are potentially unrealistic for a wide-spread lower mantle composition. FeSiO3 is unstable in the perovskite structure and decomposes to magnesiowüstite and stishovite [Jeanloz and Thompson, 1983] in which case the calculations of bulk densities performed here using the MgPv EOS are no longer appropriate; although the presence of aluminum has been shown to expand both ferrous [Kesson et al., 1995] and ferric stability in the perovskite and post-perovskite (pPv) phases at high pressure [Frost and McCammon, 2008; III-29 Frost et al., 2008; Zhang and Oganov, 2006].Additionally, experiments and calculations have indicated that an Fe-rich pPv phase (with Mg# ≤ 0.8) may be stable at 130 GPa [Mao et al., 2006; Mao et al., 2005], yet the temperature within a ULVZ remains sufficiently underconstrained that it is unclear whether Pv or pPv would be the predominate stable phase. Similarly, although despite potentially destabilizing Pv, elevated Fe contents are likely necessary to depress the melting point of Fp to reasonable CMB temperatures, considering the melting temperature of pure MgO at ambient conditions is already very high (3060 K) [Poirier, 2000], granting its high pressure melting curve remains uncertain. In contrast, this partial melt calculation could yield more reasonable results with less Fe in the bulk starting composition if carried out at a lower temperature (since liquids increase in density more rapidly than solids with decreasing temperature), but at some point the conditions would drop below the solidus. The correct mantle solidus temperature is not well known and better constraints on the phase diagrams of the pertinent compositions at high pressure and temperature are needed. Therefore consequently, if the model results are correct and high Fe contents are required to produce the necessary conditions for a ULVZ, partial melting may not even be necessary as suggested in Wicks et al. [2010], which cites the possibility of solid-state ULVZ composed of a large proportion of Fe-rich (Fe, Mg)O to produce the observed drops in S-wave velocity. We can conclude from this exercise that the ambient mantle (80% MgPv +20% Fp) cannot produce a partial melt and equilibrium residue that could be a ULVZ mush regardless of DFeMgPv/melt or eutectic composition. There exists a starting composition which can produce a ULVZ mush with the correct aggregate density and melt-residue density contrast, but only if subject to DFeMgPv/melt and eutectic composition constraints that lie outside of what has been determined experimentally. The partial melt hypothesis is therefore only plausible if the melt is III-30 fractionally segregated from its residue and combined with some other denser component. Only this process would permit both an aggregate density equivalent to the bulk ULVZ mush constrained by Rost et al. [2006] and at the same time a have a melt-residue density contrast that is consistent with assumed values used in seismic modeling (~1%) [Williams and Garnero, 1996], assuming both these physical constraints are robust. And although not explicitly calculated here, another potential way to increase the aggregate density would be to transform the residue to post-perovskite structure, but partitioning of Fe in pPv is not well constrained [Kobayashi et al., 2005; Sinmyo et al., 2011], the location of the Pv-pPv transition in Fe-bearing systems is controversial, and it would require a special scenario to place the solid Pv-pPv transition and the onset of partial melting at the same horizon. CONCLUSIONS We have completed new shock wave experiments on molten Mg2SiO4 at 2273K, a significant technical advance in initial temperature for preheated experiments of this kind, allowing study of liquid compositions and temperatures previously inaccessible to ultrasonic or other experimental methods. This new data enables a revision of the previous liquid Mg2SiO4 EOS [Mosenfelder et al., 2009]. The Grüneisen parameter is now in better agreement with values obtained from molecular dynamics simulation [de Koker et al., 2008] and is more similar to the γ behavior of other silicate liquids such as fayalite and diopside compositions. The shock-derived ambientpressure bulk sound speed at 2273 K, compared to values estimated from lower-temperature ultrasonic data, indicates that sound speed decreases with increasing temperature for Fo liquid, contrary to the positive [Ghiorso and Kress, 2004] and zero [Ai and Lange, 2008] temperature dependences previously reported. The EOS of liquid MgSiO3 and CaAl2Si2O8 were also revised III-31 to exclude experiments that may have only been partially melted upon shock compression and potentially un-relaxed at low pressure, respectively, resulting in EOS models of the five components Fa, Fo, En, Di, and An that all share well-behaved, concave-down isentropes in P-T space. These were used in an ideal mixing model to construct new isentropes for whole mantle magma oceans of selected compositions in CMASF. The results are similar to those presented previously [Thomas et al., 2012], in that MgPv may begin crystallizing in the mid-lower mantle or at the base of the mantle for different compositions or estimates of their liquidus curves but in all cases MgPv is negatively buoyant at the site of initial crystallization. This does not favor the formation of a basal magma ocean at early stages of evolution. The revised EOS of MgSiO3 and Mg2SiO4 were also used to calculate the plausibility of producing a ULVZ from a partial melt of the ambient mantle. Results indicate that any MgPv-Fp assemblage, including compositions typically assumed for the ambient mantle (80% MgPv and 20% Fp), cannot produce an equilibrium assemblage with a density comparable to that which has been inferred for a ULVZ mush. A ULVZ must be composed of fractionally segregated liquids from elsewhere in the mantle in association with a solid component denser than the equilibrium residue. III-32 ACKNOWLEDGEMENTS The authors would like to thank the following: the shock wave lab technical staff—Oleg Fat’yanov, Eprapodito Gelle, and Russel Oliver. This work was supported by the National Science Foundation through award EAR-1119522. III-33 REFERENCES Abe, Y. (1997), Thermal and chemical evolution of the terrestrial magma ocean, Physics of The Earth and Planetary Interiors, 100(1-4), 27-39. Adjaoud, O., G. 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Le Bihan (1999), Equation of state of MgSiO3 with the perovskite structure based on experimental measurement, American Mineralogist, 84(3), 226-232. Simakov, G. V., and R. F. Trunin (1973), On the existence of overdense perovskite structures in magnesium silicates under conditions of high pressure (English translation), Izvestiya Earth Phys, 9, 603-604. Sinmyo, R., K. Hirose, S. Muto, Y. Ohishi, and A. Yasuhara (2011), The valence state and partitioning of iron in the Earth's lowermost mantle, J. Geophys. Res., 116(B7), B07205. Stebbins, J. F., I. S. E. Carmichael, and L. K. Moret (1984), Heat capacities and entropies of silicate liquids and glasses, Contributions to Mineralogy and Petrology, 86(2), 131-148. Stixrude, L., and B. Karki (2005), Structure and Freezing of MgSiO3 Liquid in Earth's Lower III-37 Mantle, Science, 310(5746), 297-299. Stixrude, L., and C. Lithgow-Bertelloni (2005), Thermodynamics of mantle minerals – I. 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Fang (1982), Simplified model evaluation of cooling rates for glass-containing lunar compositions, Lunar and Planetary Science Conference, 12B, 281-288. Urbain, G., Y. Bottinga, and P. Richet (1982), Viscosity of liquid silica, silicates and alumino silicates, Geochimica et Cosmochimica Acta, 46(6), 1061-1072. Wicks, J. K., J. M. Jackson, and W. Sturhahn (2010), Very low sound velocities in iron-rich (Mg, Fe) O: Implications for the core-mantle boundary region, Geophysical research letters, 37(15), 15304. Williams, Q., and E. J. Garnero (1996), Seismic Evidence for Partial Melt at the Base of Earth's Mantle, Science, 273(5281), 1528-1530. Williams, Q., J. Revenaugh, and E. J. Garnero (1998), A Correlation Between Ultra-Low Basal Velocities in the Mantle and Hot Spots, Science, 281(5376), 546-549. Wolf, A. S., P. D. Asimow, and D. J. Stevenson (2012), A Simplified Cation Speciation Model for Silicate Liquids at Mantle Pressures and Temperatures, AGU Fall Meeting Abstracts, 1, abstract MR43A-2300. Wood, B. J., and D. C. Rubie (1996), The Effect of Alumina on Phase Transformations at the 660-Kilometer Discontinuity from Fe-Mg Partitioning Experiments, Science, 273(5281), 1522-1524. Zerr, A., and R. Boehler (1993), Melting of (Mg,Fe)SiO3 Perovskite to 625 Kilobars: Indication of a High Melting Temperature in the Lower Mantle, Science, 262(5133), 553-555. Zhang, F. W., and A. R. Oganov (2006), Mechanisms of Al3+ incorporation in MgSiO3 postperovskite at high pressures, Earth and planetary science letters, 248(1-2), 69-76. III-38 Table 1. Shock compression data Shot # Flyer material 1077 a Mo Forsterite 1075 a 455 457 458 459 Anorthite 665† ρΗ km s–1 PH Mg m–3 GPa 2.019 0.003 1.644 0.001 5.38 0.02 3.74 0.01 22.98 0.07 Mo 2273 5.426 0.007 4.08 0.02 9.14 0.17 4.68 0.09 96.7 1.4 2273 4.522 0.008 Cu 1922.2 1.8 0.03 1.325 0.025 4.68 0.01 Mo 1932 0.003 4.501 0.005 9.56 0.05 Mo Mo Mo 379 Mo Diopside Us km s–1 2274 Mo 382 km s–1 Mo 364 380 up ufp T (K) Mo Mo 2273 2273 2273 1.04 3.32 5.964 0.02 0.01 0.002 1930 4.997 0.005 1932 6.007 0.025 1932 1781 5.434 6.533 4.984 0.013 0.003 0.886 2.59 3.41 4.43 3.816 4.094 4.868 3.697 0.004 0.03 0.01 0.02 0.003 0.007 4.00 7.10 8.47 9.95 8.49 9.09 0.024 10.06 .008 9.33 0.11 3.34 0.44 4.10 0.12 5.29 0.17 5.42 0.04 75.0 2.3 0.8 3.57 0.03 15.90 0.30 4.84 0.03 95.30 0.63 4.97 0.10 0.05 47.6 0.2 114.3 4.66 0.23 0.17 9.21 0.08 4.64 0.08 0.03 4.34 0.02 82.80 0.04 110.00 0.03 90.84 0.13 126.00 a Caltech 40mm propellant gun † Anorthite and diopside shots are from Asimow and Ahrens [2010] revised in Thomas et al. [2012] but not previously published except for 665 which was revised in Asimow and Ahrens [2010] and originally from Rigden et al. [1989]. 1.5 0.28 0.44 2.30 0.72 Table 2. Parameters used ρo Co s Mg m-3 km s-1 Molybdenum (2273 K) Molybdenum (300 K) 9.785 10.21 4.858 5.033 1.288 1.289 Asimow et al. [2008]; Chase [1998] III-39 Table 3. Equation of State fits for molten Mg2SiO4 To ρo Co s γo q KoS KS' KS'' 𝜒2 KT α Cp Cv Units K kg m–3 m s–1 GPa GPa–1 GPa K-1 J kg–1 J kg–1 1 SWEOS 2273 2597 9 11 2674 ± 19 1.64 ± 0.06 0.448 -1.51† 18.57 5.58 ± 0.24 16.54 1.21-04 1926.18 1715.23 BM3 2273 2597 ± 11 0.396 –2.02 ± 1.03 16.41* 7.37 ± 0.81* 1.65 14.80 1.21-04 1926.18 11737.36 BM4 2273 2597 ± 11 0.396 0.47 ± 9.86 16.41* 4.27 ± 2.02* 1.31 ± 1.83 1.11 14.80 1.21-04 1926.18 11737.36 Source L97 fitted fitted derived fitted derived/*fitted derived/*fitted fitted derived L97 LN92 derived SWEOS= shock wave equation of state. 3BM/MG = 3rd-order Birch-Murnaghan/Mie-Grüneisen EOS 4BM/MG = 4th-order Birch-Murnaghan/Mie-Grüneisen EOS. † derived from shot 350 Fo Mosenfelder et al. [2007] Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992] Table 4. Equation of State fits for molten silicate liquids To ρo γo q KoS KS' KS'' Cp Cv Units K kg m–3 GPa GPa–1 J kg–1 K–1 J kg–1 K–1 3BM/MG Di† 1673 2643 0.493 -1.28 24.57 6.77 1612.56 1506.21 3BM/MG Fa† 1573 3699 0.412 -0.95 21.99 7.28 1182.35 1122.73 4BM/MG An 1932 2560 0.174 -1.86 19.77 3.73 0.38 1528.72 1511.28 4BM/MG En 1673 2617 0.365 -0.88 24.66 10.07 -2.35 1761.15 1690.53 Source L97 derived fitted derived fitted fitted LN92 derived 3BM/MG = 3rd-order Birch-Murnaghan/Mie-Grüneisen EOS 4BM/MG = 4th-order/Birch-Murnaghan Mie-Grüneisen EOS. Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992], †Thomas et al [2012] III-40 FIGURE CAPTIONS Figure 1. The set of experiments used for modeling sound speeds of silicate liquids in Ghiorso and Kress [2004] and Ai and Lange[2008] compared to the initial temperature and composition of Mg2SiO4 liquid. Figure 2. Preheated (2273K) forsterite Hugoniot in shock velocity (Us)-particle velocity (up) space. The thick black line represents the unconstrained linear Hugoniot; the orange and blue lines are fixed at the sound speeds calculated using the models from Ghiorso and Kress [2004] and Ai and Lange[2008], respectively. Figure 3. The behavior of Grüneisen parameter for Mg2SiO4 liquid (thick solid green) compared to its previously reported value (dotted) [Mosenfelder et al., 2009]. The thin red and blue lines are the behavior of γ for diopside and fayalite, respectively. The EOS parameters used are given in Table 4. The squares and black line and are the results and linear fit from MD simulations of Mg2SiO4 liquid [de Koker et al., 2008] Figure 4. Mg2SiO4 liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 3. Abbreviations: SWEOS- shock wave equation of state; BM/MG – Birch-Murnaghan/Mie-Grüneisen EOS Figure 5. MgSiO3 Hugoniot data used in the global fit of Mosenfelder et al. [2009] for MgSiO3 melt. The volume for each experimental point is normalized to the volume of its starting III-41 material. The data set includes enstatite glass [Mosenfelder et al., 2009] (triangle), enstatite crystal [Akins et al., 2004] (diamond), oxide mix [Marsh, 1980] (circle), and porous enstatite [Simakov and Trunin, 1973] (square). The gray line is the high pressure state assignment from Mosenfelder et al. [2009] to either perovskite, post-perovskite, or melt. Only the hollow points were included in our fit. Solid points within the melt field were excluded. Error bars not shown. Figure 6. Estimates of the Mg2SiO4 liquidus from de Koker et al. [2008] (black curve with shaded gray error), Presnall and Walter [1993](green curve) and Ohtani and Kumazawa [1981] purple curve including the experimentally determinations liquid (circles) and solid (squares) quench. The estimated T-P path of the Hugoniot (dashed line) and shot #1077 (diamond) (see Table 1) lie within the liquid field. Figure 7. Models for the early mantle upon cooling showing the comparison of the liquidus curves (black solid line) with the isentropic temperature profile (dotted line) of the same composition. (a) Simplified “chondritic” composition and liquidus from Andrault et al. [2011]. (b) Peridotite KLB-1 composition and liquidus from Fiquet et al. [2010]. The critical isentrope with its mantle potential temperature is shown in red. The tangent point of the critical isentrope and the liquidus gives the pressure and temperature at the onset of crystallization. The EOS parameters used for calculating the isentropes are given in Table 3 and 4. Figure 8. Estimated densities for silicate liquids Fa (blue curve), Fo (green curve) and En (orange curve) and mantle solids Fo, wadsleyite (Wd), ringwoodite (Rw), periclase (Pe), Mg-perovskite (Pv), and post-perovskite (pPv) (black curves) as function of pressure along a 10K/GPa mantle III-42 geotherm. The dashed lines are the previous density estimates for Fo liquid (green) and En liquid (orange) from EOS parameters given in Mosenfelder et al. [2009].The density at base of mantle (135 GPa) for each composition is thermally corrected from 3073K to 4400K (shaded cones). Also shown is the density profile from PREM [Dziewonski and Anderson, 1981], the inferred density for a ULVZ from seismic reflection coefficients [Rost et al., 2006](square), and an estimated ambient mantle composition of 80% (Mg0.85.Fe0.15)SiO3 perovskite+ 20% (Mg0.85Fe0.15)O (diamond) at 3000K and 129 GPa. The EOS parameters for the liquids are taken from Tables 3 and 4. EOS for Mg-Pv and pPv are from Mosenfelder et al. [2009]; other solid EOS are from Stixrude and Lithgow-Bertelloni [2005]. Figure 9. Results for partial melting calculation at 3700 K and 135 GPa. The solid curves represent lines of constant density contrast for the melt and residue, where positive percentages are a partial melt denser than the residue and negative percentages are a residue denser than the melt for a given starting composition. The y-axis if the fraction of MgPv in the starting material (fpv) where the remainder is Fp; the x-axis is the Mg# of the starting composition, where Mg#.= XMg/(XFe+XMg).The density of the ULVZ is the blue region delineated by dotted lines representing a density 6-14% denser than PREM (~6123±223 kg/m3). An estimate for ambient mantle composition of 80% MgPv +20% Fp (fpv =0.8) with an Mg# of 0.85 is given with its calculated aggregate density (agg ρ) with 30% volume fraction of partial melt. The unphysical region on the graph signifies where Mg# for the partial melt is negative and hence represents aggregate densities that are unachievable with the given amount of Fe in the system) The eutectic liquid is fixed at 70% MgPv (fpvE = 0.7) and the DFeMgPv/melt = 0.47. b) The eutectic liquid is fixed at 90% MgPv (fpvE = 0.97) and the DFeMgPv/melt = 0.8. III-43 FIGURES Figure 1 III-44 Figure 2 III-45 Figure 3 III-46 Figure 4 III-47 Figure 5 III-48 Figure 6 III-49 III-50 Figure 8 Figure 7 Figure 9 III-51 IV-1 Chapter IV Pre-heated shock experiments in the molten CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 ternary: a test for linear mixing of liquid volumes at high pressure and temperature Claire W. Thomas Paul D. Asimow For submission to Journal of Geophysical Research April 4, 2013 IV-2 ABSTRACT We performed seventeen new shock wave experiments on pre-heated (1673 K) hedenbergite liquid (CaFeSi2O6) and two model basalt liquids (an equimolar binary mix of CaAl2Si2O8 + CaFeSi2O6 and an equimolar ternary mix of CaAl2Si2O8 + CaFeSi2O6 +CaMgSi2O6) in order to determine their equations of state (EOS). Ambient-pressure density measurements on these and other Fe-bearing silicate liquids indicate that FeO has a partial molar volume that is highly dependent on composition, which leads to large errors in estimates of the densities of Fe-bearing liquids at ambient pressure based on an ideal mixing of any fixed set of end-member liquids. We formulated a series of mixing tests using the EOS determined in this study to examine whether ideal mixing of volumes might nevertheless suffice to describe the ternary system CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 at high temperature and pressure. The ideal mixing null hypothesis is rejected; compositional variations in partial molar volume of FeO appear to extend to high pressure. Only densities of Fe-bearing liquid mixtures with oxide mole fraction of FeO less than 0.06 can be adequately approximated using an ideal solution. IV-3 INTRODUCTION Silicate liquids play a large role in our understanding of both the early and modern Earth. Most widely-accepted scenarios for the evolution of the early Earth involve at least one stage of extensive or complete melting, for example, in the aftermath of impacts with large protoplanetary sized objects such as the presumed moon-forming event [Canup, 2004; 2012; Canup and Asphaug, 2001; Ćuk and Stewart, 2012]. The chemical and thermal evolution of the early molten mantle and how it affects the chemistry and physical properties of the modern mantle are not well understood. Furthermore, seismic observations of the 410km discontinuity in the upper mantle [Revenaugh and Sipkin, 1994; Song et al., 2004] and ultralow velocity zones (ULVZ) in the lowermost mantle [Garnero and Helmberger, 1995; Williams and Garnero, 1996] have been interpreted to indicate the presence of a neutrally buoyant partial melt, yet there is currently limited knowledge of how the density and compressibility of complex silicate liquids behave at high temperature and pressure. It is therefore essential to have experimental constraints on the equation of state (EOS) of the complex multicomponent silicate liquids that would be actually present in nature. The difficulty lies in that natural silicate liquids comprise a potentially infinite, multi-dimensional continuum of compositions. Understanding their range of volumetric behavior requires either data on an arbitrarily large number of experiments or data on a minimum set of end-member compositions along with a reliable tool of interpolating among them. Choosing the correct method of interpolation is key to producing a predictive model for silicate liquid density and compressibility over a wide range of pressure, temperature, and composition space. Experimental data that provide constraints on end-member volumes as well as a few intermediate compositions offer the opportunity to test whether available methods for such interpolation are reliable. IV-4 Ideal mixing of volumes has proven to be a reliable method of interpolation for ambientpressure experiments within a given compositition and temperature range [Bottinga and Weill, 1970; Bottinga et al., 1982; Bottinga et al., 1983; Guo et al., 2013a submitted; Lange, 1997; Lange and Carmichael, 1990; Mo et al., 1982; Nelson and Carmichael, 1979], although exceptions from ideality exist and are discussed below. Likewise, ideal mixing silicate liquid volumes at high pressure has been employed as a simplifying assumption, but it in fact remains poorly tested and constrained [Asimow and Ahrens, 2010]. In this study, we report seventeen new shock wave experiments to pressures up to 146 GPa on preheated silicate liquids of three separate compositions: hedenbergite (Hd; CaFeSi2O6), a 50-50 mixture of anorthite and hedenbergite (An-Hd; CaAl2Si2O8-CaFeSi2O6 ), and an equal mixture of anorthite, hedenbergite, and diopside (An-Hd-Di; CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 ). Determination of the EOS of Hd and the two mixtures (or “model basalts”) permits us to test the validity of linear mixing of volumes and ideal configurational mixing of entropies — two assumptions which were both used in Thomas et al. [2012] and Chapter III of this text to predict the volumes of multicomponent liquids at high T and P. METHODS Sample preparation The synthesis and analysis for the Hd and model basalt samples used in this study are described in Guo et al. [2013a submitted] and Guo et al. [2013b in prep], respectively. These studies also describe the 1-bar double-bob Archimedean density measurements and the ultrasonic sound speed measurements that were completed on these materials. The sample materials used for the shock wave experiments were cored directly from the lower buffer rod crucible used in IV-5 the sound speed determinations. The cores were sliced into disks, which were then lapped down to under-fill the volume of the molybdenum shock-experiment sample holder. The Mo sample holders are assembled from two machined pieces: 1) the driver plate, which has a hollow cylindrical projection on one side that becomes the sample well and 2) the cap, which is a 1 mmthick, 8 mm-diameter disk that is welded on after filling (for complete description of the welding process see Thomas et al. [2012]). The interior volume of the sample well is 8 mm in diameter and 2.5 mm deep. The samples under-filled this volume by 7-13% at room temperature to prevent the welded cap from bowing during heating [c.f. Thomas et al., 2012] The surfaces within the sample well including the inside surface of the cap were polished to a mirror finish with 1-micron alumina grit. Smoothing all the surfaces prevents bubbles from clinging to the walls of the sample holder during heating and potentially interfering with the imaged shock wave. The outer surface of the cap was left with the finish quality as-machined (~9-micron finish), in order to match the quality of polish on the inner portion of the rear-face of the driver plate, which was polished on a rotating stage. The speed of the polishing pad decreases towards the center, hence the quality of mirror finish also decreases. Approximately similar finishes on the cap and driver reflecting surfaces ease acquisition of evenly illuminated images of the cap and driver during the shot and greatly increased the precision of the Hd data given below. On the contrary, the model basalt capsules were polished as in the manner as described in Chapter II due to the larger quantity of light needed for the digital recording system (c.f. Chapter III). Hence this dataset is plagued with larger errors derived from determining the driver and cap arrivals with different exposure levels. An improved technique for polishing for driver to a mirror finish would greatly aid data acquisition in the future. Additional sample preparations are detailed in Asimow and Ahrens [2010], Thomas et al. [2012], and Chapter III. IV-6 Experimental setup The pioneering work for shock studies on molten materials is Rigden et al. [1984]. The description of our methods below builds on this and the work of Rigden et al. [1988; 1989], Miller et al. [1988; 1991a], Chen and Ahrens [1998], and Chen et al. [2002]. The most recent changes to experimental techniques and data analysis are in Asimow [2008], Asimow and Ahrens [2010], Thomas et al. [2012], and Chapter III. For this study, seventeen total experiments were performed—twelve in the Caltech 90/25-mm two-stage light-gas gun and five using the Caltech 40-mm propellant gun—eight performed on Hd, five performed on An-Hd, and four performed on An-Hd-Di (see Table 1). All shots were preheated to 1673 K ± 4 K (1400 °C) and used either molybdenum (Mo), tantalum (Ta), or aluminum alloy (Al2024) flyer plates. Flyer plate velocities (ufp) ranged from 1.46 – 6.19 km/s; the methods used for measuring ufp are given in Thomas and Asimow [submitted]. During each experiment, the rear face of the target is illuminated by a Specialised Imaging xenon spark lamp and filmed by a Hadland Imacon 790 streak camera through a narrow (25 μm) slit focused horizontally across the center of the driver and sample cap. Hd shots were recorded on Polaroid film; the model basalts were recorded using a digital system (c.f. Chapter III). As the shock wave reaches the free surface of the driver or the cap the reflected light from the polished surface is extinguished (or shows a sharp change in intensity). The offset time of these two cutoffs allows for the precise measurement of the shock transit time through the sample and molybdenum cap. To correct for non-uniform streaking rate, the camera was calibrated using a test streak image modulated by a radio-frequency tuner at 148.9875 MHz for a 1500 ns streak length and by a pulse generator at 1.7870 MHz for a 5000 ns streak (shot 1068). Each pixel line on the shot IV-7 streak record can then be converted to a time during the experiment, which permits the shock transit time through the sample to be calculated from the pixel distance between the two cutoffs on the film (or digital image). A detailed method for picking cutoffs and reducing the data is described in Thomas et al. [2012]. The final calculated shock state — i.e., shock pressure (PH), shock density (ρH), particle velocity (up), and shock velocity (Us) is solved for iteratively [Rigden et al., 1988] using measured shock wave travel time, impedance matching, the Rankine-Hugoniot equations, Hugoniot data (ρo, Co, s) for the metal flyer plates and hot Mo driver (Table 2), and the initial density (ρo) for each of the silicate liquids (see Tables 3, 4, and 5). An initial guess for the bulk sound speed (Co) and Hugoniot slope s is required to seed the iteration, but the converged result is independent of this assumption. RESULTS The shock wave data are reported in Table 1, including shot number, flyer material, the temperature prior to firing, ufp, up, US, ρH, and PH. A Hugoniot is a series of peak shock states achieved by progressively stronger shocks in a material from the same initial conditions [Ahrens, 1987]. The Hugoniot of a well-behaved material empirically forms a line in US-up space, given to third order in strain by US = Co + s up [Jeanloz, 1989]. The bulk sound speed of the material at room pressure, Co, is given by the intercept; the slope (s) is related to K’S, the isentropic pressure derivative of the isentropic bulk modulus (KoS) by [Ruoff, 1967], ( K S' + 1) . s= 4 (1) IV-8 Hedenbergite (CaFeSi2O6) liquid Figure 1 shows linear fits to the Hd shock data in US-up space. An unweighted linear fit to all eight preheated hedenbergite liquid data points yields US = 2.587±0.142 + 1.55±0.04 up km/s (r2 = 0.996). The experimentally determined ultrasonic value of the bulk sound speed, 2.613±0.016 km/s [Guo et al., 2013a submitted], lies within the stated uncertainty of the bulk sound speed derived from the intercept of this fit, 2.587±0.142 km/s. This relationship indicates relaxed (liquid-like) — as opposed to un-relaxed (glass-like) — behavior upon shock compression [Rigden et al., 1988]. Therefore, to reduce the error on the linear fit, we fixed the intercept at the mean ultrasonic value yielding an unweighted constrained linear fit of US = 2.613±0.016 + 1.54±0.01 up km/s (r2 = 0.999). It is necessary to select and apply a thermal equation of state formalism for investigating material properties that lie off the Hugoniot, since the Hugoniot achieves temperatures and energies much higher than those of geophysical interest at lower mantle pressures (even for processes during early Earth differentiation). We have defined the entire P-V-E surface of hedenbergite liquid using various formalisms: the shock wave equation of state (SWEOS) and the 3rd- and 4th-order Birch-Murnaghan Mie-Grüneisen equations of state (3BM/MG and 4BM/MG). For a full description of the thermal EOS equations and fitting procedures, the reader is directed to Asimow and Ahrens [2010] and Thomas et al. [2012]. The results and uncertainties for each fit on Hd are given in Table 3, and the Hugoniots are plotted in Figure 2. The SWEOS is define by a linear Hugoniot in US – up space, converted to P - ρ space using the first and second Rankine-Hugoniot equations [e.g,. Ahrens, 1987]. States that lie off the Hugoniot are found using the Mie-Grüneisen thermal pressure approximation with a temperature-independent power law expression for the thermodynamic Grüneisen parameter IV-9 (2) . Although a value for q to be used with the SWEOS was not independently determined in this study, experimental results thus far indicate that silicate liquids—including iron-bearing liquids [Thomas et al., 2012]—appear to have q values of –0.88 to –2 for compressions of 1 > ρo/ρ > 0.49. The slope and intercept of the constrained Hugoniot fit given above corresponds to a K’S value of 5.16±0.04 derived from (1) and a KoS value of 19.98±0.24 GPa derived from the expression Co = K oS ρo . (3) The 3rd- or 4th-order BM/MG EOS is defined by a 3rd- or 4th-order Birch-Murnaghan isentrope centered at 1 bar and 1673K and a Mie-Grüneisen thermal pressure approximation. The 3BM/MG fit result is KS’ = 6.22±0.55, q = –1.93±0.41, and reduced χ2= 4.20. The 4BM/MG fit result is KS’ = 3.15±1.67, KS” = 0.78±0.60 GPa-1, q = 0.47±9.86 and reduced χ 2 = 3.26. The high precision for most of the points in the Hd dataset propagates into unusually high χ 2 values for both fits. Nevertheless, comparatively the 4BM/MG has very large error bars on and strong correlations among the output parameters indicating a very unstable fitting routine. The 3BM/MG fit, by contrast, has less severe correlation between the parameters and appears justified by the fitting statistics. Hence for this data set we prefer the 3rd-order fit. IV-10 An-Hd liquid The An-Hd Hugoniot in US-up space is shown in Figure 3. The unweighted linear fit is US = 2.742±0.282 + 1.40±0.09 up km/s (r2 = 0.986). As observed above for Hd liquid, the An-Hd liquid appears to be relaxed upon compression as the experimentally determined bulk sound speed, 2.772±0.011 km/s [Guo et al., 2013b in prep], and the intercept are within error. We can then fix the intercept at the mean ultrasonic value yielding an unweighted constrained linear fit of US = 2.772±0.011 + 1.39±0.03 up km/s (r2 = 0.998). This slope and intercept constrain the SWEOS K’S and KoS values to be 4.54±0.12 and 21.30±0.25 GPa, respectively. The 3BM/MG fit result is KS’ = 5.49±0.65, q = –0.18±2.25, and reduced χ2 = 6.37. The 4BM/MG fit result is KS’ = 5.10±1.61, KS” = –0.17±0.60 GPa-1, q = 1.67±2.79 and reduced χ2 = 12.53. Both fits have relatively large error bars and large reduced χ2 due to the low number of data points and fairly significant scatter in P-ρ space (Figure 4). Yet both fits give similar K’S to SWEOS, which is well defined by the linear fit. We therefore prefer the 3rd-order fit despite having large errors, since it fits the data sufficiently well, has a lower reduced χ2, and exhibits a negative q value, which has been shown to be typical for silicate liquids (see Chapter III). The EOS parameters and their uncertainties for An-Hd liquid are given in Table 4. An-Hd-Di liquid The An-Hd Hugoniot in US-up space is shown in Figure 