I. Remarks on the Archibald Technique for Determining Molecular Weights in the Ultracentrifuge. II. The Hydrodynamic Alignment of Rod-like Molecules in Centrifugal Fields Citation Peterson, James Macon (1963) I. Remarks on the Archibald Technique for Determining Molecular Weights in the Ultracentrifuge. II. The Hydrodynamic Alignment of Rod-like Molecules in Centrifugal Fields. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VZXF-V320. https://resolver.caltech.edu/CaltechTHESIS:04262012-145551769 Abstract I. The theoretical validity of the linear extrapolation of concentration gradient usually used in Archibald measurements of molecular weights is discussed. II. Intrinsic and concentration-dependent hydrodynamic alignment of rod-like molecules are considered in an attempt to explain the observed increased sedimentation coefficients with field and concentration. An entropy mechanism is also discussed. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry and Physics) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Mazo, Robert M. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1963 Record Number: CaltechTHESIS:04262012-145551769 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04262012-145551769 DOI: 10.7907/VZXF-V320 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6981 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 30 Apr 2012 21:11 Last Modified: 09 Jan 2024 00:11 Thesis Files Preview PDF - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

I. REMARKS ON THE ARCHIBALD TECHNIQUE FOR DETERMINING MOLECULAR WEIGHTS IN THE ULTRACENTRIFUGE Wi. THE HYDRODYNAMIC ALIGNMENT OF ROD-LIKE MOLECULES IN CENTRIFUGAL FIELDS Thesis by James Macon Peterson In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1963 ACKNOWLEDGMENTS Thanks are due to Dr. Robert Mame for his help, guidance, and patience. I would like to express my gratitude to the graduate students with whom I associated for many stimulating discussions. I am indebted to the National Science Foundation, the duPont Company, the Shell Corporation, and the Institute for financial support. ABSTRACT I The theoretical validity of the linear extrapolation of con- centration gradient usually used in Archibald measurements of molecular weights is discussed. pt Intrinsic and concentration-dependent hydrodynamic align- ment of rod-like molecules are considered in an attempt tc explain the observed increased sedimentation coefficients with fleld and concentration. An entropy mechanism ts also discussed. TABLE OF CONTENTS Page PARTI. REMARKS ON THE ARCHIBALD TECHNINQUE FOR DETERMINING MOLECULAR WEIGHTS IN THE UL TRACENTRIFUGE . . . . . i I. Introduction . . . . . . : . . . 1 1. Basic Theory . . . . : . . . . 2 i. Use of the Experirnental Data . . . . . . 8 Iv. The Approach to Equilibrium . . . . . . ll Vv. The Work of Fujita and MacCosham . . . . 16 vi. Numerical Integration . . . . . . . 20 viz. Comparison of Computer and Fujita-MacCosham Results . . . : . . . . . . 25 Vil. Expansion of the Fujita-MacCosham Formula and Conclusions . . . . . ° . . 28 PART I. THE HYDRODYNAMIC INTERACTIONS OF RIGID RODS IN CENTRIFUGAL FIELDS . . ‘ . . 34 I. Introduction . . . : . . . . . . 34 NI. Spheres ander Stokes Flow in an Infinite Medium . 36 ill. Rotatory Brownian Motion Theory . : . ‘ . 39 Iv. Intrinsic Distribution Functions and Sedimentation Coefficients . . . . . . . . 41 A. Stokes flow in an Infinite Medium . . . . 41 B. Rods under Oseen Flow in an Infinite Medium e e e e e e e e e 49 C. Stokes Flow in a Medium Bounded by a Wall e ° e e e ° e . e 54 D. Particle ina Radial Field . . . . . 73 Vv. Concentration Dependent Phenomena . . . . 77 A. Spheres and the Wall . ° . . . . 82 B. Spheres and the Concentric Hy pothetical Sphere. ’ . : . . . . . 85 C. The Hydrodynamic Interactions of vI. Preliminary Comments on Liquid Crystals . . . 93 Vi. Summary and Diseussion . . . . . . . 100 PROPOSITIONS . . . . ‘ . . . . . . 114 I. INTRODUCTION Ultracentrifugal measurements afford a very useful and prac- tical way of determining certain physical properties of macro- molecules (1). The measurement of molecular weights, with which this work is concerned, falls among these. The theory of the method is based on a relationship connecting molecular weight with the ratio of sedimentation and diffusion coefficients. The ratio of sedimenta- tion and diffusion coefficients can, in turn, be deduced from the dis- tribution of particle concentration in the centrifuge cell (2). If the centrifuge is allowed to run long enough at constant epeed to establish a thermodynamic equilibrium between sedimentation and diffusion, the distribution of particle concentration about any point in the cell is sufficient, along with other very sasily obtained parameters, to calculate the molecular weight of a sample. In the absence of thermodynamic equilibrium between sedimentation and diffusion, the useful data are the particle distributions either at the inner or at the outer boundary of the cell; however, particle concentrations at other points within the cell are also of interest, since such data are nec- essary for accurate determinations of particle distributions at the boundaries (3). Experimentally it is much more convenient to work in the absence of thermodynamic equilibrium, chiefly because it takes so long to achieve it (a matter of days is common). A difficulty with non-equilibrium systems, however, is that available methods to measure particle distributions do not function properly at the boundaries. This {se due to the fact that they are based on optics and are often plagued with interference effects there (4). To ascer- tain the conditions at the boundary, the usual procedure is a linear extrapolation of the quantities of interest from those found in the regions where they can be measured to their values at the boundaries. The problem investigated here was specifically the theoretical validity of this extrapolation. According to an analytical solution, the distribution of particles near the inner boundary in the absence of thermodynamic equilibrium does not behave in a simple way as a function of distance from the boundary. Results from a numerical solution, however, indicate that linear extrapolation is a good thing todo. From this it was concluded that the validity of the Mnear extrapolation is not easy to understand theoretically. A possible means of improving the extrapolation is suggested. YW. BASIC THEORY The derivation of the equation connecting the ratio of sedi- mentation and diffusion coefficients with molecular weight is most conveniently accomplished for these purposes by considering the mathematical formulation of those processes occurring within the cell before and during equilibrium. Schematically the cell can be represented as in Figure 1. fe Y, Vi. Figure | The coordinates chosen are the usual polar oneg--+ r, 8, z » where rand @ have clear significance, and z refers to the thickness of the cell. e. is a anit vector pointing outward from the axis of rotation 0. The dots within the solid lines represent macromolecules. _ None of the phenomena which will be considered possess any dependence on 6 and 2z. When the centrifuge begins operation, the centrifugal force of magnitude wer per unit masa (where © is the angular velocity of the rotation in radians per second, and ms rs tr, ) operates on the solute macromolecules. This will induce a net flow of these particles toward the outer boundary T° The particle velocities so induced in these macromolecular systems is srmall enough to allow use of the hydrodynamic result that this velocity is proportional to the external force applied. The proportionality constant is called the sedimentation coefficient, denoted by Ss TaN Yelocity=« SU Y fy (1) s » of course, depends on the size, shape, and mass of the particles of interest. If the concentration of particles at r is C(r,t) , then the sedimenting number of particles or flow crossing unit area per unit time is aw 2 sedimenting flow = CU, t) sw Y ev (2) S.ch a sedimentation flow will, of course, cause particles to be pulled away irom the inner boundary at ris and to be packed against the outer wall at To A concentration gradientis therefore set up in these two general areas, which, according to Fick's first law will induce a diffusion flow of the solute particles. This current isa act az diffusion flow= — D oY ey (3) where D is the diffusion coefficient. The total flow is the sum of the sedimenting and diifusing flows. acct) total flow = (s wey Ccy¥,t)- D Jr gu) @ ev (4) ' Since these currents of particles are in opposite directions, they will tend to counterbalance each other, and, if the centrifuge ie run long enough, an equilibrium between the two currents will be established such that the total flow vanishes at each point within the cell. The concentration is nonuniform but time independent at equilibrium. We then have, since the flow is zero, act) suv tcy) - D By = © (5) This can be rearranged to give S$ | dCi) D wer CM) dy (5') If the weak dependences of s and D on concentration are neglected, the solution to this differential equation is ~ Ss * ~ 2 a _ Su Sn 2 CO) = Co SEG [exe (Stew) exe EH *~-)) « 6 where Cc. is the original uniform concentration throughout the cell. Equation 6 would then represent the concentration profile at equ- librium. Another use may be made of equation 5', stating that the total flow is zero everywhere at equilibrium. Itis possible to apply thermodynamics to the cell when 5' holds. For example, the total thermodynamic potential per mole of solute macromolecules which is the sum of the chemical and centrifugal potentials, F tor = Pewem t+ BPeenrmevcae (7) = fo(T,P)+RTIm ¥ Cow) -M uty where y is the activity coefficient and M is the molecular weight, must at equilibrium be constant throughout the cell. Differentiating: OL D ALT P? = _ ol fs Oa mnCo? Oo Ln A) vs 2 =, OO. 5 y + Rr( Bincow a oye FEE) Mew Y _ OA (TP) OP 7 kent t_/ 2 Lny 2 - OP oY +R av \+ 2 hnk |) Mw Y where pis the pressure. From a thermodynamic identity ° (T,P) — where v is the partial specific volume of solute particles. The quantity =e can be calculated by considering the increase in force per unit area at r due to the external field. If the pressure atris p(r), then the pressure at r + dr is P(yY) + BPO dy =P) + Pure dlvelome) Ay ea = Piw4+ purr dv where p is the solution density. Therefore, dP _ 2 ae PT (2) Substituting this we get Mus -Mo Pury = geeh (1+ 2 ae Con which can be rearranged to RT Co) _ oft | (9) M = “a (1-vp) Or Cw ae If this is combined with equation 5', the result is _ KTS Ln Ss __ M= Soap (* aL.co “ This is the thermodynamic relationship already referred to, which connects the molecular weight with the ratio of the sedimentation and diffusion coefficients. In the case of infinite dilution, 10 becomes At infinite dilution, s and D are independent of concentration, while the activity coefficient becomes unity. Equation 11 was first derived by Svedberg (2), and along with equation 9 forms the theoretical base for the determination of molecular weights in the ultracentrifuge. Wl. USE OF THE EXPERIMENTAL DATA Experimentally the usual quantity obtained is the refractive index gradient (4). This is measured by taking photographs of the cell using Schlieren optics while the centrifuge is in operation. The actual mechanical details are rather involved (5), but for the present purposes are unimportant and will not be considered. The refractive index is assumed to be proportional to the concentration gradient. It will be assumed that this proportionality constant is avatlable and henceforth only concentration gradients will be used in the discussion. If the concentration gradient ia known throughout the cell, the concentration can be found by the following formula (6): Cy,4)=C. + f 2gae acct 4, + f Yenv) acest) yy 7 YAM “VW oy To derive this, we note that the total amount of material in the cell is constant and independent of time. 2 4 2 Y¥3 -Y¥, _ d dy = (E-*) = 62 (tiv v or Co (% -v7) = 2 [carvdy Integrate by parts ¥2 oC Cr,t) d _ y,* C%,#) [er ————_ av C,(%-%*) = [ys Cova t) J - 4? (13) Use the mathematical identity Y; * aceyt) gy + Com,t) C (v2 ,t) = 6 4 Y + ? to cbtain, when combined with equation 13, CW, = Co + ['E) EE ay an Combine equation 14 with the mathematical identity VY OCW F) C(Gt) = y as dv’ + C(%, t) to obtain equation 12 Cl H=C. +f P28 4. +f (Se ) re (12) Yi or ay, ac(r, t) The two quantities ar and C(r,t) are now available for the calculation of molecular weights, the former being measured and 10 ac(r, t) Pa 64 or the latter being calculated from measured values of an equillbrium situation is present, the ratio 3/D can be computed ac(r ,t) or from any conjugate pair of C(r,t) and ( ) via equation 9. In the absence of thermodynamic equilibrium, which, as stated earlier, {3 much more convenient experimentally, the ratio 5s/D can be computed ). C(r,. t) and ( or ). r=r r=r from the conjugate pairs C(r y*) and { Sr This is due to the non-vanishing of the total flow total flow= (swry C (vt) -) 26 Y (4) under non-equilibrium conditions at all points in the cell except at the boundaries r = Tye At these points the total flow must clearly 2° vanish since there