5. The unweighted linear fit is US = 2.823±0.197 + 1.52±0.09 up km/s (r2 = 0.995). Behaving as the two liquids above, An-Hd-Di liquid is relaxed upon compression [Guo et al., 2013b in prep], which justifies an unweighted constrained linear fit of US = 2.846±0.014 + 1.54±0.02 up km/s (r2 = 0.999). This slope and IV-11 intercept constrain the SWEOS K’S and KoS values to be 5.15±0.06 and 22.03±0.25 GPa, respectively. The An-Hd-Di liquid was fit only with the 3BM/MG, as fitting the 4BM/MG to so few data points is under-constrained and does not yield a meaningful reduced χ2. The 3BM/MG result is KS’ = 6.20±0.17, q = –1.14±0.79, and reduced χ2 = 0.39. The results and uncertainties for each fit on An-Hd-Di are given in Table 5, and the Hugoniots are plotted in Figure 6. DISCUSSION Linear mixing assumptions To make progress on calculating the volumetric properties of the natural multicomponent silicate liquids that may occur in the lower mantle, the plan has been to constrain the EOS for a number of important end-member liquids using shock wave studies and then to attempt to reliably interpolate among them. End-member liquids that have been determined previously using shock wave methods are: Fe2SiO4 (Fa, fayalite), Mg2SiO4 (Fo, forsterite), MgSiO3 (En, enstatite), CaAl2Si2O8 (An, anorthite), and CaMgSi2O6 (Di, diopside). The recommended EOS parameters for each of these end members are given in Chapter III. In Thomas et al. [2012], these five end members were used to interpolate within the CaO-MgOAl2O3-SiO2-FeO major element component space to constrain the volume of a liquid along an isentropic temperature-pressure path for a desired composition, yet there was not a clear test of the validity of the two underlying assumptions: (1) linear mixing of volumes and (2) constant entropy of mixing for a given composition. The assumption of linear or ideal mixing is simplest assumption that can be made for mixtures, IV-12 Vtot = ∑i X iV i + Vmix , (4) where the volume of mixing (Vmix) is assumed to be zero, and hence the total volume (Vtot) is just a linear combination of the end-member partial molar volumes (Vi) multiplied by their respective mole fractions (Xi). The second assumption is that the configurational entropy of a liquid solution is only a function of composition and therefore does not change with variations in temperature and pressure, such that Stot = ∑i X i S i + Smix , (5) where the entropy of mixing (Smix) is assumed to be a constant and hence an isentrope or constant entropy path in T-P space can be found from =0, (6) where the T and P derivatives of entropy come from standard thermodynamic identities. Our data do not provide a test of this method of computing isentropes, since we lack experimentally defined isentropes for reference. We therefore focus on evaluating the volume of mixing assumption by comparing isentropes for intermediate compositions constructed with different sets of end members, while acknowledging that this exercise may be compromised to some extent by the configurational entropy issue. Glass and 1-bar melt data Both of these assumptions have historically performed fairly well at 1 bar. Linear mixing of temperature-independent partial molar heat capacities has been a successful model for most silicate liquids [Stebbins et al., 1984]. Similarly, linear mixing of volumes has been successful in IV-13 determining silicate liquid densities in most major element space (SiO2-Al2O3-MgO-CaO-Na2OK2O) [Lange, 1997; Lange and Carmichael, 1990]. However, exceptions to both rules do exist. Heat capacities for the Fe2O3 oxide component were shown to vary strongly with composition [Lange and Navrotsky, 1992] and the partial molar volumes of Fe2O3 and TiO2 components are both variable with composition. These exceptions are attributed to the Fe3+ and Ti4+ cations having multiple coordination states at ambient pressure that are highly dependent on the composition, i.e., fraction of non-SiO2 components [Liu and Lange, 2001; 2006]. More recent results have found this to be potentially true for Fe2+ as well, in that the partial molar volume of FeO (VFeO) in a melt is observed to depend on composition in some cases [Guo et al., 2013a submitted; 2013b in prep]. Such behavior of Fe2+ is well documented in the glass literature, which is important insofar as glasses provide a proxy for structural properties of melt at the glass transition temperature. When discussing melt structure, language is typically borrowed from random network theory [Zachariasen, 1932], which distinguishes oxides as network formers, network modifiers, or intermediates. Network-forming cations (e.g., Si4+) occur typically in tetrahedral ([4]) oxygen coordination in various connected units in the melt, whereas network modifiers (e.g., CaO, MgO) typically occupy octahedrally ([6]) coordinated sites and disrupt the tetrahedral network. Intermediates (e.g., Al2O3, TiO2) can take on either role, may occupy multiple coordination sites, and may either disrupt or stabilize the polymerization of a melt [Bottinga and Weill, 1972; Mysen, 1983; 1988]. This is a simplified view of glasses and melts, but it acts to guide our intuition concerning the behavior of cations and the contributions of Fe3+ and Ti4+ to the density, heat capacity, and other thermodynamic properties of silicate melts [Burns, 1993; Mysen, 1988]. IV-14 FeO is an intermediate glass former and has been documented to take on a range of [4] to [6] coordination dependent on composition. Fe2+ is predominantly a network modifier in basaltic glasses [Burns, 1993; Jackson et al., 2005; Mysen, 1983; 1988] and also likely in melts with similar SiO2 mole fraction. This is supported by the relatively low 1-bar VFeO value (~ 12.1 cm3/mol) for An-Hd, An-Di-Hd, and Hd-Di model basalts at 1732 K. This VFeO has been inferred to indicate an average Fe2+ coordination of ~[5.7] due to its similarity to the molar volume of wüstite, crystalline FeO (in which Fe2+ is octahedrally coordinated) [Guo et al., 2013b in prep]. On the contrary, Fe2SiO4 glasses contain very low-coordinated Fe2+ and are surprisingly polymerized [Cooney and Sharma, 1990]. X-ray absorption spectroscopy studies of both glass and melt of fayalite composition infer an average coordination of ~[4] [Jackson et al., 1993]. This is again supported by melt densities, where VFeO increases to ~14-17 cm3/mol [Mo et al., 1982; Shiraishi et al., 1978; Thomas et al., 2012], similar to the molar volume of crystalline CaFeO2, a structure in which Fe2+ is [4] coordinated [Guo et al., 2013a submitted]. Between these two extremes, Hd and the CaO-FeO-SiO2 (CFS) melts measured in Guo et al. [2013a submitted] display FeO in an average coordination state intermediate between [4] and [6]. The VFeO for Hd melt (~ 15.47 cm3/mol) is inferred to indicate [4.7] coordination, similar to the values in Hd glasses (~4.3) and in molecular dynamics simulation of this melt [Rossano et al., 2000]. The inferred coordination for CFS glasses range from 4.6-5.2. Guo et al. [2013b in prep] present an inverse correlation between CaO concentration and average Fe2+ coordination number. This is contrary to expectation from the model in Jackson et al. [2005] based on valence bond (VB) theory, which suggests a direct correlation between CaO concentration and FeO coordination. In the VB model, the Ca would have a greater “share” of the O valence due to the IV-15 higher bond valence of the Ca-O bond and force Fe to find charge balance with other oxygens and hence increase coordination. The composition dependence of Fe coordination in CFS melts appears to be more strongly related to the degree of polymerization (Si/O ratio) [Burns, 1993]. For example in Hd, where SiO2 is high (Si/O =0.32) and there are equal proportions of CaO and FeO, FeO fills both roles of network former and modifier, with an average value of [4.7]. For the family of CFS melts studies by Guo et al. [2013b in prep], the Fe content is fixed (40 mol %) and so increasing CaO is coupled to decreasing SiO2. The Fe2+ may be energetically forced to play the role of network former as CaO competes more effectively for octahedral sites. For fayalite liquid, the Si/O value is quite low (0.25), and FeO is again energetically favored as a network former to accommodate the charge balance of the oxygen, creating a highly polymerized glass as observed by Cooney and Sharma [1990]. Qualitatively, one can understand the importance of FeO as an intermediate behavior cation when Fe2SiO4 is compared to Mg2SiO4 melt, which has the same Si/O ratio. Since MgO is unable to act as a network former, Mg-rich melts typically have fully depolymerized structure and display very low viscosities. The dependence of FeO coordination and molar volume on other cation concentration invalidates the assumption of linear mixing of volumes at 1 bar for FeO-bearing liquids. It is unclear, however, what effect pressure and temperature may have. For glasses, higher temperatures are believed to decrease the coordination of transition metals [Jackson et al., 2005], and there is some evidence that cations in melts will favor a coordination state lower than that of the glass [Jackson et al., 1993]. At low pressure, network forming SiO4 tetrahedra predominate, but as P increases, Si4+ coordination increases (~6.5 at 150 GPa) [Karki et al., 2007]. Less is known about the coordination of network modifiers at high pressure. Preliminary modeling of IV-16 Fe2SiO4 liquid shows decreasing Fe-O bond length and increasing coordination of Fe with pressure, but this does not constrain the effect of other cations [Muñoz Ramo and Stixrude, in review]. Guo et al. [2013a submitted] and Guo et al. [Guo et al., 2013b in prep] provide indirect evidence that coordination of Fe2+ in basaltic melts may decrease towards [5] at moderate pressures (~5GPa). The only previous test of linear mixing at high pressure and temperature was for Fe-free anorthite-diopside liquids [Asimow and Ahrens, 2010], which found the linear approximation suitable at elevated T-P conditions along the measured Hugoniots. Linear mixing appeared to fail at low temperatures, but this may have been an artifact of uncertainties in the large extrapolation downwards from the Hugoniot temperature. Another partial success for linear mixing was found in MD simulations of An-Di and also in MD simulations of the MgO-SiO2 binary, each of which showed well-behaved mixing at high pressure which began to break down at lower pressure [de Koker et al., 2013; Martin et al., 2012]. Tests of linear mixing Our first test examines whether the VFeO difference between Fe2SiO4 and CaFeSi2O6 at 1 bar is still significant at high temperature and pressure. That is, can VFeO for either end member substitute for the other, as it would in the case of ideal mixing of volumes? To test this we compare the Hd isentrope from shock wave experiments determined in this study to a modeled linear combination of end-member liquids, where HdmodelFa = Di + 0.5Fa – 0.5Fo. The equations and details of creating isentropes of mixtures are given in Thomas et al. [2012]. The results are shown in Figure 7, where it is apparent that the model does not capture the behavior of the Hd isentrope. The errors for temperature and density are highly correlated such that if the model IV-17 predicts a lower temperature than the actual isentrope, it will over predict the density and vice versa. Such is the case for Hd, where the errors at high pressure (~120 GPa) and temperature are maximized, and the model underestimates the temperature by 30% and overestimates the density by 20%. In this test, comparing the use of two Fe-rich end members, we find a failure of linear mixing and this is presumably due to complexity in the coordination state of Fe2+. Our second test examines the question: Which model will perform better in creating an Fe-rich model basalt, An-Hd — using VFeO derived from Fa or VFeO derived from Hd? Our expectation from 1 bar coordination numbers is that since Hd should be a better end member for modeling An-Hd. Two mixing models, AnHdmodelFa = 0.5An + 0.25 Fa – 0.25 Fo + 0.5Di and AnHdmodelHd = 0.5 An +0.5 Hd, are compared to the An-Hd isentrope determined in this study. It can be seen in Figure 8 that neither model is able to recover the An-Hd isentrope, but that the Fa model does significantly worse. The Fa model also becomes unphysical at high pressure and lower temperature (Fig. 8a) and this breaks the anticorrelation between errors in T and ρ, which are both underestimated, by 15% and 10% at 120 GPa, respectively (Fig. 8b). The Hd model does significantly better with maximum errors at 120 GPa that overestimate T by 6% and underestimate ρ by 7% (Fig. 8d). These errors worsen systematically for isentropes with progressively higher potential temperature (Fig. 8c). These results are in line with our expectations based on 1-bar results: neither Fa or Hd models can fully recover the behavior of the An-Hd isentrope, but closer similarity of VFeO at 1-bar for Hd and An-Hd are significant enough to improve the fit at high pressure and temperature. Our third test examines the question: As FeO content decreases, does the difference between the end-member VFeO and that of the model basalt still play as significant of a role in the final volume of the mixture at high pressure and temperature? We attempt to recover the IV-18 isentrope of the model basalt, An-Hd-Di (which has less FeO than the previous An-Hd basalt), again using Fa and Hd end-member models: AnHdDimodelFa = 0.33An + 0.667Di + 0.167 Fa – 0.167 Fo and AnHdDimodelHd = 0.33 An + 0.33Hd + 0.33Di. The results in Figure 9 show improvement for both models with the Fa model having high-pressure errors that underestimate T by 12% and underestimate ρ by 4% and the Hd model overestimating T by only 3% and underestimating ρ by 3%. Again, the Hd model does significantly better than the Fa model, likely due to the greater structural similarity of FeO in Hd and in An-Hd-Di at 1 bar. This test also suggests, at least in the studied region of CaO-FeO-MgO-Al2O3-SiO2, that the linear mixing approximation becomes better as the amount of FeO in the mixture decreases. That is, despite the variable behavior of VFeO, there may be a threshold concentration of FeO below which this is negligible and the ideal solution approximation is adequate. It still remains to define best choice for VFeO (i.e., Fe2+ coordination state) for modeling a particular liquid composition in the mantle and to define the threshold FeO concentration below which this choice is unimportant. Our final test attempts to answer these questions by comparing the original chondrite model given in Chapter III that uses Fa as the VFeO component (ChmodelFa= 0.62En + 0.24Fo + 0.08Fa + 0.04An + 0.02Di), and a model which uses An-Hd-Di as the VFeO component (ChmodelAnHdDi = 0.62En + 0.32Fo + 0.45AnHdDi – 0.12An –0.28Di). The results in Fig. 10 show that the An-Hd-Di model predicts a slightly higher temperature but that models are within ±1.5% for both T and ρ. Hence, the models are effectively indistinguishable, indicating that liquids with XFeO ≤ 0.06 (chondrite model [Andrault et al., 2011]) can be suitably approximated with linear mixing. Mole fractions greater than this amount, such as An-Hd-Di (XFeO = 0.08), can be approximated within reasonable error (±3%), but only when using an FeO component that has a similar VFeO at 1 bar. IV-19 Direct knowledge of the coordination state of Fe2+ in a given liquid thus is an important constraint on the volume behavior of that liquid, especially for high FeO contents. Conversely, precise data on the equation of state of an Fe-bearing liquid can be interpreted primarily as a constraint on the coordination state of Fe2+ in that liquid. However, liquids containing Fe2O3, Na2O, K2O, varying amounts of SiO2, and other oxide species are far more complex than the five-component system studied so far. Other cations may influence the Fe2+ coordination state and may have variable coordination states themselves. In situ experiments analyzing cation coordination of complex glasses and melts at high pressure and temperature would greatly aid in elucidating these consequences and complementing macroscopic constraints from shock wave or other equation of state measurements. CONCLUSIONS We completed seventeen new pre-heated shock wave experiments on Hd, An-Hd, and An-Hd-Di liquids to determine the EOS of each composition. Having data on more compositions than are necessary to span the multicomponent oxide system, we applied these data to testing the validity of ideal mixing of volumes as a method of interpolation among end-member liquids. Ambient-pressure density measurements on these and other Fe-bearing silicate liquids indicate that FeO has a molar volume (and therefore coordination state) that is highly dependent on composition [Guo et al., 2013a submitted; 2013b in prep], and our results show that this behavior extends to high pressure. An Fe-bearing end member close in composition to the liquid of interest is necessary for adequately predicting temperature and density along its isentrope. We find that deviations from linear mixing of volumes are significant at the level of several percent for liquid mixtures with molar XFeO >0.06. On the other hand, densities and isentropic gradients IV-20 for liquids in the system CaO-FeO-MgO-Al2O3-SiO2 with less than this threshold concentration of FeO can be approximated both at ambient and at high pressure using ideal volume of mixing. IV-21 ACKNOWLEDGEMENTS The authors would like to thank the following: the shock wave lab technical staff—Oleg Fat’yanov, Eprapodito Gelle, Russel Oliver, and Emma Dodd. Thanks to X. Guo and R. A. Lange for their ongoing collaboration and willingness to share their manuscripts. This work was supported by the National Science Foundation through award EAR-1119522. IV-22 REFERENCES Ahrens, T. J. (1987), 6. 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W., and P. D. Asimow (submitted ), Direct shock compression experiments on pre molten Mg2SiO4 and progress towards a consistent high-pressure equation of state for CaO-MgO-Al2O3-SiO2-FeO liquids. Thomas, C. W., Q. Liu, C. B. Agee, P. D. Asimow, and R. A. Lange (2012), Multi-technique equation of state for Fe2SiO4 melt and the density of Fe-bearing silicate melts from 0 to 161 GPa, J. Geophys. Res., 117(B10), B10206. Williams, Q., and E. J. Garnero (1996), Seismic Evidence for Partial Melt at the Base of Earth's Mantle, Science, 273(5281), 1528-1530. Zachariasen, W. H. (1932), The atomic arrangement of glass, Journal of the American Chemical Society, 54(10), 3841-3851. IV-25 TABLES Table 1. Shock compression data Shot # Flyer material 413 Al Hedenbergite 1068 a T (K) ufp up Us ρΗ PH km s–1 km s–1 km s–1 Mg m–3 GPa Mo 1673 1.976 0.002 419 Mo 1674 3.497 0.001 417 Ta 418 416 415 414 An-Hd Mo Ta Ta Ta 1673 1673 1677 1672 1673 1675 4.471 4.066 4.535 5.144 5.613 6.187 0.005 0.012 0.001 0.020 Mo 1675 3.618 0.003 471 An-Hd-Di Mo 1673 1674 4.700 5.815 23.8 0.4 0.02 6.59 0.24 4.99 0.17 52.2 1.5 0.01 466 Mo 0.05 3.05 0.003 469 4.27 0.148 2.006 1.456 0.11 0.005 1673 1673 5.12 2.006 Mo Mo 0.01 0.002 1070 a 1074 a 1.58 2.68 3.65 4.10 4.46 4.88 1.62 0.01 0.01 0.01 9.01 9.46 0.10 0.09 0.05 5.29 5.42 5.59 0.05 67.6 89.1 0.06 109.1 0.03 145.6 0.03 124.5 0.3 0.4 0.6 0.9 0.5 0.12 4.04 0.05 23.2 0.4 0.01 6.56 0.15 4.87 0.10 51.3 0.9 0.003 3.55 0.01 0.01 7.99 8.66 0.28 0.10 0.11 5.72 0.03 32.9 5.17 4.27 0.05 4.98 0.04 0.01 0.014 4.37 8.29 0.06 4.62 10.10 1.211 0.016 7.50 0.07 0.01 0.002 2.82 5.56 3.87 4.99 5.59 0.13 0.06 14.3 78.7 0.08 104.8 0.9 0.8 0.8 1.0 1069 a Mo 1673 2.016 0.003 1.63 0.03 5.48 0.47 3.88 0.10 24.4 2.1 468 Mo 1673 3.503 0.008 2.70 0.01 7.00 0.07 4.43 0.03 51.5 0.4 1071 a 470 Mo Mo a Caltech 40mm propellant 1673 1674 1.470 4.796 0.005 0.006 1.20 3.61 0.01 0.02 4.77 8.31 0.30 0.16 3.64 4.81 0.09 0.09 15.6 81.7 0.8 1.3 IV-26 Table 2. Parameters used ρo Mg m-3 km s-1 Co s Molybdenuma (2273 K) Molybdenuma (300 K) Tantalumb (300 K) Aluminum 2024c (300 K) 16.65 2.78 9.785 10.21 3.293 5.330 4.858 5.033 1.288 1.289 1.307 1.34 aAsimow et al. [2008], Chase[1998]; b Mitchell and Nellis [1981b]; c Marsh [1980] Table 3. Equation of State fits for molten CaFeSi2O6 To ρo Co s γo q KSo KS' KS'' 𝜒2 KT α Cp Cv Units K kg m–3 m s–1 GPa GPa–1 GPa K-1 J kg–1 1 J kg–1 1 SWEOS 1673 2913 ± 11 2613 ± 16 1.54 ± 0.01 0.300 19.89± 0.24 5.16 ± 0.04 19.32 5.91-05 1345.48 11306.70 BM3 1673 2913 ± 11 0.300 –1.93 ± 0.41 19.89± 0.24 6.22 ± 0.55* 4.20 19.32 5.91-05 1345.48 11306.70 BM4 1673 2913 ± 11 0.300 0.47 ± 9.86 19.89± 0.24 3.15 ±1.61* 0.78± 0.60 3.26 19.32 5.91-05 1345.48 11306.70 Source G13a G13a fitted derived fitted derived derived/*fitted fitted derived G13a LN92 derived SWEOS= shock wave equation of state. 3BM/MG = 3rd-order Birch-Murnaghan/Mie-Grüneisen EOS 4BM/MG = 4th-order Birch-Murnaghan/Mie-Grüneisen EOS Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992], G13a is Guo et al. [2013a] IV-27 Table 4. Equation of State fits for molten An-Hd To ρo Co s γo q KSo KS' KS'' 𝜒2 KT α Cp Cv Units K kg m–3 m s–1 GPa GPa–1 GPa K-1 J kg–1 J kg–1 1 SWEOS 1673 2772 ± 3 2772 ± 11 1.39 ± 0.03 0.310 21.30 ± 0.25 4.54 ± 0.12 20.68 5.81-05 1442.34 11400.19 BM3 1673 2772 ± 3 0.310 –0.18 ± 2.25 21.30 ± 0.25 5.49 ± 0.65* 6.37 20.68 5.81-05 1442.34 11400.19 BM4 1673 2772 ± 3 0.310 1.67 ± 2.79 21.30 ± 0.25 5.10 ±1.61* -0.17± 0.82 12.53 20.68 5.81-05 1442.34 11400.19 Source G13b G13b fitted derived fitted derived derived/*fitted fitted derived G13b LN92 derived Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992], G13b is Guo et al. [2013b] Table 5. Equation of State fits for molten An-Hd-Di To ρo Co s Units K kg m–3 m s–1 SWEOS BM3 1673 1673 2722 ± 2 2722 ± 2 2846 ± 14 1.54 ± 0.02 0.361 0.361 γo q –1.14 ± 0.79 KSo GPa 22.03 ± 0.25 22.03 ± 0.25 KS’ 5.15 ± 0.06 6.20 ± 0.17* –1 KS’’ Gpa 𝜒2 0.39 KT Gpa 21.19 21.19 -1 α K 6.64-05 6.64-05 J kg–1 Cp 1491.96 1491.96 1 J kg–1 Cv 11434.47 11434.47 1 For abbreviations see Tables 3 and 4 Source G13b G13b fitted derived fitted derived derived/*fitted fitted derived G13b LN92 derived IV-28 FIGURE CAPTIONS Figure 1. Preheated (1673K) CaFeSi2O6 liquid Hugoniot in shock velocity (Us)-particle velocity (up) space. The dotted line represents the unconstrained linear Hugoniot, G13a is Guo et al.[2013a submitted]. Figure 2. CaFeSi2O6 liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 1. Abbreviations: SWEOS- shock wave equation of state; BM/MG – Birch-Murnaghan/Mie-Grüneisen EOS, G13a is Guo et al. [2013a submitted]. Figure 3. Preheated (1673K) An-Hd liquid Hugoniot in shock velocity (Us)-particle velocity (up) space. The dotted line represents the unconstrained linear Hugoniot. G13b is Guo et al.[2013b in prep] Figure 4. An-Hd liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 3. The abbreviations are the same as Figure 2 and 3. Figure 5. Preheated (1673K) An-Hd-Di liquid Hugoniot in shock velocity (Us)-particle velocity (up) space. The dotted line represents the unconstrained linear Hugoniot G13b is Guo et al. [2013b in prep] Figure 6. An-Hd-Di liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 6. The abbreviations are the same as Figure 2 and 5. IV-29 Figure 7. A comparison of the mixing model (HdmodelFa = Di +0.5Fa -0.5Fo) and the Hd isentrope at various potential temperatures (TP). Figure 8. a) A comparison of the mixing model using a Fa end member (AnHdmodelFa = 0.5An +0.25 Fa -0.25 Fo +0.5Di) and the An-Hd isentrope at various potential temperatures (TP) and b) its % errors in temperature and density in estimating the An-Hd isentrope. c) A comparison of the mixing model using a Hd end member (AnHdmodelHd = 0.5 An +0.5Hd) and the An-Hd isentrope at various potential temperatures (TP) d) its % errors in temperature and density in estimating the An-Hd isentrope. Errors shown in text are the maximum at 120 GPa. Figure 9. a) A comparison of the mixing model using a Fa end member (AnHdDimodelFa = 0.33An +0.667Di +0.167 Fa -0.167 Fo) and the An-Hd-Di isentrope at various potential temperatures (TP) and b) its % errors in temperature and density in estimating the An-Hd-Di isentrope. c) A comparison of the mixing model using a Hd end member (AnHdDimodelHd = 0.33 An +0.33Hd +0.33Di) and the An-Hd-Di isentrope at various potential temperatures (TP) d) its % errors in temperature and density in estimating the An-Hd-Di isentrope. Errors shown in text are the maximum at 120 GPa. Figure 10. a) A comparison of the two mixing models for chondrite liquid (ChmodelF a= 0.62E n+0.24Fo +0.08Fa +0.04An +0.02Di and ChmodelAnHdDi = 0.62En +0.32Fo +0.45AnHdDi - IV-30 0.12An -0.28Di) and b) their % error difference in temperature and density. Errors shown in text are the maximum at 120 GPa. IV-31 FIGURES Figure 1 IV-32 Figure 2 IV-33 Figure 3 IV-34 Figure 4 IV-35 Figure 5 IV-36 Figure 6 IV-37 Figure 7 IV-38 Figure 8 IV-39 Figure 9 IV-40 Figure 10  Liquid silicate equation of state: Using shock waves to understand the properties of the deep Earth Thesis by Claire Waller Thomas In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2013 (Defended April 11, 2013) ii  2013 Claire Waller Thomas All Rights Reserved iii ACKNOWLEDGEMENTS There are many to whom I owe thanks in the creation of this thesis. It takes a lot of people to blow up so many things—first thanks to Paul Asimow for being a pretty fantastic advisor, teacher, and mentor. He is likely the smartest person I know, but also manages to be the most patient and good-humored as well. That is a rare combination for which I am abundantly grateful. Thanks to the rest my committee: George Rossman, for academic and general life guidance; Jennifer Jackson, for an incredibly friendly and all-around pleasant demeanor, which makes me regret not throwing in a DAC chapter just for fun; and Ed Stolper, who despite having an incredibly full schedule makes time to serve on my committee and also provide both insightful and entertaining contributions to PRG each week. Thanks to the shock wave staff: Mike Long, Papo Gelle, Russ Oliver, and Oleg Fat’yanov for advice, patience, and insight as I learn from the masters of making things explode. My officemates Kristin Bergmann, Ajay Limaye, and Katie Stack are amazing people, but I can’t really thank them for helping me with my thesis as we would rarely get work done when we were all in the office at the same time. Thanks to the delightful Dodd family and the Caltech swim team for making me have a life outside of lab. Thanks to my roommates/neighbors of the past 5 years, Fay Bi and Clara O’Farrell, for being such great friends despite being only thrown together by lovely chance of the Caltech housing lottery. Thanks to my parents as I am sure raising me was unpleasant as was having to deal the with the confusion of what it is I have been doing the past 5 years,“ I thought you were a geologist.” And finally thanks to my husband Daniel. He is the best person I know. iv ABSTRACT The equations of state (EOS) of several geologically important silicate liquids have been constrained via preheated shock wave techniques. Results on molten Fe2SiO4 (fayalite), Mg2SiO4 (forsterite), CaFeSi2O6 (hedenbergite), an equimolar mixture of CaAl2Si2O8-CaFeSi2O6 (anorthite-hedenbergite), and an equimolar mixture of CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 (anorthite-hedenbergite-diopside) are presented. This work represents the first ever direct EOS measurements of an iron-bearing liquid or of a forsterite liquid at pressures relevant to the deep Earth (> 135 GPa). Additionally, revised EOS for molten CaMgSi2O6 (diopside), CaAl2Si2O8 (anorthite), and MgSiO3 (enstatite), which were previously determined by shock wave methods, are also presented. The liquid EOS are incorporated into a model, which employs linear mixing of volumes to determine the density of compositionally intermediate liquids in the CaOMgO-Al2O3-SiO2-FeO major element space. Liquid volumes are calculated for temperature and pressure conditions that are currently present at the core-mantle boundary or that may have occurred during differentiation of a fully molten mantle magma ocean. The most significant implications of our results include: (1) a magma ocean of either chondrite or peridotite composition is less dense than its first crystallizing solid, which is not conducive to the formation of a basal mantle magma ocean, (2) the ambient mantle cannot produce a partial melt and an equilibrium residue sufficiently dense to form an ultralow velocity zone mush, and (3) due to the compositional dependence of Fe2+ coordination, there is a threshold of Fe concentration (molar X Fe ≤ 0.06) permitted v in a liquid for which its density can still be approximated by linear mixing of endmember volumes. vi TABLE OF CONTENTS Acknowledgements ............................................................................................ iii Abstract ............................................................................................................... iv Table of Contents ................................................................................................ vi List of Tables and Figures ................................................................................ viii Chapter I Introduction................................................................................................. I-1 References................................................................................................... I-4 Chapter II Multi-technique equation of state for Fe2SiO4 melt and the density of Fe-bearing silicate melts from 0-161 GPa ............................... II-1 Abstract ...................................................................................................... II-2 Introduction................................................................................................ II-3 Methods ..................................................................................................... II-6 1-atm double-bob Archimedean methods .......................................... II-6 Sink/float methods............................................................................... II-7 Shock wave methods ........................................................................... II-8 Sample synthesis and preparation of preheated (1573 K) shots .. II-8 Preparation of room temperature (300 K) shots ......................... II-11 Experimental setup ....................................................................... II-11 Results ...................................................................................................... II-13 Density and ultrasonic results at ambient pressure .......................... II-14 Sink/float results ................................................................................ II-14 Shock wave results ............................................................................ II-14 Comparison to previous data ....................................................... II-14 Solid Hugoniot shocked to liquid / χ2 test of weighted values .... II-17 Grüneisen parameter .................................................................... II-17 Thermal equation of state fitting .................................................. II-20 Correction and re-analysis of silicate liquids.............................. II-26 Implications ............................................................................................. II-26 Mixing of oxide volumes at high pressure and temperature; application to a magma ocean liquidus ............................................ II-26 Buoyancy in the current temperature regime; application to the solidus and ULVZ .................................................................. II-32 Conclusions.............................................................................................. II-35 Acknowledgements ................................................................................. II-36 References................................................................................................ II-37 Chapter III Direct shock compression experiments on pre-molten Mg2SiO4 and progress towards a consistent high-pressure equation of state for CaO-MgO-Al2O3-SiO2-FeO liquids .................................................. III-1 vii Abstract ..................................................................................................... III-2 Introduction............................................................................................... III-3 Methods .................................................................................................... III-5 Sample preparation of preheated (2273 K) shots .............................. III-5 Experimental setup ............................................................................. III-6 Results ....................................................................................................... III-8 Linear Hugoniot ................................................................................. III-8 Grüneisen parameter ........................................................................ III-10 Thermal equation of state fitting ...................................................... III-12 Revised Equations of State for MgSiO3 and CaAl2Si2O8 liquids .. III-14 Enstatite, MgSiO3 ........................................................................ III-14 Anorthite, CaAl2Si2O8 .................................................................. III-15 Discussion ............................................................................................... III-18 Sound speed of Mg2SiO4 liquid ...................................................... III-18 Isentropes for a full mantle magma ocean ...................................... III-20 Evaluating the partial melt hypothesis for Ultralow velocity zones ................................................................... III-23 Conclusions............................................................................................. III-30 Acknowledgements ................................................................................ III-32 References............................................................................................... III-33 Chapter IV Preheated shock liquids Hd, An-Hd-Di, An-Di: a test for linear mixing at high pressure and temperature ................................................ IV-1 Abstract .................................................................................................... IV-2 Introduction.............................................................................................. IV-3 Methods ................................................................................................... IV-4 Sample preparation............................................................................ IV-4 Experimental setup ............................................................................ IV-6 Results ...................................................................................................... IV-7 Hedenbergite (CaFeSi2O6) liquid...................................................... IV-8 An-Hd liquid................................................................................... IV-10 An-Hd-Di liquid .............................................................................. IV-10 Discussion .............................................................................................. IV-11 Linear mixing assumptions ............................................................. IV-11 Glass and 1-bar melt data ................................................................ IV-12 Tests of linear mixing ..................................................................... IV-16 Conclusions............................................................................................ IV-19 Acknowledgements ............................................................................... IV-21 References.............................................................................................. IV-22 viii LIST OF TABLES AND FIGURES Chapter II Page Table 1: Double-bob density measurements........................................... II-41 Table 2: Shock compression data ........................................................... II-42 Table 3: Parameters used......................................................................... II-43 Table 4: Equation of state fits for molten Fe2SiO4 ................................. II-43 Table 5: BM/MG fits for silicate liquid .................................................. II-44 Table 6: Major oxide proportions for model mantle .............................. II-44 Figure Captions........................................................................................ II-45 Figure 1: Comparison of previous sink/float and shock wave results ... II-49 Figure 2: Schematic of shock wave arrival selection ............................. II-50 Figure 3: Double-bob density linear fit ................................................... II-51 Figure 4: Density-pressure diagram of sink/float results ....................... II-52 Figure 5: US-up preheated fayalite Hugoniot .......................................... II-53 Figure 6: Grüneisen parameter of liquid and solid silicates .................. II-54 Figure 7: P-ρ preheated fayalite Hugoniot.............................................. II-55 Figure 8: Model mantle isentropic temperature profiles ........................ II-56 Figure 9: Silicate liquid densities along a modern geotherm ................. II-57 Figure 10: Density of liquid mixtures for ULVZ ................................... II-58 Chapter III Page Table 1: Shock compression data........................................................... III-38 Table 2: Parameters used........................................................................ III-38 Table 3: Equation of state fits for molten Mg2SiO4............................... III-39 Table 4: Equation of state fits for molten silicate liquids ...................... III-39 Figure Captions....................................................................................... III-40 Figure 1:Modeled sound speed parameter space ................................... III-43 Figure 2: US-up preheated forsterite Hugoniot....................................... III-44 Figure 3: Grüneisen parameter of liquid silicates .................................. III-45 Figure 4: P-ρ preheated fayalite Hugoniot............................................. III-46 Figure 5: MgSiO3 Hugoniot dataset ....................................................... III-47 Figure 6: Estimates of Mg2SiO4 liquidus .............................................. III-48 Figure 7: Silicate liquid and solid densities along a geotherm ............. III-49 Figure 8: Re-modeled mantle isentropic temperature profiles .............. III-50 Figure 9: Partial melting of ULVZ composition .................................. III-51 Chapter IV Page Table 1: Shock compression data.......................................................... IV-25 Table 2: Parameters used....................................................................... IV-26 ix Table 3: Equation of state fits for molten CaFeSi2O6........................... IV-26 Table 4: Equation of state fits for molten An-Hd ................................. IV-27 Table 5: Equation of state fits for molten An-Hd-Di ........................... IV-27 Figure Captions...................................................................................... IV-28 Figure 1: US-up preheated CaFeSi2O6 Hugoniot................................... IV-31 Figure 2: P-ρ preheated CaFeSi2O6 Hugoniot ...................................... IV-32 Figure 3: US-up preheated An-Hd Hugoniot ......................................... IV-33 Figure 4: P-ρ preheated An-Hd Hugoniot ............................................ IV-34 Figure 5: US-up preheated An-Hd-Di Hugoniot ................................... IV-35 Figure 6: P-ρ preheated An-Hd-Di Hugoniot ....................................... IV-36 Figure 7: Hd mixing model .................................................................. IV-37 Figure 8: Comparison of An-Hd mixing models .................................. IV-38 Figure 9: Comparison of An-Hd-Di mixing models ............................ IV-39 Figure 10: Comparison of mantle mixing models ................................ IV-40  I-1 Chapter I Introduction Claire W. Thomas I-2 An equation of state (EOS) (i.e., density as a function of pressure and temperature) for silicate melts at high pressure is of primary importance for understanding processes occurring within terrestrial interiors. There has previously existed only limited knowledge of melt density and compressibility determined at low pressures (< 40 GPa) on a restricted range of compositions. The goal of my graduate work documented by this thesis was to constrain the EOS for a number of important end-member liquids using shock wave studies and to incorporate them into an empirical, predictive model of melt density as a function of temperature, pressure and composition. Chapter II describes the multi-technique collaboration of the University of Michigan, the University of New Mexico, and Caltech to constrain the EOS of molten Fe2SiO4 (fayalite). Shock wave experiments further extended the fayalite Hugoniot of Chen et al. [2002], which aided in resolving previously discrepant results for that study and the sink/float data set of Agee [1992]. The majority of the shock wave methods used for pre-heated experiments are described in this chapter. Subsequent chapters cover only changes made during each new set of experiments. The equations and derivation of the model used for creating isentropes of liquid mixtures is presented in detail in this chapter and is employed throughout the rest of the thesis. The determination of the molten Mg2SiO4 (forsterite) EOS is presented in Chapter III, along with revised EOS parameters for CaMgSi2O6 (diopside), CaAl2Si2O8 (anorthite) [Asimow and Ahrens, 2010], and MgSiO3 (enstatite) melts [Mosenfelder et al., 2009] . This new set of end member EOS, including fayalite, is used to determine the isentropic temperature profile of a fully molten magma ocean of two hypothetical bulk mantle compositions, chondrite [Andrault et al., 2011] and peridotite[Fiquet et al., 2010], by way of the isentrope mixture model presented in Chapter II. The temperature and pressure of first crystallization was determined from the I-3 intersection of the isentrope and the experimental liquidus. This chapter also describes a melting model which employs linear mixing of liquid volumes to determine whether a partial melt and its equilibrium residue could produce an aggregate density that is comparable to that estimated by Rost et al.[2006] for a ultralow velocity zone and still maintain a low melt-residue density contrast (~1%) employed by the seismic modeling of Williams and Garnero [1996]. The EOS for CaFeSi2O6 (hedenbergite), a 50-50 mixture of CaAl2Si2O8-CaFeSi2O6 (anorthite-hedenbergite), and an equimolar mixture of CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 (anorthite-hedenbergite-diopside) are determined and described in Chapter IV. A series of tests on the validity of using linear mixing of volumes to predict the densities of multicomponent liquids at high temperature and pressure are presented. Results for these tests indicate that Febearing silicate liquid densities can only be approximated as ideal under certain compositional restrictions. I-4 REFERENCES Agee, C. B. (1992), Isothermal compression of molten Fe2SiO4, Geophys. Res. Lett., 19(11), 1169–1172. Andrault, D., N. Bolfan-Casanova, G. L. Nigro, M. A. Bouhifd, G. Garbarino, and M. Mezouar (2011), Solidus and liquidus profiles of chondritic mantle: Implication for melting of the Earth across its history, Earth and planetary science letters, 304(1 – 2 ), 251 – 259. Asimow, P. D., and T. J. Ahrens (2010), Shock compression of liquid silicates to 125 GPa: The anorthite-diopside join, J. Geophys. Res., 115(B10), B10209. Chen, G. Q., T. J. Ahrens, and E. M. Stolper (2002), Shock-wave equation of state of molten and solid fayalite, Physics of The Earth and Planetary Interiors, 134(1 – 2), 35 – 52. Fiquet, G., A. L. Auzende, J. Siebert, A. Corgne, H. Bureau, H. Ozawa, and G. Garbarino (2010), Melting of Peridotite to 140 Gigapascals, Science, 329(5998), 1516-1518. Mosenfelder, J. L., P. D. Asimow, D. J. Frost, D. C. Rubie, and T. J. Ahrens (2009), The MgSiO3 system at high pressure: Thermodynamic properties of perovskite, postperovskite, and melt from global inversion of shock and static compression data, J. Geophys. Res., 114. Rost, S., E. J. Garnero, and Q. Williams (2006), Fine-scale ultralow-velocity zone structure from high-frequency seismic array data, J. Geophys. Res., 111(B9), B09310. Williams, Q., and E. J. Garnero (1996), Seismic Evidence for Partial Melt at the Base of Earth's Mantle, Science, 273(5281), 1528 – 1530.  II-1 Chapter II Multi-technique equation of state for Fe2SiO4 melt and the density of Fe-bearing silicate melts from 0 – 161 GPa Claire W. Thomas Qiong Liu Carl B. Agee Paul D. Asimow Rebecca A. Lange Accepted to Journal of Geophysical Research August 27, 2012 II-2 ABSTRACT New equation of state measurements on liquid Fe2SiO4 have been conducted in a collaborative, multi-technique study. The liquid density (ρ), the bulk modulus (K), and its pressure derivative (K’) were measured from 1 atm to 161 GPa using 1-atm double-bob Archimedean, multi-anvil sink/float, and shock wave techniques. Shock compression results on initially molten Fe2SiO4 (1573 K) fitted with previous work and the ultrasonically- measured bulk sound speed (Co) in shock velocity (US)-particle velocity (up) space yields the Hugoniot: US =1.58±0.03 up+ 2.438±0.005 km/s. Sink/float results are in agreement with shock wave and ultrasonic data, consistent with an isothermal KT=19.4 GPa and K’=5.33 at 1500°C. Shock melting of initially solid Fe2SiO4 (300 K) confirms that the Grüneisen parameter (γ) of this liquid increases upon compression where γ= γo(ρo/ρ)q yields a q value of –1.45. Constraints on the liquid fayalite EOS permit the calculation of isentropes for silicate liquids of general composition in the multi-component system CaO-MgO-Al2O3-SiO2-FeO at elevated temperatures and pressures. In our model a whole-mantle magma ocean would first crystallize in the mid-lower mantle or at the base of the mantle were it composed of either peridotite or simplified “chondrite” liquid, respectively. In regards to the partial melt hypothesis to explain the occurrence and characteristics of ultra-low velocity zones, neither of these candidate liquids would be dense enough to remain at the core mantle boundary on geologic timescales, but our model defines a compositional range of liquids that would be gravitationally stable. II-3 INTRODUCTION The formulation of an accurate equation of state (EOS) for silicate melts at high pressure and temperature is necessary to understand fully the important role of liquids in the differentiation and formation of terrestrial planets. The early history of the Earth may well have included one or more deep magma ocean events, perhaps extending to the core-mantle boundary (CMB) [Labrosse et al., 2007]. During crystallization of magma oceans and also during partial melting events in predominantly solid mantle environments, the formation of chemically distinct reservoirs is controlled by gravitational differentiation and therefore intimately linked to the solid and liquid EOS. In particular, the possibility of crystal-liquid density cross-overs — that is, for a given liquid and a given mineral or assemblage of minerals, a horizon above which the liquid rises but below which it sinks [Agee, 1988; Stolper et al., 1981] — implies dramatic bifurcation points in the possible paths of differentiation, leading to formation of either rising or sinking enriched reservoirs. These rising or sinking liquids will either potentially form crust or else potentially be sequestered at depth. Furthermore, changes in chemical equilibria with increasing pressure are defined by molar volume, the pressure derivative of the Gibbs free energy [Asimow and Ahrens, 2010]. Thus, accurate models of both the thermodynamic and fluid dynamic evolution of the Earth demand a fundamental knowledge of silicate liquid volumes at extreme pressure and temperature conditions. In addition, detection of ultra-low velocity zones (ULVZ) [Garnero and Helmberger, 1995] at the base of the mantle may indicate the presence of silicate partial melts [Lay et al., 2004; Mosenfelder et al., 2009; Williams et al., 1998] or more specifically iron-bearing melts [Labrosse et al., 2007]. Seismic observations such as S- and P-wave speeds and inferred density have given clues as to the nature of the materials in ULVZs, but there are a number of candidate II-4 iron-bearing solid phases that can produce the correct drops in wave speed and increased density without partial melting. Proposed phases include ferropericlase (Fe, Mg)O [Wicks et al., 2010] and iron‐enriched post‐perovskite [Mao et al., 2006]. Thus, improved constraints on iron-bearing silicate liquid densities at elevated pressure and temperature will aid in answering whether a liquid or liquid-solid mush could be dynamically stable at the CMB and similarly what the relative buoyancy of melt and its co-existing solids would be. There are presently no published experimental constraints on the density of iron-bearing silicate liquids beyond 47 GPa [Chen et al., 2002; Miller et al., 1991a]. Silicate liquids occupy a large, multi-dimensional continuum of compositions and therefore understanding their range of behavior requires either data on a very large number of compositions or data on a minimum set of end-member compositions and a reliable means of interpolating in composition space. At low pressure, ideal mixing of volumes has proven to be a reliable tool [Lange and Carmichael, 1987], but it remains poorly tested at high pressure [Asimow and Ahrens, 2010]. Here we present results on a pure iron-bearing end-member, fayalite (Fe2SiO4). In the future, this will allow assessment of intermediate compositions with lower iron contents without the need for extrapolation and provide an essential tie-point for the assessment of linear mixing along iron-magnesium mixing joins. Previous shock experiments on fayalite (Fe2SiO4) liquid achieved pressures up to 47 GPa [Chen et al., 2002], which corresponds to about mid-mantle range [Dziewonski and Anderson, 1981]. For this work, we extended this pressure range to 161 GPa, beyond CMB pressures, so we may apply our results to understanding the effects of iron on deep mantle systems. Additionally, this collaborative work sought to resolve inconsistencies in previous reports on Fe2SiO4 liquid compressibility and its change with pressure. Published isothermal static compression data from II-5 sink/float experiments [Agee, 1992] and shock wave results [Chen et al., 2002] are in close agreement for bulk modulus (Ko) values but differ greatly on the value of the pressure derivative (K’), 10.1 (isothermal) and 5.36 (isentropic), respectively, which give very divergent predictions for density at higher pressure. Isothermal and isentropic equations of state need not be the same, but a larger K’ for the isotherm implies a negative coefficient of thermal expansion, which is contrary to expectations. The anomaly is even more apparent when the Hugoniot data at estimated shock temperatures are compared to the extrapolated isotherm (Figure 1). For instance, at 15 GPa, the estimated shock state is at temperature (TS) 1783K and density ~4.75 g cm–3, somewhat denser than the extrapolated isotherm at the same P and T, ~4.5 g cm–3. At 50 GPa however, the estimated Hugoniot state is at temperature 2943 K and density ~6.0 g cm–3, much denser than the extrapolated 1773 K isotherm at 50 GPa, 4.75 g cm–3, whereas one expects density to decrease with increasing temperature. This discrepancy was also noted by Ghiorso (2004), whose equation of state fitting exercise produced an internally consistent prediction of all fusion curve data and Hugoniot volume data, but underestimated the density of neutral buoyancy experiments from Agee [1992] by 1%. We have conducted new measurements of the equation of state of molten fayalite (Fe2SiO4) using the following techniques: (i) double-bob Archimedean method for melt density and thermal expansion at ambient pressure (University of Michigan), (ii) multi-anvil sink/float technique to measure melt density to 4.5 GPa (University of New Mexico), and (iii) shock wave measurements of P-V-E equation of state to 161 GPa (Caltech). The aim of this multi-technique study has been to measure the density of molten Fe2SiO4, to resolve the discrepancy in published EOS for this material, and to use the measured equation of state to address processes of early earth differentiation and stability of a liquid at the CMB. II-6 METHODS 1- atm double-bob Archimedean methods The shared starting material for the double-bob density measurements and the shockwave experiments was synthesized at the University of California, Berkeley, by Mark Rivers and contains 64.48 mol% Fe as FeO, 34.97 mol% SiO2, and 0.56 mol% Al2O3 (wet chemical analysis is reported in Rivers and Carmichael [1987]), slightly off fayalitic composition (66.66 mol% FeO, 33.33 mol% SiO2). This is the same material used for ambient ultrasonic sound speeds reported in Ai and Lange [2004]. The starting material was a brick-red powder with iron predominantly in the form of ferric iron (Fe3+). To produce a starting composition with iron as Fe2+, approximately 75 g was loaded step-wise in four batches (~18-20 g each) into a straight-wall molybdenum crucible (7.8 cm deep and 3.6 cm inner diameter) placed in a Deltech furnace at 1300°C in a reducing stream of 1%CO-99%Ar. Each batch of melt was held at 1300°C for 12 hours until all 75 grams were loaded into the crucible and reduced. The Mo crucible was then placed on a platform in the Deltech furnace such that the top 3 cm of the liquid corresponded with the isothermal hotspot in the furnace. A thermal profile from the top of the crucible down into the sample had been previously documented for a Mo crucible filled with alumina powder for each temperature of measurement. Temperature was measured at the top of the crucible with an S-type thermocouple, calibrated against the melting temperature of gold. Liquid density measurements were made on the Fe2SiO4 liquid using the double-bob Archimedean method, described in detail in Lange and Carmichael [1987]. The method was adapted for a reducing atmosphere (1%CO-99%Ar) in this study by using molybdenum instead of platinum for the two bobs and the crucible. An electronic balance with a precision of ±0.0001 II-7 g was mounted on an aluminum platform above the furnace to measure the weights of the Mo bobs before and after immersion into the Fe2SiO4 liquid. By using two Mo bobs of different mass (~20 and ~7 g), but identical stem diameters (1 mm), the effect of surface tension on the stem was eliminated. The density of the liquid is calculated from the following equation: ρ (T ) = B L (T ) − B S (T ) V L (T ) − V S (T ) (1) where BL(T) and BS(T) are the buoyancy of the large and small bob, respectively, and VL(T) and VS(T) are the immersed volume of the large and small bob, respectively. The buoyancy is defined as the mass of the bob in the gas stream (corrected to vacuum) minus the mass of the bob during the immersion in the Fe2SiO4 melt. Four density measurements were made at each temperature with two different large bobs and two different small bobs. The accuracy of using Mo bobs instead of Pt bobs was tested by measuring the density of NaCl liquid at 1286 K and the result (1.445 g/cm3) closely matches that recommended by NIST [Janz, 1980] at this temperature (1.441 g/cm3). Sink/float methods Fayalite used for sink-float experiments was the same sample used in the original Agee [1992] study. This is a different starting material than used in the double-bob density and shockwave measurements of this study. The fayalite was synthesized in a Deltech gas-mixing furnace for 24 hours at controlled fO2 (IW-1). All sink-float experiments were carried out in a Walker-style multi-anvil device at the University of New Mexico, as opposed to a pistoncylinder device at the Bayerisches Geoinstitut used in Agee [1992]. Two types of gem-quality marker spheres were used: a spessartine garnet from the Malagasy Republic with composition Alm10.6Sps89.4 (also used in Agee [1992]) and a San Bernardino almandine-rich garnet II-8 Alm80.0Grs8.9Sps6.4Pyr4.7. Experiments were run in high purity molybdenum capsules, with run times of approximately 30 seconds at 1500°C. For further details of the experimental method, see Circone and Agee [1996]. Shock wave methods Sample synthesis and preparation of preheated (1573 K) shots The shared starting material used for the shock wave and double-bob experiments was reduced in a similar manner as described above in the double-bob methods section to produce a starting composition with the correct oxidation state (Fe2+). The powder was melted in a deep (3.5 cm deep, 1.7 cm inner diameter) molybdenum crucible hung on molybdenum wires in a Deltech furnace at 1300°C in a reducing gas stream of 1%CO-99%Ar. The powder was melted in short 30-45 minute runs; quenched and re-filled several times to reach the crucible’s capacity; and then left to equilibrate for 18 hours at temperature. The quenching process used was unfortunately slow to avoid CO gas release or rapid oxidation on hot exposure to air. An extralong ceramic hanging rod was used to push the crucible out of the hot spot to the bottom of the furnace tube to cool within the reducing atmosphere. Once the crucible was no longer glowing red (approximately 10 minutes), nitrogen was streamed for 10 minutes followed by a finishing quench in water. A combination of volume reduction on crystallization and surface tension at the crucible walls typically left a 5mm diameter triangular pit in the center, which reached nearly half way to the bottom of the crucible. Mössbauer spectra measured by Bjorn Mysen at the Carnegie Institution of Washington confirmed that Fe in the run products was reduced and dominantly in the form of fayalite and wüstite (FeO) with minor amounts of an iron-bearing accessory mineral, most likely magnetite. This aggregate was too fragile to be lapped into disks to fit into shock wave experiment sample II-9 holders, so it was re-ground and hot-pressed in a piston-cylinder device at 10 kbar and 725°C using a graphite sleeve and CaF2 base plug. The maximum fO2 in our graphite sample assembly is controlled by the CCO buffer[Holloway et al., 1992; Médard et al., 2008] with a calculated log fO2 of -16 at our P-T conditions [Jakobsson and Oskarsson, 1994; Ulmer and Luth, 1991].Experimental results from Médard et al. indicate that the fO2 within graphite capsules is likely more reducing than the CCO buffer (average value: CCO-0.07; IW+1.5), therefore the amount of Fe3+ introduced during this process is assumed to be negligible. The sample underwent synchronized slow decompression (1 MPa/minute) and cooling (1 K/minute) to reduce cracking during recovery, but usually the cylinder of fayalite naturally separated into rough 2–3 mm-thick disks. The hot-pressing increased the bulk density and sintered the aggregate into a machinable form; the sample disks were cored to 8 mm outer diameter. The composition was checked for carbon contamination from the graphite hot-pressing sleeve on a scanning electron microscope using energy-dispersive x-ray spectroscopy, but none was found. There were minor amounts of molybdenum contamination in the form of Fe-Mo metal oxides, but total bulk Mo content was constrained to less than 2 wt% with electron microprobe analyses. The disks were then lapped down to under-fill the volume of the molybdenum sample holder (see below) by 7-13%, such that — accounting for thermal expansion of the Mo holder [Miller et al., 1988] and the Fe2SiO4 liquid density at ambient pressure and 1300°C — a void space 1–1.5 mm-deep would remain at the top of the heated capsule. This small void space volume prevents blow out, excessive warping, or failure of the welded cap and yet remains far enough above the imaged area of the shock wave to preserve one-dimensional wave propagation. The molybdenum sample holders are comprised of a driver plate, sample well, and cap. The outer and inner portions of the rear-surface of the driver plate and both sides of the cap were II-10 polished to a mirror-finish with 1-micron alumina grit. For the two completed preheated shots, both caps had a thickness of 1mm, and all sample wells were 8mm in diameter and approximately 2.5mm deep. Shot 409 had a driver plate 1mm thick, and shot 396 had a 2 mmthick driver plate. After polishing, a line profile of the driver plate topography inside the sample well was measured with a depth gauge micrometer using a 0.7 mm flat-end tip in 0.2 mm increments along the centerline, which is aligned to the streak camera slit position. An outside contractor (Electron Beam Engineering Inc., Anaheim, CA) electron-beam welded the caps to the sample holder wells enclosing the sample to prevent leaking during experimental heating. After bake-out at 100±3 °C, an initial circular pass of a defocused, lowpower beam under 10–4 torr vacuum helps residual gases to escape; trapped volatiles could lead to bowing of the sample cap during experimental heating. Sample holders are leak-checked at room temperature in a helium bomb and additionally in a heat test chamber at 1573 K by visual inspection for escaping fayalite liquid. We note that fayalite is an especially troublesome liquid to weld into capsules due to its low melting point, high surface tension, and poor glass-forming; ten capsules were lost to bad welds in obtaining the two successful capsules that were used for our experiments. The final driver and top hat shape were measured with a depth gauge micrometer on a digital mill table in 0.1 mm increments along the bright vertical line in Figure 2a. The sample capsules were carbon coated on both sides to ~80-200 nm thickness using a Cressington model 208C coater except for a thin rectangular masked area where the streak camera slit images the shock arrival. The layer of carbon provides an oxygen sink to reduce oxidation the molybdenum during heating, which can degrade the reflection into the streak camera [Asimow and Ahrens, 2010]. II-11 Preparation of room temperature (300 K) shots One solid room-temperature (300 K) shot was performed, extending the pressure range of previous room-temperature shocks on fayalite [Chen et al., 2002] from 212 GPa to 229 GPa. This was done to verify that the highest-pressure 300 K shots (107, 161, 126, 412; See Table 2) melted during impact, which is necessary for calculation of the liquid Grüneisen parameter (γ). Details of this calculation are in the Results section under Grüneisen parameter. The sample used was a single crystal of fayalite synthesized by H. Takei [Takei, 1978]. The boule was cut perpendicular to the growth axis into an elliptical disk with a minimum diameter of 7.8 mm and a maximum diameter of 8.8 mm. The disk was doubly polished to 2 mm thickness (within ±0.002 mm of uniform thickness). The crystal contained some glassy inclusions, but their estimated volume from electron backscatter images was less than 4%. The sample was directly mounted on a 0.5 mm tantalum driver plate and was not encapsulated. A series of fused quartz mirrors were glued to the down-range side of the driver plate and sample [cf. Mosenfelder et al., 2007] for recording of the shock arrival. Experimental setup The experimental methods used in our study for molten materials were originally developed by Rigden et al. [1984] and refined by Rigden et al. [1988; 1989], Miller et al. [1988; 1991a], Chen et al. [1998; 2002] and Asimow et al. (2008). The experimental configuration is summarized here; for additional description, see Asimow and Ahrens [2010]. The experiments for this study were performed in a 90/25- mm two-stage light-gas gun; previous shots from Chen et al. (2002) were performed using a 40- mm propellant gun. Shots 396 and 409 were preheated with a H2O-cooled copper induction coil powered by a 10 kW Lepel radio frequency generator [Chen and Ahrens, 1998] to 1300°C (1573 K), 95°C above the melting II-12 point of pure Fe2SiO4. For preheated experiments, the sample holders are mounted through a hole drilled in a Zircar™ alumina board and are held in place by a “guy-wire” set up in which W-Re alloy wire is threaded through two Mo posts screwed into the sample side of the target. Tantalum flyer plates were pressed into Lexan projectiles and launched at velocities (ufp) of 4.997-7.085 km/s, measured by a double-flash x-ray system and a two-magnet induction detector [Asimow et al., 2008]. For all three shots reported here, the two measurements agreed to within the stated uncertainty, and the average of the two measurements was used for the ufp. For room-temperature experiments (shot 412), the streak camera is electrically triggered by signals from