are physical boundaries present which material cannot cross. Archibald (3) first pointed out the usefulness of data extracted from the boundary. Archibald's method represents a valu- able tool in the determination of molecular weights. Accurate values of concentration and concentration gradient at the boundaries are clearly necessary for Archibald calculations. As has been stated previously, however, the optical devices available for these measurements are inefficient at a boundary (4). The measured concentration gradientis estimated at the boundary by extrapolation from the ragion nearby by means of a straight line. The inner boundary at ty is customarily used to avoid activity effects as much as possible. Typical pictures of concentration gradients ll obtained experimentally for short times of centrifugation have a linear look about them. The problem investigated here is specifically the validity of a linear extrapolation, to remove any nagging doubts. IV. THE APPROACH TO EQUILIBRIUM To establish the linearity or non-linearity of the concentration gradient curve, itis necessary to investigate the behavior of this quantity near the boundary under non-equilibrium conditions. The equation governing approach to equilibrium in these systems isa simple expression of conservation of mass. ac? wi =— div (swe Cot)_-D OCCyt) ey (15) OY with boundary conditions BC Gt) Sw7v C 4) = D aw Y2Vi,%e (16) Cy, t) = Co tio In words, this says that the change in number of particles per fixed unit volume per unit time is equal to the number of particles flowing into (or out of) this volume per unit ime. This equation was first derived by Lamm (7). It is instructive to investigate physically what happens during a short Archibald run. Att= 0, the concentration is uniform throughout 12 the cell and( 6C (r,t) ) 20. The concentration profile is like or rer) Co ct Y, Y> Ya Figure 2 If, for t> o, diffusion is inoperative, the solute particles will move away ‘rom the inner boundary, r,, and pack against the outer boundary, 1 Tos due to the action of the centrifugal field. The concentration would then be zero in a region beginning at ry and ending at sorne r > Ty. It would be some constant value corresponding to packed molecules in another region ending at the outer boundary r, and beginning at 2 < e somer< r, Between these two regions, there is initially no concentration gradient. That, in fact, no gradient will be set up for pure sedimenta- tlon can be seen as follows: Consider a band of solute molecules of width dr atr att 0 Y dv Ye Figure 3 13 The volume of this bandis @zardr att 0. From equation 1 z “~ sedimentation velocity = SWw*V ev (1) it follows that after a time dt the boundary at r has moved to a point r+ 36 edt , and the boundary at r ¢ dr has moved to (r+dr)(l+ sr “at. Hence the new width of the band is (rade) (14 sumdt) —v (itsu*dt) = dv (1+sur dt) The new volume of the band is Oz (itsurdt) dv ~ 62 dy (i425 wrdt) The relative increase in volume during the time dtis ozvdy (425 whdt? = |+2swrdt bzvdv This is independent of r and demonstrates thatin the case of pure sedimentation no concentration gradientis setup. Therefore, in the region between the zones of zero concentration near the inner boundary and packed molecules near the outer boundary, the concentration . is constant, independent of r, and dependent only on ttme. The time dependence must be such that the concentration decreases, since the volume available to a band increases. The concentration profile shown in Figure 2 becomes CT vi v> Figure 4 Yo 14 The flat portion between the regions of zero concentration and packed molecules is called the plateau region. Diffusion does occur, of course, at the boundaries of this central region, but some time is required before the plateau is destroyed. The concentration profile actually looks more like os ) cT v v—~ Yo Figure 5 In the plateau region, therefore, there is no diffusion because there is no concentration gradient, and the Lamm equation becomes act + div (surv Cu, t) ) ev (17) at or act) ot p,i2tyrsuw*y Cit) =9 v av act) os sweet) at ~ (18) C = Co exp (-254"t) 15 It might be mentioned that the decrease of particles in the time interval 0-t within a region bounded by ry and so:ne point in the plateau region, "> say, has to be equal to the number of particles crosaing the surface r tp + zf [Co - Corp) v dv = Sw Vp if C (vp, t) dt 6ZY, or After a small amount of manipulation, this becomes (¥,t) Cv, t)= Co - oe te This formula can be used instead of equation 12 if a plateau is present. It was first derived by Klainer and Kegeles (8). The importance of the existence of a plateau in a treatment of the Lamm equation is that it greatly simplifies the boundary con- ditions. The outer boundary can be disregarded for phenomena occur- ring near the inner boundary, if a plateauis present, and its conditions replaced by zt ~2sw C (co, t) - Co e = constant in space The conditions are now 16 acy, t) _ Sw Y; Ctv,t) = D oY Y2”% - at C (e,t)= Cs € ae = constant in Space (20) C(yt) = Ce | a are) Vv. THE WORK OF FUJITA AND MacCCOSHAM It was decided to attempt to investigate the linearity of the concentration gradient curve by performing a numerical integration of the Lamm equation subject to plateau boundary conditions, and 2 8-C(r,t) taking a careful look at the constancy of the quantity near or the boundary Pye None of the available analytic solutions found were particularly useful. The exact analytic solution of Archibald (9) was too tedious to work with (10). Cka's solution is a formal one of no practical value (11). Faxen (22) used an approximation 2P— <<1 Ssw* Y* which does not fit the conditions of the present situation. When the numerical work was well under way, the work of Fujita and MacCosham (13) was diacovered in the Kterature. These i7 authors have presented an approximate solution to the Lamm equation subject to plateau boundary conditions. The analytic forrn of the results is not unduly complicated. Subject to the severity of the approximation involved, it should be possible to deduce from their work the concentration gradient near the boundary and thus investigate its linearity. It was decided to go through with the numerical work to find out how good this approximation was. Fujita and MacCosham began with the Lamm equation aCty,t) a) 2Cuy.w "eee +> = 2 Paton Saas = a+ . x (sw vy Ci.t) —D ar } oO New variables were defined _t z & = @e io & = e % ct -2sui°*t 2D _ 42 € = V2 w? S Substitution of these into the Lamm equation gave _2 atu Ou u ~ 2@& 2 = €e 9 # z The approximation made by Fujita and MacCosham was to neglect the =Z : A : variation of the factor e from unity for mathematical convenience. _ ane e?.-_() =1 18 As to order of magnitude, r, ~ 5.6cm., r~ 5.6- cm. in the region 1 of interest. So at most 2 -2_ (S&) += o e-?- ( st ) 935 or about 7% variation. Atter this approximation, the equation becomes a au _ au _ Bu ot € 32 oz or ina more condensed notation Ue = € Ugg — Uy A new variable was defined Zz Tt uz ¥ ep ( Re GE whereupon the equation became subject to the boundary conditions yu = exp(- = ) O<2<@ tHe This equation is of the heat flow type, which has been intensely studied by Carslaw and Jaeger (14), from which the solution was determined to be 19 Veo = c= fen(Z oF 09 SEE SS) Z+T, tte \, L/e (21) + (14 2 ven -tT- #/2 2/¢ , e Z+T zs) ate = 2 [oy ($e) (a4 4) (c- 2)" “2 _[E & oF) 2 fe grfe(x) = |- ef = Im ee 20 VI. NUMERICAL INTEGRATION The Lamm equation is ac ac 1 2 vlsurve -D37) =? ot VY oY with boundary conditions CoCo t<o 2 ~ suw*C = DS¥ yay -2sw*t C (w,t) = Coe 2SW = constantin space Dimensionless variables are defined _ Su,” W =F A= > Then oY a ( y — or )j)_-o or a | er at + A - ase ph =0 56 +2rY¥ Ps Pa 3p (23) subject to 2 A\py at pel a —> © 21 since we are interested in a solution with plateau present. This differential equation is now replaced by a finite difference equation, i.e., the derivatives are replaced by finite differences. The basic idea behind this (13) approach is to set up a grid, or net, in space-time, and treaty, 9, and® as discrete rather than con- tinuous variables. Quantities Ap, A@ are defined such that p assumes the values 1, 1+Ap, 1+ 2Ap,...-, 1 mAp,...., and & assumes values 0, 480, 246, 340,...., n&@,.... Therefore, ve refers to(1+mAp, nd). Schematically this can be represented eT yz x. ; e oe e bd eo e Cd e ° ° e o ° o . Yo a vi , @e-0 o o ° ° e e r ° ° Py Pa e Py o Viv ’ ‘ 22 AP and A@ should be as small as convenient to approximate as closely aa possible the actual values of the derivatives. A necessary restriction on these quantities to prevent propagation of errors introduced by the use of finite, rather than infinitesimal, differences (15,16), is that < d(apy The derivatives become 2 e se (24) OM _. Bien 7 Sime op 7 2a P yr - Ym Ym - fons) 12 p 2% = (Foes Smo OO" Lp C2 pee? Rap (Ime ap AP Since Fm ™ the above becomes ar 2 [( + tena) be mr 72% + (1- $F) % m- a | (25) Lamm's equation then takes the form 23 Sn -~ on wnt -¥, as - 2A Ym — A (14map) Zap + ap 2 [4 ep) Yau ~ 2 Ie + (\ ~ npn] The following parameters are defined AS ~ K = |-2@pr — 248 whereupon Lamm's equation becomes yar y” K + vn, { p+ * (ase — A E (ismap)) } (26) Sinn { p- (Fay _) (itm opi) This is a type of recursion relationship. It allows us to advance in time, or gives us the values of all Ye atn+1 if we know the y's atn. m™ The above formula does not hold for the calculation of Y, for which the boundary condition must be used. The condition is (24) - \yge oP pe! d¥o To make use of this, we write Yo as a power series in Ap or about ¥ l 24 Now Yo = WK API = Ak a ¥2 -8o ZAP nv . y-2%+% bo GP When these equations are solved for y 0° the result is Y= 4% + Se (27) where 3.2 = ' 3+ 2AaP J2 ~ B+2Asf Lamm's equation ia essentially now in the form of a routine recursion relationship of the type admirably suited for digital com- puters. A program was written for the Burroughs 220 at the Institute, and the solution of the equation obtained there. The scheme of calculation is straightforward. The parameters in braces in equation 26 were calculated and stored in memory for 0< mz 1000. Then the memory cells were loaded with Yo » or unity. The proper new y's were computed according to equations 26 and 27. When ‘9 and y 10 differed in the th decimal place, the integration was extended by 10 units. The same was true for 49 and 50 ’ v9 and Y30 » and go forth. Physically this just means that the system was integrated out to the plateau and stopped, but continued 25 when the plateau was pushed farther out by diffusion. After the integration had proceeded. the equivalent of 5 minutes centrifuge time, the computer was allowed to print out the concentration and concen- tration gradient data contained in memory, and then allowed to proceed to do 5 more minutes worth. The system picked for study was the hemoglobin system. The data used were obtained from Dr. William Hutchinson of these labor- atories. a #4.50x 1073 s6c 4t #1 sec M * 66,800 gm/mole &6@ = 2.01875450 « 107° v = 0.749 em/gm h #8 21,.32450842 density = 0.998 gm/cm Ap = 2.409852114 x 1074 r= S7em € 2 4,18854438 x 107° o * 9.340 rev/min B = 0.347683752 K = 0.3047623586 AE = 8.931864990 x 1074 VW. COMPARISON OF COMPUTER AND FUJITA-MacCOSHAM RESULTS The results given by the computer were compared with those predicted by the Fujita-MacCosham formulae. They agreed remark- ably well, being within 0.1% near the inner boundary after reasonable ran times. 26 The concentration gradient profile at 45 minutes 19 from the numerical solution D n i 1 a 1 i 1.0024 1.0048 1.0072 r/r “9 Figure 7 1.0096 1.0120 1.0195 1.0169 27 v dy / ap Fujita- Fujita - m MacCosham Computer MacCosham Computer 30 min. Qo - 8617 - 8621 18.36 18.38 - 8826 - 8832 16.64 16. 66 10 -9016 - 9023 14.88 14. 89 15 - 9186 9191 13.10 13.12 20 - 9332 9339 11.40 11.38 50 « 9839 - 9844 3.47 3.51] 100 - 9980 - 9981 O01 - 00 ao - 9984 - 9984 0 0 -_ 45 min. 1) "9325 «8331 17.75 17.76 - 8532 - 8536 16. 46 16. 47 10 - 9054 - 90B9 12.38 12. 40 15 9197 - 9200 11.06 11.06 20 9319 - 9326 9.74 9.76 50 - 9679 - 9684 5.35 5.32 100 - 9956 - 9956 55 52 fev) ©9977 - 9976 0 0 The excellence of agreement leads to the conclusion that the Fujita-MacCosham formulae can serve as an excellent guide to the theoretical line shape near the boundary of the concentration gradient curve. The validity of the customary linear extrapolation of &aC/ér from the region of good Schlieren definition back to the boundary can be investigated by expanding the Fujita-MacCosham formula ina 28 Taylor series aboutgp*# il. The coefficients will, of course, be time 2 dependent. If a and higher terms are arnall relative to C(r) or and x for times of interest, then theoretical justification of the br extrapolation will be established. VII. EXPANSION OF THE FUJITA-MacCOSHAM FORMULA AND CONCLUSIONS Taylor's formula is ' z S(p,t) = HP24 T+ (UTP tar" (yD (pY tty c)(p-yir Since the Fujita-MacCosham formula is most conveniently expressed in terms of the variables z and t, it is advantageous to perform some mathematical transformations before calculating the necessary derivadves with respect top. 2/2 2 p* 2 2* az Os 9.2% 22 5-7 2 3% 92 2 p3 e d2z3 ~/2e ger t4e 52 22 a pt Ibe o 2+ 48e J23 + 4.4 e7** 552 —/2e- bz and so forth. In addition, the equation Ur = EUs, - Us 29 where y ® ena is useful, as well as the boundary condition u{il,r ) se u (7). Since the derivatives with respect to p are expressible in terms of derivatives with respect to £, itis convenient to set up a table of z derivatives evaluated at Z 20 (or p# 1). Define A «2 . Then ato = 1, or £ = 0, Up rreu 2 Uzact Au, thu Ua22 = a Ur ra “ Aarzg = ¢ Aer pt Az pat a and so forth. In addition, 2 _% Ultzor) -t{2+FeplE ~~ fe fe} a Tc (Me U, (2-207) = 2¢ orfe [fo - Yerne © { _ ae Ur (#207) = zee e 2T rer and so forth. These expressions can be cornbined to give atp=1, or z= 0, U ep = 2A Ur +A CA-1)u 30 Wrep = A(A-i)(A-2) « + 2A(20-3) Ue U peep = d ()-1)(A-2)(4~3) u —2, (bA-thuy t ed Uce and so forth. Using the data from the hemoglobin system, we have after a 45 minute run ee” 20.9977 au # 0,8344 u,, * 8163. 