shorting pins. For preheated shots, the camera and xenon spark lamp (which illuminates the target) are triggered using a “non-contact” system from the flyer-magnet signals and an up-down counter [Asimow and Ahrens, 2010]. The light is filmed by a Hadland Imacon 790 streak camera equipped with a 25 μm slit focused horizontally across the center of the sample holder (Figure 2a). As the shock wave reaches the free surface of the driver and later the sample cover, it changes the reflectance of the polished surface, creating a visible cutoff on the film. The shock wave velocity (Us) through the sample is calculated from the time difference of these two cutoffs, corrected for the known shape of the driver face and cap and thermal expansion of the assembly, and subtracting the shock transit time through the Mo cap. The final shock state is calculated iteratively using the technique developed in Ridgen et al. [1988], which takes into account the thickness and EOS of the hot molybdenum cap. The camera writing rate was calibrated using a test streak modulated by a radio-frequency tuner at 147.9993 MHz, which provides a time signal every 6.757 ns. The streak photo was scanned at 2400 dpi in 8-bit gray scale and matched to the calibration streak scanned at the same resolution to assign each column of pixels a time value. Driver and sample cutoffs were picked by selecting the inflection point of II-13 an arctangent function fit by cross-correlation to the gray scale values of each row of pixels parallel to the time axis (Figure 2c, shows only 3 pixel rows). If the correlation coefficient for the arctangent fit for a given pixel line was less than 0.95, then that pick was discarded. Discarding of pixels usually occurred in dark portions of the photo that were either poorly exposed or dark in the pre-experiment static photo due to imperfections in the polish. A few hundred points on either side usually constrain the driver cutoff (except in shot 396, see below); these picks are coregistered with the previously measured driver topography profiles and corrected for deviations from flat. Although tilt and bowing of the impactor and hence the shock front are often approximated as parabolic [Mitchell and Nellis, 1981a], we found that fourth order polynomial fits were suitable for extrapolating the shape of the shock front within the sample (Figure 2b). Only the center 3mm portion of the sample arrival was used for the calculation of shock transit time to reduce edge effects. The iterative solutions [Rigden et al., 1988] to particle velocity (up), pressure (PH), and density (ρH) of the shock state were determined from impedance matching, the Rankine-Hugoniot equations, and the following input parameters: sample geometry; standard Hugoniot data (ρo, Co, s) for the metal flyer, driver and cover plates [Asimow et al., 2008; Mitchell and Nellis, 1981b]; and the initial density (ρo) and best-initial guess values for 1 bar bulk sound speed (Co) and Hugoniot slope s for fayalite liquid at 1300 °C. It should be noted that regardless of the initial guess for the Co and s the same resulting shock state converges after three iterations). Uncertainties in all derived parameters (US, up, PH, ρH) were obtained from the uncertainties in the measured and standard parameters by analytical error propagation [cf. Jackson and Ahrens, 1979]. RESULTS II-14 Density and ultrasonic results at ambient pressure The double-bob density measurements between 1551 and 1831 K are reported in Table 1 and plotted against temperature in Figure 3. A linear equation: ρ(T) = 3699±1 – 0.30±0.01*(T1573 K) was fit to the data. Sound speed (C) data on the same sample between 1516 and 1879 K are reported in Ai and Lange [2004] using a frequency-sweep acoustic interferometer. Those data can also be described as a linear function of temperature: C(T) = 2438±5 – 0.33±0.03*(T-1573 K) m/s. Sink/ float results The results from the sink/float experiments are shown in Figure 4. We observed floating of the almandine garnets at 3.5 and 4.5 GPa and floating of spessartine garnets at 4.0 GPa. We were unsuccessful in observing sinks or floats at 3.0 GPa because of marker sphere dissolution or breakdown during the run. Therefore, our new data do not bracket the fayalite liquid density. However the observed floats do give a minimum density of the liquid within the experimental pressure range 3.5-4.5 GPa, and thus the new data do not agree with the sink/float measurements of Agee [1992] requiring that the compression curve for fayalite liquid be at higher density values in the range 3.5-4.5 GPa. On the other hand, the measurements of Agee [1992] at lower pressure using ruby spheres are still consistent with the new multi-anvil measurements. Reasons for possible inconsistencies in the Agee [1992] are discussed below. Shockwave results Comparison to previous data The shock wave data are reported in Table 2, including shot number, flyer/driver material, temperature prior to firing, ufp, up, US, PH, and peak density (ρH). Figure 5 shows the Hugoniot of this data in US-up space. A Hugoniot is a family of shock states achieved in a II-15 material from the same starting conditions (in this case, liquid fayalite at 1573 K and 1 bar). Empirically the Hugoniot of a well-behaved material forms a line in US-up space, given to third order in strain by US = Co + s up [Jeanloz, 1989] where the intercept Co = (KoS / ρo)1/2 is the bulk sound speed of fayalite liquid at room pressure and the slope s = (K’S + 1)/4 is related to K’S, the isentropic pressure derivative of the isentropic bulk modulus (KoS) [Ruoff, 1967]. Previous work from Chen et al. [2002] consists of 7 experiments completed on the Caltech 40-mm propellant gun. All Chen et al. data are taken from the original publication and were not re-analyzed. These data have widely varying reported uncertainties; Chen et al.’s linear regression was weighted by errors on Us only: Us = 2.63±0.02 + 1.59±0.01 up km/s. To these data, we added two preheated shocks, extending the Hugoniot to 161 GPa. Shot 409 from this study had clear cutoffs and well-defined shock arrivals. Conversely, shot 396 had a poorly exposed driver arrival, but showed a simple, un-tilted planar shock arrival at the top hat. The driver arrival for this shot was approximated as linear and un-tilted, which introduced error in the travel time measurement but is a suitable assumption due to the flatness of the top hat. An un-weighted linear fit to all nine preheated fayalite liquid data points from Chen et al. [2002] and this study yields Us = 2.67±0.08 + 1.50±0.03 up km/s (r2= 0.996), whereas a York regression (a regression which accounts for errors in both Us and up) gives Us = 2.80±0.18 + 1.488±0.071 up km/s (MSWD = 8.9). The unconstrained intercept of the York fit to the full data set is consistent within the 99% confidence interval with the bulk sound speed determined to much higher precision by ultrasonic techniques, Co= 2.438±0.005 km/s [Ai and Lange, 2004]. This suggests we cannot confidently reject the null hypothesis that this material, like other silicate liquids measured in the same way, displays relaxed (liquid-like), as opposed to un-relaxed (glass-like), behavior upon II-16 shock compression [Rigden et al., 1988]. Therefore, to reduce the error on the linear fit, we fixed the intercept at the ultrasonic value measured in this work. The un-weighted, constrained linear fit in US-up space yields US = 2.438±0.005 + 1.58±0.03 up km/s (r2=0.998). A weighted, constrained fit yields poor results because the uncertainties of US reported for some of Chen et al.’s data are anomalously small. The selected slope s corresponds to a K’S value of 5.33±0.10, and overall this is in reasonable agreement with the previous work by Chen et al. [2002], which estimated K’S of 5.39±0.04. Figure 5 includes both the unconstrained and constrained linear fits to the entire data set, as well as the previous fit by Chen et al. [2002]. It is worth noting that the slopes of the new and old linear fits are very similar despite having slightly different though nominally fayalitic compositions. Thus within the errors of the experiments, our results seem to be insensitive to minor compositional differences. Our newest shockwave and sink/float results also resolve previously discrepant measurements for K’S. Isothermal static compression data from sink/float [Agee, 1992] and shock wave [Chen et al., 2002] experiments gave K’ values of 10.1 and 5.36, respectively. The newest sink/float measurements are shown in Figure 4 alongside two 3rd-order Birch-Murnaghan isotherms (dashed and dotted lines).The isotherms are fit with ρo from this study, the bulk modulus from ultrasonic experiments (KT=19.4) [Ai and Lange, 2004], and K’ from the slope of the newest constrained linear Hugoniot (K’=5.33)(dashed in Fig. 4) or K’ from the 3rd-order Birch-Murnaghan/Mie-Grüneisen (3BM/MG) (K’=7.28) from this study (dotted in Fig. 4). The details of the 3BM/MG fit are given below in the Thermal equation of state fitting section. The solid line is calculated using the previous data from Agee [1992]. The three experiments showing flotation are consistent with the dashed and dotted lines and discount the previous measurements (>3 GPa) on identical material used in the 1992 study. It is conceivable that piston “bottoming- II-17 out” or other unknown friction effects at high compressions and high temperatures produced lower pressures than expected from calibration runs. In any case, the new multi-anvil sink/float data are in good agreement with the ultrasonic and shockwave results. Solid Hugoniot shocked to liquid / χ2 test of weighted values Shot 412 was carried out on single crystal fayalite and is believed to have undergone a phase transition to liquid during the shock. A phase transition to liquid along the Hugoniot can often be detected by an offset in pressure-volume [Akins et al., 2004] or US-up space. To establish where in US-up space the phase transition occurs, we used a χ2 test of weighted values on linear fits to all room-temperature shock data (Table 1). The goal was to minimize the χ2 value of fitting two separate linear Hugoniots to solid and shock-melted data. Excluding shot 116, which has an anomalously low Us, the best fit (χ2= 0.0014867) occurred when shots 107, 161, 126, and 412 were calculated as liquids. The pressure limit of 190 GPa for shock melting implied by placing the boundary between shots 99 and 107 is also consistent with the melting thresholds along the Hugoniots of many iron-free silicate minerals shocked from room temperature: quartz at 120 GPa [J. A. Akins, 2002; Lyzenga, 1983], diopside at ≤144 GPa [Svendsen and Ahrens, 1990], and diopside-anorthite (Di64An36) at 133 GPa [Asimow and Ahrens, 2010] all of which have higher ambient melting temperature than that of fayalite (previous descriptions of shock melting of forsterite at 142 GPa [Mosenfelder et al., 2007] enstatite at 174 GPa [Mosenfelder et al., 2009] may not be accurate and are discussed in Chapter III). We have therefore used the four highest-pressure data points, presumed to have melted on shock, to calculate a value for the liquid Grüneisen parameter. Grüneisen parameter The thermodynamic Grüneisen parameter (γ) is a macroscopic parameter that relates II-18 thermal pressure to the thermal energy per unit volume, where thermal pressure is defined as the increase in pressure due to heating at a constant volume. Because a solid shocked to a liquid and an initially molten sample obtain different pressures and internal energy states at equal density (ρH) upon compression, γ can be directly determined by comparison of the two Hugoniots (e.g., [Luo et al., 2002; Mosenfelder et al., 2007, Asimow & Ahrens, 2010]). We calculated the Grüneisen parameter from the following expressions; the parameter definitions and the values used are given in Table 3. By definition, γ= 1  ∂P    . ρ  ∂E  ρ (2) The Mie-Grüneisen approximation defines γ as a function of ρ only, which simplifies the above expression to a finite difference: γ= 1 ∆Pth 1 ( PH −liquid − PH − solid ) = , ∆E ρ ∆E ρ (3) where ρ is the density of interest and ΔPth and ΔE are the offsets in pressure and internal energy between the two Hugoniot states at that density. The subscripts H-liquid and H-solid correspond to states on the Hugoniots that are both liquid in the shock state but that were initially a liquid at 1573 K and a solid at 300 K, respectively. The energy offset is calculated from ∆E = ∆E H − solid − ∆E H −liquid + ∆E tr , (4) where ∆Etr , the transition energy, is the internal energy difference at ambient pressure conditions and is estimated by the enthalpy difference ( ∆H tr ) because the transition takes place at nearly zero pressure [Asimow and Ahrens, 2010]. ∆H tr is composed of two parts, the integral II-19 of the isobaric heat capacity (CP) and the enthalpy of fusion, and was calculated with MELTS [Ghiorso and Sack, 1995]. The internal energy change across each shock front (ΔEH) was calculated using the 3rd Rankine-Hugoniot relation, ∆E H = PH  1 1    − 2  ρ o ρ H  . (5) There are several empirical forms for describing the density dependence of γ(ρ) consistent with the Mie-Grüneisen assumption [Stixrude and Lithgow-Bertelloni, 2005]. We will look only at the power law form, which is both the simplest and most prevalent in the literature: γ (ρ ) = γ o (ρ o / ρ ) q . The Grüneisen parameter at initial conditions is derived from the expressions (6) αK T ρCV (7) CP = 1 + αγT , CV (8) γ= and where α is the isobaric coefficient of thermal expansion, CV is isochoric heat capacity, and CP is isobaric heat capacity. Fitting the offset between the Hugoniots of fayalite liquid shocked from liquid and solid initial states to equation (5) yields a q value of –1.45 based on data in the vicinity of ρo/ρ =0.49. In solid materials γ generally decreases upon compression (Figure 6), but a negative q value demonstrates that γ for this liquid increases upon compression. Moreover, this q value is similar II-20 to values obtained on other silicate liquid compositions using Hugoniot offsets, such as forsterite (q = –1.47) (corrected from the Mosenfelder et al. value, see Table 5; Chapter III includes a new EOS from preheated forsterite), enstatite (q = –1.71) [Mosenfelder et al., 2009] (revised in Chapter III), and anorthite-diopside eutectic (q = –1.63) [Asimow and Ahrens, 2010]. We take this result as strong evidence that iron-bearing silicate liquids follow the same general behavior as iron-free silicate liquids, which are all described by –2.0 ≤ q ≤ –1.45 for the compression range 1 ≥ ρo/ρ ≥ 0.49. Figure 6 shows the similarities in volume dependence of γ for liquids of composition MgSiO3 (En), Mg2SiO4 (Fo), and Fe2SiO4 (Fa) as well as the opposite dependence displayed by mantle solids. Negative q values for liquids have been interpreted qualitatively as a result of increasing cation coordination by analogy to structural phase changes in related solids [Stixrude and Karki, 2005] and, more recently, quantitatively in terms of liquid configurational entropy in the context of a hard-sphere model of atomic packing in liquids [Jing and Karato, 2011a]. The consistent behavior of fayalite with other liquids so far studied does not obviously resolve the question of the microscopic origin of the behavior, but it shows that the behavior does not depend very much on the major cations in the liquid. Thermal equation of state fitting A Hugoniot is a one-parameter curve through a P-V-E thermodynamic space of two independent variables, in this case V(P, E). However, we are generally interested in defining the behavior of liquids in regions that lie off the Hugoniot curve. In particular, at high compressions, the Hugoniot reaches temperatures and energies much higher than mantle conditions of geophysical interest, even for the early earth. In order to investigate material properties that lie off the Hugoniot, we attempt to define the entire P-V-E surface for a given composition by fitting II-21 a thermal equation of state. There are several formalisms for thermal EOS; we have chosen to investigate three of them (all of which are plotted in Figure 7): (1) the shock wave equation of state (SWEOS, derived from both the constrained and unconstrained linear fit above), (2) 3rd and 4th order Birch-Murnaghan Mie-Grüneisen equation of state (3BM/MG and 4BM/MG), and (3) de Koker-Stixrude fundamental relation (dKSFR) [de Koker and Stixrude, 2009]. The SWEOS is defined by a linear Hugoniot in US – up space, transformed into pressure (P)- density (ρ) space using the first and second Rankine-Hugoniot equations [e.g., Ahrens, 1987]: ρH = ρ oU S US − up PH = ρ oU S u p (9) . (10) States that lie off the Hugoniot are found using the Mie-Grüneisen thermal pressure approximation with temperature-independent γ(ρ), as calculated above (6). The 3rd- and 4th-order Birch-Murnaghan/Mie-Grüneisen equations of state are defined by 3rd or 4th order Birch-Murnaghan isentropes centered at 1 bar and 1573K plus a Mie-Grüneisen thermal pressure approximation. Thus, total pressure P at any volume and internal energy is given by P = PS + Pth . (11) Ps is the Birch-Murnaghan isentropic pressure given to 4th order in strain by 5 2 PS = 3K oS f (1 + 2 f ) (1 + 3 3 143 2 ( K ' S −4) f + ( K " S K oS + K ' S ( K ' S −7) + ) f ) (12) 2 2 9 II-22 where f is the Eulerian finite strain parameter 2    1  ρ  3 f =    − 1 . 2  ρo     (13) [Birch, 1947]. The thermal pressure term (Pth) is given by Pth = γρ ( E − E S ) (14) where E is the energy of interest. The internal energy along the isentrope (ES) is found by integration of dES = –PSdV: ES = 9 K oS 2 3 143 4 ( f + ( K ' S −4) f 3 + ( K " S K oS + K ' S ( K ' S −7) + )f ) 2 ρo 4 9 (15) For 3rd-order BM isentropic pressure and energy, K” in the previous expressions is fixed to the value that makes the 4th-order term vanish: K " S = −( K ' S ( K ' S −7) + 143 ) / K oS 9 (16) In order to infer a temperature, we need to define an isochoric heat capacity (CV). Currently no shock temperature measurements exist to constrain an expression for CV(ρ) for an iron-bearing silicate liquid, so CV was fixed at the constant value determined from constant CP at ambient pressure and 1573 K [Lange and Navrotsky, 1992]. The temperature of interest is then calculated with the equation ρ T = To E γ (ρ ) 1 + d ρ ∫ρ ρ ∫E CV dE o o S , (17) II-23 which is composed of two parts: (1) integration along the reference isentrope to the density of interest with To = 1573 K as the foot temperature and (2) addition of the energy offset proportional to CV. A method for fitting shock wave data to the 3BM/MG and 4BM/MG forms is found in Asimow and Ahrens [2010]. To account for uncertainties in both pressure and volume, the function  P i ( ρ i ) − P i  2  ρ i − ρ i  2  H  χ 2 = ∑i =1  * * i + * i H     σ    σρ ρ      n (18) is minimized, where pressure is evaluated along the model Hugoniot at the trial density P*i = PS − γρ *i ( E S − E tri )  γ  ρi 1 −  *i − 1 2  ρo  . (19) The subscript * denotes a trial pressure or density, and σ denotes the respective uncertainties for each pressure-density experimental pair, (PiH, ρiH). Uncertainties in the fitted parameters are estimated by bootstrap iteration [Efron, 1982]. The input parameters that remain fixed are KoS, CV, ρo, and γo; the best-fit parameters (K’S and q, plus K’’ for 4BM) are given in Table 4. The 3BM/MG fit result is KS’ = 7.28±0.35, q = –0.95±0.51, and a reduced χ2 = 1.526. The 4BM/MG fit result is KS’ = 6.64±0.58, KS” = –0.47±0.27 GPa-1, q = –0.54±1.49 and reduced χ2 = 0.941. The 4BM/MG chi-squared value indicates probable over-fitting, and the fairly large error bars on all output parameters seem to indicate a very unstable fitting routine. On the other hand, the 3BM/MG seems relatively stable. The unconstrained-SWEOS appears to have similar form to the 3BM/MG and matches the highest pressure data better than the constrained SWEOS. The II-24 stiffness in the constrained SWEOS is likely due to being tied to the intercept and the most precise of Chen et al.’s data points—the precision of which is likely overestimated. The q value for the SWEOS is just within the stated errors of the 3BM/MG value, and the 3BM/MG reduced chi-squared value is reasonable. Additionally with little justification to implement the 4th-order fit, we recommend the 3rd-order fit for fayalite liquid. The initially solid data are included in the fits to the 3BM/MG and 4BM/MG thermal EOS for the liquid (figure 7). The 4th-order fit recovers these data very well, but only the 3rdorder fit appears justified by the fitting statistics. The Eulerian finite strain Birch-Murnaghan EOS has been used widely in the literature for describing the volumetric behavior of liquids [Agee, 1998; Y Ai and Lange, 2008; Asimow and Ahrens, 2010; Bottinga, 1985; Lange, 2007; Rigden et al., 1989; Suzuki et al., 1995; Tenner et al., 2007]; it has thus far proven to be adequate for fitting although it lacks any rigorous basis in theory as an EOS for liquids. The BM formalism has been criticized for its theoretical weakness in giving unphysical interatomic potentials for high K’ materials (gases and liquids) [Hofmeister, 1993] and for singularities that occur at high temperature and low pressure [Ghiorso, 2004]. Consequently, although the BM EOS can fit our data and allows interpolation between experimental points, it should be used only with caution for extrapolation beyond experimental constraints. It is a difficult task to capture the nature of liquids, which have short-range order but lack long-range order, in a single EOS. Two recent EOS formalisms have been developed specifically for silicate liquids that are both thermodynamically and theoretically self-consistent in their approach to describing the volume behavior at elevated temperatures and pressure: (1) the deKoker and Stixrude fundamental relation (dKSFR) [de Koker and Stixrude, 2009] and (2) the II-25 hard sphere equation of state (HSEOS) [Jing and Karato, 2011a]. The dKSFR is an extension of the finite strain description given in Birch [1952; 1978] but additionally takes into account the thermal free energy contribution and temperature dependence of liquid properties. The thermodynamic variables are self-consistent in that they are derived from a single equation for the Helmholtz energy, and the contributions to the Helmholtz energy are separable [McQuarrie, 1984] into an ideal gas term, an excess term, and an electronic term. Recent work by Muñoz Ramo and Stixrude [in review] using first principles molecular dynamics (FPMD, GGA+U) of Fe2SiO4 liquid obtained a fit with a fourth-order expansion in finite strain and first-order expansion in the reduced temperature using the dKSFR. Their Hugoniot, shown in Figure 7, is in good agreement with our shockwave data. However, contrary to the monotonic increase in γ with increasing pressure inferred from shock wave studies, their results show γ increasing up to a compression of ρo/ρ ≈ 0.49 followed by a decrease in γ coinciding with a change from the high-spin state of Fe. The HSEOS assigns a hard sphere for each cation species, which move freely in the liquid. The geometrical arrangement of these spheres gives the contribution of entropy to the compression of the liquid [Jing and Karato, 2011a]. This formalism does well in explaining the complex behaviors of melts, such as the dependence of γ with pressure. However currently, the calibration of the Fe sphere diameter is based on two experiments within a close range of temperature. Therefore, the Fe sphere diameter dependence on temperature remains somewhat poorly constrained for extreme temperature conditions along the Hugoniot. Calibration would be improved with shock temperature measurements or any other independent measurements of Febearing silicate liquids within a wider temperature range. II-26 Corrections and re-analysis of silicate liquids Also in this study, the anorthite and diopside liquid data of Asimow and Ahrens [2010] were re-analyzed, only sampling the center 3 mm of the arrival top hat. This was motivated by a hydro-code model of shock propagation through our capsule geometry carried out at Lawrence Livermore National Laboratory by J. Nguyen, which indicated that the outer part of the top hat arrival is influenced by edge effects. The corrected BM/MG fits are given in Table 5, and the reanalyzed data points are given in Table 1 of Chapter III. The 4th-order fit for anorthite liquid excludes 3 shots: 382, the highest pressure shot from Asimow and Ahrens [2010], due to an abnormally shaped shock arrival; and shots 663 and 672, the highest pressure shots from Rigden [1988] due to possible un-relaxed behavior upon compression. For further discussion of glasslike behavior in anorthite under some shock conditions, see Asimow and Ahrens [2010], section 4.4. The re-analysis of diopside liquid excludes 378, the lowest pressure shot from Asimow and Ahrens [2010] due to irregular and over-exposed top hat arrival. Also presented in Table 5 is the corrected forsterite liquid γo (which previously contained a sign error), and its subsequent re-fitted 4BM parameters. The entire EOS of forsterite is revised in Chapter III. IMPLICATIONS Mixing of oxide volumes at high pressure and temperature; application to a magma ocean liquidus Our extension of the EOS of fayalite liquid completes the basis set needed for estimation, subject to the assumption of linear mixing, of liquid densities in the full CMASF (CaO-MgOAl2O3-SiO2-FeO) major element space for mafic to ultramafic liquids at elevated temperatures and pressures. Shockwave experiments completed thus far that span this component space are II-27 forsterite (Fo, Mg2SiO4) [Mosenfelder et al., 2007], enstatite (En, MgSiO3) [Mosenfelder et al., 2009], anorthite (An, CaAl2Si2O8), diopside (Di, CaMgSi2O6) [Asimow and Ahrens, 2010], and now also fayalite (Fa, Fe2SiO4). All of the EOS parameters for each of these compositions, corrected (see above) or recommended by the original authors, are given in Table 5.The revised EOS are discussed in detail in the subsequent chapter (Tables 3 and 4) In this study we examine only one simple model of many possible scenarios for the crystallization of an early terrestrial magma ocean. Our initial condition is a well-mixed, convecting liquid composed of one of two plausible bulk silicate Earth compositions — “chondrite” or peridotite — that reaches the base of the mantle. To learn at what depth and critical potential temperature such a whole-mantle magma ocean would begin to crystallize, we used mixtures of silicate liquids in the CMASF system and calculated the intersections or tangency points between isentropes of these two liquid compositions and their respective liquidus curves. The basis of our model is derived from an alternative definition of the Grüneisen parameter, γ = (∂ ln T ) /(∂ ln ρ ) S , (20) which expresses the adiabatic temperature gradient with increasing density [Miller et al., 1991b]. Figure 8 shows the two mantle compositions that we examined. The first (Fig. 8a) is a synthetic “chondrite” mix [Andrault et al., 2011] with relative proportions of the five major oxides akin to the primitive mantle but lacking any minor components (TiO2, Na2O, K2O… etc.); its major element ratios are similar to pyrolite composition [Ringwood, 1975]. The second composition (Fig. 8b) approximates the major oxide composition of KLB-1 peridotite [Fiquet et al., 2010], a fertile spinel lherzolite. We also considered the KLB-1 composition given by Davis et al. [2009], but found that the isentropes were nearly indistinguishable from those calculated from the II-28 published composition in Fiquet et al. [2010]. Table 6 gives the major oxide proportions for each composition as well as the normalized percent of each end-member liquid used in the model. We also were concerned with what the density contrast of our model liquid would be with its first forming solid—whether it would sink or float. As a conservative measure of where solid flotation is possible, we assumed the liquidus solid to be pure Pv (MgSiO3, magnesium perovskite) using the EOS from Mosenfelder et al. [2009]. Mg-rich Pv was the liquidus solid in both Andrault et al. [2011] and Fiquet et al.[2010] at the pressures of interest. We determined each CMASF model isentrope for a given potential temperature (TP) in a stepwise fashion, solving for temperature (T) at each fixed increment of pressure (P) from 0 to 140 GPa. In this calculation, each CMASF isentrope is a combination of the pure end-member isentropes of An, Di, Fo, En, and Fa; and each of the end-member isentropes is defined by its own potential temperature TiP (superscript i denotes an end-member mineral), where TiP is a separately defined value from the TP of the CMASF model isentrope. The separate TP and TiP values permit the mixture of the end-member isentropes to define a point on the final CMASF model isentrope in which the total entropy (Stot) remains a constant, such that S tot = ∑i f i S i + S mix = constant, (21) where f is the mole fraction (Table 6) and Si is the molar entropy for each end-member in the model liquid. By assuming the entropy of mixing (Smix) is constant for a given liquid composition, the potential temperature of each end-member liquid isentrope (TiP) was solved by setting the derivative of the total entropy equal to zero: Ti  d ∑ f i S i = ∑ f i C Pi ln P  = 0 . i i  TP  The potential temperature for each end-member liquid isentrope is further defined as (22) II-29  γ i V i q  i q  V   i TP = T exp oi  Pi  −  i   .  q  Vo   Vo    i i (23) This expression is derived from (20) by integrating q ρ  T ∂T = γ o  o  ∂ρ .  