7896 X = 23.3245 a, = 033.1029 Hence C/Co = eu @ 0.8325 + 17.7530 (p <1) ~ 523.8765 (p -1)* -8, 115.9355 (p -1)> + 617,303. 3892 (p <1)" 4+ --- After a run of 90 minutes, the parameters become C/Co = 0.7689 + 16.3974 (p 1) - 300. 64 (p -1)* ~ 5. 102. 22 (p -1)° 4 + 225,877 (p -1) + °° A cursory examination of these expressions will indicate that the terms of higher order in the Taylor expansion are not small, and < curves are not linear. The higher order terms alternate in sign in a rather random fashion and apparently tend to cancel out. It would seem that a much safer, although more laborious, 31 procedure than the linear extrapolation to find =< at the boundary would be to put the measurable experimental data into the Fujita- MacCosham formula and provide some sort of best fit to determine the parameter <. This would most lkely require aid from computers. Only ¢ is needed because it is equal to = > and thus contains the ry Sm ratio a/D plus other known parameters. A possible scheme is to do a linear extrapolation of the = curve to get 3 » calculate ¢, and then fiddle with this value of e¢ until good agreement with the Fujita- MacCosham formula is reached. i. should be mentioned here that the concentration dependence of s and D has not been taken into account. Also, the effects of poly~ disper sity have not been investigated. be 10. 1}. 12. 13. 32 REFERENCES Schachman, H. K., Ultracentrifugation in Biochemistry, Academic Press, New York, 1959. Svedberg, T., Kolloid Zeitschrift, 36, Erg-Bd., 53 (1925). Archibald, W. J., J. Phys. & Colloid Chem., 51, 1204-1214 (1947). Schachman, H. K., op. cit., p. 190. Thid., p. 35. Tbid., p. 187. Lamm, ©. Arkiv Mat., Astron., och Fysik, 21B, No. 2 (1929) (no page number available). Klainer, S. M., and Kogeles, G. J., J. Phys. Chem., 59, 952-955 (1955). Archibald, W. J., Phys. Rev., 54, 371-374 (1938); Ann. N. Y¥. Acad. Sci., 43, Art. 5, 211-227 (1942). Yphantis, D. A., and Waugh, D. F., J. Phys. Chem., 60, 622-629 (1956). Oka, S., Proc. Physico-Math. Soc. Japan (3) 18, 519-523 (1936 (3) 19, 1094-1104 (1937). Faxen, H., Arkiv Mat., Astron., och Fysik, 21B, No. 3, 1-6 (1929); 25B, No. 13, 1-4 (1936). Fujita, H., and MacCosham, V. J., J. Chem. Phys., 39, 291-297 (1959). 14, is. 16. 33 Carslaw, H. 5., and Jaeger, J. C., Conduction of Heat in Solids, 2nd ed., Oxford, Clarendon Press, Ch. 12, 1959. Kunz, K. S., Numerical Analysis, McGraw-Hill Book Co., Inc., New York, N. ¥., 1957, Ch. 14. Private diacussion with Dr. Joel Franklin. 34 I. Introduction Hearst and Vinograd have discovered that the sedimentation coefficients of deoxyribonucleic acid and tobacco moaaic virus increase as the angular velocity of the centrifuge increases (1). The phenomenon is highly concentration dependent, and appears to vanish at zero con- centration. T-4 phage DNA displaye a field independent sedimentation coefficient of about 550 S at a concentration of op*°? 0.122, but at op?°° 0.458, the sedimentation coefficient varies from about 280 S at 4,908 RPM to about 450 S at 20,410 RPM. E. Coli DNA at op?°® 0.36 has a field independent sedimentation coefficient of 2505S, but at op? 2.7 the sedimentation coefficient variee from about 130 S at 9,945 RPM to about 1505 at 56,100 RPM. TMV at 0.2 mg/ml showa a sedimenta- tion coefficient of about 178 S at 4,908 RPM and 185 S at 20,410 RPM; at 0.6 mg/ml the coefficient is about 185 S at 4,908 RPM and 193 S at 20,410 RPM. Although the TMV data indicate that this effect does not extrapolate out at zero concentration, Hearst and Vinograd point out that more work needs to be done in the low concentration region to determine just what happens at zero concentration. It is not unreason- able to expect the sedimentation coefficients at different fields to coin- cide at sero concentration. | Hearst and Vinograd were unable to offer any definite theoretical explanation of the effect. They concluded that a likely origin of the anomaly is an alignment of the molecules. A long rigid rod-like particle 35 oriented parallel to an applied field sediments twice as fast through the medium asa particle aligned perpendicular to the field, so since both particles possess rod-like characteristics, the magnitude of the effect ies within this range. The two most obvious mechanisms for alignment, as mentioned by Hearst and Vinograd, are concentration- dependent hydrodynamic alignment and an entropy alignment such aa that operative in the formation of liquid crystals. Hydrodynamic alignment is considered in detail here. The model used corresponds to a rigid inflexible rod, whose properties are much more characteristic of TMV than DNA. This work can thue be considered to be restricted somewhatin scope. The resulta of the calculations are applied solely to TMV. They predict a dependence on concentration for the sedimentation coefficient which is within 6% of experiment, and indicate that the expected field and concentration dependent hydrodynamic alignment is sufficiently small that it cannot be detected exper{mentalily and also that the molecules tend to align perpendicular, not parallel, to the fleld. Therefore, according to these calculations, any hydrodynamic alignment present would decreage vather than increase the sedimentation coefficient with an increase in field. A few preliminary comments concerning tactold formation are presented. Itis shown that the effects of an angular anisotropy due to the excluded volume effect are too small to be observed. This is 36 true, of course, in regions .oflow concentration where no liquid cry stala are formed. Before concentration dependent phenomena are conaidered, the consequences of a sphere moving through an infinite medium and a brief sketch of rotatory brownian motion will be discussed in order to form a framework for later theory. Also, as a matter of complete- ness, certain types of intrinsic alignments and sedimentation coeffici- ents will be presented, all but one of which were worked out in the course of thia investigation. Il. Spheres under Stokes Flow in an Infinite Medium The equations governing the flow of incompressible viscous fluids are the Navier-Stokes equations (2) (22 40.0%) = -vpeny Vv: v29 where p is the density, p the pressure, and 1} the viscosity. If the flow is steady state, ~ 20. In cage the velocities are low, the - quadratic terms pve av are neglected. To this approximation the hydrodynamic equations are linear 2 Tvvevgp = gev 20 For a sphere moving through the medium with velocity U @ the exact solutions to these equations are (2) 37 2 3¥4 (4) a ‘“.' Qi 2 } sin® cos8 cos (la) 3a Pi v 3V/a (1 -—=) sin@ cosé sing (1b) y ar 2 3Ua 2 a? 2 vo See 1 (1 + cos 6) t+—~—(1 - 3 cos 8)} (ic) ‘2 4r 3r° U e is the velocity of the sphere with respect to fluid infinitely far away, ais the radius of the sphere, r is the distance from the center of the sphere to an external point of interest, and @ and © are polar angles defined in the usual manner. Atr = a, it can be easily verified that these expressions give v* ‘y 2 0; 7 U so that the requirement of no alip on the sphere surface is satisfied. If the quantity a/r becomes considerably less than unity, the leading terms are 3Va Ua zx ‘i ie sin® cos® cosg = = “2 (2a) 3Ua . 3Ua By y . > ain® cos6 sing = - 5 (2b) r FA 3Va 2 30a Z v,* iz (1 + coa 8)= dr (1+ = | (2c) If the sphere had been moving in the ‘es rather than ee direction, the result for a/r << 1 would have been, by symmetry, 2 3Va xy ‘y" ae 2 (3b) 3Y¥a xz vy” an “2 (3¢) 38 Likewise, the results for a sphere moving in the e direction are wa ‘a Gp we (40) 2 vs 22 (14d) (4b) Fy le (+0) ? According to Stokes’ law the sphere exerts a force upon the fluid in the direction of its motion of magnitude 6n jal . The effect of neglecting the higher order terms is equivalent to treating spheres of vanishingly small radii, or point forces. It is convenient to define a tensor T (r,r'} such that the fluid velocity at any point t due to the action of a point force at 7?’ is obtained as indicated in equation 5, V(r) © T(r, E> FEY (5) From equations 2, 3, and 4, this tensor is seen to be 2/ / / ‘ l+ sine cos © ein6 sine cos , gin® cou cosd 1 / / / / y aes sine ‘cose’ sing i+ ain*s ain-e sin® cos® sing 8nf | rer | T(z, ?') = / / n@ cos ‘cosp / sind cost sing 1+ cose ° (6) The primes on the angles refer to the coordinates of the vector Tier. Itis sometimes convenient to write this in dyadic form 4, i aA - A - 3G.7) + (A +s, +o, ~) (7) 39 MW. Rotatory Brownian Motion Theory In the first part of thie thesis, an expression called the Lamm equation, denoting conservation of matter, was deduced. In words, this states that the decrease in number of particles per untt volume per unit time equals the divergence of current of particles across the boundaries of unit volume. glee) zs « div (flow) The same sort of relationship can be deduced for angular orientations of rigid rod-like molecules. Call {(@,¢,t) the angular distribution function, such that the fraction of rods having their long axes aligned in the solid angle sin? d9 dw at tlme t is £(@,«, t) sind d§9dq@. There then is the quite analogous equation aoc!) ~div(angular flew) If there is an external force on the particle which gives rise to a net torque, 7(6 »9,t), there will tend to be a rotational angular flow. F (@.¢,t) § 53 7 Vv, F (8 ost) The angular velocity which this torque induces is ~~ we = (8,9,t) ‘ R7 (8) 40 where R is the rotational friction tensor, which for rods is clearly diagonalin a coordinate sy stem using the long axis of the rod as an axis (this means that the rod tends to rotate only in the direction of the applied force). This flow will cause points to move with velocity ms x e. so that the flow due to external forces is —- - an 9.960) (T(2 0.) RY xe, If £(3 ,¢,t) is nonuniform, Brownian motion will tend to destroy the nonuniformity (4) by setting up a counterflow of velocity -D, . Vo £(8 ®t) Saad The symbol VV means gradient with respect to angular coordinates only. De is the rotational diffusion tensor, which is also diagonal mn for rods in the same type of coordinate system. The total angular flow is then ~< ol (6.9.0) (T(9.¢,t)) RO xe - DR VU, owed) ~ and . > g = (2 st) ot) ew » div (total angular flow) In steady states of interest here, {(° ,»,t) is time independent and the total flow vanishes(4). Therefore, (7(9.0) -B x, = DQ + V bntls .) (9) ~~ 41 From a generalization of a result due to Einstein (5) Dp ®kT rR”! so that 7(8 ») xe, = kT Vo fin (5 ,¢) (9') IV. Intrinsic Distribution Functions and Sedimentation Coefficients Using the ideas just developed on brownian motion, four types of angular diatributions in a dme-independent external field for a single rod will be developed. The last three are new. The four cases will be: A. Stokes flow in an infinite medium B. Oseen flow in an infinite medium C. Stokes flow in the presence of a wall D. Stokes flow in a radial centrifugal Meld. IVa. Stokes Flow in an Infinite Medium According to the method of Kirkwood and Riseman (6), a rod- like molecule is represented as far as hydrodynamic interactions are concerned by a system of dirmensionless hydrodynamic elements, being 2n + 1 in number, with 0 in the center, lying all on the same straight line a distance b apart. This model is so constructed that 42 its length and the actual physical length of the molecule are the same. Each hydrodynamic element is assumed to possess the same charac- teristic friction constant ¢. 3b oe 202 The force which element i exerts upon the fluid is ~e ¥ =Cw (10) i where w, is the velocity of the element with respect to the fluid velocity which would exist at the location of element i if i were not present. Itis convenient to write ~~: ~— Cad Wry ty (12) where % is the velocity of the Pha element with respect to the lab, and a is the velocity which the fluid would possess with respect to the lab if element i were not present. If u, arises solely trom velocities i induced by the other elements of the chain B® > 30°F; (12) if j Then w sy - ) ‘FF 43 and FeeCweet vy -6 > 2° F; (13) if j For an isolated particle euch as this ina space and tlme constant external field, the only source of a torque is a hydrodynamic one induced internally. However, since Aj does not have an antisym- metric part, it follows there is no such torque. This means that the angular distribution function ia uniform and that there is no tendency toward rotation irrespective of the orientation of the molecule. | Therefore, V,2v.27 ij and Foetv-c) Ti" F; (14) if j It is now advantageous to proceed according to Saito (7). Ix stead of viewing the action from some external, fixed coordinate system, a position on the molecule itself is assumed, defining the s' axis along the long axis of the rod, and the others mutually perpendicular to it and to each other. Figure 1 The matrix of this transformation between laboratory and rod-fixed coordinates is cos? cos@ easing sin?’ cose an cos? sing cose sin@ sing «sind ) . cos? Then CR alt i on tR S$ Me YR —| o) u~ DW oy i] wn f " Qt al ta et, or aly Uu~ G4 \ ~ eo ~ 9 © oO (15) 45 where \* Sip Equations 16, along with the further condition 0 oO oe | =) @, (17) suffice to determine the individual Fe F is the force magnitude per unit mass corrected for buoyancy, M is the molecular weight, and N | is Avogadro's number. The sum in equation 16 is now approximated by an integral. Here a * i/n, t@ j/n. oo ww Ome oO a i mn . Fs) © ¢v' +a | | - f EQ at - | t~6f with the condition on the kernel 1 l tes! gn zero otherwise For convenience, two new functions are defined 46 FX)-6 a¢ . Wy vie (18a) Fqy-3 = coy o ry (18b) Fy-S 8 +My o 3, (18¢) The integral equation becomes @) (,) 1 100) (sey eM ey) = (ad af (oro (6 (y 7 - (19) t Bet 4) (a) i) 7 foo zl belay ° 1 fn | get! The vector components here do not mix, as opposed to equation 14, The solutions for ¢ (My) and $ (2), 3) were given by Kirkwood and Riseman (6) oo Ys) = { }+ 2h ) Sc ark coorks | + 2A (Ly 2» -)} ot ke }-2) als (20a) Ys) 2 { CE Sc awk coonks | [kaw It o ye | AN CTE) J(29%) where 47 Sx (x) = i a dy ° te) Cc (ye | SEL ay a ¥ from the definitions of the ¢'s, it follows that F(s) = a ECs) PS) oo ° -/ -- =, = . ) ol . = 4 x ° y'¢s) O a-ak Vv (21) ° 0 yes The force on the particle is taken in the minus s direction, 60 that from equation 17 0 1 0) 0 0 JIM 6 enca”- f 0 64) 0 desa-¥ N ~ (2) 1 «1 rt) tt) ¢°"(s) (1) Qa 0 0 (2) “ 2 n¢ a” ° 0 Day 6 "av 0 aay where x J ¢(3) de = 2(x) “xX Multiplying both sides on the right by 48 aay 9 0 “iy “1 ~ 0 2 Very : ~ 0 0 “Ty, M1) it follows that y ; fo) ay) 0 0 0 -t Ve iM ve 0) fal] 0 >* at ¢9 4 n€N | 0 0 (2a -2 } [J -1 (22) The quantity (2) 2 n¢ om) *r Sr ( On ) (29) is the mobility tensor. ~ (1) um 2 ef um 1 + 4. (fn 2n ~ 1) a 2 mo 2 21) 0 1+ 2. (fn 2n - I) From the relationship follows 49 When the particle is aligned parallel to its translational motion, the 22 component of this tensor is twice as large as that for a particle aligned perpendicular to its motion. This is in agreement with results obtained by previous methods (8). The force which an individual element exerts on the fluid in terms of laboratory coordinates is obtained by substituting equation 22 into equation 21. The result is i> oO ta a) pars) yrs) Els ) = ara a - Dey 6 -/ 24) It ia instructive to note that this function is even in s F(s) © F(-s) This is quite to be expected, since there is no hydrodynamic external torque on the particle. IVb. Rods under Oseen Flow in Infinite Medium The Navier-Stokes equations are ~~ (5 +¥-9%)s-a pend v-eve20 (24) or 50 — For steady-state phenomena of interest here, e 20. In Section , the terms ve vv were completely neglected. This linearized the equations, for which solutions had long since been worked out for a sphere. From the sphere analysis it was possible to deduce the Oseen tensor. Itis possible, however, to proceed somewhat differently. It is now convenient to consider the sphere at rest with respect to lab- oratory coordinates, and allow fluid infinitely far away to be moving with velocity ue. (9). Then, instead of neglecting completely the inertial terms v- vv, these are approximated by U = ve Thus, for steady-state flow, the Navier-Stokes equations become pus: Vecgpenyey (25) Vev29 These equations are called the linearized Oseen equations. Cseen gave the solutions for spheres (3). A more or less detailed analysis can be found in Lamb's Hydrodynamics (9). Define ¢o = 3 - The approximate solutions for a sphere, given in Lamb, are orm oY crt y? 40? Ba. sewacel (oh + ze eee” — Hope) _er(1-@o?) 4 tae *y sv OY” £ 51 3uva | fl , wwe + 2x2) -7v- oe) gn 2 Uz= 4 (2 Y ort JE ~ S220? 45; (1-3 teste) (26c) It is readily verified that as a~ 0, these solutions coincide with Stokes solutions. Upon neglecting terms in ate? and expanding the exponentials to first order ino, the resultis va. cows oc, gue, ; Coe O x a Uy = "GZ Ln 6 Sow { ye? vo | (278) l 2 3Uoa} — , cos'S so o~ Ua= “| [++ yO TO Hy Coed +7 Cone (27¢) The Oseen tensor constructed from this is +6 6 ~ ror 1 a Be Tr 8r7 2 T = t oo w ow Oo vo Oo + "| AT tet ss Yb n ) m = (28) amt ZS coee & Se mk It is easily seen that this tensor is to order o° the same as the pure Stokes Oseen tensor. To determine the motion of a molecule in an infinite medium to this approximation, itis necessary to write an equation analogous to equation 13 52 Seed ~~ . x . + l Foecy-C) Ty°F, (13) iF j The quantity Ys ‘can be written viz vt ibe xe. (29) where v is the velocity of the center of mass and & is the angular velocity of rotation. The latter term, which was not present in the pure Stokes flow, considered in the last section, arises because there is an antisymmetric part to TS; here which gives rise to aninternally aon" induced hydrodynamic torque in an isolated molecule for certain orientations. From the symmetry of the situation, it follows that ® A can possess only an e., component. According to equation 8 o=t(s)- Rr (8) The main interest here, however, lies not in how a single molecule behaves instantaneously but in how a particle behaves on the average. Assuming ergodicity this is equivalent to considering how an ensemble of hydrodynamically noninteracting particles behave . In this case the motions on brownian motion apply, and G(r) -@7) xd oD, + ven ea) (9) R and equation 239 becomes 53 - - er Dan {(*) vi tvtibb, Gin fl) 2v+ibD se (30) Itis important to realize that a diffusion back flow has been added here to satisfy the requirements of brownian motion. This flow exactly cancels the flow induced by the hydrodynamic torque. Hence on the average a rnolecule does not rotate in any orientation. Combining equations 30 and 13 gives ~- ~ ” + 9 £ Brcv-¢) T° +¢¢pv° danse) (31) i¥j | 0 n ZM | 4 =) ¥F (17) N ‘ i «i izen and n A . ? Aa (ex) ibF.)xe =#kT vin £(@ ) (32) ie-n suffice to determine the F's and f(@). Equation 32 {s merely a state- ment of equation 9'. It is convenient to separate out the symmetric and antisym- metric parts of equation 31. P+? 220v-¢ ) rr Feet ‘Fd (33a) i jo wedi-j ¢ -j ifj 54 ~ ~~ 86 re) fin f(A \ ~ +. - = - ° - .” . b Fo-F_,=2i¢bp cto th) e.-¢) (3° F- T_y Fj] (338) if j The Oseen Cseen tensors are here 1+é@,@, tos (: ° 2) I 3¢ _ — (e3 0 Cowo U1 (c-5) Te pay bel Pry z a + Pm 2 Se" Ce -) 4. x An es — Lo A A nA “ (34a) tot Fowl C€ Sqe (0-3) mm? (22 2 +% &) Sqm ley) Tey - 26% 2 ti gfgo o\ 1 ~~ Pumblej) Pay 7154 2)7 30 Pry yp woe LS5n c.'-j) (34b) o Oo _ <a Tr AA _: _fL o/g 2 AN oe ea COV Er Be SCENE oT Ee. + 80.) Sqm (e-j) where sgn x ® tl ifx> 0, and sgnx 2 -lifx< 0. The signum func- tion comes in because the actual angles of interest in these tensors are those of the vectore t - je and 3 -i, respectively. The angles 9 and » which appear in the formulae are those which the molecule makes with external axes. When i - j or } -i change direction, the direction cosines of the angles of interest change sign, but © and ¢ remain the same. To take this into account requires the factor sgn({ - j) for terms which change sign on inversion of the molecular long axis. Combining equations 34a and 34b gives { _ f A Tee =TS - Fm 30 cove TL Seles) a, Tcws ee Syn Cé-5) (35) 55 ‘The distribution function f(®) will now be computed. Equation 33b becomes, upon combination with equation 35, (eo) A a étj , > > Define (F, - 9 74;- Equation 36 then becomes to order Fi) So dn @, = 22 typ? 75 oo are Teo é 4) ¢ (37) -—< { sino @, +3 cows 25 |: ) _F, Sam Lé-j) 81} et; The quantity a) F j sgn(i - j) can be evaluated properly to order o we by substituting the functions worked out in section IVA, equation 24. x x Cc. ) Foy syn bens) 2 nf Beds - nf{ Fads = [Bars ,, x a o (38) = $M )2"a) 4.3 ao nP) [3 N 29) Ye Ast 2 Ma) a, ead Allowing @, to vary continuously as @(x) rather than discretely, substituting 38 into 37, and approximating the sum by an integral leads to '@w), dy Ux-9h yl Ox) ex2n0bD 4+ _£M o$ pase OK) ~ 3 on cin 6 COU 8 gry (new 2G) (39) 56 This equation is of the form Oy) | <= (x) = Lux) - \ {4 gi ~ 7% (40) @ ), Tx-91 [x-9J for which a general solution was given by Kirkwood and Riseman (6). Co He. ¢ tex bx) f, (fy He tl ¢ (41) x} = ~ . lel 1+2A (An2n-1) TL /-2) Gi (= ) +0 Where { _¢ yrfe et zf fix) e K dk (42a) ke He C1) a" A Se 21the (42b) fl Explicitly 2 bak Fue (aed _— atx) i =X 2nb ¢ Dp? + FTN 2.2%) “3 AD W Sin 8-cod 9 (43) Since £(x) is an odd function of x here, is zero. Thus the solution formally reduces to - fe @ (x) = 7. }-2A _ (44) deco $0 Now 57 “xy 2. € y* tthe sum rth (45a) OG) 44 ! Le zmk sew 7 andinsand (eet) SE eb et , kori 2" (x) = 30 that f, = + coe Fm oS {gp 28 cok Se ark ) Pry N 2. x 2k Gi)" Seam sem rhe (455) [eA (Rn2n-i) T* > [-2A cz (He) knit =iCE 9,4 Cpe? dda fla) 4 rr) Tk qrk we k™ |-4) Ce: (TS) (46) 3( 1k Te ke =z) The second terms in the fo coming from the k' sums in the Q(x)'s will be neglected, since they are considerably smaller than the first terms. This follows, aince for small k, Ci(tk/n) goes like ftn(y +k/n) (6), where y is Euler's constant, and thus is about -2.5 for n about 50; for small k then (1 - 2\.Ci(mk/n)) is larger than one. If k is large, Ci(7 k/n) goes like (sina k/n)/(*k/n), but then the extra 58 factor of 1/7k in the second terms reduces their size significantly. $i(27 k) is about 1.5 for any value of k allowed. In addition, A’ < 1. With this simplification, equation 46 becomes Ing coos (47) ¢, 4S a ( ee 0 2bek) _2¥Me$ 87HN and ‘ .. fe Ff C. Li) Su rhe X. OG) =4 nb GD °° bY - Ol sino cove ) >. The [ -2AG (48) k=; “a ~ ~ This gives the e, component of or -F , toorderso. There is also ~i —~ ~ an e. component of a -F i? but it gives no contribution to the torque. The quantity n n Vcbe= ) ch(R-Fe) (49) c=-n <=e must now be computed in order to use equation 32 to determine £(9). Only the Om part just calculated is of interest here. This gives { nvb fx Oc) d y = 4nrb b p®? ahntto) - gaos coe | (50) n An 4 26 | 9179 NV Sun G 3 na 7m Dra or, using the function defined by Kirkwood and Riseman (¢), 6 ot —~ fr (51) 59 equation 50 becomes nb {x@cedx Go- FFG rtb(nbSD° Unit " B1QN Sem Oo29)% (52) and 32 gives ZE(A)n*b nb Cp Odnf (oe) _ FM oS sino cove) = £T Date) 08 sT nN (53 Solving equation 53 for gén {(9 gives oe abS (a) ame SF(B) ah /2 3120" 90 } (54) ae [27 N (3 Fea nb SD - 7} sino cow 6 Assuming the value of D which was given by Kirkwood and Riseman (6) De =F OER es leads to 2 nF = Stee rb some cove (56) which gives upon integration Fo -Hege(- MOSER eoutg) 7 60 where Kis a normalizing factor. Now z » 2% where s is the sedimentation coefficient. Also Pa nb= > (59) where £ is the length of the molecule. Combining equations 58 and 59 with 57 gives - — Z*Mspn CL F(A) 20 fe=K exp(- 48? NRT cow } (60) In the centrifuge F=a%r (1-7 Pp) (61) where r is the distance from the axis of rotation to the point of interest and 60 becomes Fee ene at IrmeR SEE coure) tn This function peaks at 6= 1/2, so there is a preferential orientation perpendicular to the field. For TMV, the parameters are (10) 61 3 "y = 30,000 rpm p =1lgm/cm 2 vi - - 3 » r 26x10 cm/sec vy £0.74 cm" /gm M =z=4x 10! grm/mole £ 24x 107? cm (63) - «1 q = 10 gm/cm sec kT =3x 10 torg 6 = 1.85x 107)" sec N =6x10°9 Let d be the diameter of the rod. Assuming n = {/d, and = 6nNd/2, and taking note that 0< F(A /n)< 1, the argument of the exponential is less than 2x 107°. Using this number, the distribution function is approximately £ =F (1- 2x10° (coe*O -3) ) (64) From this no observable orientation is expected to occur as a result of this mechanism. To find the velocity of sedimentation, equation 33a must be considered. > > > > Fit Fa, =2¢%-¢) [WF + Tinj FJ] (33a) éti This is conveniently rewritten Fi +F.ii =2 CF -C) Ty -(F; +F.;) +f) [Te -Ti-j] °F; (65) . ij ixy The last term on the right side has already been evaluated to order o and found te be odd ini. Since the function F, +F i is necessarily 62 even ini, the odd part of £ ) Ti CF, + Fo ; must exactly cancel . “adj OAT > Dy -j Hence the interesting part of 33; is the even ij J J J A A 3 0 0 Tes (even) = H+eeG lo £/33 6 (66) ~ Saynble-j) 8mm 2% \ oo 2 After removing the cancelling parts, equation 65 becomes Nt eee, = = (a 2¢t- ») ae Tear (Fy + Fs) cH " 3.0 0 omy +(: 2 :) DG -8) (67) j -¢(et-3M22,)-) ) A. (F+F,) - ayy le-J! ° ° CRS This, however, has precisely the same form as equation 14, with ~. Fi Me _ ein e. playing the role of V. Consequently, from 2 2 { { ° >» £Mo 2 ae ( arr e, (a ~ )- ° u 9m7N Cea 2a rey (ay am) _| (68) or st tM LL an A t _ l f- ° + nok g = netN am t Cy Cy (aaa ay =! gry (69) The average sedimentation velccity is 63 <¥) -/ f (e) F sin ode (70) If a very long molecule is assumed such that al 4 oQeay, the