ρ  ρ (24) and is dependent on the following three variables: (1) the temperature at each iteration along the model isentrope (T), (2) the volume of each end member at T and P (Vi), and (3) the volume of each end member at TiP and 0 GPa (ViP). These three variables are solved iteratively to satisfy equation (22) and to minimize the expression ∑ ( P − P) + ( P ) , i S i P 2 (25) i where PiP is the BM isentropic pressure at TiP and equals 0 GPa at the target volume ViP and PiS is the BM isentropic pressure for each end member at T. Minimization of this expression ensures that the isentropic pressure for each end member converges to the defined pressure in question and that PiP is in fact equal to 0 GPa. For Fa, we used the BM3 parameters given in Table 4; for all other end members, we used the BM parameters in Table 5. In total, 11 parameters (T and two volumes, Vi and ViP, for each of the 5 end members) were solved iteratively for each step increment in pressure along a model isentrope. The densities of the mixtures were calculated using linear mixing of volumes. This assumption has been previously tested for mixing of anorthite and diopside liquids [Asimow and Ahrens, 2010] and was found to be well-behaved at high temperatures and pressures but began to break down when applied at low temperatures requiring very large extrapolation downwards from the Hugoniot temperature. Furthermore, anorthite liquid may be a special case due to its highly polymerized structure at low pressure [Mysen and Richet, 2005], although we examine II-30 this further in Chapter III. Hence, despite examination of three compositions along the anorthitediopside join, it remains unclear whether or not linear mixing of volume at high pressure is valid for most mafic to ultramafic silicate liquids. Nevertheless, given five measured compositions in a five-oxide system, the assumption of linear mixing is necessary to make progress at this time and remains consistent with data presently available. We examine this assumption in detail in Chapter IV. The resulting model for the Andrault et al. [2011] “chondrite” mix shows that crystallization would begin at the base of present-day mantle (~135 GPa) with a critical adiabat of TP = 2600K (Fig 8a). Andrault et al. found Pv to be the liquidus phase at all pressures above 60 GPa; therefore we calculated the density for a first forming crystal and found it to be denser than the surrounding liquid (ρPv, 5.38 g cm–3 > ρliquid, 5.28 g cm–3). Thus a basal magma ocean similar to that described in Labrosse et al. [2007] in which a large portion of the lowermost mantle is zone-refined to concentrate incompatible elements would be unlikely as crystallization of the mantle would occur from the bottom up. A chondritic basal magma ocean is likely only able to exist within the thermal boundary layer (highest estimated outer core of 4150K) [Hernlund et al., 2005] or at the elevated basal temperatures that occurred during early Earth accretion [Canup, 2004]. The critical adiabat for the peridotite composition of Fiquet et al. (2010), on the other hand, occurs at a higher TP of 2800 K and, notably, is tangent to the liquidus at 85 GPa (Fig 8b). Consequently crystallization would begin in the middle of the lower mantle. A Pv liquidus phase would, like the model above, sink upon formation (ρPv, 4.94 g cm–3 > ρliquid, 4.63 g cm–3). This model is more akin to the picture described in Labrosse et al. [2007] as the basal magma ocean would include a portion of the lower mantle which could be used to distill incompatible elements II-31 into a potential present-day ULVZ, yet the sinking liquidus phase if not conducive to the dynamic formation of a basal ocean. It is worth mentioning that the surface temperatures in both of these models could be too hot for simple, realistic magma ocean models; Andrault et al. [2011] suggest that such high foot temperatures would lead to large portions of the mantle vaporizing and rapid cooling if thermal blanketing is inefficient. Within this exercise, the location of first crystallization is likely dominantly controlled by two factors: (1) the topology of the liquidus curve and (2) the amount of Fo end-member in the model liquid. It is unclear whether different shapes of liquidus curves stem from differing experimental techniques in the two synchrotron-based laser-heated diamond anvil studies considered or instead if they reflect thermodynamic differences within that range of composition space. However, Fig. 8 makes clear that without a notable decrease in slope in the liquidus curve compared to the isentrope at high pressure, crystallization would always be predicted at the base of the mantle (Fig. 8b). Secondly, it has been observed that the forsterite component has a large control on the spacing and steepness of the isentropes. This behavior can be seen to be true in peridotite, which has a greater fraction of Fo component (Table 6) and has a much steeper isentrope series than the En-rich chondrite liquid (Fig. 8). The effect that Fo has on the isentrope mixing can be predicted from its 1-bar thermodynamic parameters (Table 5) with larger γo , α, and KS. These larger parameters are inversely proportional to the amount of SiO2 in Fo, since the dV/dT value of SiO2 at ambient pressure (from which these parameters are derived) is zero [Lange and Carmichael, 1990]. The bulk modulus of SiO2 is notably low compared to other oxides likely because of topological degrees of freedom in tetrahedral framework liquids. Thus the component with the II-32 least SiO2 gives the largest γo, α, and KS and so the steepest isentrope, which contributes directly to the steepness of the isentrope of the mixture. In Chapter III, we used the revised EOS of Fo, An, and En to re-model these liquid isentropes (Figure 7). The results are fairly similar, but it is important to note that the γo , α, and KS of forsterite liquids are not as large as modeled by the 1-bar data, and thus forsterite does not play such a dominant role on the steepness of the isentrope. A full discussion is given in Chapter III, section Isentropes for a full mantle magma ocean. We can conclude from this exercise in modeling the Earth’s accretion, acknowledging the simplifying assumptions and presumed bulk compositions, that the first crystallizing solid will likely occur within the lower mantle or at the base of the mantle. This is made more likely if early earth core temperatures are assumed to be elevated above current estimates [Canup, 2004]. Although largely controlled by the chosen bulk composition, within our exercise the liquidus phase would be Pv and would sink upon formation. Whether pure Mg-Pv is a correct approximation to the liquidus phase, as opposed to a more Fe-rich (Mg,Fe)-Pv, is unclear based on the spread of high T-P Fe partitioning values in the current literature (DFe = 0.6 to > 0.1) [Andrault et al., 2012; Nomura et al., 2011]. Regardless, since pure Pv is the least dense endmember of the (Mg, Fe)SiO3 solid-solution, greater partitioning of Fe into the solid would only further increase the density and magnitude of the already observed negative buoyancy. Buoyancy in the current temperature regime; application to the solidus and ULVZ For long-term gravitational stability, a presumed molten silicate liquid of the ULVZ must be denser than or at least neutrally buoyant compared to the ambient lower-most mantle bulk composition that exists today. Shown in Figure 9 are the densities of fayalite, enstatite and forsterite liquids along a plausible modern mantle adiabat including a thermal boundary II-33 correction, with a potential temperature of 1673 K and a gradient of 10 K/GPa. A pure molten fayalite composition with density of 7 g cm–3 would pond at the core mantle boundary if present, as it is much denser than any likely solid assemblage at lowermost mantle conditions. However the model of a pure fayalite liquid ULVZ is inconsistent both with the density of 6.12 g cm–3 in the ULVZ inferred from seismic reflection coefficients [Rost et al., 2006] (Figure 9) and with the non-zero shear velocity of this layer. Instead, a less-dense multicomponent partial melt needs to be considered as a plausible state for the ULVZ. Assuming linear mixing of the oxide volumes calculated from the BM EOS given in Tables 4 and 5, Figure 10a displays lines of constant density within a Fa-Fo-En-Fs space (where ferrosilite (Fs) = En+ .5Fo -.5Fa). Figure 10b is the same plot, but uses the revised EOS in Chapter III Tables 3 and 4. The y-axis is Mg# (Mg# = MgO/FeO+MgO), and the x-axis is (MgO+FeO)/SiO2. The solid curves delineate compositions that are neutrally buoyant with respect to a pure Pv solid at ~135GPa (roughly the pressure at the CMB) at the given temperature conditions (3900 K–4500 K). The dashed lines indicate a constant density of 6.12 g/cm3, which represents the estimated upper limit of a 10% density jump of an ULVZ [Rost et al., 2006] with respect to PREM [Dziewonski and Anderson, 1981]. Any liquid mixture that lies above the solid lines where Pv sinks (white in Fig. 10) is inferred to be gravitationally unstable with respect to upwards percolation into the solid mantle, and any mixture below the dashed lines is denser than has been inferred for ULVZs. In this exercise, potential liquid compositions for a high-melt fraction ULVZ must span a narrow range in composition space (striped blue in Figure 10), eliminating pure enstatite, and fayalite end-members as candidate compositions. Previously (Fig. 10a), pure forsterite was not denser than coexisting Pv crystal, but in the revised EOS (Fig. 10b), this is no longer shown to be true. For reference, the simplified chondrite and peridotite liquids modeled in the section above II-34 are also plotted on Figure 10. These results indicate that these liquid compositions are unlikely to be gravitationally stable at the CMB today (although peridotite liquid is denser than previously modeled Fig 10b); if residual liquids were to be derived from magma oceans of either composition, they would need to be enriched in Fe, depleted in Si, or both during some process of re-crystallization and distillation from the original bulk mantle. The gray arrows from each composition indicate the direction of compositional evolution if Pv (MgSiO3, En) were to crystallize and be removed. It also gives a rough estimate of what the final liquid composition would need to be for Pv to be buoyant. When evaluating whether a liquid would be gravitationally stable, Figure 10 assumes that a pure Pv end member is an acceptable estimation for bulk lower mantle density. Adding Fe to the pertinent solid phase in this scenario (modeled as Pv) would qualitatively narrow the region of gravitationally stable liquids to only low Mg# (<0.5) compositions in Fig. 10 (blue-lined area). Also the densities shown here are for fixed compositions that are unlikely to be in chemical equilibrium. For a complete examination of stability, we would need a better grasp on the actual bulk composition of the solid and an understanding of Fe/Mg partitioning at lower mantle conditions. Moreover, it is likely that the bulk lower mantle contains some amount of ironbearing Pv and calcium-silicate Pv, which would be denser than pure Mg-Pv and also effectively narrow the range of acceptable, stable compositions for a ULVZ liquid shown in Figure 10. Similarly, incorporation of Al2O3 and CaO into the model partial melt using the EOS of anorthite and diopside would narrow this same region of Fig. 10 by decreasing the density of the liquid. Therefore, our simplified model in Fig. 10 indicates the minimum amount of Fe enrichment that the liquid must have to be a high-fraction, gravitationally stable silicate melt at the CMB, but the amount of enrichment is likely to be much greater. II-35 CONCLUSIONS The density for Fe2SiO4 liquid as well as values for K and K’ have been constrained. Additionally, the Grüneisen parameter for Fe-bearing silicate liquids is shown to change with compression in a manner similar to non-Fe bearing liquids and the opposite to that of solids. Although the Grüneisen parameter has been measured for formulation of a thermal equation state, as with any thermal EOS formalism, there are trade-offs especially when examining states far from the Hugoniot including extrapolations in pressure beyond experimental pressures (161 GPa). Temperature estimates could be greatly improved by either shock temperature measurements or any other off-Hugoniot temperature measurements to constrain Cv. Constraints on the end-member liquid fayalite EOS permit interpolation between CMASF silicate liquids densities, assuming linear mixing of volumes. The calculation of isentropes at elevated temperatures and pressures for model whole-mantle magma oceans composed of peridotite and simplified “chondrite” liquid indicate crystallization initiating at the mid-lower mantle (85 GPa) and the base of the mantle, respectively. Finally, a compositional range of gravitationally stable liquids was determined to examine the likelihood of a gravitationally stable ULVZ composed of a large degree of partial melt. There exists a region of melt composition that is denser than Pv at the CMB, but neither chondrite nor peridotite is sufficiently enriched in Fe to be stable. Fractional crystallization of Pv would need to take place before either liquid would be dense enough to remain at the CMB. To elucidate fully the proper stable liquid composition, formulation of multicomponent phase equilibria or partition coefficients at high pressure and temperature are necessary for accurate description of the CMB. II-36 ACKNOWLEDGEMENTS The authors would like to thank the following: the shock wave lab technical staff—Michael Long, Eprapodito Gelle, and Russel Oliver; additionally, Bjorn Mysen for conducting Mössbauer measurements, and Jeff Nguyen for running simulations of our shock wave experiments. This work was supported by the National Science Foundation through award EAR-0855774. II-37 REFERENCES Agee, C. B. (1988), Mass balance and phase density constraints on early differentiation of chondritic mantle, Earth and planetary science letters, 90(2), 144. Agee, C. B. (1992), Isothermal compression of molten Fe2SiO4, Geophys. Res. Lett., 19(11), 1169–1172. Agee, C. B. (1998), Crystal-liquid density inversions in terrestrial and lunar magmas, Physics of The Earth and Planetary Interiors, 107(1-3), 63-74. Ahrens, T. J. (1987), 6. Shock Wave Techniques for Geophysics and Planetary Physics, in Methods in Experimental Physics, edited by G. S. Charles and L. H. Thomas, pp. 185235, Academic Press. Ai, Y and R. A. Lange (2004), An ultrasonic frequency sweep interferometer for liquids at high temperature: 2. Mechanical assembly, signal processing, and application, J. Geophys. Res., 109(B12204). Ai, Y., and R. A. 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Svendsen, B., and T. J. Ahrens (1990), Shock-Induced Temperatures of CaMgSi2O6, J. Geophys. Res., 95(B5), 6943-6953. Takei, H. (1978), Growth of fayalite (Fe2SiO4) single crystals by the floating-zone method, Journal of Crystal Growth, 43(4), 463-468. Tenner, T. J., R. A. Lange, and R. T. Downs (2007), The albite fusion curve re-examined: New experiments and the high-pressure density and compressibility of high albite and NaAlSi3O8 liquid, American Mineralogist, 92(10), 1573-1585. Ulmer, P., and R. Luth (1991), The graphite-COH fluid equilibrium in P, T, & fO2 space, Contributions to Mineralogy and Petrology, 106(3), 265-272. Wicks, J. K., J. M. Jackson, and W. Sturhahn (2010), Very low sound velocities in iron-rich (Mg, Fe) O: Implications for the core-mantle boundary region, Geophysical research letters, 37(15), 15304. Williams, Q., J. Revenaugh, and E. J. Garnero (1998), A Correlation Between Ultra-Low Basal Velocities in the Mantle and Hot Spots, Science, 281(5376), 546-549. II-41 TABLES Table 1. Double-bob density measurements T (K) 1551 1632 1723 1818 ρ g cm–3 3.710 3.679 3.652 3.630 ± 0.002 0.002 0.002 0.001 Table 2. Shock compression data Shot # Chen et al. (2002) 999a a This work 1000 110 120 116 113 099 107 161 126 412 Chen et al. (2002) 995 996 This work 981 993 990 994 998 396 409 Flyer/ ufp T (°C) Driver km s–1 Al2024/ Al2024 W/Al2024 Cu/Cu Cu/Cu Ta/Ta Ta/Ta Ta/Ta Ta/Ta Ta/Ta Ta/Ta Ta/Ta Al2024/ Mo Al2024/ Mo Ta/Mo Ta/Mo Ta/Mo Ta/Mo W/Mo Ta/Mo Ta/Mo ± up km s–1 ± 25 1.96 0.001 0.754 0.26 25 25 25 25 25 25 25 25 25 25 2.03 4.63 5.30 5.171 5.38 5.77 6.25 6.494 6.51 7.085 0.01 0.01 0.002 0.006 0.005 0.01 0.001 0.001 0.005 0.012 Us km s–1 ρH g cm–3 ± PH GPa ± 7.05 0.26 4.90 0.03 23.3 0.4 1.51 0.18 2.87 0.08 3.29 0.08 3.58 0.11 3.69 0.17 3.95 0.05 4.27 0.1 4.44 0.11 4.57 0.1 4.864 0.019 6.95 8.2 8.71 9.02 9.44 9.79 10.15 10.33 10.57 10.90 5.61 6.76 7.00 7.26 7.21 7.34 7.59 7.65 7.73 7.81 46.2 103.3 124.7 141.4 153.0 169.4 190.5 200.3 211.9 229.2 0.9 0.8 0.8 1.7 2.1 0.7 1.5 2.1 1.6 2.1 ± 0.18 0.08 0.08 0.11 0.17 0.05 0.1 0.11 0.1 0.04 0.05 0.05 0.06 0.06 0.1 0.03 0.07 0.06 0.08 0.07 1300 1.05 0.004 0.43 0.02 3.23 0.02 4.33 0.12 5.25 0.03 1300 1.95 0.81 0.01 3.91 0.31 4.73 0.05 11.9 0.7 1300 1300 1300 1300 1300 1299 1300 1.52 0.02 1.890 0.002 2.00 0.02 2.52 0.01 2.50 0.01 4.997 0.003 6.050 0.002 4.72 4.94 4.92 5.82 5.85 8.45 9.50 5.09 5.43 5.61 5.70 5.96 6.76 7.15 0.03 21.9 0.14 28.3 0.34 30.1 0.02 43.4 0.1 47.6 0.02 119.5 0.04 161.0 0.3 1.5 2.5 0.2 0.9 0.3 0.6 0.01 1.24 0.02 1.53 0.08 1.63 0.04 1.99 0.01 2.17 0.02 3.82 0.003 4.58 0.01 a 0.01 0.01 0.54 0.01 0.14 0.03 0.05 40mm shots on Caltech polycrystalline fayalite II-42 II-43 Table 3. Parameters used Units ΔEtr ≈ ΔHtr α Cp Cv ρo Co s kJ kg-1 K-1 –1 –1 J kg K J kg–1 K–1 g cm-3 km s-1 Fe2SiO4 liquid Molybdenumd (1573 K) (1573 K) -1584.2194a 8.1949e-05b 1182.35c 1123.13c - Tantalume (300 K) 9.96 4.927 1.288 16.65 3.293 1.307 a Ghiorso and Sack [1995] bderived from this work, (see text and Table 1 for dV/dT) cLange and Navrotsky [1992] dAsimow et al. [2008] eMitchell and Nellis [1981] Table 4. Equation of state fits for molten Fe2SiO4 To ρo Co s γo q KoS KS' KS'' 𝜒2 Units SWEOS SWEOS* 3BM/MG 4BM/MG K g cm–3 m s–1 1573 3.699 2438±5 1.58 ± 0.03 0.412 -1.45 21.99 5.33 1573 3.699 2670±8 1.50±0.03 0.412 -1.04 26.38 4.99 1573 3.699 1573 3.699 0.412 -0.95 21.99 7.28 0.412 -0.55 21.99 6.64 -0.47 0.941 GPa GPa–1 1.526 Source this work AL04 fitted derived fitted derived fitted fitted SWEOS= shock wave equation of state. SWEOS*= shockwave EOS derived from the unconstrained linear fit (see text). 3BM/MG = 3rd order Birch-Murnaghan isentrope. 4BM/MG = 4th order BirchMurnaghan isentrope. Sources: Fitted indicates an adjustable parameter, AL04 is Ai and Lange [2004] II-44 Table 5. BM/MG fits for silicate liquid To ρo γo q KS KS' KS' CP CV Units Fo 4BMa En 4BMb An 4BMc Di 3BMc K g cm–3 1673 2.825 0.816 -1.47 42.19 4.73 -0.11 1926.18 1684.30 1673 2.618 0.365 -1.71 24.66 9.18 -1.87 1761.15 1690.54 1673 2.584 0.18 -1.71d 20.39 2.31 0.60 1528.72 1512.56 1673 2.643 0.49 -1.28 24.57 6.98 GPa GPa–1 J K–1 kg–1 J K–1 kg–1 1612.56 1506.21 Source LC90; L97 LC90 fitted derived derived fitted LN92 derived corrected γo and refit to 4BM b from Mosenfelder et al. [2009] c resampled shock arrivals and refit to BM and excluding some shots, see text. d fixed q value LC90 is Lange and Carmichael [1990]; L97 is Lange [1997]; LN92 is Lange and Navrotsky [1992]. a Table 6. Major oxide proportions for model mantle SiO2 FeO “chondrite” Andrault et al. (2011) mol % 2.6 45.9 1.8 43.5 6.2 KLB-1 Fiquet et al. (2010) mol % 1.5 53.7 1.2 38.3 5.3 KLB-1 Davis et al. (2010) mol % 2.8 50.8 1.8 38.7 5.9 Di An Fa Fo En mol % 1.95 4.40 7.57 24.2 61.9 mol % 0.70 3.32 7.16 56.1 32.7 mol % 2.93 4.98 8.25 55.0 28.8 CaO MgO Al2O II-45 FIGURE CAPTIONS Figure 1. A comparison of previous static sink/float and shock wave compression methods on fayalite liquid. The purple dashed line is an extrapolation of the 1773K isotherm from Agee [1992] with KT’=10.1. The blue solid line is the fayalite liquid Hugoniot from Chen et al. [2002] with KS’=5.36. Estimated shock temperatures are from Chen et al. [2002]. Figure 2. (a) Static image of the back of the molybdenum sample holder inverted 90 degrees clockwise such that the top is to the left. The centermost circle is the sample chamber or “top hat”; the guy wire posts are the smaller circles to the left and right. The vertical yellow lines correspond to the portion of the drivers (two outside lines) and the top hat (center line) that are sampled for shock wave arrival cutoffs. The bright vertical line is a superposed static image of the 25 micron slit in position. (b) On the streak image, the y-coordinate corresponds to position along the slit image in (a) where the x-coordinate is time increasing to the right. Yellow lines denote observed shock arrivals. Blue dashed lines are the 4th-order polynomial extrapolation of the shape of shock front from the drivers. The calibration streak shown at bottom converts an xcoordinate on the photo to a time during the experiment, giving the time between the shock wave passing from driver into sample (red line) and reaching the free surface of the top hat (rightmost yellow line). (c) An enlarged view of the top hat arrival with examples of pixel gray scale value vs. distance along the time axis profiles on the right. The arrival point is picked as the inflection point of the arc tangent function fit (black lines). The amplitude of the arctangent functions is arbitrary. II-46 Figure 3. The double-bob density measurements between 1551 and 1831 K, plotted against temperature, are fit by the line ρ(T) = 3699±1 – 0.30±0.01*(T-1573 K). Figure 4. Density versus pressure diagram showing the new multi-anvil sink/float results for fayalite liquid at 1500°C. “Up” triangles represent observed floats, in which the fayalite liquid was denser than the mineral marker buoy. No sinks were accomplished in the new study. The green curve is the calculated compression for the spessartine garnet marker; the blue curve is the calculated compression for the almandine-rich garnet marker. Fayalite liquid density at 1-bar is from this study (purple diamond) determined by the Archimedean technique. The dashed and dotted curves are the compression curves calculated for fayalite liquid at 1500°C using the K from ultrasonic data [Ai and Lange, 2004] and the K’ derived from the shock wave EOS and the 3rd-order Birch-Murnaghan EOS in this study, respectively (see Table 4). The thin purple solid curve is the compression curve for liquid fayalite from Agee [1992]. Float observations should plot below the correct liquid compression curve. Figure 5. Preheated (1573K) fayalite Hugoniot in shock velocity (Us)-particle velocity (up) space. Blue circles at the highest velocities were performed in this study. The red filled squares are taken from Chen et al. [2002] and were fit with the dotted blue line given in that study. The red solid line is the unconstrained un-weighted linear fit for all data points, and the light green solid line is the constrained un-weighted linear fit for all data points where the intercept is fixed at the bulk sound speed of liquid fayalite measured at 1573K. The hollow symbols are the initially solid experiments (300K) that were shocked to liquid from Chen et al. [2002] (hollow red II-47 squares) and this work (hollow blue circle). The dashed green line represents a cold Hugoniot offset from the fitted hot liquid Hugoniot using the derived model for the liquid Grüneisen parameter. Figure 6. The behavior of Grüneisen parameter for liquid and solid silicates with compression. The reference densities for the liquids (solid lines) are at 1673K for MgSiO3 [Mosenfelder et al., 2009] and Mg2SiO4 [Mosenfelder et al. 2009, corrected in this chapter] and at 1573 K for Fe2SiO4 (this study). The gamma functions for the solid materials are shown by dashed lines: magnesium perovskite (Pv) [Mosenfelder et al. 2009]-“BM3S model”; majorite (Mj) and enstatite (En) [Stixrude and Lithgow-Bertelloni, 2005]. Figure 7. Hugoniot data plotted in pressure-density space with thermal EOS fits for fayalite liquid and shock-melted single crystal all fixed at the Archimedean density at zero pressure. Data symbols are the same as Figure 5. Abbreviations: SWEOS- shock wave equation of state; BM/MG – Birch-Murnaghan/Mie-Grüneisen EOS; dKSFR- de Koker –Stixrude fundamental relation Figure 8. Models for the early mantle upon cooling showing the comparison of the liquidus curves (black solid line) with the isentropic temperature profile (dotted line) of the same composition. (a) “Chondritic” composition and liquidus from Andrault et al. [2011]. (b) Peridotite KLB-1 composition and liquidus from Fiquet et al. [2010]. The critical isentrope with II-48 its mantle potential temperature is shown in red. The EOS parameters used for En, Fo, An, and Di are given in Table 5 and Fa 3BM/MG parameters in Table 4. This figure is redone with the revised EOS in Chapter III Figure 8. Figure 9. Calculated densities as function of pressure along a 10K/GPa mantle geotherm. The density at base of mantle (135 GPa) for each composition is thermally corrected from 3073K to 4400K (shaded cones). Also shown is the density profile from PREM [Dziewonski and Anderson, 1981] and the inferred density for a ULVZ from seismic reflection coefficients [Rost et al. 2006]. This figure is redone with the revised EOS in Chapter III Figure 9. Figure 10. The density of liquid mixtures within En-Fs-Fo-Fa space at 135 GPa. The solids lines represent liquid mixtures that are neutrally buoyant with respect to solid magnesium perovskite (Pv) at different core-mantle-boundary temperatures from 3700-4500K. Any liquid compositions above these lines are less dense than Pv (white area) and any below are denser (blue striped area). Constant density lines for liquid compositions at 6.12 g/cc (dashed lines) represent the upper density limit for ULVZ’s inferred from Rost et al. [2006]. The blue C represents the “chondrite” composition from used in Andrault et al. [2011], and the yellow P is the peridotite composition from Fiquet et al. [2010]. Both compositions are given in Table 6. The gray arrows indicate the direction of liquid composition change with fractional crystallization of Pv (disregarding partitioning of Fe). a) The EOS parameters used for En and Fo are given in Table 5 and Fa 3BM parameters in Table 4. b) The EOS parameters used for En and Fa are given in Table 4 and Fo 3BM parameters in Table 3 of Chapter III. II-49 Figure 1 II-50 Figure 2 II-51 Figure 3 II-52 Figure 4 II-53 Figure 5 II-54 Figure 6 II-55 Figure 7 II-56 Figure 8 II-57 Figure 9 II-58 Figure 10  III-1 Chapter III Direct shock compression experiments on pre-molten Mg2SiO4 and progress towards a consistent high-pressure equation of state for CaO-MgO-Al2O3-SiO2-FeO liquids Claire W. Thomas Paul D. Asimow For submission to Journal of Geophysical Research April 4, 2013 III-2 ABSTRACT We performed shock compression experiments on preheated forsterite liquid (Mg2SiO4) at an initial temperature of 2273 K and have revised the equation of state (EOS) that was previously determined by shock melting of initially solid Mg2SiO4 (300 K). The linear Hugoniot, US = 2.674±0.188 + 1.64±0.06 up km/s, constrains the bulk sound speed within a temperature and composition space as yet unexplored by 1-bar ultrasonic experiments. We have also revised the EOS for enstatite liquid (MgSiO3) to exclude experiments that may have been only partially melted upon shock compression and also the EOS for anorthite liquid, which now excludes potentially unrelaxed experiments at low pressure. The revised fits and the previously determined EOS of fayalite and diopside were used to produce isentropes in the multicomponent CaO-MgO-Al2O3-SiO2-FeO system at elevated temperatures and pressures. Our results are similar to those previously presented for peridotite and simplified “chondrite” liquids such that regardless of where crystallization first occurs, the liquidus solid sinks upon formation. This process is not conducive to the formation of a basal magma ocean. We also examined the chemical and physical plausibility of the partial melt hypothesis to explain the occurrence and characteristics of ultralow velocity zones. We determined that the ambient mantle cannot produce an equilibrium partial melt and residue that is sufficiently dense to be a ULVZ mush. The partial melt would need to be segregated from its equilibrium residue and combined with a denser solid component to achieve a sufficiently large aggregate density. III-3 INTRODUCTION The volumetric behavior of silicate melts at high pressures and temperatures is not well constrained despite its importance for accurately modeling the formation and early evolution of the Earth. Our ability to predict the chemical and physical consequences of the Earth’s differentiation, for example, after the putative moon-forming impact, depends on a formulation of a liquid equation of state (EOS) accurate enough to determine the buoyancy of liquids relative to coexisting solids. Furthermore, the liquid EOS is also needed for rigorous interpretation of present-day deep Earth seismic observations: detection of ultra-low velocity zones (ULVZ) [Garnero and Helmberger, 1995] at the base of the mantle have been interpreted to possibly indicate the presence of liquid—potentially a partial melt [Lay et al., 2004; Williams et al., 1998] or a dense liquid that has remained un-sampled at the core-mantle boundary since differentiation of the planet [Labrosse et al., 2007]. This latter idea is based in part on the simplified modeling of the mantle as an isochemical MgSiO3 system [Mosenfelder et al., 2009], which would begin crystallization in the mid-mantle for a full mantle magma ocean. This concept provides the potential for creating two separately evolving chemical reservoirs in the upper and lower mantle that would allow distillation of a presumed dense melt. Yet applying this same idea to slightly more complex multicomponent mantle compositions in the CaO-MgO-Al2O3-SiO2-FeO (CMASF) system demonstrates that mid-mantle crystallization is not very likely or at least unlikely to lead to dynamically separate reservoirs [Thomas et al., 2012]. Similarly, our current knowledge of the density differences between complex multicomponent silicate liquids and the present-day ambient mantle are not adequate to make precise predictions of the gravitational stability of a liquid-mush ULVZ. III-4 Most of our understanding on the volume and elastic behavior of silicate liquids has been limited to pressures and temperatures within the upper mantle. Data are derived by density and sound speed measurements at 1 bar [Ai and Lange, 2008; Ghiorso and Kress, 2004; Lange and Carmichael, 1990] or by sink-float experiments at higher pressures up to a few GPa [Agee, 1992; 1993; Ohtani et al., 1995; Suzuki et al., 1998]. Diamond anvil cell (DAC) experiments are conducted at temperature and pressure regimes comparable to the lower mantle, but even distinguishing whether melting has occurred remains a non-trivial exercise (techniques for detecting melt vary among laboratories and results can often differ [Andrault et al., 2011; Fiquet et al., 2010]) and measuring the density of silicate melts in situ in the DAC remains an elusive goal. Shock experiments that start at ambient conditions and produce melt upon dynamic compression have achieved pressures equivalent to those in the deep mantle but can often be difficult to interpret due to solid-solid phase transitions along the Hugoniot or challenges in deciphering how far the melt reaction has been over-stepped [Akins, 2003; Akins et al., 2004; Luo et al., 2002; Mosenfelder et al., 2007; Mosenfelder et al., 2009]. Recently, preheated shock techniques have achieved pressures equivalent to those in the lower mantle for fully molten anorthite (An; CaAl2Si2O8), diopside (Di; CaMgSi2O6), An36-Di64 eutectic [Asimow and Ahrens, 2010], and fayalite (Fa; Fe2SiO4) [Thomas et al., 2012] compositions. To this shock dataset of liquids we have added a series of novel experiments, which preheat Mg2SiO4 (forsterite, Fo) beyond its melting point at ambient conditions to 2273 K prior to dynamic compression. These experiments compliment the shock experiments performed on initially solid forsterite and wadsleyite (300K) [Mosenfelder et al., 2007], and permit reevaluation of the previous EOS without the ambiguities of distinguishing solid-melt transformations along the Hugoniot. III-5 These experiments are also are the first to measure thermodynamic properties of Mg2SiO4 melt directly. The high melting temperature of forsterite (2163±20 K; [Bowen and Andersen, 1914]) has made in situ experimental measurements of melt properties difficult for this composition. They are instead typically calculated using linear mixing of partial molar properties extrapolated from lower temperatures and higher SiO2 compositions [Lange, 1997; Lange and Carmichael, 1990; Lange and Navrotsky, 1992; Stebbins et al., 1984]. Our newest measurements constrain the sound speed of silicate liquid within temperature and composition space yet unexplored by ambient-pressure ultrasonic experiments [Ai and Lange, 2008; Ghiorso and Kress, 2004]. METHODS Sample Preparation of Preheated (2273 K) Shots The Mg2SiO4 used for shock wave experiments was un-doped, single crystal forsterite (Morion Co. Gems) cored and lapped into disks. The sample disks under-filled the volume of the molybdenum sample holders by 7-13% at room temperature [cf. Thomas et al., 2012]. The room temperature under-filling—accounting for thermal expansion of Mo and the thermal expansion and volume of fusion of forsterite—results in a 1 – 1.5 mm-deep meniscus at the top of the heated capsule at 2273 K. This bubble is above the area imaged by the streak camera and therefore still allows observation of simple one-dimensional wave propagation. A void space is preferred to over-filling of the capsule, which can result in warping or failure of the welded cap. The surfaces within and outside the sample well including both sides of the cap were polished to a mirror finish with 1-micron alumina grit. Smoothing all the surfaces prevents bubbles from clinging to the walls of the sample holder during heating and potentially interfering with the imaged shock wave. The caps were electron-beam welded (Electron Beam Engineering III-6 Inc., Anaheim, CA) to the sample holder wells enclosing the sample. The details of this process are described in Thomas et al. [2012]; we note that forsterite, unlike fayalite, welded easily with only occasional and minor oxidation of the cap. Before and after welding, a profile of the topography aligned to the streak camera slit was taken with a depth gauge micrometer using a 0.7 mm flat-end tip. The inner portion of the sample well was measured in 0.2 mm increments, and the final driver and welded cap (or “top hat”) were measured in 0.1 mm increments. Typically the topography varied no more than ±0.008mm from flat for the inside of the sample well and ±0.015mm for the outside driver and top hat. Sample capsules were carbon coated on both sides to reduce oxidation of the reflecting molybdenum surface during heating [cf. Asimow and Ahrens, 2010; Asimow et al., 2008] except for shots completed on the 40-mm propellant gun. Oxidation of the target was less of a problem in the 40-mm gun possibly due to better vacuum for the smaller target chamber and catch tank volume compared to that of the 90mm/25mm two-stage light-gas gun (LGG). Experimental Setup The pioneering work for shock studies on molten materials is Rigden et al. [1984]. The description of our methods below builds on this and the work of Rigden et al. [1988; 1989], Miller et al. [1988; 1991a], Chen and Ahrens [1998], and Chen et al. [2002]. For more recent changes in experimental techniques and data analysis, the reader is directed to Asimow et al. [2008], Asimow and Ahrens [2010], and Thomas et al. [2012]. For this study, six total experiments were performed — four in the Caltech 90/25-mm twostage LGG and two using the Caltech 40-mm propellant gun. All shots were preheated to 2000°C (2273 K) using a H2O-cooled copper induction coil powered by a 10 kW Lepel radio frequency heater [Chen and Ahrens, 1998]. The targets were held in place by a “guy-wire” set up and III-7 mounted through a hole drilled in a high-temperature Zircar™ zirconia board as opposed to the alumina board used for lower-temperature experiments [cf. Asimow and Ahrens, 2010].The boards were also cut with a slit from the center hole through to the bottom including two separate inch-deep slits at the top (a Y shape) in order to relieve mechanical stresses that arose due to thermal expansion of the board while at high temperatures. Lexan projectiles with molybdenum flyer plates (see Table 1) were used in this study; flyer plate velocities (ufp) ranged from 1.04 5.963 km s-1. For the LGG shots, ufp was measured redundantly by a double-flash x-ray system and a two-magnet induction detector as described in Asimow et al. [2008]. The ufp for the 40-mm gun shot was also measured by two methods: double-exposure x-ray image and laser cutoffs. The 40-mm x-ray system utilizes a double-flash x-ray, which doubly exposes the flyer image on a single piece of film. Velocity is determined by the separation of the two flyer images — taking into account the magnification on film and the previously calibrated parallax distance — divided by the time between the two x-ray discharges [Rigden et al., 1988]. The 40-mm gun was also outfitted with a new laser timing system using 3 lasers aimed across the projectile path onto small active-area photodiode detectors. The distance between the lasers was measured to within ±0.05 cm, and the time between cutoffs is recorded on a 1 GHz oscilloscope. The velocities measured by the two techniques for each gun agreed to within the stated uncertainty for all the shots, and the average of the two values was used for the ufp given in Table 1. During each experiment, the rear face of the target was illuminated by a Specialised Imaging xenon spark lamp and filmed by a Hadland Imacon 790 streak camera through a narrow (25 μm) slit focused horizontally across the center of the driver and sample cap. The image of the shockwave transit was recorded using a new Specialised Imaging 2DR digital readout system. The system collects the streak image using a 4008 x 2688 pixel CCD, which was specifically III-8 designed to affix to the streak camera tube. The image is captured digitally, in contrast to the previous method, which relied on digital scanning of analog images captured on Polaroid film [cf. Asimow and Ahrens, 2010; Thomas et al., 2012]. The precise measurement of the shock transit time through the sample and molybdenum cap was measured from the extinguished (or sharp intensity change in) reflected light as the shock wave reached the free surface of the driver plate followed by the sample cap. The procedure for picking cutoffs and the calibration of the streak rate is described in Thomas et al. [2012]. The selected cutoffs were co-registered with the previously measured driver topography profiles and corrected for deviations from flat. The shape of the shock front as it enters the sample is determined by interpolating the visible driver cutoffs using a 4th-order polynomial. The 95% confidence interval for the fit was used as the upper and lower bounds for the location of the shock front and was the greatest source of error in the final calculated shock state — i.e., the error bars for shock pressure (PH), shock density (ρH), particle velocity (up), and shock wave velocity (Us). The time that the shock wave spends in the Mo cap is estimated using the iterative method of Rigden et al. [1988] and subtracted from the offset in time between the two cutoffs to calculate the shock wave velocity Us in the sample. Then up, PH, and ρH follow from impedance matching, the Rankine-Hugoniot equations, and the following input parameters: standard Hugoniot data (ρo, Co, s) for the cold metal flyer and hot Mo driver plates [Asimow et al., 2008; Chase, 1998] (see Table 2), the initial sample density (ρo) [Lange, 1997], and an initial guess for Co and the s parameter of Mg2SiO4 liquid to seed the iteration (the converged result is independent of this guess). RESULTS Linear Hugoniot The shock wave data for Mg2SiO4 liquid are reported in Table 1, including shot number, flyer/driver material, temperature prior to firing, ufp, up, US, ρH, and PH. III-9 A Hugoniot is a family of peak shock states achieved in a material by progressively stronger shocks from the same starting conditions [Ahrens, 1987] (in this case, liquid Mg2SiO4 at 2273 K and 1 bar). Empirically, the Hugoniot of a well-behaved material forms a line in US-up space, given to third order in strain by US = Co + s up [Jeanloz, 1989]. The slope (s) is related to K’S, the pressure derivative of the isentropic bulk modulus (KoS) by s = (K’S + 1)/4; the intercept, Co, is the bulk sound speed of the material at room pressure (Co2 = KoS / ρo) [Ruoff, 1967]. Typically, the intercept of the un-weighted linear Hugoniot in US-up space falls within error of the 1-bar value of the sound speed measured by ultrasonic techniques indicating relaxed (as opposed to “glass-like”) behavior upon compression. Currently there are no 1-bar sound speed measurements on CaO-MgO-Al2O3-SiO2 (CMAS) liquids with temperatures in excess of 1727K and mole fractions of MgO greater than 0.46 (Figure 1). Sound speed values can be calculated from the models of Ghiorso and Kress [2004] or Ai and Lange [2008] assuming linear addition of oxide component Co and dCo/dT but must be extrapolated well beyond the composition and temperature space used in either calibration (which is not recommended by the authors). The sound speed for Mg2SiO4 liquid at 2773K from Ghiorso and Kress [2004] is 3.195 km/s and from Ai and Lange [2008] is 3.126 km/s. Figure 2 shows linear fits to shock data in US-up space fixed at these 1-bar model values as well as the unconstrained fit. The unconstrained, un-weighted linear fit to all preheated Mg2SiO4 liquid data points yields US = 2.674±0.188 + 1.64±0.06 up km/s (r2 = 0.995). The unconstrained Co falls below the derived model values potentially indicating that the dCo/dT value is negative for Mg2SiO4 liquid, contrary to the positive [Ghiorso and Kress, 2004] and zero [Ai and Lange, 2008] values given in the previous studies. This is not evidence for unrelaxed behavior in the shock compression experiments; this would be expected to yield a higher bulk III-10 modulus than expected from relaxed ultrasonic data. Instead it is evidence that the temperature or composition dependence of those models cannot be extrapolated to the current conditions; see discussion below. Consequently, the KoS of 18.56±2.61 GPa (derived from the expression KoS = ρo Co2 and the unconstrained Co) is lower than the bulk moduli derived from either the Ghiorso and Kress [2004] or the Ai and Lange [2008] values, 26.55 and 25.38 GPa, respectively. This KoS is also much lower than the previously reported value given in shock wave studies, 41 GPa at To=1673K [Mosenfelder et al., 2009], which was derived from the Lange and Carmichael [1990] data set. If corrected to an initial temperature of 2273K using dV/dT and d2V/dPdT [Lange and Carmichael, 1990], KoS is 27.87 GPa and still much stiffer than our newest value of 18.56 GPa. The slope of our unconstrained Hugoniot fit corresponds to a K’ of 5.58±0.24, compared to that of the previous estimate of 4.73 [Thomas et al., 2012]. Grüneisen parameter The thermodynamic Grüneisen parameter (γ) is a macroscopic parameter that relates thermal pressure to the thermal energy per unit volume. The Mie-Grüneisen approximation this can be expressed as γ= 1 ∂Pth ρ ∂E (1) where thermal pressure (Pth) is defined as the increase in pressure due to heating at a constant volume [Poirier, 2000]. The Grüneisen parameter can be directly determined by the comparison of two Hugoniots with different initial densities [e.g. Asimow and Ahrens, 2010; Luo et al., 2002]. When compared at an equal density, a liquid obtained by shock compression of an initially solid material will have a distinct pressure and internal energy state than that obtained by direct shock compression of the liquid of the same composition. This method also applies for III-11 other changes in the initial state of shocked samples, including differences in initial solid phase and variations in initial porosity [Mosenfelder et al., 2009]. We determined the Mg2SiO4 liquid γ by comparison of our 2273K liquid Hugoniot with the initially solid 300K forsterite Hugoniot from Mosenfelder et al. [2007]. Mosenfelder et al. used two Mg2SiO4 starting compositions, polycrystalline forsterite and wadsleyite, and they asserted that the two highest pressure shots from each dataset were shock melted (shots #350 and #349 forsterite and #350 and #349 wadsleyite). However, the calculations of de Koker et al. [2008] indicate that shock compression of crystalline forsterite along the 300K principal Hugoniot produces incongruent melting to periclase (MgO) and a more silica rich liquid in the pressure range of 150-170 GPa; only above 170 GPa is pure Mg2SiO4 liquid present on the calculated Hugoniot. Correspondingly, their calculated wadsleyite Hugoniot was found to be 1000-1400 K cooler than the forsterite Hugoniot. Therefore even the highest-pressure wadsleyite datum (shot #350) is unlikely to be fully molten and does not provide a secure point for evaluation of liquid properties. We therefore have used only the highest forsterite point (#350, 188.5 GPa) from the Mosenfelder et al. [2007] dataset for determination of the liquid Grüneisen parameter. For the equations and details of this calculation, the reader is directed to Asimow and Ahrens [2010] and Thomas et al. [2012]. Using the power law form for γ ρ  γ ( ρ ) =  o   ρ  q (2) and γo = αK oS , ρoCP (3) III-12 yields a q value of –1.51 and γo = 0.448. This corroborates the general trend that has been observed by both calculation [Adjaoud et al., 2008; Boehler and Kennedy, 1977; de Koker et al., 2008; Stixrude and Karki, 2005] and experiment [Asimow and Ahrens, 2010; Mosenfelder et al., 2009; Thomas et al., 2012] that liquid γ increases upon compression, opposite to the behavior of solids. This q value is also fairly similar to the Mg2SiO4 fit from Thomas et al. [2012] (which corrected a sign error in the re-fit from Mosenfelder et al. [2009]), but differs greatly in the value for γo. The difference in these fits can been seen most clearly in Figure 3, which displays the previous γ curve (dotted) and the newest fit tied at a much lower γo. This offset is due to the very discrepant values for KoS used in (3), 41 GPa and 18.56 GPa for the previous and current study, respectively. Our newest fit for γ is also in much closer agreement to the linear fit given in de Koker et al. [2008] (Figure 3) and other silicate liquid γ functions derived from shock experiments. Thermal Equation of State Fitting The Hugoniot reaches temperatures and energies much higher than those of geophysical interest at lower mantle pressures (even for early Earth processes). It is therefore pertinent to select and apply a thermal equation of state formalism to investigate material properties that lie off the Hugoniot. We attempt to define the entire P-V-E surface of Mg2SiO4 liquid using the shock wave equation of state (SWEOS) and the 3rd- and 4th-order Birch-Murnaghan/MieGrüneisen equations of state (3BM/MG and 4BM/MG). The results and uncertainties for each fit are given in Table 3, and the Hugoniots are plotted in Figure 4. The SWEOS is defined by a linear Hugoniot in US – up space, converted to P - ρ space using the first and second Rankine-Hugoniot equations [e.g., Ahrens, 1987]. The slope of our linear Hugoniot, US = 2.674±0.188 + 1.64±0.06 up km/s, corresponds to a K’S value of III-13 5.58±0.24. States that lie off the Hugoniot are found using the Mie-Grüneisen thermal pressure approximation with a temperature-independent power law expression for the thermodynamic Grüneisen parameter (2). A q value of –1.51 was independently determined in the above Section 3.2 using selected experimental results from Mosenfelder et al. [2007]. The 3rd- or 4th-order BM/MG EOS is defined by a 3rd- or 4th-order Birch-Murnaghan isentrope centered at 1 bar and 2273K and a Mie-Grüneisen thermal pressure approximation. The 3BM/MG and 4BM/MG EOS are global fits, which include the six newest initially molten data points (Table 1) and the initially solid shot #350 forsterite from Mosenfelder et al. [2007]. Including the initially solid point (#350) reduces the error bars slightly for both fits; however, excluding the point returns nearly identical EOS parameters. We estimated uncertainties in these fits using bootstrap resampling of the data set; however, with only seven data points each at a distinct pressure, rather than many random samples from a population of Hugoniot constraints, this exercise yields rather artificial results. We applied two sets of 3rd- and 4th-order BM/MG fits at To= 2273 K, one where KoS was fixed to 18.56 GPa (derived from the unconstrained Co) and a second where KoS was permitted to vary within the stated error of the unconstrained Co. For the first set of fits, the 3BM/MG EOS result is KoS= 18.56 GPa, K’S = 6.37±0.04, q = –2.16±0.14, and reduced χ2 = 2.13. The 4BM/MG fit result is KoS= 18.56 GPa, K’S = 3.12±2.39, KS” = 1.26±1.47 GPa-1, q = 1.31±6.74 and reduced χ 2 = 1.18. The 4BM/MG evidently has very large error bars and strong correlations among the output parameters indicating a very unstable fitting routine. The 3BM/MG fit by contrast has less severe correlation between the parameters and appears justified by the fitting statistics. For the second set of fits in which we allowed KoS to vary, the best KoS value was 16.41 GPa. This KoS is the limit at which the derived Co still yields a linear Hugoniot in US-up space. III-14 The 3BM/MG fit with this KoS value is K’S = 7.37±0.81, q = –2.02±1.03, and reduced χ2 = 1.65. The 4BM/MG fit is KoS = 16.41 GPa, K’S = 4.27±2.02, KS” = 1.31±1.83 GPa-1, q = 0.47±9.86, and reduced χ 2= 1.11. This 4BM/MG fit is also unstable with large error bars and strong correlations among the parameters, but the 3BM/MG fit is much more reasonable. This 3BM/MG fit has a smaller reduced χ2 than the 3BM/MG fit above; hence for this dataset, we prefer the 3rd-order fit with KoS = 16.41 GPa. Revised Equations of State for MgSiO3 and CaAl2Si2O8 liquids Enstatite, MgSiO3 The isentrope of MgSiO3 derived from the recommended global EOS fit to shock wave data from all initial polymorphs and porosities of MgSiO3 [Mosenfelder et al., 2009] displays an inflection point (concave down to concave up) in T-P space. Although this topology is not necessarily unphysical, it conflicts with the simple concave down behavior of isentropes derived from shocked liquids only (Fa, Di, Di64An36 and Fo). It is possible that the volume decreases along the Hugoniot documented for the MgSiO3 shock wave data [Mosenfelder et al., 2009] may indicate that the sample is only partially molten as opposed to fully molten upon shock. Such an explanation would be consistent with the interpretation of discrepancies between shock data and MD simulations noted above for some Mg2SiO4 experiments [de Koker et al., 2008]. However, this idea is contrary to the previous assumption that silicate minerals require such large overstepping of the liquidus that partial melting does not occur [Akins et al., 2004]. To test whether the enstatite global fit may include non-liquid data, we exclude all but six of the highest pressure points from the glass [Mosenfelder et al., 2009], enstatite crystal [Akins et al., 2004], oxide mix [Marsh, 1980], and porous [Simakov and Trunin, 1973] data sets. These III-15 points are the most likely to be unambiguously molten during passage of the shock wave. The hollow points in Figure 5 shows the data used in this fit. The resulting 4BM/MG fit yields KoS= 24.66 GPa, K’S = 10.06±0.95, KS” = 2.34±0.82 GPa-1, q= –0.88±0.68, and reduced χ2= 1.72 at To = 1673.15 K. Due to the low number of points being fit, the bootstrap trial values for the output parameters for K’ and q were highly correlated and were in two very distinct parameter populations — one with reasonable q ≤ 1, and another with very high K’ (~14) and unreasonably high q values (40-90). This higher set is disregarded when determining the error bars on q. This fit has a similar reduced χ2 value to the fit given in Mosenfelder et al. [2009] (χ 2 = 1.89) but overall displays a simple concave down isentrope in TP space. Although not conclusive, this result may indicate that the inflection point on the model isentrope in dT/dP space was an artifact of fitting both molten and non-molten data as opposed to showing a unique physical behavior of enstatite liquid. Therefore this result also demonstrates that the technique of detecting shock melting via significant (~3%) volume [Akins et al., 2004] and temperature drops [Luo et al., 2004] may not be as straightforward as previously assumed and may only indicate partial melting of silicate samples. Anorthite, CaAl2Si2O8 In Thomas et al. [2012], the anorthite and diopside liquid data of Asimow and Ahrens [2010] were re-analyzed only sampling the center 3 mm of the arrival top hat (see Table 1). This re-sampling was motivated by a hydro-code model of shock propagation through the capsule geometry that indicated that the outer part of the top hat arrival was influenced by edge effects. The Di EOS and the An EOS parameters were both adjusted, but only the An fit displayed an inflection point for the isentrope in T-P space. As discussed in the MgSiO3 section above, it is III-16 believed this topology is more likely an artifact of fitting non-similar experimental states as opposed to representing the actual isentropic behavior of the liquid. The anorthite liquid Hugoniot is composed of two separate experimental data sets—one at low pressure (< 36 GPa) [Rigden et al., 1989] and one at high pressure (< 126 GPa) [Asimow and Ahrens, 2010]. The Rigden et al. [1989] experiments were re-analyzed in Asimow and Ahrens [2010], but the Co derived from this Hugoniot in US-up space still did not fall within error of the Co determined in 1-bar ultrasonic measurements [Lange and Carmichael, 1990]. In fact, anorthite liquid is the only composition studied thus far by preheated shock wave experiments (for which an ultrasonic velocity is available at similar temperature) that displays un-relaxed behavior. In the original Rigden et al. study, the data was fit with two separate Hugoniots due to a sharp offset observed in US-up space for the two highest pressure points. These same data points within the 25-36 GPa range have since been interpreted to be un-relaxed during compression [Asimow and Ahrens, 2010; de Koker, 2010; Ghiorso et al., 2009] despite being in the middle of the full pressure range now studied. Yet excluding these points alone does not resolve the discrepant behavior obtained by simultaneously fitting the lowest pressure points of Ridgen et al., the high pressure data of Asimow and Ahrens, and the 1 bar bulk sound speed [Lange and Carmichael, 1990]. One potential reason for this discrepancy is that the two shock data sets should not actually lie on the same Hugoniot, meaning the difference in the initial temperatures for two sets of experiments may be greater than stated. The Asimow and Ahrens experiments were heated to initially 1932 K and measured using a pyrometer; the Ridgen et al. experiments ranged from 1902 K to 1919 K and were measured using a thermocouple (actual experimental temperatures of Ridgen et al. are published in Asimow and Ahrens [2010]). Typically a 30 K temperature III-17 difference should not be significant for fitting the Hugoniot, but perhaps temperature errors using the thermocouple are larger than expected with consequently some of the experiments being conducted at significantly cooler conditions. Other possibilities include the experiments not being held at a high enough temperature above the melting point (1826 K) or held long enough above the liquidus before firing. Finally, it is also possible that there are no experimental discrepancies but that the volumetric and elastic behavior of anorthite at low pressure may be difficult to probe due to its highly polymerized structure. The rise time for the shock may be comparable to the relaxation time at least until much stronger shocks and higher temperature conditions along the Hugoniot are achieved. Although several plausible explanations can be put forward for the seemingly unusual character of the Ridgen et al. anorthite data, preliminary shock results on mixed liquids with high mole fractions of anorthite component (An50Hd50 and An33Hd33Di33) (see Chapter IV), in contrast, display predictable, relaxed behavior upon compression. Although it does not provide conclusive evidence, this newest data does support the likelihood of experimental differences between the two studies as opposed to unique low-pressure behavior for anorthite liquid. In summary, although we lack a definite understanding of the source of the low-pressure anorthite anomaly, we proceed here to derive a new fit for anorthite that can be used in conjunction with other compositions to synthesize the overall behavior of silicate liquids over a broad composition range. This is obtained from Ridgen et al. shot 665, the re-analyzed Asimow and Ahrens [2010] points [Thomas et al., 2012] and the experimental 1-bar sound speed at 1932 K [Ai and Lange, 2008]. The parameters are for the 4BM/MG fit are KoS = 19.77 GPa, K’S = 3.72±2.14, KS” = 0 .37±2.21 GPa-1, q = –1.86±1.14, and reduced χ2= 3.09 at To = 1932 K. This fit displays the same two highly correlated output parameter populations due to the low number III-18 of points included in the fitting as seen in the MgSiO3 fitting in the section above. The errors are determined using only the population of bootstrap trials with q ≤ 1. The errors for this fit are still unavoidably large, but the fit recovers the data well and also displays a concave down isentrope in T-P space. DISCUSSION Sound Speed of Mg2SiO4 liquid Although our derived Co is not within the error of the 1-bar models, it is unlikely that our liquid is un-relaxed or crystallized upon compression especially at the lowest experimental pressure (9.1 GPa, see Table 1). First, it would be typical for glass-like behavior to display bulk moduli higher than the ultrasonic values [Rivers and Carmichael, 1987]. Although the model values are extrapolated beyond their parameter space, it is at least encouraging to observe that the liquid does not have stiffer behavior than expected from these models. Second, it has been demonstrated that crystallization during the time of the experiment is improbable since the rise time of the shock wave in the sample is much less than the time necessary for crystallization to occur [Rigden et al., 1988]. Rise time (τr) can be estimated from τr = uPη U S ∆Pmax (4) [Jeanloz and Ahrens, 1979], where η is the effective viscosity and ΔPmax is the maximum pressure offset between the equilibrium Hugoniot and the Rayleigh line at PH(Us, up). Although η is not well constrained, the experimental 1-bar viscosity at ~2000°C for near Mg2SiO4 liquid composition (65.1 mol% MgO) is less than 0.1 Pa s [Urbain et al., 1982]. Using conservative estimates of 0.1-3 Pa s for the change in viscosity with elevated temperature and pressure [Adjaoud et al., 2008], τr calculated at our lowest pressure point (9.21 GPa) is on order of 10-10 III-19 seconds. This rise time is more than five orders of magnitude less than the time needed for crystallization for basaltic liquids derived from the time-temperature-transformation curves given in Uhlmann et al. [1982] (~10-5 s). Fang et al. [1983] further show that crystallization times increase with decreased polymerization; therefore crystallization is most likely kinetically impeded on the timescales of the experimental shock in Mg2SiO4 liquid. Finally, the most compelling evidence that crystallization did not occur during shock compression is that the temperature-pressure path of the Hugoniot is unlikely to have crossed into the solid forsterite stability field. The Mie-Grüneisen model to estimate shock temperature (TH) is given by TH = PH − PS + TS γCV ρ (5) where TS and PS are temperature and pressure along the BM isentrope, CV is the specific heat at constant volume, and γ is the Grüneisen parameter defined as a function of volume only (1). Using the 3BM/MG EOS fit (calculated in Section 3.3), Figure 6 shows that for even the highest temperature estimates of the Mg2SiO4 liquidus curve [de Koker et al., 2008; Ohtani and Kumazawa, 1981] the 2273 K Hugoniot lies within the liquid field. Consequently, we are confident that our experiments are probing relaxed liquids and that the unconstrained intercept of the linear Hugoniot of 2.674±0.188 km/s is the most accurate description of the bulk sound speed at 1 bar and 2273 K. This result implies that the dCo/dT for Mg2SiO4 is negative, i.e., sound speed decreases as temperature increases. This result is actually more in line with intuition than the previous estimates of zero [Ai and Lange, 2008] and positive dCo/dT values [Ghiorso and Kress, 2004]. Unless there are special structural rearrangements in liquid that overcome the general tendency of materials to become more anharmonic with increasing temperature and vibrational energy, one would expect liquids to act like solids and display decreasing sounds speed as temperature III-20 increases. Results from de Koker et al. [2008] also indicate a decrease in sound speed with increasing temperatures from 3000 K to 6000 K; yet extrapolation of these results to 2273 K gives a value of 3124 m/s, which does not however agree with our measured value. Preliminary modeling of this apparent complex behavior for sound speed suggests it may be due to cation coordination change and the greater degrees of structural freedom of liquids compared to solids [Wolf et al., 2012]. Isentropes for full mantle magma oceans The examination of liquidus topologies has been previously employed to describe dynamics of a solidifying magma ocean [Andrault et al., 2011; Miller et al., 1991b; Mosenfelder et al., 2009; Thomas et al., 2012]. Crystal/liquid density crossover points and isentrope-liquidus intersections are both key phenomena, and the relationship between these points is important to understand. Originally treated by Miller et al. [1991b], the crystallization of a chondritic (irondepleted CI composition—see Agee and Walker [1988]) magma ocean was depicted as a sequence of isentropes with decreasing specific entropy (or potential temperature, TP). The tangency of the liquid isentrope with the liquidus (maximum in S-P space) gives the pressure of first crystallization. The liquidus solid is the composition of the first forming crystal, and its density contrast with respect to the ambient liquid (whether is sinks, floats, or is neutrally buoyant) has important ramifications for understanding the first steps in the dynamic modeling of the Earth’s differentiation. The one-component liquid systems enstatite and forsterite [Mosenfelder et al., 2009] have been used as compositional analogues for the mantle and exhibited deep (>80 GPa) maxima in SP space. The liquidus-isentrope intersections for two multicomponent systems, KLB-1 peridotite III-21 [Fiquet et al., 2010] and synthetic chondrite [Andrault et al., 2011], have also been previously examined [Thomas et al., 2012]. The liquidi were taken from the above referenced studies; the details for calculating the isentropes of liquid mixtures in CaO-MgO-Al2O3-SiO2-FeO composition space are given in Thomas et al. [2012]. Isentropes derived using the newest Mg2SiO4 EOS (Table 3) and other re-analyzed end-member EOS of MgSiO3 and CaAl2Si2O8 (Table 4) yield slightly different results yet similar consequences to those previously presented. The critical isentrope for simplified chondrite composition shown in Figure 7 (Ch=.62En+.24Fo+.08Fa+.04An+.02Di) has a potential temperature (Tp) of 2960 K (compared to the previous estimate of Tp =2600 K), and the intersection occurs similarly at the base of the mantle (135 GPa). The peridotite (P=.33En+.56Fo+.07Fa+.03An+.007Di) critical isentrope is Tp = 3050 K and intersects the liquidus at 105 GPa, a slightly higher temperature (Tp =2900 K) and higher pressure than previously reported (~85 GPa). The first crystallizing phase observed on the liquidus in both experimental studies [Andrault et al., 2011; Fiquet et al., 2010] is Mg-perovskite (MgSiO3, Pv). Using the BM3S model given in Mosenfelder et al. [2009] (which includes the static data of Saxena et al. [1999]), the density of Pv at each intersection point can be calculated to determine whether the first forming crystals would sink or float. For chondrite, the density of Pv at 135 GPa and 4690 K is 5815 kg m-3, whereas chondrite liquid is 5150 kg m-3. Peridotite liquid is 4874 kg m-3 at 105 GPa and 5035 K, but Pv is 5401 kg m-3. In both cases, the liquidus solid is ~15% more dense than the liquid from which it formed and would therefore sink upon formation. This calculation can be redone to include partitioning of Fe, but current published values (DFe = 0.6 to 0.07) [Andrault et al., 2012; Nomura et al., 2011], which favor Fe partitioning into the liquid over high-pressure III-22 phases, would only further increase the density difference by incorporating at least some Fe into the solid Pv to form (Mg,Fe)SiO3 perovskite (MgPv). Crystallization from the middle or base of a magma ocean has implications for potentially creating a chemically stratified early mantle [Labrosse et al., 2007; Mosenfelder et al., 2009] provided that the lowermost mantle were to remain molten and mechanically separated from the upper mantle [Abe, 1997; Tonks and Melosh, 1993]. It has been postulated that a dense liquid at the core-mantle boundary (CMB), isolated from convection, could serve as a location for an un-degassed, primordial reservoir, which could hold a missing budget of incompatible elements [Labrosse et al., 2007; Lee et al., 2010]. Yet, similar to the previous conclusion given in Thomas et al. [2012], because the first crystals sink upon forming, this does not support the dynamic creation of a chemically stratified mantle. This conclusion is highly dependent on the choice of composition for the early bulk mantle and the liquidus topology of that composition. As shown here, despite the simplified chondrite and peridotite liquids having very similar bulk composition, the liquidus of each composition has a different maximum in T-P space. This discrepancy in liquidus shape in turn gives rise to very different predictions of where crystallization would begin. It is unclear whether these very different liquidus topologies are reflective of nature or are potentially derived from the different procedures used in identifying melt in the DAC [Andrault et al., 2011; Fiquet et al., 2010]. Additionally, choosing a bulk earth mineralogy that was more Fe-rich than either of these compositions would result in a smaller density difference with respect to the liquid and liquidus solid, but such a composition would need to be justified theoretically with respect to the timing and sequestration of the core. III-23 Evaluating the Partial Melt Hypothesis for Ultralow Velocity Zones Ultra low velocity zones (ULVZs) occur in thin (5-40 km) patches above the core-mantle boundary (CMB) [Garnero and Helmberger, 1995]. They are distinguished by several key features, the first being that they are best fit with a 3:1 ratio of S-wave to P-wave velocity anomaly, which has been inferred to indicate of the presence of liquid [Williams and Garnero, 1996]. They are also discontinuous along the CMB and can be often associated with the edges of large low shear velocity provinces and the source areas of plumes and large igneous provinces [McNamara et al., 2010; Rost et al., 2005; Williams et al., 1998]. They exhibit a non-zero shear wave velocity, so the assumed liquid cannot be present as a pure melt but must be within a liquid-solid “mush.” Modeling of different melt structures for this mush permits melt fractions of 0.01 % to 30% [Williams and Garnero, 1996], although more than 20% could lead to the loss of structural integrity for some melt configurations [Hernlund and Jellinek, 2010]. At select locations where reflection coefficients have been constrained, the ULVZ is inferred to be 6-14% denser than the overlying mantle, and there is an increase in S-wave velocity with depth within the ULVZ itself [Rost et al., 2006]. In order for a molten silicate liquid to comprise some portion of a ULVZ, it must be denser than or at least neutrally buoyant compared to the ambient lower-most mantle; otherwise the liquid would percolate upwards away from the CMB over geologic time. The question is whether such a melt can exist. The presence of liquid in ULVZ’s as a partial melt was first inferred by Williams and Garnero [1996] with the assumption that the density contrast between the liquid and solid would be relatively small (~1%) so that the variations in inferred seismic velocities would be small. This idea of low density contrast is based on shock-loading experiments [Brown et al., 1987; Rigden et al., 1984], which showed liquid densities overtaking III-24 their equivalent solids at moderate pressures (6-10 GPa for basaltic liquid). Figure 8 shows the densities of silicate liquids Fa, En, and Fo with major mantle solids, including periclase (Pe) and Pv, along a modern mantle adiabat (10 K/ GPa) with a TP of 1673K. This figure indicates that the assumption of density crossovers is highly dependent on chemistry as MgSiO3 liquid remains less dense than MgPv at all pressures. The density curve for each liquid in Figure 8 is shown at the base of the mantle with a range of uncertainty accounting for thermal expansion across a boundary layer for CMB temperatures of 3073-4400 K. Notably, the liquid compositions have a greater sensitivity to temperature than the solids, which is in accord with their large γ and CV values (α = γρCV/KT). It is easy to imagine that a simple mixture of any of the liquids could produce a density comparable to that of a ULVZ. In fact Mg2SiO4 liquid without the additional Fe at 3073K is 10% denser than PREM [Dziewonski and Anderson, 1981], but the important questions are whether a chosen liquid mixture is probable both 1) chemically based on our current knowledge of lower mantle chemistry and temperatures and 2) physically based on a plausible process by which it could be produced. There are two proposed processes for creating the putative ULVZ melt. The first is a melt left over from the differentiation of the Earth, perhaps distilled from the chemistry of the lower mantle, having been segregated from the surface by mid-mantle crystallization [Labrosse et al., 2007]. A second method of producing the ULVZ melt is by partial melting of the ambient mantle. We discuss these each in turn. As shown above, our current calculations appear to oppose segregation of a magma ocean into distinct reservoirs, but in fact our ability to constrain this problem so far applies only to the very initial stage of crystallization at the liquidus and is strongly dependent on the chosen composition. Development of a solid septum at later stages of magma ocean crystallization III-25 cannot be entirely ruled out. In detail it remains difficult test the distillation hypothesis for chemical and physical plausibility. The liquid chemistry, initially or during progressive differentiation, could occupy a wide range of compositions within CMASF space. Furthermore, the model of Hernlund and Jellinek [2010] suggests that stirring of the ULVZ piles by viscous coupling to the overlying mantle could create a stable mush layer for all geologic time without ongoing melt production. This model is also rather insensitive to density contrast, requiring only that the liquid be at least slightly denser than coexisting solids and disregards any chemical interaction with the mantle. Hence it is difficult to bring much insight to bear on this model using new constraints on the composition-dependent equation of state of silicate melts. Turning to the partial melting hypothesis, its plausibility has been supported by experiments showing that magnesiowüstite and perovskite assemblages melt at temperatures comparable to estimates for the CMB — e.g., (Mg0.9,Fe0.1)2SiO4 has a solidus temperature of 4300±270 K at 130±3 GPa [Holland and Ahrens, 1997]. Yet presently the melting temperatures for both Mg-Pv [Sweeney and Heinz, 1998; Zerr and Boehler, 1993] and MgO [Asimow and Fat'yanov, 2011] remain controversial, and the effect of Fe on the melting curve remains poorly constrained. Here, we limit our discussion to the plausibility of generating a stable ULVZ via partial melting of the ambient mantle. We will construct a model of phase and bulk densities under conditions of chemical equilibrium, which searches for an initial bulk mantle composition that would yield an equilibrium assemblage of 30% volume fraction of melt and 70% volume fraction residue that could form a gravitationally stable liquid mush (i.e., melt ≤ 1% denser than residue) with bulk density 10±4% denser than PREM (~ 6123±223 kg/m3). To calculate the densities of the mantle solids, (Mg, Fe)SiO3 perovskite (MgPv) and (Mg,Fe)O ferropericlase (Fp), we used the EOS provided in Mosenfelder et al.[2009] for pure III-26 Mg-Pv and pure Mg-Pe and calculated the addition of small amount of Fe2+ by ignoring changes in molar volume upon substitution. This is certainly an oversimplification but has already been shown to be a good approximation for estimating densities of (Mg, Fe)O solid solutions (for Fe contents up to ~60 mol%) at pressures greater than 70 GPa where Fe2+ is in the low-spin state and has an effective ionic radius similar to Mg2+ (~0.72 Å) [Fei et al., 2007]. Given the low Fe contents expected in Pv solid solutions in equilibrium with melt and the substantial uncertainty that remains in the liquid EOS, this is an adequate approximation for assessment of relative melt/solid buoyancy. For calculation of liquid densities, we assume linear mixing of the oxide volumes derived from the BM/MG EOS given in Tables 3 and 4. To define the distribution of Fe and Mg at chemical equilibrium among melt, Fp, and MgPv we need to define two partition coefficients, for example a solid-solid partition coefficient DFeMgPv/Fp = XFeMgPv/XFeFp (where XFe is molar iron content of the phase) and one solid-melt partition coefficient DFeMgPv/melt = XFeMgPv/XFemelt. The solid-solid DFeMgPv/Fp can be as low 0.09 [Auzende et al., 2008] in Al-free systems but approaches unity for Al-bearing systems at high pressure [Wood and Rubie, 1996]. A similar range of experimental values exists for the solidmelt DFeMgPv/melt for non-Al bearing [Nomura et al., 2011] (~0.07) and Al-bearing systems [Andrault et al., 2012] (~0.47-0.6). We have chosen to use the values derived from high pressure Al-bearing experiments as they are likely a closer approximation to natural systems. Hence for our calculations, we assumed DFeMgPv/Fp= 1and DFeMgPv/melt = 0.47 (we also explore the sensitivity of our result to this choice of solid-melt partition coefficient for Fe). In our calculations, we also fix the SiO2 concentration of the MgPv-Fp eutectic liquid. We chose two compositions: 1) a eutectic liquid with an SiO2 concentration of 41 mol % such that 70% of the melt is MgPv composition (fpvE = 0.7) and 30% is Fp composition and 2) a liquid III-27 with 47 mol% SiO2 concentration (fpvE = 0.9). These two estimates bracket the eutectic composition determined by Liebske and Frost [2012] for the pure MgO-MgSiO3 system such that SiO2 is 44 mol % (fpvE = 0.8). Ambient mantle is often approximated as 80% (Mg, Fe)SiO3 and 20% (Mg, Fe)O with some proportion of Fe [Boehler, 2000]. For our model, we evaluate whether this composition as well as other more exotic compositions would be able to produce a partial melt and an equilibrium residue with a low density contrast (a melt 1% denser than the residue, equivalent to the assumption in Williams and Garnero [1996]) and also produce an aggregate density comparable to a ULVZ allowing 30% volume fraction of melt. Results for this calculation at 3700 K and 135 GPa for fpvE =0.7 are given in Figure 9a. The solid lines represent curves of constant density contrast for the melt and residue, where positive percentages are a partial melt denser than the residue and negative percentages are a residue denser than the melt .The ULVZ density region is delineated by dotted lines representing a density 6-14% denser than PREM (~6123±223 kg/m3). An estimate for ambient mantle composition of 80% MgPv +20% Fp (fpv =0.8) with a Mg# of 0.85 (Mg# = XMg/(XFe+XMg)) would produce a partial melt 5% denser than its equilibrium residue, but the aggregate density would be 5503 kg/m3, which is actually 1% less dense than PREM. To produce an aggregate density comparable to PREM, the ULVZ bulk composition would need a very high Fe concentration (Mg# ≤ 0.6), but this would yield a very large (at least 14%) melt-residue density contrast. In fact for a eutectic composition of fpvE =0.7 and the selected partition coefficients, there is no starting composition that can fulfill both the aggregate density and density-contrast constraints. The unphysical region on the graph represents aggregate densities that are unachievable with the given amount of Fe in the system (Mg# for the partial melt is negative III-28 within this space). If we allow DFeMgPv/melt to approach unity (higher than the current experimentally determined values), thereby decreasing the extent to which Fe favors the melt and also thereby decreasing the density contrast between melt and residue, the unphysical region expands and does not permit aggregate densities within the ULVZ region. This result is also insensitive to percent of partial melting, as decreased volume fraction of melt also allows for a diminished melt-residue density contrast, but likewise further limits the highest aggregate densities that are achievable. It is an important conclusion therefore that if the eutectic composition is fpvE=0.7, then there is no composition in the ternary MgO-FeO-SiO2 system (that is, any combination of Mg# or fraction of MgPv or Fp), regardless of DFeMgPv/melt or melt fraction, that will produce an equilibrium assemblage equivalent to what is believed to be present in a ULVZ. In fact, this is also true for eutectic compositions fpvE = 0.8 [Liebske and Frost, 2012] and fpvE =0.9 with any experimentally determined DFeMgPv/melt value [Andrault et al., 2011]. Only at DFeMgPv/melt = 0.8 and a eutectic liquid of fpvE = 0.9 is the minimum estimate of the aggregate density achievable with a melt 1% denser than its residue (Figure 9b). The initial composition would also need to be exotic compared to typical ambient mantle compositions: richer in Fp (fpv = 0.2 -0.5) and Fe (Mg# = 0.6-0.7). Even disregarding the lack of experimental evidence for such high DFe or MgPv rich eutectic liquids, such high Fe contents are potentially unrealistic for a wide-spread lower mantle composition. FeSiO3 is unstable in the perovskite structure and decomposes to magnesiowüstite and stishovite [Jeanloz and Thompson, 1983] in which case the calculations of bulk densities performed here using the MgPv EOS are no longer appropriate; although the presence of aluminum has been shown to expand both ferrous [Kesson et al., 1995] and ferric stability in the perovskite and post-perovskite (pPv) phases at high pressure [Frost and McCammon, 2008; III-29 Frost et al., 2008; Zhang and Oganov, 2006].Additionally, experiments and calculations have indicated that an Fe-rich pPv phase (with Mg# ≤ 0.8) may be stable at 130 GPa [Mao et al., 2006; Mao et al., 2005], yet the temperature within a ULVZ remains sufficiently underconstrained that it is unclear whether Pv or pPv would be the predominate stable phase. Similarly, although despite potentially destabilizing Pv, elevated Fe contents are likely necessary to depress the melting point of Fp to reasonable CMB temperatures, considering the melting temperature of pure MgO at ambient conditions is already very high (3060 K) [Poirier, 2000], granting its high pressure melting curve remains uncertain. In contrast, this partial melt calculation could yield more reasonable results with less Fe in the bulk starting composition if carried out at a lower temperature (since liquids increase in density more rapidly than solids with decreasing temperature), but at some point the conditions would drop below the solidus. The correct mantle solidus temperature is not well known and better constraints on the phase diagrams of the pertinent compositions at high pressure and temperature are needed. Therefore consequently, if the model results are correct and high Fe contents are required to produce the necessary conditions for a ULVZ, partial melting may not even be necessary as suggested in Wicks et al. [2010], which cites the possibility of solid-state ULVZ composed of a large proportion of Fe-rich (Fe, Mg)O to produce the observed drops in S-wave velocity. We can conclude from this exercise that the ambient mantle (80% MgPv +20% Fp) cannot produce a partial melt and equilibrium residue that could be a ULVZ mush regardless of DFeMgPv/melt or eutectic composition. There exists a starting composition which can produce a ULVZ mush with the correct aggregate density and melt-residue density contrast, but only if subject to DFeMgPv/melt and eutectic composition constraints that lie outside of what has been determined experimentally. The partial melt hypothesis is therefore only plausible if the melt is III-30 fractionally segregated from its residue and combined with some other denser component. Only this process would permit both an aggregate density equivalent to the bulk ULVZ mush constrained by Rost et al. [2006] and at the same time a have a melt-residue density contrast that is consistent with assumed values used in seismic modeling (~1%) [Williams and Garnero, 1996], assuming both these physical constraints are robust. And although not explicitly calculated here, another potential way to increase the aggregate density would be to transform the residue to post-perovskite structure, but partitioning of Fe in pPv is not well constrained [Kobayashi et al., 2005; Sinmyo et al., 2011], the location of the Pv-pPv transition in Fe-bearing systems is controversial, and it would require a special scenario to place the solid Pv-pPv transition and the onset of partial melting at the same horizon. CONCLUSIONS We have completed new shock wave experiments on molten Mg2SiO4 at 2273K, a significant technical advance in initial temperature for preheated experiments of this kind, allowing study of liquid compositions and temperatures previously inaccessible to ultrasonic or other experimental methods. This new data enables a revision of the previous liquid Mg2SiO4 EOS [Mosenfelder et al., 2009]. The Grüneisen parameter is now in better agreement with values obtained from molecular dynamics simulation [de Koker et al., 2008] and is more similar to the γ behavior of other silicate liquids such as fayalite and diopside compositions. The shock-derived ambientpressure bulk sound speed at 2273 K, compared to values estimated from lower-temperature ultrasonic data, indicates that sound speed decreases with increasing temperature for Fo liquid, contrary to the positive [Ghiorso and Kress, 2004] and zero [Ai and Lange, 2008] temperature dependences previously reported. The EOS of liquid MgSiO3 and CaAl2Si2O8 were also revised III-31 to exclude experiments that may have only been partially melted upon shock compression and potentially un-relaxed at low pressure, respectively, resulting in EOS models of the five components Fa, Fo, En, Di, and An that all share well-behaved, concave-down isentropes in P-T space. These were used in an ideal mixing model to construct new isentropes for whole mantle magma oceans of selected compositions in CMASF. The results are similar to those presented previously [Thomas et al., 2012], in that MgPv may begin crystallizing in the mid-lower mantle or at the base of the mantle for different compositions or estimates of their liquidus curves but in all cases MgPv is negatively buoyant at the site of initial crystallization. This does not favor the formation of a basal magma ocean at early stages of evolution. The revised EOS of MgSiO3 and Mg2SiO4 were also used to calculate the plausibility of producing a ULVZ from a partial melt of the ambient mantle. Results indicate that any MgPv-Fp assemblage, including compositions typically assumed for the ambient mantle (80% MgPv and 20% Fp), cannot produce an equilibrium assemblage with a density comparable to that which has been inferred for a ULVZ mush. A ULVZ must be composed of fractionally segregated liquids from elsewhere in the mantle in association with a solid component denser than the equilibrium residue. III-32 ACKNOWLEDGEMENTS The authors would like to thank the following: the shock wave lab technical staff—Oleg Fat’yanov, Eprapodito Gelle, and Russel Oliver. This work was supported by the National Science Foundation through award EAR-1119522. III-33 REFERENCES Abe, Y. (1997), Thermal and chemical evolution of the terrestrial magma ocean, Physics of The Earth and Planetary Interiors, 100(1-4), 27-39. Adjaoud, O., G. 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Shock compression data Shot # Flyer material 1077 a Mo Forsterite 1075 a 455 457 458 459 Anorthite 665† ρΗ km s–1 PH Mg m–3 GPa 2.019 0.003 1.644 0.001 5.38 0.02 3.74 0.01 22.98 0.07 Mo 2273 5.426 0.007 4.08 0.02 9.14 0.17 4.68 0.09 96.7 1.4 2273 4.522 0.008 Cu 1922.2 1.8 0.03 1.325 0.025 4.68 0.01 Mo 1932 0.003 4.501 0.005 9.56 0.05 Mo Mo Mo 379 Mo Diopside Us km s–1 2274 Mo 382 km s–1 Mo 364 380 up ufp T (K) Mo Mo 2273 2273 2273 1.04 3.32 5.964 0.02 0.01 0.002 1930 4.997 0.005 1932 6.007 0.025 1932 1781 5.434 6.533 4.984 0.013 0.003 0.886 2.59 3.41 4.43 3.816 4.094 4.868 3.697 0.004 0.03 0.01 0.02 0.003 0.007 4.00 7.10 8.47 9.95 8.49 9.09 0.024 10.06 .008 9.33 0.11 3.34 0.44 4.10 0.12 5.29 0.17 5.42 0.04 75.0 2.3 0.8 3.57 0.03 15.90 0.30 4.84 0.03 95.30 0.63 4.97 0.10 0.05 47.6 0.2 114.3 4.66 0.23 0.17 9.21 0.08 4.64 0.08 0.03 4.34 0.02 82.80 0.04 110.00 0.03 90.84 0.13 126.00 a Caltech 40mm propellant gun † Anorthite and diopside shots are from Asimow and Ahrens [2010] revised in Thomas et al. [2012] but not previously published except for 665 which was revised in Asimow and Ahrens [2010] and originally from Rigden et al. [1989]. 1.5 0.28 0.44 2.30 0.72 Table 2. Parameters used ρo Co s Mg m-3 km s-1 Molybdenum (2273 K) Molybdenum (300 K) 9.785 10.21 4.858 5.033 1.288 1.289 Asimow et al. [2008]; Chase [1998] III-39 Table 3. Equation of State fits for molten Mg2SiO4 To ρo Co s γo q KoS KS' KS'' 𝜒2 KT α Cp Cv Units K kg m–3 m s–1 GPa GPa–1 GPa K-1 J kg–1 J kg–1 1 SWEOS 2273 2597 9 11 2674 ± 19 1.64 ± 0.06 0.448 -1.51† 18.57 5.58 ± 0.24 16.54 1.21-04 1926.18 1715.23 BM3 2273 2597 ± 11 0.396 –2.02 ± 1.03 16.41* 7.37 ± 0.81* 1.65 14.80 1.21-04 1926.18 11737.36 BM4 2273 2597 ± 11 0.396 0.47 ± 9.86 16.41* 4.27 ± 2.02* 1.31 ± 1.83 1.11 14.80 1.21-04 1926.18 11737.36 Source L97 fitted fitted derived fitted derived/*fitted derived/*fitted fitted derived L97 LN92 derived SWEOS= shock wave equation of state. 3BM/MG = 3rd-order Birch-Murnaghan/Mie-Grüneisen EOS 4BM/MG = 4th-order Birch-Murnaghan/Mie-Grüneisen EOS. † derived from shot 350 Fo Mosenfelder et al. [2007] Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992] Table 4. Equation of State fits for molten silicate liquids To ρo γo q KoS KS' KS'' Cp Cv Units K kg m–3 GPa GPa–1 J kg–1 K–1 J kg–1 K–1 3BM/MG Di† 1673 2643 0.493 -1.28 24.57 6.77 1612.56 1506.21 3BM/MG Fa† 1573 3699 0.412 -0.95 21.99 7.28 1182.35 1122.73 4BM/MG An 1932 2560 0.174 -1.86 19.77 3.73 0.38 1528.72 1511.28 4BM/MG En 1673 2617 0.365 -0.88 24.66 10.07 -2.35 1761.15 1690.53 Source L97 derived fitted derived fitted fitted LN92 derived 3BM/MG = 3rd-order Birch-Murnaghan/Mie-Grüneisen EOS 4BM/MG = 4th-order/Birch-Murnaghan Mie-Grüneisen EOS. Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992], †Thomas et al [2012] III-40 FIGURE CAPTIONS Figure 1. The set of experiments used for modeling sound speeds of silicate liquids in Ghiorso and Kress [2004] and Ai and Lange[2008] compared to the initial temperature and composition of Mg2SiO4 liquid. Figure 2. Preheated (2273K) forsterite Hugoniot in shock velocity (Us)-particle velocity (up) space. The thick black line represents the unconstrained linear Hugoniot; the orange and blue lines are fixed at the sound speeds calculated using the models from Ghiorso and Kress [2004] and Ai and Lange[2008], respectively. Figure 3. The behavior of Grüneisen parameter for Mg2SiO4 liquid (thick solid green) compared to its previously reported value (dotted) [Mosenfelder et al., 2009]. The thin red and blue lines are the behavior of γ for diopside and fayalite, respectively. The EOS parameters used are given in Table 4. The squares and black line and are the results and linear fit from MD simulations of Mg2SiO4 liquid [de Koker et al., 2008] Figure 4. Mg2SiO4 liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 3. Abbreviations: SWEOS- shock wave equation of state; BM/MG – Birch-Murnaghan/Mie-Grüneisen EOS Figure 5. MgSiO3 Hugoniot data used in the global fit of Mosenfelder et al. [2009] for MgSiO3 melt. The volume for each experimental point is normalized to the volume of its starting III-41 material. The data set includes enstatite glass [Mosenfelder et al., 2009] (triangle), enstatite crystal [Akins et al., 2004] (diamond), oxide mix [Marsh, 1980] (circle), and porous enstatite [Simakov and Trunin, 1973] (square). The gray line is the high pressure state assignment from Mosenfelder et al. [2009] to either perovskite, post-perovskite, or melt. Only the hollow points were included in our fit. Solid points within the melt field were excluded. Error bars not shown. Figure 6. Estimates of the Mg2SiO4 liquidus from de Koker et al. [2008] (black curve with shaded gray error), Presnall and Walter [1993](green curve) and Ohtani and Kumazawa [1981] purple curve including the experimentally determinations liquid (circles) and solid (squares) quench. The estimated T-P path of the Hugoniot (dashed line) and shot #1077 (diamond) (see Table 1) lie within the liquid field. Figure 7. Models for the early mantle upon cooling showing the comparison of the liquidus curves (black solid line) with the isentropic temperature profile (dotted line) of the same composition. (a) Simplified “chondritic” composition and liquidus from Andrault et al. [2011]. (b) Peridotite KLB-1 composition and liquidus from Fiquet et al. [2010]. The critical isentrope with its mantle potential temperature is shown in red. The tangent point of the critical isentrope and the liquidus gives the pressure and temperature at the onset of crystallization. The EOS parameters used for calculating the isentropes are given in Table 3 and 4. Figure 8. Estimated densities for silicate liquids Fa (blue curve), Fo (green curve) and En (orange curve) and mantle solids Fo, wadsleyite (Wd), ringwoodite (Rw), periclase (Pe), Mg-perovskite (Pv), and post-perovskite (pPv) (black curves) as function of pressure along a 10K/GPa mantle III-42 geotherm. The dashed lines are the previous density estimates for Fo liquid (green) and En liquid (orange) from EOS parameters given in Mosenfelder et al. [2009].The density at base of mantle (135 GPa) for each composition is thermally corrected from 3073K to 4400K (shaded cones). Also shown is the density profile from PREM [Dziewonski and Anderson, 1981], the inferred density for a ULVZ from seismic reflection coefficients [Rost et al., 2006](square), and an estimated ambient mantle composition of 80% (Mg0.85.Fe0.15)SiO3 perovskite+ 20% (Mg0.85Fe0.15)O (diamond) at 3000K and 129 GPa. The EOS parameters for the liquids are taken from Tables 3 and 4. EOS for Mg-Pv and pPv are from Mosenfelder et al. [2009]; other solid EOS are from Stixrude and Lithgow-Bertelloni [2005]. Figure 9. Results for partial melting calculation at 3700 K and 135 GPa. The solid curves represent lines of constant density contrast for the melt and residue, where positive percentages are a partial melt denser than the residue and negative percentages are a residue denser than the melt for a given starting composition. The y-axis if the fraction of MgPv in the starting material (fpv) where the remainder is Fp; the x-axis is the Mg# of the starting composition, where Mg#.= XMg/(XFe+XMg).The density of the ULVZ is the blue region delineated by dotted lines representing a density 6-14% denser than PREM (~6123±223 kg/m3). An estimate for ambient mantle composition of 80% MgPv +20% Fp (fpv =0.8) with an Mg# of 0.85 is given with its calculated aggregate density (agg ρ) with 30% volume fraction of partial melt. The unphysical region on the graph signifies where Mg# for the partial melt is negative and hence represents aggregate densities that are unachievable with the given amount of Fe in the system) The eutectic liquid is fixed at 70% MgPv (fpvE = 0.7) and the DFeMgPv/melt = 0.47. b) The eutectic liquid is fixed at 90% MgPv (fpvE = 0.97) and the DFeMgPv/melt = 0.8. III-43 FIGURES Figure 1 III-44 Figure 2 III-45 Figure 3 III-46 Figure 4 III-47 Figure 5 III-48 Figure 6 III-49 III-50 Figure 8 Figure 7 Figure 9 III-51  IV-1 Chapter IV Pre-heated shock experiments in the molten CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 ternary: a test for linear mixing of liquid volumes at high pressure and temperature Claire W. Thomas Paul D. Asimow For submission to Journal of Geophysical Research April 4, 2013 IV-2 ABSTRACT We performed seventeen new shock wave experiments on pre-heated (1673 K) hedenbergite liquid (CaFeSi2O6) and two model basalt liquids (an equimolar binary mix of CaAl2Si2O8 + CaFeSi2O6 and an equimolar ternary mix of CaAl2Si2O8 + CaFeSi2O6 +CaMgSi2O6) in order to determine their equations of state (EOS). Ambient-pressure density measurements on these and other Fe-bearing silicate liquids indicate that FeO has a partial molar volume that is highly dependent on composition, which leads to large errors in estimates of the densities of Fe-bearing liquids at ambient pressure based on an ideal mixing of any fixed set of end-member liquids. We formulated a series of mixing tests using the EOS determined in this study to examine whether ideal mixing of volumes might nevertheless suffice to describe the ternary system CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 at high temperature and pressure. The ideal mixing null hypothesis is rejected; compositional variations in partial molar volume of FeO appear to extend to high pressure. Only densities of Fe-bearing liquid mixtures with oxide mole fraction of FeO less than 0.06 can be adequately approximated using an ideal solution. IV-3 INTRODUCTION Silicate liquids play a large role in our understanding of both the early and modern Earth. Most widely-accepted scenarios for the evolution of the early Earth involve at least one stage of extensive or complete melting, for example, in the aftermath of impacts with large protoplanetary sized objects such as the presumed moon-forming event [Canup, 2004; 2012; Canup and Asphaug, 2001; Ćuk and Stewart, 2012]. The chemical and thermal evolution of the early molten mantle and how it affects the chemistry and physical properties of the modern mantle are not well understood. Furthermore, seismic observations of the 410km discontinuity in the upper mantle [Revenaugh and Sipkin, 1994; Song et al., 2004] and ultralow velocity zones (ULVZ) in the lowermost mantle [Garnero and Helmberger, 1995; Williams and Garnero, 1996] have been interpreted to indicate the presence of a neutrally buoyant partial melt, yet there is currently limited knowledge of how the density and compressibility of complex silicate liquids behave at high temperature and pressure. It is therefore essential to have experimental constraints on the equation of state (EOS) of the complex multicomponent silicate liquids that would be actually present in nature. The difficulty lies in that natural silicate liquids comprise a potentially infinite, multi-dimensional continuum of compositions. Understanding their range of volumetric behavior requires either data on an arbitrarily large number of experiments or data on a minimum set of end-member compositions along with a reliable tool of interpolating among them. Choosing the correct method of interpolation is key to producing a predictive model for silicate liquid density and compressibility over a wide range of pressure, temperature, and composition space. Experimental data that provide constraints on end-member volumes as well as a few intermediate compositions offer the opportunity to test whether available methods for such interpolation are reliable. IV-4 Ideal mixing of volumes has proven to be a reliable method of interpolation for ambientpressure experiments within a given compositition and temperature range [Bottinga and Weill, 1970; Bottinga et al., 1982; Bottinga et al., 1983; Guo et al., 2013a submitted; Lange, 1997; Lange and Carmichael, 1990; Mo et al., 1982; Nelson and Carmichael, 1979], although exceptions from ideality exist and are discussed below. Likewise, ideal mixing silicate liquid volumes at high pressure has been employed as a simplifying assumption, but it in fact remains poorly tested and constrained [Asimow and Ahrens, 2010]. In this study, we report seventeen new shock wave experiments to pressures up to 146 GPa on preheated silicate liquids of three separate compositions: hedenbergite (Hd; CaFeSi2O6), a 50-50 mixture of anorthite and hedenbergite (An-Hd; CaAl2Si2O8-CaFeSi2O6 ), and an equal mixture of anorthite, hedenbergite, and diopside (An-Hd-Di; CaAl2Si2O8-CaFeSi2O6-CaMgSi2O6 ). Determination of the EOS of Hd and the two mixtures (or “model basalts”) permits us to test the validity of linear mixing of volumes and ideal configurational mixing of entropies — two assumptions which were both used in Thomas et al. [2012] and Chapter III of this text to predict the volumes of multicomponent liquids at high T and P. METHODS Sample preparation The synthesis and analysis for the Hd and model basalt samples used in this study are described in Guo et al. [2013a submitted] and Guo et al. [2013b in prep], respectively. These studies also describe the 1-bar double-bob Archimedean density measurements and the ultrasonic sound speed measurements that were completed on these materials. The sample materials used for the shock wave experiments were cored directly from the lower buffer rod crucible used in IV-5 the sound speed determinations. The cores were sliced into disks, which were then lapped down to under-fill the volume of the molybdenum shock-experiment sample holder. The Mo sample holders are assembled from two machined pieces: 1) the driver plate, which has a hollow cylindrical projection on one side that becomes the sample well and 2) the cap, which is a 1 mmthick, 8 mm-diameter disk that is welded on after filling (for complete description of the welding process see Thomas et al. [2012]). The interior volume of the sample well is 8 mm in diameter and 2.5 mm deep. The samples under-filled this volume by 7-13% at room temperature to prevent the welded cap from bowing during heating [c.f. Thomas et al., 2012] The surfaces within the sample well including the inside surface of the cap were polished to a mirror finish with 1-micron alumina grit. Smoothing all the surfaces prevents bubbles from clinging to the walls of the sample holder during heating and potentially interfering with the imaged shock wave. The outer surface of the cap was left with the finish quality as-machined (~9-micron finish), in order to match the quality of polish on the inner portion of the rear-face of the driver plate, which was polished on a rotating stage. The speed of the polishing pad decreases towards the center, hence the quality of mirror finish also decreases. Approximately similar finishes on the cap and driver reflecting surfaces ease acquisition of evenly illuminated images of the cap and driver during the shot and greatly increased the precision of the Hd data given below. On the contrary, the model basalt capsules were polished as in the manner as described in Chapter II due to the larger quantity of light needed for the digital recording system (c.f. Chapter III). Hence this dataset is plagued with larger errors derived from determining the driver and cap arrivals with different exposure levels. An improved technique for polishing for driver to a mirror finish would greatly aid data acquisition in