result of this integration is at eM (4. BOOTHS _ 7) (t D> = Q% non (4 81 » 2x10 (71) From equation 23 it is clear that the quantity n¢ ayy is to order 0” just aM (> 2) ) since the average of the tensorial terms in the brackets ie 4/3 ee, and the mobility tensor is directly related to the sedimentation coefficient by the formula M _ S = CM ez) 4 (I- Vf) (72) The average velocity then becomes PK { FM 4 _ —7 = - Say At ($0 ) (rs) If there were no effect irom the Oseen correction, the average velocity would be CH= - 55H ara y (F ) (74) It is then obvious that the order of magnitude of this effect is quite unobservable experimentally, and is of only academic interest, since 64 -5 the sedimentation coefficient decreases by only about 10 percent. IVC. Stokes Flow in a Medium Bounded by a Wall The components of fluid velocity for a point force operating on a fluid in an infinite medium were stated in Section II. These components satisfied the equations 2=a AV uv =-Vp VU =O If now an infinite plane wall is inserted in the medium, the flow is of course altered since the fluid velocity must vanish at the wall. This modified flow was computed by Lorentz (11) by noting that if the quantities vi), V O), V C), and pi) satiafy the hydrodynamic equations, then so do the quantities ave (2 ge 22 i) vy = -u, -2(z+€) Sax Ft (z+) v*v, (75a) «) ov _ (i ne -5 2 ‘ uy? =-v,0-2(2+ §) dy + (z+) vey’ (750) (1) , av 2 Ue =u” -2(2+ 8) Soe + (2+E)” v7 ug” (75¢) 65 7) org’ p? = pl +2 (a+b) $2 - 47 Se (754) These formulae 75 are general for the coordinates x, y, or 8, so that in place of ver either Vv OF y could be substituted. The usefulness of the formulae is shown as follows: z _ (0,90) a 4 4 (image) Consider the point force above the plane to be situated at the origin of a right-handed cartesian coordinate system. The z direction points upward perpendicular to the plane, whichis situated at 2 = -% and extenda infinitely in the x and y directions. If the plane were absent, the solutions for velocity components would be just those for pure Stokes flow U= F XZ f z 2b ies (2c) Va ~ gr7 (rrA 66 Upon addition of the plane, the velocity components must vanish atz = -§. The component v can clearly be made to vanish by adding an image point force on the other side of the plane, in the same manner as electrostatic potentials for point charges in front of infinite conducting planes are constructed. The an and ‘y components, however, are seen to add rather than cancel, but if the components are transformed according to equations 37, the boundary conditions ats a -£ will be satisfied upon the addition of the thus transformed point force velocities. The total flow for this situation is ec (xz XZ _ 6x §(#+5) (2428) Va gry\ rr? 7? Fe (76a) _YZ 6yE(z +E) (2428) YN am rr? Fe Be E)* ( Vz = (lr zy- [e+ = 2. ]+ 25+ +E 28") (76c) (76b) > aL When Z=-§, Ux = Vy = Uz =O - The Oseen tensor is yz ay 67 1 oy ty wt ot at _ {3K rd ae: 5-35 2 (e+) (5; = f fa -% 4 exySlee $) } xy Fim r? r3 ye rT s-! {x2 _ xe - bxSlerasxars) | xe ?™ y3 73 y~ T eT yz xy T - ft ¢si-L-a' 2 g(a) (25-40 ‘)} (77) YY gry CV rs oY ¥? Y y* ai f92 92 _ yf C2 22 J(ees) | Fm & 3 7? ¥> = 5, [48 -se 4 oxS(eresfert) ary y3 iad ys L_{ae 3% 463 S(ero fLer$) | muy yr? ¥3 sso 7 \ ty,at tl (tg) 2 g(ars)( di, - 30220 )f Now conaider a long rod in front of the plane. 68 To compute the distribution function in this case, equation 31 is again employed 66 alm, (6) @ = SF - Ti: F, + SibD ee (32) which leads to equation 33b oe = 2i pp ta, - > [Ty-F- Ej -F,] (33) It is convenient to expand the Oseen tensor to first order in b/€ =a. Each force element as shown in the figure is treated asa separate entity. Immediate relationships are « ® ue pe 8 (¢-j) b cos (78) E( [+ Xf cos6) 25 [I4e(4)) coabe (i4j%2i) soe2 0)” * m2 (ln & (iti)e008) a] # Ss a ij With these relationships, itis possible to develop the Oseen tensor to first order ina. 69 yi ae (+ 453 4 3X Ce +5) coz ) 3™ sé y3 4€ yoo gt (4) xy 3m y? ij 1 x2 3x Soy ee t im (% - (c-j) Sino conf) xz 7 § ij ft y* 3 4 3a C +i) cove ) Tt Fy + 4. ity yy an ye +5) fF (79) rs y% - r3 re eA xe + 25 ne (ei) az sem ax P17 g& rg 1 (24.4 34 (¢-j) sro sim 4) ay oy v3 8 § ij a fly e* -2% 2. Bt ery avs) Ta. «= (y 7 26 4¢ These lead to the relations rid 2 pt 1 3H (045) core wi-j ij Py . Ty wi-j ij 1 34 /¢- 6 TC" z — 4¢ (63) Sean 8 cow “5 gf 3&7; 45) cove vy lyy tm OF (2 +5) (80) 80 yz ye 9m 4§ “i-j ij oS) Ba (fj) vim com Tox * TO g 4F (¢-J) rig gil 1 3x (¢-j) peo seh By ay om 4¢ Tet Te -— 2 (étj) come or / s (t tj) cove o - (6-j) Sem 8 Lord tw -Tii- rth O (6453) C008 -(C~}) sem sim 2(81) (é-j )sinotesp (-j)sno sing Llc+j) tore Equation 33b now becomes ante) > e+ j (82) +2) sed (Cej)Cove -(S-j) Smo xd 5 Co -j) F~60924 (ij Simo gen 2(c+j) Cove 71 This equation is treated in the same way as the expressions in section IVB. F.-F:: ) *@6 = OW Yh X= ‘ h arpnb ey (1 +é re, ) y= Ip ¥F . are functions worked out in IVA, and given in equation 24. Thus :y @(x) = x(2nb5D% Saf (e)_ oat sun 8 cove ) J | x-¥] Ix-y/ The solution to this equation has essentially been worked in (83) section IVB. Itis only necessary to adapt equation 48 to the present purposes. The proper resultis clearly f(@) %wFME Sin TR Ow) = = 4(nSbD°? 8 2 - Sar stat) ts 2AG(es)) (84) Procedures exactly analogous to those previous give Z e0atnf(8)_ Qu FME s; = pr Unfia) (85) Fan re-b(nbS'D oa - garg 8") Ar date and _ ZIM SF(AL™ £ (@)=Kegp( HASTE, coa*8) 72 JIMTE DL Using TMV dat, assuming §3 lemandd*7.5x 107! cm, this ~7 exponential argument is less than 3x10 . The distribution function ia approximately fees (1-3x16" (c0<?6 -3)) (87) No observable orientation is expected. To determine the sedimentation velocity, the analog of equation 33a gives, after cancelling odd parts and keeping terms to order 1/£ ie (H+ é Grr): \-(Fe +Fy) _ 35FM & 2 Using the functions of equation 24 to evaluate the last term on the right, the result is | 0 3ns 4 y= 1 ° + e - A ie Lista t® er ar 9 By ™)| (2) 32775 = | (89) Again assuming a very long molecule such that (2), 1) = 20 Va), the average sedimentation velocity is wT a Te oP ~ (4_3n5Q.w 8 (F=f (0) F sin. 0 dO =~ omy 3 3279F ~ 35x10 ) = Fa) (4- 10 *) (90) 73 This change in sedi mentation velocity is also unobservable experi- mentally, and establishes that wall effects are negligible for these molecules if they are far enough from a wall that their length relative to the distance to the wall is very small. It may be of some academic interest, however, to note that there is a very slight tendency toward orientation parallel to the wall toward which the particles are sedi- menting, and the net effect on the average sedimentation velocity is to decrease it by about 10°? percent. IVD. Particle in a Radial Field A radial field is produced in a rotating system such as the ultracentrifuge. The more distant a point is from the axis of rotation, the greater the centrifugal force. In the case of a rigid rod-like particle, it may be suspected that a torque may be induced by the external field, since some parts of the particle for certain orientations are farther from the axis of rotation than others. It can be shown that there may indeed exist a torque due to the centrifugal fleld for particles which do not He in a plane perpendicular to the axis of rotation. The external torque due to the centrifugal Meld will be computed asa function of particle orientation with respect to the axis of rotation. Once this is obtained, equation 93' TZ (o,$) X ey 2kT Va L000, 4) (9') will be used to determine the distribution function. 74 A Define an axis of rotation in the e, direction, and place a rod-like molecule some distance away. Define the a axis to pass through the middle of the molecule. v \A ey The centrifugal force acting on elementiis 25. = wR, m(I-VvfP) where m = mass of each individual element. The force on -iis The torque on the particle is clearly on : . OA a4 — T =mbw* ) ie, x Ri (I-Vp) (93) i=-n This can be simplified somewhat. From the figure it is clear that —_ 2bié, + Ri -2B-R: =0 (92) and p=bi (é,- éy) ey =D; sum OSing Ey (93) 75 Thus Re-R: =2bié, - 2B = 2bi (é -SInO Sing Ey) (94) The torque can now be written t(g ¢) = uth). i, x(R-Ri) (I-Vp) =- Tes* sind sing (G xéy)(I-Vp) 230 n where] =2 Mb ) 4a - ve) = moment of inertia of the rod corrected i=Q for buoyancy. Now e = ; rs) SA a . “a AN 4 = Sin ® Sindh Cy + CowO Sing Eg + Cord &g ; A A A A “A so £ (6,4) =-Tutsin 0 sim (Cot Sin poy — cosh by) (95) and T (6,4) x Ep = -1 Ww Sind sind ( cow sing Eo + cod dh 24) Then equation 8 leads to 2a ie é, + ottae by = LW" (SinB cord Sin’ PE, t5inb snp coef Ey) The * part gives Iw 2 fn (5) = 4 rT (cos”* aint 4+ A(c) tank The ¢ part gives Im Z dn «(¢) 23 a (sin2o cos*@ + B(@)) + én Kk 76 Equating these leads immediately to A(c) = cos“ B(@) = cos’e Therefore (6,4)= Kerp | Zar Se? Osist4] 09 Hence there is a slight tendency to align parallel to the fleld. To first order in tw" » thia becomes kT £(04) =K (1- LE sitOse*6) If ain-y is replaced Te its average value &, the result is fle) =K(I- 22 sine) The normalization condition gives to order Iw 21KT - + 2 f(e)= tf1+2 ce 3 cos*O- I) (97) Saito and Hasegawa began work on this problem after they found out that the present investigation had already gotten under way. They report a similar distribution function for the same conditions (12). _ ; f (9) = Lfi+Z aor * (ScesO- -|) They have given no details of calculation. The origin of the discrep- ancy of the factor # 1s unknown at present. It might be mentioned that the term in < in equation 96 is 77 exactly half of what is obtained if one assumes the field to be cartesian rather than radial. The moment of inertia corrected for buoyancy for a rigid rod mL “(1 -~ vp) 12 to the distribution function is is » Hence ior TMV itis 1.6 x 10747, and the correction £ (0) = 4 ( 1+4.5 Xlo ? (30% -1)) (98) This is too small to see experimentally. Vv. . Concentration Dependent Phenomena The motion of a particle through a viscous medium disturbs the fluid at points removed from it. In certain cases itis possible to calculate approximately the magnitude of this disturbance, as was done, for example, by Stokes for the case of spheres. This will lead to a concentration dependence as follows: Consider a particle at the origin of a coordinate system. Allow an external force F to act on the particle such that it moves with respect to the medium with a velocity VeM-r where M is the mobility tensor, or the reciprocal of the friction tensor. This quantity has tensorial character because the velocity of motion in the case of asymmetric particles is not necessarily in the direction of applied external force (See section IVA). 78 If the medium itself is moving with respect to an external observer, the motion of the particles will appear to change, or alter- natively the mobility tensor will appear to be different. The medium, however, is indeed in motion if there are other particles present which are perturbing the fluid velocity. The degree of this perturbation will depend on how many particles are acting on the fluid and on their posi- tions, or, in short, on the concentration of particles. In the case of asymmetric particles, the orientation of a particle has a great deal to do with the velocity with which it moves through the medium. Very long rods, oriented parallel to the applied field, move twice as fast as those aligned perpendicular to the field, as was shown in SectionIVA. Therefore, if there exists a preferential alignment of particles, the net average sedimentation coefficient will change over that for a syaterm of randomly oriented particles. A nonuniform align- ment with respect to the field will occur if the perturbed velocity at the site of a long rod includes a shear, for then a hydrodynamic torque will be exerted on the particle, the magnitude of which will be a function of the orientation of the rod. As explained in Section I, brownian motion will counterbalance this in such a way as to induce a nonuniform angular distribution function. The degree of shear, if it exists, will be a function of the concentration of particles and the magnitude of the external force acting on thern; in turn, the angular distribution function will depend on the same parametere. 79 In this connection itis of interest to investigate the effect, ata single point, of the velocity perturbations due to a distribution of spheres in a medium, all of the same size and moving in the « direction with equal speeds. This perturbation is given by —~ > — T(F)= 2 T (GA) FF) tl where T(r, cA ) is an appropriate Oseen tensor, and the sum is over all spheres, or point forces. In the case of an infinite medium, representing the spheres by point forces gives A A ~~ A + On erp é = — " 'z u(F) = gr7 7] (99) ri If the distribution of spheres is uniform througheut all space, and the sum replaced by an integral nt e » o> y OA “A F Terr err. e, d (vol) =poés v(F)= any ) [PoP] (100) This infinite result is not obtained physically. The model above which gives the infinite result must therefore be faulty. The mathematical origin of the infinity is the assumed additivity of effects from different force sources, and since these are all positive in the z direction and die off like l/r, the integral diverges like r’. The reason this divergence does not occur physically is the inevitable presence of a ‘return flow'' or "back flow.’ These 80 terminologies mean that on the fluid velocity given by infinite medium Stokes solutions must be superposed additional velocities of opposite egigns in order that the total fluid velocity vanishes on the likewise inevitable walls of the containing veasel, or on some other convenient surface. Of course, the walls of the vessel are not the only inevitable objects; there are also many other spheres surrounding any given one, and it is not unreasonable to expect these surrounding spheres to shield relatively distant regions of the medium from the effects of a given sphere, due to their finite rigid volume. Thus, sufficiently far away from any given sphere one would anticipate the medium to be ‘completely and totaly unaware of the actions of this given sphere. The velocities induced by it are reflected’ off its neighbors, and this gives rise to a "back flow."’ This latter effect may be approximated by inserting a hypothetical sphere of some convenient radius L around any sphere of interest, with the fluid velocity arising from the perturb- ing influence of the sphere made to vanish on the concentric hypo- thetical sphere and assigned the value zero outside the hypothetical sphere. It will be shown that either mechanitam tor back flow gives essentially equivalent results. Therefore, there are two effects which tend to counterbalance each other in the perturbed fluid velocities. Gne is the tendency of a sphere to drag the fluid along with it, pushiag forward the spheres ahead and pulling along those spheres behind. This effect is the infinite 8) medium Stokes part and leads to the divergence discussed. The other effect is the reflected flow, either from the wall or neighboring particles, which cancels the divergence. Kxperimentally, there is a slight excess of back flow, which fact is attested by a decrease of sedimentation coefficients for spheres with concentration. The results of several studies done on glass beads hover about Vege (1-50) (101) where ¢ is the volume fraction of spheres (14). For spheres sedimenting in a vessel, Burgers has made a theoretical study of the expected sedimentation velocity. The analysis 4a rather lengthy, so only the results will be given here. The predicted velocity is F 55 | An amount 15/8 ¢ of the correction factor comes from higher inter actions between spheres, such as induced rotations and double velocity reflections. Omitting this leads to vess (1 - 5@) (103) This term 5q@ in addition contains consideration of the finlte size of the spheres and the average perturbation velocity on the surface of a test sphere in ita absence, rather than the perturbation at the center. If, using Burgers approach, the average perturbation at the canter of 82 a test sphere due to the action of all other spheres, approximated by point forces, is calculated assuming a uniform distribution of the point forces about the test sphere but taking account of the excluded volume, the result is ge V "tate (1 ~- 6) (104) Thia value differs from experiment by only about 20% for small ¢. Ths primary interest in this work is in rigid rods, not spheres. The general scheme for calculation is to treat all particles having hydrodynamic interactions as point forces, but to take account of excluded volumes. The epace and orientation averaging processes for rods are considerably more difficult than for spheres, and the approximations used to do this do not warrant any more accurate treatment. VA. Spheres and the Wall It will now be dernonstrated how the value gr arises for a sphere moving perpendicularly to a plane wall of infinite extent. Equations 7? give the Oseen tensor for this case. 83 Consider a point force -F(s) e acting at r as shown in the diagram. For generality itis considered to be a function of z. The velocity at (0,0, 0) due to this point force is V(0,0,0) = -T(0,r) -F(e) © (105) The location of the force is now averaged over all possible positions external to the volume excluded by the test sphere at the origin, and the average velocity perturbation at (0,0,0) thus computed. The dis« tribution of forces is assumed uniform, and ranges in geometrical extent from f= V¥ x2 +4 = ZQ—y» 0 =— F->f where ais the radius of a sphere, andh is arbitrary. Clearly, the on and e, components vanish on the average. The average velocity at (0,0,0) is then am Pe) h F=-cfad| pap dz Taz Fz 6, o “2a AF 2% peo rh A 20 2a =-¢ Jad [edp[ de Taz FRE +¢ fp fanaa r las Fz)éz (106) 84 From equation 77 Toe = gag [Ee ERE — artes) (e- ED) 847 r (107) --- 1 2 pa -f+r— es(e+t)) 8nn f ap @ r It {s convenient to convert the first integral in equation 106 into a surface integral os rap} Jz Tee Fle) = 9 [ae [rg (-r+F- 2i(e+8))) = : The second integral in equation 106 is, neglecting terms of (108) order a/*, V (0) = avr 7 22. _ clap Gin0de de Tae woe (EP) (z% +7 co#8) (109) ° o oO 4.Q2CF 32 2 where F = = (Z, + rcos® (1 - vp) is the force due toa centrifugal field on the perturbing particle. Note that the radial field has been approximated by a cartesian one, but this should sufficlently approxtmate a centrifuge cell so as to lead to correct results. 30) is a net fluid velocity opposite in direction to the motion of the test particle, and 85 will thus tend to slow it down. The net velocity is then VC. Spheres and the Concentric Hypothetical Sphere Cunningham (15) worked out the solutions to this problem many years ago, using the boundary conditions O=v* ‘y sv atre L, v @y eQatrera, andv = -vatr 2a, x y z & Define e = a/y (11a) 2 j-e7 a = oe (1ilb) 5 (\-€*) 2_ S . . 9-Se*-4e (1114) 3L(1-€5) 86 Then, if an external force acts on the small inner sphere in the minus z direction, the hydrodynamic velocities are (15) v3 — Stu Beowe conb(B% y*4 7 Lt 2) (112a) c— 22 ‘y Sim 8 coo 0 coup (—3 y* += ~2£-) —F— or (112b) v = (sv* 445 royar_t , #)\& (112c) +34 10 + (\-3coare)( F -s7 + 7) P17 ac If terms of order “jz are neglected, the Gseen tensor is then Yr T 6 con? h)/ XY — ex am oye Bh + (1-3? $)(2 -=)) po oeolw Sia 28 Seu $ cou (— By y+ +) xy vm T =—— sino cones cor h [— 3X y* +4) XZ aes) Tye” Txy 4 26 sim?) x) (113) T ai~( 4 F (1-328 Sem? yy Ir Yt by t (1-3 He ( pt et sing Core S~> - Bh yt) y% an” Ss Tax” Jxez Tay * Tye Th" sh (ene fo th 4(l- ~3eea>9)(XX" - — ) The analog of equation 109 is 87 F(0)=- oar | dé sing de [Tae (20+ Cow6) rdr ez =~ Cwm cet Nd [se [sas | a + root) rdr @z + a4 2n pw A 2 F Zz cwM [44 f .0d6/ Ten (2.410000) dre, =-20° 4 4acF N fs ee de ae ( 9% 3% This is the same result as was obtained for the effect of a wall with 2 the exception of the additional term - at - The velocity is 2 F 2 F F jl a a Fo Lael 4ael _ (i-z ¢) V= “672 9” 3% “6m7a VC. The Hydrodynamic Interactions of Rigid Rods Consider the following figure: 88 axis of rotation >a Q Force on 8 is N“ - wim (-se)f2,—r bocowe -KtowD +N be ‘cow o’§ N wall The velocity perturbations about the rod shown in box A due to the hydrodynamic elements of the rod in box B will be calculated. This will involve an average over regions outside the volume excluded by the particle in box A. To enable the mathematics to be carried out, the region excluded will be assumed to be a sphere of diameter Znb, where 2nb is the length of the rod. The results so obtained will be multiplied by the factor 3d/8nb, where d is the diameter of the rod, in order to attempt to scale down the value to correspond more closely 89 to that of the actual excluded volume, which is more like a disc of radius 2nb and thickness d. The quantity 3d/8nb is the ratio of disc volume to sphere volume. Just how drastic this approximation is will be teated by comparing the result to experimentally determined values for TMV. Call o' the fractional distance from the center of rod B toa point off center. The force exerted by nbo' on the NMuid is denoted Fle '). I(o,e') clearly depends on ", 9, “, ¢', @,%,0,9', andR. To get the effectata due to the entire particle in B, integrate over o' 7 . af T(5') Fle')de" 2 To get the hydrodynamic perturbation ato due to the particle B aver» aged over its possible orientations, integrate over “'and y', weighted with the proper angular distribution function f(* '). | _ Jaa! $0) [n fr 629-F le do") coh “ To get the average hydrodynamic perturbation ato dire to all the other particles like B, average the position of the center of mass of B over all allowable positions (assuming the radial distribution function to be uniform for simplicity) and multiply by the total number of particles. cfd (on | dn'f C nf Tee -F(e!) do! 90 It can be shown that a condition analogous to equation 109 ia also true here ! —_ Vie=c Jaca fast fo) nf t050') Fe) do’ (115) excluded ; Volume / =-c {d(vol) fds! fie) n Teo!) *F (6') do! —1 This integral is evaluated in Appendix A. It is interesting to consider the results when the Oseen tensor for the hypothetical sphere-surrounded hydrodynamic elements is used instead of the Oseen tensor for a rigid wall. The additional term over the excluded volume term, which was wa CF /9N for spheres (see equation 114)is _ arc (20-nbe cows) w*M(I-vP) 97N where a is the radius of a hydrodynamic element. This is worked out for this case, in Appendix B. This term is neglected since the excluded volume term is much larger. The quantity of interest then is the excluded volume integral in equation 115. The external force operating on particle B is 2M (I-VP - aa ~ *(z_-nbeenas-Reee @ +nba'eow 6’) (116) 91 The quantities Fe ') are approximated by the functions of equation 24. According to Appendix A, the value of this integral is V(e)= ap ate (z.- SET cose) vp) é Es (117) From this the average perturbation over the entire molecule is i 2 J Ve 3) do CW MddZo(/ VF) 4 2 2ntl (118) so that the velocity of the molecule on the average, in the absence of any preferred orientation, is ~r_ _ Mat Zoe) . Cui Md 20 (I-Vy) tome A where (Cy is the average friction constant of the entire molecule. It is possible to compare this with experiment. For example, the sedimentation coefficient for TMV at very low concentration varies with concentration like 0 + 4C/7) where C is given in weight percent TMV (16). The ratio of this correction term to 118 is 4500" Ze C /OOMc _42N = 0.94 7 NC cw*M<dz,(/-vP) (120) This agreement with experiment is extremely gratifying, and indicates that the approximation of scaling down the excluded volume possibly 92 possesses some validity. The torque on the particle A is given by _ 1a” T=5 er XN bla Veo)de (121) cw2tMtidn f(1-7y~) é, “SG cod “EO GN The factor of § comes from the fact that the field considered is cartesian rather than radial. See the comment at the end of Section IVd. Equation 3' now gives oat g 2txé, = cel Nan Ee Sin8 C0088, (122) f (= ' Cu*Mt dn S(I-ve) , 2 ) )= Ket 3209 NET cow @ (128) Assuming n = = » € = énNd/2, the argument of this exponential is of order 5 x 1078 for TMV. This quite strongly indicates there is no- hydrodynamic alignment which is measurable. The sign of the argu- ment shows 2 slight preference to align perpendicular to the field. Such an alignment would in any case cause the sedimentation coefficient to decrease, rather than increase, with an increase in field. It is thus doubtful, on the basis of this model, that the mechanism for speed dependence as observed by Hearst and Vinograd haa anything to do with hydrodynamic alignment, at least for TMV. 93 VI. Preliminary C.rnments on Liquid Crystals It Ls well known that TMY solutions in water will separate into two phases if the concentration becomes high enough. This phenomenon occurs at about 2% volume fraction of TMV (17). No such phase separa- tion is known for DNA, but these cornments may throw some light on the situation anyway. in the case of TMV at 2% concentration, the two phases are characterized by different concentrations and different degrees of alignment for the rod-like molecules. In the more concentrated phase the particles are seen to be oriented in a specific direction prefer -~ entially such that a fair degree of order is exhibited (hence the name quid crystal), while the more dilute phase displays long-range angular isotropy. Onsager was able to explain this on a statistical mechanical basis, assuming pure repulsive forces of the hard core type (18). Now itis clear that if a situation should occur wherein a large number of molecules should line up with other and parallel to the applied fleld via a liquid crystal mechanism, the change in mobility factors with direction of motion would cause an observed increase in sedimentation coefficient. However, in the systems studied by Hearst and Vinograd, concentrations of TMV on the order of half a milligram per milliliter, or about 0.5% were used. This is considerably below the concentration level required for the separation of phases; hence it could be argued that no significant order of the liquid-crystal type will 94 be said about this prospect shortly. Even in the absence of liquid cry stal formmtion there ia a pose sibility of induced order worth considering, notably the excluded volume effect (19). In dilute solutions, particles possessing asymmetry have a@ alight tendency to assume configurations