the future. Additional sample preparations are detailed in Asimow and Ahrens [2010], Thomas et al. [2012], and Chapter III. IV-6 Experimental setup The pioneering work for shock studies on molten materials is Rigden et al. [1984]. The description of our methods below builds on this and the work of Rigden et al. [1988; 1989], Miller et al. [1988; 1991a], Chen and Ahrens [1998], and Chen et al. [2002]. The most recent changes to experimental techniques and data analysis are in Asimow [2008], Asimow and Ahrens [2010], Thomas et al. [2012], and Chapter III. For this study, seventeen total experiments were performed—twelve in the Caltech 90/25-mm two-stage light-gas gun and five using the Caltech 40-mm propellant gun—eight performed on Hd, five performed on An-Hd, and four performed on An-Hd-Di (see Table 1). All shots were preheated to 1673 K ± 4 K (1400 °C) and used either molybdenum (Mo), tantalum (Ta), or aluminum alloy (Al2024) flyer plates. Flyer plate velocities (ufp) ranged from 1.46 – 6.19 km/s; the methods used for measuring ufp are given in Thomas and Asimow [submitted]. During each experiment, the rear face of the target is illuminated by a Specialised Imaging xenon spark lamp and filmed by a Hadland Imacon 790 streak camera through a narrow (25 μm) slit focused horizontally across the center of the driver and sample cap. Hd shots were recorded on Polaroid film; the model basalts were recorded using a digital system (c.f. Chapter III). As the shock wave reaches the free surface of the driver or the cap the reflected light from the polished surface is extinguished (or shows a sharp change in intensity). The offset time of these two cutoffs allows for the precise measurement of the shock transit time through the sample and molybdenum cap. To correct for non-uniform streaking rate, the camera was calibrated using a test streak image modulated by a radio-frequency tuner at 148.9875 MHz for a 1500 ns streak length and by a pulse generator at 1.7870 MHz for a 5000 ns streak (shot 1068). Each pixel line on the shot IV-7 streak record can then be converted to a time during the experiment, which permits the shock transit time through the sample to be calculated from the pixel distance between the two cutoffs on the film (or digital image). A detailed method for picking cutoffs and reducing the data is described in Thomas et al. [2012]. The final calculated shock state — i.e., shock pressure (PH), shock density (ρH), particle velocity (up), and shock velocity (Us) is solved for iteratively [Rigden et al., 1988] using measured shock wave travel time, impedance matching, the Rankine-Hugoniot equations, Hugoniot data (ρo, Co, s) for the metal flyer plates and hot Mo driver (Table 2), and the initial density (ρo) for each of the silicate liquids (see Tables 3, 4, and 5). An initial guess for the bulk sound speed (Co) and Hugoniot slope s is required to seed the iteration, but the converged result is independent of this assumption. RESULTS The shock wave data are reported in Table 1, including shot number, flyer material, the temperature prior to firing, ufp, up, US, ρH, and PH. A Hugoniot is a series of peak shock states achieved by progressively stronger shocks in a material from the same initial conditions [Ahrens, 1987]. The Hugoniot of a well-behaved material empirically forms a line in US-up space, given to third order in strain by US = Co + s up [Jeanloz, 1989]. The bulk sound speed of the material at room pressure, Co, is given by the intercept; the slope (s) is related to K’S, the isentropic pressure derivative of the isentropic bulk modulus (KoS) by [Ruoff, 1967], ( K S' + 1) . s= 4 (1) IV-8 Hedenbergite (CaFeSi2O6) liquid Figure 1 shows linear fits to the Hd shock data in US-up space. An unweighted linear fit to all eight preheated hedenbergite liquid data points yields US = 2.587±0.142 + 1.55±0.04 up km/s (r2 = 0.996). The experimentally determined ultrasonic value of the bulk sound speed, 2.613±0.016 km/s [Guo et al., 2013a submitted], lies within the stated uncertainty of the bulk sound speed derived from the intercept of this fit, 2.587±0.142 km/s. This relationship indicates relaxed (liquid-like) — as opposed to un-relaxed (glass-like) — behavior upon shock compression [Rigden et al., 1988]. Therefore, to reduce the error on the linear fit, we fixed the intercept at the mean ultrasonic value yielding an unweighted constrained linear fit of US = 2.613±0.016 + 1.54±0.01 up km/s (r2 = 0.999). It is necessary to select and apply a thermal equation of state formalism for investigating material properties that lie off the Hugoniot, since the Hugoniot achieves temperatures and energies much higher than those of geophysical interest at lower mantle pressures (even for processes during early Earth differentiation). We have defined the entire P-V-E surface of hedenbergite liquid using various formalisms: the shock wave equation of state (SWEOS) and the 3rd- and 4th-order Birch-Murnaghan Mie-Grüneisen equations of state (3BM/MG and 4BM/MG). For a full description of the thermal EOS equations and fitting procedures, the reader is directed to Asimow and Ahrens [2010] and Thomas et al. [2012]. The results and uncertainties for each fit on Hd are given in Table 3, and the Hugoniots are plotted in Figure 2. The SWEOS is define by a linear Hugoniot in US – up space, converted to P - ρ space using the first and second Rankine-Hugoniot equations [e.g,. Ahrens, 1987]. States that lie off the Hugoniot are found using the Mie-Grüneisen thermal pressure approximation with a temperature-independent power law expression for the thermodynamic Grüneisen parameter IV-9 (2) . Although a value for q to be used with the SWEOS was not independently determined in this study, experimental results thus far indicate that silicate liquids—including iron-bearing liquids [Thomas et al., 2012]—appear to have q values of –0.88 to –2 for compressions of 1 > ρo/ρ > 0.49. The slope and intercept of the constrained Hugoniot fit given above corresponds to a K’S value of 5.16±0.04 derived from (1) and a KoS value of 19.98±0.24 GPa derived from the expression Co = K oS ρo . (3) The 3rd- or 4th-order BM/MG EOS is defined by a 3rd- or 4th-order Birch-Murnaghan isentrope centered at 1 bar and 1673K and a Mie-Grüneisen thermal pressure approximation. The 3BM/MG fit result is KS’ = 6.22±0.55, q = –1.93±0.41, and reduced χ2= 4.20. The 4BM/MG fit result is KS’ = 3.15±1.67, KS” = 0.78±0.60 GPa-1, q = 0.47±9.86 and reduced χ 2 = 3.26. The high precision for most of the points in the Hd dataset propagates into unusually high χ 2 values for both fits. Nevertheless, comparatively the 4BM/MG has very large error bars on and strong correlations among the output parameters indicating a very unstable fitting routine. The 3BM/MG fit, by contrast, has less severe correlation between the parameters and appears justified by the fitting statistics. Hence for this data set we prefer the 3rd-order fit. IV-10 An-Hd liquid The An-Hd Hugoniot in US-up space is shown in Figure 3. The unweighted linear fit is US = 2.742±0.282 + 1.40±0.09 up km/s (r2 = 0.986). As observed above for Hd liquid, the An-Hd liquid appears to be relaxed upon compression as the experimentally determined bulk sound speed, 2.772±0.011 km/s [Guo et al., 2013b in prep], and the intercept are within error. We can then fix the intercept at the mean ultrasonic value yielding an unweighted constrained linear fit of US = 2.772±0.011 + 1.39±0.03 up km/s (r2 = 0.998). This slope and intercept constrain the SWEOS K’S and KoS values to be 4.54±0.12 and 21.30±0.25 GPa, respectively. The 3BM/MG fit result is KS’ = 5.49±0.65, q = –0.18±2.25, and reduced χ2 = 6.37. The 4BM/MG fit result is KS’ = 5.10±1.61, KS” = –0.17±0.60 GPa-1, q = 1.67±2.79 and reduced χ2 = 12.53. Both fits have relatively large error bars and large reduced χ2 due to the low number of data points and fairly significant scatter in P-ρ space (Figure 4). Yet both fits give similar K’S to SWEOS, which is well defined by the linear fit. We therefore prefer the 3rd-order fit despite having large errors, since it fits the data sufficiently well, has a lower reduced χ2, and exhibits a negative q value, which has been shown to be typical for silicate liquids (see Chapter III). The EOS parameters and their uncertainties for An-Hd liquid are given in Table 4. An-Hd-Di liquid The An-Hd Hugoniot in US-up space is shown in Figure 5. The unweighted linear fit is US = 2.823±0.197 + 1.52±0.09 up km/s (r2 = 0.995). Behaving as the two liquids above, An-Hd-Di liquid is relaxed upon compression [Guo et al., 2013b in prep], which justifies an unweighted constrained linear fit of US = 2.846±0.014 + 1.54±0.02 up km/s (r2 = 0.999). This slope and IV-11 intercept constrain the SWEOS K’S and KoS values to be 5.15±0.06 and 22.03±0.25 GPa, respectively. The An-Hd-Di liquid was fit only with the 3BM/MG, as fitting the 4BM/MG to so few data points is under-constrained and does not yield a meaningful reduced χ2. The 3BM/MG result is KS’ = 6.20±0.17, q = –1.14±0.79, and reduced χ2 = 0.39. The results and uncertainties for each fit on An-Hd-Di are given in Table 5, and the Hugoniots are plotted in Figure 6. DISCUSSION Linear mixing assumptions To make progress on calculating the volumetric properties of the natural multicomponent silicate liquids that may occur in the lower mantle, the plan has been to constrain the EOS for a number of important end-member liquids using shock wave studies and then to attempt to reliably interpolate among them. End-member liquids that have been determined previously using shock wave methods are: Fe2SiO4 (Fa, fayalite), Mg2SiO4 (Fo, forsterite), MgSiO3 (En, enstatite), CaAl2Si2O8 (An, anorthite), and CaMgSi2O6 (Di, diopside). The recommended EOS parameters for each of these end members are given in Chapter III. In Thomas et al. [2012], these five end members were used to interpolate within the CaO-MgOAl2O3-SiO2-FeO major element component space to constrain the volume of a liquid along an isentropic temperature-pressure path for a desired composition, yet there was not a clear test of the validity of the two underlying assumptions: (1) linear mixing of volumes and (2) constant entropy of mixing for a given composition. The assumption of linear or ideal mixing is simplest assumption that can be made for mixtures, IV-12 Vtot = ∑i X iV i + Vmix , (4) where the volume of mixing (Vmix) is assumed to be zero, and hence the total volume (Vtot) is just a linear combination of the end-member partial molar volumes (Vi) multiplied by their respective mole fractions (Xi). The second assumption is that the configurational entropy of a liquid solution is only a function of composition and therefore does not change with variations in temperature and pressure, such that Stot = ∑i X i S i + Smix , (5) where the entropy of mixing (Smix) is assumed to be a constant and hence an isentrope or constant entropy path in T-P space can be found from =0, (6) where the T and P derivatives of entropy come from standard thermodynamic identities. Our data do not provide a test of this method of computing isentropes, since we lack experimentally defined isentropes for reference. We therefore focus on evaluating the volume of mixing assumption by comparing isentropes for intermediate compositions constructed with different sets of end members, while acknowledging that this exercise may be compromised to some extent by the configurational entropy issue. Glass and 1-bar melt data Both of these assumptions have historically performed fairly well at 1 bar. Linear mixing of temperature-independent partial molar heat capacities has been a successful model for most silicate liquids [Stebbins et al., 1984]. Similarly, linear mixing of volumes has been successful in IV-13 determining silicate liquid densities in most major element space (SiO2-Al2O3-MgO-CaO-Na2OK2O) [Lange, 1997; Lange and Carmichael, 1990]. However, exceptions to both rules do exist. Heat capacities for the Fe2O3 oxide component were shown to vary strongly with composition [Lange and Navrotsky, 1992] and the partial molar volumes of Fe2O3 and TiO2 components are both variable with composition. These exceptions are attributed to the Fe3+ and Ti4+ cations having multiple coordination states at ambient pressure that are highly dependent on the composition, i.e., fraction of non-SiO2 components [Liu and Lange, 2001; 2006]. More recent results have found this to be potentially true for Fe2+ as well, in that the partial molar volume of FeO (VFeO) in a melt is observed to depend on composition in some cases [Guo et al., 2013a submitted; 2013b in prep]. Such behavior of Fe2+ is well documented in the glass literature, which is important insofar as glasses provide a proxy for structural properties of melt at the glass transition temperature. When discussing melt structure, language is typically borrowed from random network theory [Zachariasen, 1932], which distinguishes oxides as network formers, network modifiers, or intermediates. Network-forming cations (e.g., Si4+) occur typically in tetrahedral ([4]) oxygen coordination in various connected units in the melt, whereas network modifiers (e.g., CaO, MgO) typically occupy octahedrally ([6]) coordinated sites and disrupt the tetrahedral network. Intermediates (e.g., Al2O3, TiO2) can take on either role, may occupy multiple coordination sites, and may either disrupt or stabilize the polymerization of a melt [Bottinga and Weill, 1972; Mysen, 1983; 1988]. This is a simplified view of glasses and melts, but it acts to guide our intuition concerning the behavior of cations and the contributions of Fe3+ and Ti4+ to the density, heat capacity, and other thermodynamic properties of silicate melts [Burns, 1993; Mysen, 1988]. IV-14 FeO is an intermediate glass former and has been documented to take on a range of [4] to [6] coordination dependent on composition. Fe2+ is predominantly a network modifier in basaltic glasses [Burns, 1993; Jackson et al., 2005; Mysen, 1983; 1988] and also likely in melts with similar SiO2 mole fraction. This is supported by the relatively low 1-bar VFeO value (~ 12.1 cm3/mol) for An-Hd, An-Di-Hd, and Hd-Di model basalts at 1732 K. This VFeO has been inferred to indicate an average Fe2+ coordination of ~[5.7] due to its similarity to the molar volume of wüstite, crystalline FeO (in which Fe2+ is octahedrally coordinated) [Guo et al., 2013b in prep]. On the contrary, Fe2SiO4 glasses contain very low-coordinated Fe2+ and are surprisingly polymerized [Cooney and Sharma, 1990]. X-ray absorption spectroscopy studies of both glass and melt of fayalite composition infer an average coordination of ~[4] [Jackson et al., 1993]. This is again supported by melt densities, where VFeO increases to ~14-17 cm3/mol [Mo et al., 1982; Shiraishi et al., 1978; Thomas et al., 2012], similar to the molar volume of crystalline CaFeO2, a structure in which Fe2+ is [4] coordinated [Guo et al., 2013a submitted]. Between these two extremes, Hd and the CaO-FeO-SiO2 (CFS) melts measured in Guo et al. [2013a submitted] display FeO in an average coordination state intermediate between [4] and [6]. The VFeO for Hd melt (~ 15.47 cm3/mol) is inferred to indicate [4.7] coordination, similar to the values in Hd glasses (~4.3) and in molecular dynamics simulation of this melt [Rossano et al., 2000]. The inferred coordination for CFS glasses range from 4.6-5.2. Guo et al. [2013b in prep] present an inverse correlation between CaO concentration and average Fe2+ coordination number. This is contrary to expectation from the model in Jackson et al. [2005] based on valence bond (VB) theory, which suggests a direct correlation between CaO concentration and FeO coordination. In the VB model, the Ca would have a greater “share” of the O valence due to the IV-15 higher bond valence of the Ca-O bond and force Fe to find charge balance with other oxygens and hence increase coordination. The composition dependence of Fe coordination in CFS melts appears to be more strongly related to the degree of polymerization (Si/O ratio) [Burns, 1993]. For example in Hd, where SiO2 is high (Si/O =0.32) and there are equal proportions of CaO and FeO, FeO fills both roles of network former and modifier, with an average value of [4.7]. For the family of CFS melts studies by Guo et al. [2013b in prep], the Fe content is fixed (40 mol %) and so increasing CaO is coupled to decreasing SiO2. The Fe2+ may be energetically forced to play the role of network former as CaO competes more effectively for octahedral sites. For fayalite liquid, the Si/O value is quite low (0.25), and FeO is again energetically favored as a network former to accommodate the charge balance of the oxygen, creating a highly polymerized glass as observed by Cooney and Sharma [1990]. Qualitatively, one can understand the importance of FeO as an intermediate behavior cation when Fe2SiO4 is compared to Mg2SiO4 melt, which has the same Si/O ratio. Since MgO is unable to act as a network former, Mg-rich melts typically have fully depolymerized structure and display very low viscosities. The dependence of FeO coordination and molar volume on other cation concentration invalidates the assumption of linear mixing of volumes at 1 bar for FeO-bearing liquids. It is unclear, however, what effect pressure and temperature may have. For glasses, higher temperatures are believed to decrease the coordination of transition metals [Jackson et al., 2005], and there is some evidence that cations in melts will favor a coordination state lower than that of the glass [Jackson et al., 1993]. At low pressure, network forming SiO4 tetrahedra predominate, but as P increases, Si4+ coordination increases (~6.5 at 150 GPa) [Karki et al., 2007]. Less is known about the coordination of network modifiers at high pressure. Preliminary modeling of IV-16 Fe2SiO4 liquid shows decreasing Fe-O bond length and increasing coordination of Fe with pressure, but this does not constrain the effect of other cations [Muñoz Ramo and Stixrude, in review]. Guo et al. [2013a submitted] and Guo et al. [Guo et al., 2013b in prep] provide indirect evidence that coordination of Fe2+ in basaltic melts may decrease towards [5] at moderate pressures (~5GPa). The only previous test of linear mixing at high pressure and temperature was for Fe-free anorthite-diopside liquids [Asimow and Ahrens, 2010], which found the linear approximation suitable at elevated T-P conditions along the measured Hugoniots. Linear mixing appeared to fail at low temperatures, but this may have been an artifact of uncertainties in the large extrapolation downwards from the Hugoniot temperature. Another partial success for linear mixing was found in MD simulations of An-Di and also in MD simulations of the MgO-SiO2 binary, each of which showed well-behaved mixing at high pressure which began to break down at lower pressure [de Koker et al., 2013; Martin et al., 2012]. Tests of linear mixing Our first test examines whether the VFeO difference between Fe2SiO4 and CaFeSi2O6 at 1 bar is still significant at high temperature and pressure. That is, can VFeO for either end member substitute for the other, as it would in the case of ideal mixing of volumes? To test this we compare the Hd isentrope from shock wave experiments determined in this study to a modeled linear combination of end-member liquids, where HdmodelFa = Di + 0.5Fa – 0.5Fo. The equations and details of creating isentropes of mixtures are given in Thomas et al. [2012]. The results are shown in Figure 7, where it is apparent that the model does not capture the behavior of the Hd isentrope. The errors for temperature and density are highly correlated such that if the model IV-17 predicts a lower temperature than the actual isentrope, it will over predict the density and vice versa. Such is the case for Hd, where the errors at high pressure (~120 GPa) and temperature are maximized, and the model underestimates the temperature by 30% and overestimates the density by 20%. In this test, comparing the use of two Fe-rich end members, we find a failure of linear mixing and this is presumably due to complexity in the coordination state of Fe2+. Our second test examines the question: Which model will perform better in creating an Fe-rich model basalt, An-Hd — using VFeO derived from Fa or VFeO derived from Hd? Our expectation from 1 bar coordination numbers is that since Hd should be a better end member for modeling An-Hd. Two mixing models, AnHdmodelFa = 0.5An + 0.25 Fa – 0.25 Fo + 0.5Di and AnHdmodelHd = 0.5 An +0.5 Hd, are compared to the An-Hd isentrope determined in this study. It can be seen in Figure 8 that neither model is able to recover the An-Hd isentrope, but that the Fa model does significantly worse. The Fa model also becomes unphysical at high pressure and lower temperature (Fig. 8a) and this breaks the anticorrelation between errors in T and ρ, which are both underestimated, by 15% and 10% at 120 GPa, respectively (Fig. 8b). The Hd model does significantly better with maximum errors at 120 GPa that overestimate T by 6% and underestimate ρ by 7% (Fig. 8d). These errors worsen systematically for isentropes with progressively higher potential temperature (Fig. 8c). These results are in line with our expectations based on 1-bar results: neither Fa or Hd models can fully recover the behavior of the An-Hd isentrope, but closer similarity of VFeO at 1-bar for Hd and An-Hd are significant enough to improve the fit at high pressure and temperature. Our third test examines the question: As FeO content decreases, does the difference between the end-member VFeO and that of the model basalt still play as significant of a role in the final volume of the mixture at high pressure and temperature? We attempt to recover the IV-18 isentrope of the model basalt, An-Hd-Di (which has less FeO than the previous An-Hd basalt), again using Fa and Hd end-member models: AnHdDimodelFa = 0.33An + 0.667Di + 0.167 Fa – 0.167 Fo and AnHdDimodelHd = 0.33 An + 0.33Hd + 0.33Di. The results in Figure 9 show improvement for both models with the Fa model having high-pressure errors that underestimate T by 12% and underestimate ρ by 4% and the Hd model overestimating T by only 3% and underestimating ρ by 3%. Again, the Hd model does significantly better than the Fa model, likely due to the greater structural similarity of FeO in Hd and in An-Hd-Di at 1 bar. This test also suggests, at least in the studied region of CaO-FeO-MgO-Al2O3-SiO2, that the linear mixing approximation becomes better as the amount of FeO in the mixture decreases. That is, despite the variable behavior of VFeO, there may be a threshold concentration of FeO below which this is negligible and the ideal solution approximation is adequate. It still remains to define best choice for VFeO (i.e., Fe2+ coordination state) for modeling a particular liquid composition in the mantle and to define the threshold FeO concentration below which this choice is unimportant. Our final test attempts to answer these questions by comparing the original chondrite model given in Chapter III that uses Fa as the VFeO component (ChmodelFa= 0.62En + 0.24Fo + 0.08Fa + 0.04An + 0.02Di), and a model which uses An-Hd-Di as the VFeO component (ChmodelAnHdDi = 0.62En + 0.32Fo + 0.45AnHdDi – 0.12An –0.28Di). The results in Fig. 10 show that the An-Hd-Di model predicts a slightly higher temperature but that models are within ±1.5% for both T and ρ. Hence, the models are effectively indistinguishable, indicating that liquids with XFeO ≤ 0.06 (chondrite model [Andrault et al., 2011]) can be suitably approximated with linear mixing. Mole fractions greater than this amount, such as An-Hd-Di (XFeO = 0.08), can be approximated within reasonable error (±3%), but only when using an FeO component that has a similar VFeO at 1 bar. IV-19 Direct knowledge of the coordination state of Fe2+ in a given liquid thus is an important constraint on the volume behavior of that liquid, especially for high FeO contents. Conversely, precise data on the equation of state of an Fe-bearing liquid can be interpreted primarily as a constraint on the coordination state of Fe2+ in that liquid. However, liquids containing Fe2O3, Na2O, K2O, varying amounts of SiO2, and other oxide species are far more complex than the five-component system studied so far. Other cations may influence the Fe2+ coordination state and may have variable coordination states themselves. In situ experiments analyzing cation coordination of complex glasses and melts at high pressure and temperature would greatly aid in elucidating these consequences and complementing macroscopic constraints from shock wave or other equation of state measurements. CONCLUSIONS We completed seventeen new pre-heated shock wave experiments on Hd, An-Hd, and An-Hd-Di liquids to determine the EOS of each composition. Having data on more compositions than are necessary to span the multicomponent oxide system, we applied these data to testing the validity of ideal mixing of volumes as a method of interpolation among end-member liquids. Ambient-pressure density measurements on these and other Fe-bearing silicate liquids indicate that FeO has a molar volume (and therefore coordination state) that is highly dependent on composition [Guo et al., 2013a submitted; 2013b in prep], and our results show that this behavior extends to high pressure. An Fe-bearing end member close in composition to the liquid of interest is necessary for adequately predicting temperature and density along its isentrope. We find that deviations from linear mixing of volumes are significant at the level of several percent for liquid mixtures with molar XFeO >0.06. On the other hand, densities and isentropic gradients IV-20 for liquids in the system CaO-FeO-MgO-Al2O3-SiO2 with less than this threshold concentration of FeO can be approximated both at ambient and at high pressure using ideal volume of mixing. IV-21 ACKNOWLEDGEMENTS The authors would like to thank the following: the shock wave lab technical staff—Oleg Fat’yanov, Eprapodito Gelle, Russel Oliver, and Emma Dodd. Thanks to X. Guo and R. A. Lange for their ongoing collaboration and willingness to share their manuscripts. This work was supported by the National Science Foundation through award EAR-1119522. IV-22 REFERENCES Ahrens, T. J. (1987), 6. 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Shock compression data Shot # Flyer material 413 Al Hedenbergite 1068 a T (K) ufp up Us ρΗ PH km s–1 km s–1 km s–1 Mg m–3 GPa Mo 1673 1.976 0.002 419 Mo 1674 3.497 0.001 417 Ta 418 416 415 414 An-Hd Mo Ta Ta Ta 1673 1673 1677 1672 1673 1675 4.471 4.066 4.535 5.144 5.613 6.187 0.005 0.012 0.001 0.020 Mo 1675 3.618 0.003 471 An-Hd-Di Mo 1673 1674 4.700 5.815 23.8 0.4 0.02 6.59 0.24 4.99 0.17 52.2 1.5 0.01 466 Mo 0.05 3.05 0.003 469 4.27 0.148 2.006 1.456 0.11 0.005 1673 1673 5.12 2.006 Mo Mo 0.01 0.002 1070 a 1074 a 1.58 2.68 3.65 4.10 4.46 4.88 1.62 0.01 0.01 0.01 9.01 9.46 0.10 0.09 0.05 5.29 5.42 5.59 0.05 67.6 89.1 0.06 109.1 0.03 145.6 0.03 124.5 0.3 0.4 0.6 0.9 0.5 0.12 4.04 0.05 23.2 0.4 0.01 6.56 0.15 4.87 0.10 51.3 0.9 0.003 3.55 0.01 0.01 7.99 8.66 0.28 0.10 0.11 5.72 0.03 32.9 5.17 4.27 0.05 4.98 0.04 0.01 0.014 4.37 8.29 0.06 4.62 10.10 1.211 0.016 7.50 0.07 0.01 0.002 2.82 5.56 3.87 4.99 5.59 0.13 0.06 14.3 78.7 0.08 104.8 0.9 0.8 0.8 1.0 1069 a Mo 1673 2.016 0.003 1.63 0.03 5.48 0.47 3.88 0.10 24.4 2.1 468 Mo 1673 3.503 0.008 2.70 0.01 7.00 0.07 4.43 0.03 51.5 0.4 1071 a 470 Mo Mo a Caltech 40mm propellant 1673 1674 1.470 4.796 0.005 0.006 1.20 3.61 0.01 0.02 4.77 8.31 0.30 0.16 3.64 4.81 0.09 0.09 15.6 81.7 0.8 1.3 IV-26 Table 2. Parameters used ρo Mg m-3 km s-1 Co s Molybdenuma (2273 K) Molybdenuma (300 K) Tantalumb (300 K) Aluminum 2024c (300 K) 16.65 2.78 9.785 10.21 3.293 5.330 4.858 5.033 1.288 1.289 1.307 1.34 aAsimow et al. [2008], Chase[1998]; b Mitchell and Nellis [1981b]; c Marsh [1980] Table 3. Equation of State fits for molten CaFeSi2O6 To ρo Co s γo q KSo KS' KS'' 𝜒2 KT α Cp Cv Units K kg m–3 m s–1 GPa GPa–1 GPa K-1 J kg–1 1 J kg–1 1 SWEOS 1673 2913 ± 11 2613 ± 16 1.54 ± 0.01 0.300 19.89± 0.24 5.16 ± 0.04 19.32 5.91-05 1345.48 11306.70 BM3 1673 2913 ± 11 0.300 –1.93 ± 0.41 19.89± 0.24 6.22 ± 0.55* 4.20 19.32 5.91-05 1345.48 11306.70 BM4 1673 2913 ± 11 0.300 0.47 ± 9.86 19.89± 0.24 3.15 ±1.61* 0.78± 0.60 3.26 19.32 5.91-05 1345.48 11306.70 Source G13a G13a fitted derived fitted derived derived/*fitted fitted derived G13a LN92 derived SWEOS= shock wave equation of state. 3BM/MG = 3rd-order Birch-Murnaghan/Mie-Grüneisen EOS 4BM/MG = 4th-order Birch-Murnaghan/Mie-Grüneisen EOS Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992], G13a is Guo et al. [2013a] IV-27 Table 4. Equation of State fits for molten An-Hd To ρo Co s γo q KSo KS' KS'' 𝜒2 KT α Cp Cv Units K kg m–3 m s–1 GPa GPa–1 GPa K-1 J kg–1 J kg–1 1 SWEOS 1673 2772 ± 3 2772 ± 11 1.39 ± 0.03 0.310 21.30 ± 0.25 4.54 ± 0.12 20.68 5.81-05 1442.34 11400.19 BM3 1673 2772 ± 3 0.310 –0.18 ± 2.25 21.30 ± 0.25 5.49 ± 0.65* 6.37 20.68 5.81-05 1442.34 11400.19 BM4 1673 2772 ± 3 0.310 1.67 ± 2.79 21.30 ± 0.25 5.10 ±1.61* -0.17± 0.82 12.53 20.68 5.81-05 1442.34 11400.19 Source G13b G13b fitted derived fitted derived derived/*fitted fitted derived G13b LN92 derived Sources: Fitted indicates an adjustable parameter, L97 is Lange [1997], LN92 is Lange and Navrotsky [1992], G13b is Guo et al. [2013b] Table 5. Equation of State fits for molten An-Hd-Di To ρo Co s Units K kg m–3 m s–1 SWEOS BM3 1673 1673 2722 ± 2 2722 ± 2 2846 ± 14 1.54 ± 0.02 0.361 0.361 γo q –1.14 ± 0.79 KSo GPa 22.03 ± 0.25 22.03 ± 0.25 KS’ 5.15 ± 0.06 6.20 ± 0.17* –1 KS’’ Gpa 𝜒2 0.39 KT Gpa 21.19 21.19 -1 α K 6.64-05 6.64-05 J kg–1 Cp 1491.96 1491.96 1 J kg–1 Cv 11434.47 11434.47 1 For abbreviations see Tables 3 and 4 Source G13b G13b fitted derived fitted derived derived/*fitted fitted derived G13b LN92 derived IV-28 FIGURE CAPTIONS Figure 1. Preheated (1673K) CaFeSi2O6 liquid Hugoniot in shock velocity (Us)-particle velocity (up) space. The dotted line represents the unconstrained linear Hugoniot, G13a is Guo et al.[2013a submitted]. Figure 2. CaFeSi2O6 liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 1. Abbreviations: SWEOS- shock wave equation of state; BM/MG – Birch-Murnaghan/Mie-Grüneisen EOS, G13a is Guo et al. [2013a submitted]. Figure 3. Preheated (1673K) An-Hd liquid Hugoniot in shock velocity (Us)-particle velocity (up) space. The dotted line represents the unconstrained linear Hugoniot. G13b is Guo et al.[2013b in prep] Figure 4. An-Hd liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 3. The abbreviations are the same as Figure 2 and 3. Figure 5. Preheated (1673K) An-Hd-Di liquid Hugoniot in shock velocity (Us)-particle velocity (up) space. The dotted line represents the unconstrained linear Hugoniot G13b is Guo et al. [2013b in prep] Figure 6. An-Hd-Di liquid Hugoniot plotted in pressure-density space with thermal EOS fits. Data symbols are the same as Figure 6. The abbreviations are the same as Figure 2 and 5. IV-29 Figure 7. A comparison of the mixing model (HdmodelFa = Di +0.5Fa -0.5Fo) and the Hd isentrope at various potential temperatures (TP). Figure 8. a) A comparison of the mixing model using a Fa end member (AnHdmodelFa = 0.5An +0.25 Fa -0.25 Fo +0.5Di) and the An-Hd isentrope at various potential temperatures (TP) and b) its % errors in temperature and density in estimating the An-Hd isentrope. c) A comparison of the mixing model using a Hd end member (AnHdmodelHd = 0.5 An +0.5Hd) and the An-Hd isentrope at various potential temperatures (TP) d) its % errors in temperature and density in estimating the An-Hd isentrope. Errors shown in text are the maximum at 120 GPa. Figure 9. a) A comparison of the mixing model using a Fa end member (AnHdDimodelFa = 0.33An +0.667Di +0.167 Fa -0.167 Fo) and the An-Hd-Di isentrope at various potential temperatures (TP) and b) its % errors in temperature and density in estimating the An-Hd-Di isentrope. c) A comparison of the mixing model using a Hd end member (AnHdDimodelHd = 0.33 An +0.33Hd +0.33Di) and the An-Hd-Di isentrope at various potential temperatures (TP) d) its % errors in temperature and density in estimating the An-Hd-Di isentrope. Errors shown in text are the maximum at 120 GPa. Figure 10. a) A comparison of the two mixing models for chondrite liquid (ChmodelF a= 0.62E n+0.24Fo +0.08Fa +0.04An +0.02Di and ChmodelAnHdDi = 0.62En +0.32Fo +0.45AnHdDi - IV-30 0.12An -0.28Di) and b) their % error difference in temperature and density. Errors shown in text are the maximum at 120 GPa. IV-31 FIGURES Figure 1 IV-32 Figure 2 IV-33 Figure 3 IV-34 Figure 4 IV-35 Figure 5 IV-36 Figure 6 IV-37 Figure 7 IV-38 Figure 8 IV-39 Figure 9 IV-40 Figure 10