such that the maximum allowable volume per particle is achieved. This is due to the increase of translational entropy with increase of available geometrical volume. In the case of long rigid reds such as TMV, the relative orientation of two rods which gives the maximum allowed volume for each is clearly a parallel one. Therefore, since (as was shown in Section IVD) there is a slight tendency for each individual rod to align parallel to the field, given by the distribution function Tu" 3.26 sim * $) (96) f (64) = Herp (- 2KT the excluded volume effect will enhance this alignment. To get the angular distribution function including the excluded volume effect, it is convenient to consider the definition of a one-body angular distribution function (20) f (8, >)= ail: fa ded ep |- x | (123) where N is the total number of particles in the system, V is the volume, 95 d'{1) is the volume element associated with the center of mass of particle 1, but not its angular coordinates, d({t) iz the volume and orientation element associated with particle i, U is the total potential energy, and mby is the configurational integral Q=[~ [dod epee] (2a The potential energy is / U= 2. Ui; + >. u Ss igj s (125) where u,. is the interaction energy between particles { and j, and ij ut is the potential energy due to an external field. From equation 96 a 1: 2 . Us = ta Sim @, Sin (126) In addition, the approximation of hard core interactions for the 4h are assumad, such that aT = 0 if particles i and j do not intersect (127a) 44, = © if particles i and j intersect (127) Following Mayer and Mayer (21), define fs wu, /kT e = 1+ fi (128) ao that equation 123 becomes 96 Eu f, (6,4) = ave [fa “W) d 2) de “hey Ft )02M) i<j By employing the techniques of Mayer's cluster integrals, it can be shown that this becomes (22, 23) f(a¢)=K > B c* (130) L200 where “ar fle’ (1) d(z)--.d (+1) [=af.;] [a e- “Ar J (131) irreducible and K is a normalizing constant. Then, to first order in the particle concentration, £6 . “Kv oe o- ad fe", dd) | (32) The integral on the right can be evaluated by keeping the angular variablee fixed and integrating first with respect to the spatial vari- ables of particle 2, since u, depends only on angular variables. 2 According to the hard core potentials, f.> = 0 if particles } and 2 do not intersect (i33a) {i> = ») if particles 1 and 2 intersect (1330) 97 This integral with respect to the coordinates of particle thus repre- sents the negative of the volume denied by particle 2 to particle lI, if their orientations are kept constant. This value, according to Onsager (18), is (2f*¢ ; ad") sin ys [ae [evo + 2E(simy) Ld? | (134) where £ is the length of a rod, d the diameter, y the angle made by the long axes of rods 1 and 2, and Z/ ok + ’ = . - 2 2 E (simy) =f (|-sin*y sin?p )*@ do The integration over the spatial coordinates of particle 1 just gives a factor of the total volume. Since £ »d, only the maximum term in equation 134 will be kept, which is 207a siny. Hence, equation 132 becomes 2 U2 ; f ep-Kve|iszck de ler siny dR, (135) From equation 96, Iw* _ 2 _% or a ae Sin @, Sin P Te (136) (3) ‘ 2 2 u,® 2 Siw” 02 Ln dp, Using the approximation Yer M wap. 2 137 e ~=j- hay £2 sun2Osi.7¢ (137) ZK 98 the integral is Tw. . {G- Ler Sin * 9, Sin * 2 ) Siw Y dQ, (138) 22 The first term is elementary, but the second is not a trivial integration. In Appendix G it ia shown to be { sintes Sin *>, Suny dn,- I (3- sme, sim? $,) (139) fhe Equation 132 is then £124) = Kv(t- FS Ste, sur d, +20*d 7? — (140) Tw? 3r*ttde 3 7rtttde Tw sin? B, sa, | kT 8 PLT In the future *y will hold no tnterest, so eines: is replaced by its average value #. Thus, L ° 2 tT Qrtd face) = KV (1- Se + 2h eden? _ ee a c + coos, £2\/\, 2n* Je) Akr 4. Normalization gives (141) 99 f.@=3 [+ ter (!4 2p ) - To, 202den% (3° Twl “de 4kT (OKT (cates) When numbers are substituted into this expression, itis immediately seen that the validity of keeping only the first order terms in the concentration is in jeopardy. The reason lies in the fact that ata concentration of $% by weight, the product of concentration and excluded volume per pair exceeds unity for TMV. 227 den a. 1.8 (143) Keeping this in mind, the previous data for TMV give, at } % by weight concentration Fil@)=z (I442.x10""Geos’e-i)) oa This correction, as it stands, is unobservable experimentally. The fact that itis slightly smaller than the intrinsic correction given by equation 38 f (6) = +( [+4.5 xX 1o"(Se0a'9-1) (98) undoubtedly reflects the invalidity of using the first order correction to the concentration, ase indicated in equation 143. Before the much 100 more difficult task of using higher order terms in the concentration is considered, however, itis prudent to reflect if auch a procedure is justified. The answer is clearly no, since concentration correction terms to the alignment of order 1 will be necessary before this mech- anism will give anything which is experimentally observable. Note 2 that the alignment term will always contain the factor Ie » which kT tor der 107? » and this then requires the concentration factor itself to be of order 10! for the result to be observable. It is inconceivable that a more accurate treatment will give a correction of this size. Therefore, since at this point the interest ia in explaining an experi+ mental phenomenon and not in academic curiosities, this alignment mechanism will be dropped as a possibility. VO. Summary and Discussion The main purpose of this investigation, notably to find a mech- anism which will cause a field alignment of rod-like molecules of suf- ficient magnitude to perceptibly increase the observed sedimentation coefficient, has been unsuccessful. Hydrodynamic alignment has been investigated in some detail, and on the basis of the results obtained, it can probably be ruled out as the origin of the phenomenon. One aspect of entropy alignment, the excluded volume effect in dilute solutions, was also shown to be insufficient in magnitude to serve aa the basic mechanism. 101 Another aspect of an entropy effect which was not considered here is the formation of small aggregates of aligned molecules. If an aggregate is formed, it should behave like a much larger molecule and thus feel the aligning effects of the field more. In addition, the larger size of an aggregate induces faster sedimentation, so if the formation of aggregates is field-dependent such that the number and size of these clusters increases with applied tield, then the sedimenta- tion coefficient will increase with applied field also. A mechanism for higher stability in a cluster with increasing field is available. It has been shown that the potential energy of a rigid rod due to an applied radial field is 2 where v is the angle made by the rod's long axis and the axis of rotation. This tends to align the rods perpendicular to the axis of rotation. The randomizing effect of brownian motion is thus less in- tense, which fact leads to a higher stability for the cluster. There is another interesting aspect to this in its own right. According to Onsager's theory of liquid crystals (18), the angular dis- tribution function is given by the solution to the equation An4n ft @ = j-i-e f BaF (a) da’ (146) lo 102 where} isa Lagrangian multiplier arising from a normalization condition, and 8 (a,4') is a factor having to do with the excluded volume. The critical concentration occurs when a value of ¢ is reached at which an angularly anisotropic function f(a) leads to a system of lower free energy than one which is isotropic at the same concentration. The inclusion of the potential energy of the radial field leads to an additional term in this equation An Arf ()=A-14 Ser iw cot: 9 - c/a) f (a) ds’ (147) The effect of this on the critical value of ¢ might be of interest and perhaps should be investigated. It is clear on physical grounds that the critical value of c will be decreased, but just how much effect the radial field will have has not aa yet been studied. 103 APPENDIX A The integral of interest is / Fee=-ef [ex feb nl Teo) Fede! excluded “52 hi volume First, itis noted that the maximum distance apart which the centers of molecule A and molecule B enjoy within the excluded volume is 2nb. It is therefore convenient to write the limits of the variable R from 0 to a value commensurate with given orientations of A, 3, and their relative positions. For this itis necessary to have R as such a function. For convenience define nbo-R cosB = eR Tn 8 Sin @ coa($-¢) +c0a@ Cow nbo’R = : J+ _ ' y, cosy nboR Sw 6 sun con (P ¢) 4 cos @ Cowe —_—_——_ — be’ : . ! / / nba nbo" _ si. 6 1. 6! cow(h-$) + coo 9 Coad i Then ed Cd =~ 2nbs {nbs +R ~ nbe ') or 2,2 22.2, .2 é 2,2 4n°b en bo +R $n b 2 o' +2nbe R cos® - 2nbg 'R cosy 2n“b “as! cos} 104 This is a quadratic equation in R R 24 R(2nboce sh -2nbs' cosy)tn-b“o-4n7b-9! 2 #4n“b*«2n*p-ca'cosé = oO f Renb a 'cosy-ccoatt 40 oa! bs*c08“e 43 !cos~y «2aa'cosscosyt2as'cos! ] As given here, R depends on 0,9; 6',9'; 8,4; 0.0. The integral of interest then becomes 27 I 27 r,t R a -c [dé [sno 4 [do dp’ | s.6' de’ Rr” de! T (0') Eloy £(e) ° & 41 0 ° o A very good approximation to F(o') should be the function given in equation 24 of the text: F(o')= ile aj l-* mr ee (Sag se") Say) “| Here Fa w*(i-7p) (Zo— nba cos - R! cow@r nba! cow) and ] —-> => The integral is then explicitly ay - avr rT l am R Mate) [ug pn Od oe (4, -nbocee8 lo o 1 SO 2 ra) STN ye) A - R’co@+ nbo!omie') fe) | Wer om a) é w(o!) _ saga O —_ (2.2, + en é., Cosy) (He - 5 acy) vet) | 3) 105 Before proceeding further, it isa convenient to expand Rasa Taylor series ing’. Define 4 - a” + 0? cos’B 2X. Then R=nb[ (-2 coo + 4) tot (Leeda Beeey)) + 2 / (4-07 +0? coal (-I + 0007 y) — x (coef eos y- of") te] +07 2x" = nb (A+ 0/6 +a*C) Further, to order ¢ 2 rn?» n*p*f a? + ZABo' + (3? + 2AC)o 24 3 Re mp fa? 2 + 3ABe'+ 3(A°c +AB o 24 Actual evaluation of the integral is now begun. Itis convenient to integrate with respect to R' first, and to expand the integrand less the ¢ functions in powers of g'. To order a', the integrand is, lesa certain terms which obviously vanish, 2[2 a2 3,345 (1) ‘) £(e') } (z.-nbeewae) 2 A. coe AEA ut (é +4! cos} It is seen that the variables o', 9', and’ can be integrated out immediately, and that to this order the form of f(§ ') is immaterial. Thus, the integral is 106 27 Tr cM white) (4g su Pd@ jhe fr (z,-nbe cove) — 2 71m N 3 A ~ cow @ Die? A } (@,+@, coe ®) To order a” [us [ss 0do A* (Ce +b0 Coo @ ) arorr Hu @ C02@ Coo = [45 [sm @de (20% ce0p + 4-0*-vocenga) HD CoD SG 1+ Cor* ® fa) > 56 ces 0 ceo $ 8 . . = (4-07) o + 20°77 TF 58 ces0 send w - 47 3 ~ 32 rot® + Lor gtey, tee 8 / To order o 2 woo Sim @ ta0e@ Coe § L486 (,3-0d604? | si.6 cow a F | coo® [+ coe @ Sem @ Cor@ Coos 1T Tw > -|20 [asl SB C82® (¢-bs* 4/20" Co2*B -/20¢e78) Sae® Cor@ sn © )/+¢e7*@® 28 aA a7” = -j20( 7s x6 Ca + at, ) 107 With these results then, the integral to sero order in g' is omar Ci-v p) nth’ (2, -nboccows)( S40 se, _ 32% eres InN 2 3 1s 3 le 7 r cows) $e [20 7a Cow ar ey, | = CM winrb” nee Fal (42 &- A Le yy le A PA tnhe (- § Owe @2 + 7 é,) The next important term after ¢ Misa 2 A calculation such ay that just demonstrated has been performed for this case and the results indicate that this contribution ia quite negligible (about 1%) compared to the @'° contribution. It and higher terms ino' will be neglected, as will small terma in the o 10 part. Thus, the velocity perturbation is approximately 4+ Z CM yb? G-se) ea + cm wrtny2 bre C-7e) ff 3 NV» FN Go cows be +f 2, ) or, since 2nb= f, Zo CM w?k* 2 4 CMwK TS wih ig saute [2 2 3>N b4N 108 To scale this down as was mentioned in the text, multiply by 3d/42f. Also, since the e. terms do not give any contribution to physical observables, they are. neglected, and the velocity perturbation becomes Vied= “Saw Mw ACCEVES (2,8 >s cove €, ) 109 APPENDIX B The integral desired is cite) fap Sedo fae: fay fo Sing de | aide’ (Z.-n bo cose Pann ~R'too ® +nbo ‘cooo') de yf ele (1+ é Oy 20%) ay, / - (6,6, +e Ecooy)/ Le? - wile) | (> ) Since L does not depend on any variables also integrated over, the angular parts can be done first. Thus, the integral becomes _ aT (7 L ; cMw* (1-¥¢) (25 -nbe cows ) [is [Hous RodRk (14Gnt@) Sry ° o 0 Since this ts the same integral as in equation 114, the result is - CMwrli -spl Zo-n be cowela~ ” 110 APPENDIX © The integral of interest is zc wT oO 7 ) where “A _ aA . A . : A Qe, = & sou 8, Cowp, € Cy Hew 8, em, + Cn CowS, 2 ya > % —_> va { { | i | | t+; ~ KO | | ~ ! \ T~ i \ oN \ A \ ! { { XN It is convenient to transform to a new coordinate system whose gs axis is along Py and to express ey in terms of this coordinate system. The transformation matrix is the same as equation 15 of the text. cos), cos ¥, - sin §, sin9, cos @, ome cos 9, sin 9, cos sing, sing - sin 6, 0 eos 8, Lil Therefore, ey cos8, cos’, - sin? sins sos ¥#, ex (0 1 0) ey =(0 1 0) / cos, sing, cos¢?, sin®&, sin %, ®y é. - sing, 0 cos O @s1 or * = (eos %. sin &, cos Y, sin, sine) and e.-*.: cos 6, sin?, sin ¥ cost + cos ?, sin y sin X x Po ‘ + sin@G, sin?, cos ¥ aT wv 2 fs x | sine ¥ 4x (8, ep 5)° = rae) - sin@9, sin“ o ) i. 2. 3. 10. 11. 12, dle BIBLIOGRAPHY Hearst, J., and Vinograd, J., Archives of Biochemistry and Biophysics, vol. 92, no. 2, 206-215, 1961. Lamb, H., Hydrodynamics, 6th Edition, page 598, Dover Publications, New York, 1945. Oseen, C. W., Hydrodynamik, Akademische Verlagsgesellschaft, Leipzig, 1927. Flow Properties of Disperse Systems, J. J. Hermans, Editor, chapter 4, Interscience Publishers, New York, 1953. Einstein, A., Zeitschrift flir Electrochemie, 14, 235-239 (1908). Kirkwood, J. G., and Riseman, J., J. Chem. Phys., 8, 512-516 (1950). Saito, N., J. Phys. Soc., Japan, 7, 297-302 (1951). Burgers, J. M., Second Report on Viscosity and Plastdcity, chapter II, Nordemann, Amsterdam, 1938. Lamb, H., Op. cit., page 610. Springall, H. D., The Structural Chemistry of Proteins, page 202, Academic Press, Inc., New York, 1954. Lorentz, H. A., Abhandlungen liber Theoretische Physik I, page 40, Leipzig, 1907. Saito, N., and Hasegawa, M., Reports on Polymer Physics in Japan, vol. V, 389-39, 1962. 13. 14. 15. 16. i7. 18. 19. 20. ak. 22. 23. 113 Burgers, J. M., Akademie Wetenschappen Amsterdam, 44, 1045-1051; and 1177-1184 (1941); 45, 9-16, and 126-128 (1942). Maude, A. D., and Whitworth, R., British Journal of Applied Physics, 9, 477-482 (1958). Cunningham, E., Royal society of London, Proceedings, Series A, 83, 357-365 (3910). Schachman, H. K., Ultracentrifugation in Biochernistry, page 95, Academic Freas, New York, 1959. Brown, H. B., and Shaw, W. G., Chemical Reviews, 57, 1049- 1157 (1957). Onsager, Lars, Annals of N. Y. Academy of Sciences, 31, 627-659 (21949). Tanford, Charles, Physical Chemistry of Macromolecules, page 192, John Wiley & Sons, New York, 1961. Hin, T. L., Statistical Mechanics, chapter 6, McGraw-Hill Book Co., New York, 1956. Mayer, J., and Mayer, M., 3tatistical Mechanics, chapter 12, John Wiley & Sons, New York, 1940. Salpeter, E. E., Annals of Physics, 5, 183-223 (1958). Private discussion with Mr. Robert G. Mortimer. 114 PROPOSITION I A new method of handling chronopotentiometric data by the method of least squares is proposed. In the theory of totally irreversible chronopotentiometric pro- cesses, the equation relating the various parameters is (1) -J£] (1) L, nFCc? k Gn Co E = ant where E = voltage o 3 transfer coefficient n_ # number of electrons transferred in rate-cetermining step roy 8 current density ty & Faraday c°= concentration of material in solution reacting at the electrode © k 2 rate constant f,n t = time + ® time at which reacting component is used up at the electrode surface. Experimentally, the measured quantities are Eandt. A characteristic of the recording apparatus is that tis more accurately measured than E, From the experimentally produced data, the 115 parameters in the equations are adjusted to fit the data as well as possible, and are thus considered to be determined. This is usually done graphically by fiddling with /+ until a sensibly Mnear plot is obtained for E versus In(1 + (th); the slope gives RT/on,F and the intercept (RT/an_F) In(nFC°K? fi). a fn o A scheme is developed here which points out the feasibility of accomplishing this more accurately by computer methods, using a least squares technique. Define y= a@®RT/on F a b= -l/ fr (2) 0,90 c3 (RT/on, F) In(nFC Kin! to) xaJ/t Then y aln(l + bx) tc (3) Suppose there aren pieces of experimental data. Itis desired to compute from these data the quantities a, b, and c such that the principle of least squares is satisfied. Define A, ty, + a In(l + bx.) - (4) where the experimental quantities Y; and x, are used in this evaluation let of 4. The least squares condition is JLAi _ agar . ater _ Sa ab ac or Ga OL or __ Axi 26 ab - 2y A. Tebe: a of” ~ ~2)' A =O c These conditions lead immediately to Fy:dntirbed 2a Le Cisbra tC yo Le Ctr bee) xX. br.) c yet za y ters +) k l4by. {tby; t+bx, ) 4 = a) hn (14 ber) + ne For conciseness of notation, define the following quantities (5) (6a) (6b) (6c) (7a) (7b) (7c) 117 asf in? (1 + bx,) y 2 In (1 + bx,) 1 + bx, "2° * 1+ bo, ga7 x, In (1 + bx,) 1 + bx i = Ey, In (1 + bx,) » aes y, Then equations 7 become ay ¢cy #2 (8a) aB +cp =€ (8b) ay ten #9 (8c) If 8b and &c are solved for a andc, the results are n€- p-2 A= ng- sy (9a) fp 2- xs When these quantities are substituted into equation 8a, the result ia «(n§-Cay+eylpo-rg)-A(ng-cyx) =o (20) 118 Equation 10 is an implicit transcendental expression for the parameter b, which equation can be solved. The most convenient way is by com- puter techniques. For computation, a more convenient form of equation 10 is ») fe Pe Liebe [ de be b+ bye) Ye Bn (i+ bre) k c (1) + (be (re baed- Lelisbyed) + % (4:94) | Using this form of the equation determining b, the tendency to accumulate very large numbers and then take differences is somewhat mitigated. A trial calculation carried out indicated that numbers with fourteen necessary significant figures may be involved unless the summation order suggested by equation ll is used. Once b has been determined by solving equation 11, a and c are easily computed via 9a and 9b. A program to solve equation ]] and to compute a and c has been written for the Burroughs 220 computer. Several trial calculations have been performed with quite reasonable results. References 1. Delahay, Paul, New Instrumental Methods in Electrochemistry, Chapter 8, Interscience Fublishers, Inc., New York, 1954. 119 PROPOSITION II A table of associated Legendre polynomials expressed as the unassociated Legendre polynomials is presented. In this form cere tain integrals which arise involving Legendre polynomials are easier to evaluate when a table such as this is available. As an example of the simplifications intended, the following - $s considered. z x & < / / From the formula to be found in Morse and Feshback (2), P + p 2 - —v_ t°T, @ (1+ t*-2t2)F** Fig 2 =} where the T (2) are Gegenbauer polynomials, can be derived the following expressions: ym m cm (Ga - ¢$ \% y ° Ly) J Geto)! P, (Com 6,) R (co20,) € 3) =o mMm=-2 120 eo on _! ay) (4) mCn-m)! Pr (ooe,) Preece) sin ml dy - bg) Iva - vel? ved Ye Cn+m)! Sim 4 Sem Og Tim (>, - bp ) Neer mee The first formula is fairly well known, the second posaibly net so . < e > f much so. Here ra * *5 if es the positions of TA, and nt in the formulae are to be interchanged. The usefulness of the table presented occurs in integrala with > > -3 aN -¥,! consider the integral in a Coreg f 4b, { Sinogdeg Ss as a factor {n the integrand. Asa specific example, The es integration gives 2-6 ode , and itis immediately apparent that the situation is complicated compared to an integral involving -%,| “1 » where the only contribution is m = 0. Since cose # PY (cos®), the integral becomes (*, odd oo n T at { m 4 Vay! m(n-m)- P.. Ceoe 6,4) ° m Vg » >) (n+m)| Sin On s,. F,° (ae) Rh, (eotse) d@g 6 Nzs met Now since m is odd, n must be even or the integral vanishes because 412i for n odd and m odd the integrand {s an odd function about 11/2. . The table givee odd m entries as a product of sin® and a sum of Legendre polynomials, so the factor sin® is restored to the integrand and allows use of the orthogonal properties of Legendre polynomials. The purpose of the example is to show how interest in the table may arive, and now that this has been done, the evaluation will be taken no further. Itis clear that an infinite series is involved, and the terms are much easier to compute when the Py (cos@) projections on the pe (cos@) are known. The table gives values from n = 1 ton # 10. All entries have been double-checked for accuracy. They were calculated by fitting the highest power of cos9 in P™(cos8) to the highest power of cos8 in the P (cove ) or the P .,(eo88). depending on whether m happened to be even or odd. Then the next highest power of cos@ was fit, and 80 On. 0 Po ™ Po 0 PiSP, 1 Py? ain’ Po 0 p°sp 2° *2 1 P= sind (3P,) 2 poe -2P, + 2P, 0 DU YU WWE NW Hw Oo Y ¢ wy MY Ph mw wb & PR O i U0 DS wo WS WUT PU Hw Uw WO oso DY @ BW YU KRONA ROWAN DE TO 122 a n at) w 8 ) a" a) sin® GP, + P,) -6P, + 6P, sine (-10P, + 10P,) 2 4 sin® (7P, + 3P,) -12P, + 10P, + 2P, sin (-42P, + 42P,) 24P , - 80P , + 56P) Ps sind (9P, +5P_ + Po) 2 -20P. + 14P, + 6P, ein® (-108P, + 52P, + 56P,) 120P . - 336P, + 2l6P, sin8 (216P, - 720P, + 504P)) zs P = 360P 6 sin6 (lip, + 7P, + 3P,) + 18P, + 10P, + 2Py + 11l2P, + 108P ,) -30P sin (-220P, 67 864P , + 360P, + 144P, sin@ (1320P. - 3696P, + 2376P 5 3 ) -720P, + 3888P , - 7920P, + 4752P 6 0 "J *} iy ~ 9 "J ¥¥) Me] AAD OR Un wb a dW ON A ee 4 v 0 OS DB WV 0 3S YY Y DOOR TONDO HOW ONO HOO 123 8 a ig ~) r " 8 " " a U sin® (13P, + IP 4 +5P,+ P)) 2 -42P_ + 2eP,. + 14P, + 6P) eing (-390P , + 126P , + 210P, + 54P)) 2 840P_, - 1760P, + 392P, + 528P, sin® (46809 | - 11016P,, + 3960P , + 2376?) -5040P_ + 23760P, - 39312P, + 20592P, ain6 (-9360P, + 50544P , - 102960P , + 61776P)) 8 sin® (15P_ + iP, + 1P, + 3P) -56P,, + 26P + 18P, + 10P, + 2P sin® (-630P, + 110P, + 322P 0 3 ¢ 198P)) 1680P , - 3120P + 216P, + 960P , + 264P,, ain® (12600P, - 25080P, + 2184P, + 10296P,) -20160P, + 84240P , - 112752P + 28080P , + 20592P,, sin8 (-75600P__ + 1029600P, - 1262880P, + 308880P ) 40320P,, - 299520P + 933120P , - 1497600P,, + 823680P, i24 P_sP 3 sin® (17P "0 a + 13P, + IP, + SP + P,) 8 2 “72P, + 30P,, + 22P,. + 14P, + 6P, ain® (-952P + 432P , + 380P, + 88P,) A J 8 + 52P = 3024P, - 5040P.. « 204P,, + 1344P, + 936P, sin® (28560P, - 48360P - 7344P, + 20280P, + 6864P,) -60480P, + 226800P_, - 245520P - 5040P, + 84240P sin® (-342720P, + 1422720P - 1866240P , + 374400P,, bl 362880P, - 2419200P_ + 6462720P, - 8225280P, + 3818880P , + 15863040P , - 25459200P ,+ 14002560P,) a) A) 0 + 41 1840P)) P Ks) WWVEONOSOWO UW HR UOWDONDOH OO sin@ (665440P_ -5091840P 8 6 0 Pio” Pio p! # sin@(l9P. + 15P, + MP, +7P,+ 3P,) 9 3 Pio” “90P 14 + 34P, + 26P , + 18P , + 10P, + 2P) P = gin®(-1368P. - 60P,, + 528P, + 588P_ + 312P,) 9 3 Pi” 5040P - 7616, - 1144P + 1584P , + 1720P PP. sine (57456P, - 83160P, - 30096P,, + 27720P, + 28080P ,) 6 P 10” -151200P, + 514080P,, « 449280P - 120960P ,+ 151200P ,+ 56160P, Pig” sind ( -1149120P 5+ 4233600P_, - 4308460P ~ 685440P + 1909440P,) 8 P 10" 1814400P, .~ 109 67040P,+ 25309440P, - 2467 5840P + 3427 200P,, + + 5091840P,, o” sind (6894720P, - 45964600F_ + 122791680P, « 156280320P, + + 72558720P i 10 Pp 10" -3628800F 9+ 3427 2000P,- 142272000P, + 334886400P, - 465120000P,+ + 241862400P,, 2 + 4l6P, 9 Pi 125 erin m (bn-60) f(b.) ddr odd , 27 bm ain’ \ odd 7! a wSm + 211Sim2, $ 2 ' dd > . even 7° sing 2m sind, Jy even 70 cose , 21 cow >, Sn. . dd >! sing cose , 21 Siw db, Cow d, ge sin“¢ .cose EF coay 5° 2r t.2 even > 2 : 2 sing {cos ®> 3 3h sin'@, 3 cos even? wm F Lind, Sm 4217 sim b, coat. § rt 2 7 AF sind, Sn, +217 send 4, even? 37 2 eyenrdre aT coed Sm FIT CoaIs, § Lh. 26 126 References Moree, P. M., and Feshback, H., Methods of Theoretical Physics, Vol. 1, page 782, McGraw-Hill Book Co., New York, 1953. Prévost, G., Tables de Fonctions Sphériques, Gautier-Villars, Paris, 1933. 127 PROPOSITION HI There are two parts to this proposition; both are concerned with the change of viscosity of solutions of long flexible molecules with rate of shear. It is first proposed that Kirkwood's theory of viscosity be applied to flexible molecules which are just beginning to deform, and secondly, itis proposed that the radius of gyration of random coil molecules can be related to the rate of shear by using viscosity versus shear rate data for DNA. Partl There exist no theoretical predictions concerning the de- pendence of viscosity on shear rate for macromolecular solutions except for the case of ellipsoidal molecules (1,2) and infinitely long rods (3). A formal theory has been developed by Kirkwood (4) which is powerful enough to handle systems with molecules of any shape, but mathematical complexity has so far discouraged any application other than to rigid-rod type molecules. Zimm and Crothers (5) have measured the change of viscosity of DNA solutions with rate of shear, and came up with resulta which look qualitatively like the results predicted by Scherega (2) for rigid ellipsoids. Both display zero slope for viscosity as a function of shear rate at zero shear. 128 The mechanism of viscosity versus shear rate in the case of rigtd-rod type molecules ia the tendency of the shear to align the molecules parallel to the flow lines, which alignment decreases the effective viscosity. The reason the viscosity decreases steadily versus increasing shear rate is the counterbalancing randomising effect of brownian motion, which becomes less intense as more torque is applied. The mechanism of viscosity variation with shear rate in DNA solutions is probably much different, for DNA is by no means a rigid molecule. It.possesses rod-like characteristics, butis rather long and flexible. Professor S. G. Mason and coworkers have performed measurements on long flexible rayon fibers and have observed their phy sical behavior as a function of rate of shear (6,7). Long flexible filaments were noted to form ''snake turns,'' "S-turns,'' and coils. Geometrically the simplest of these is the snake turn. An application of Kirkwood's theory to snake turns approximated ag ellipses looks difficult but feasible. It should be of interest as a first step toward understanding the behavior of flexible molecules such as DNA under shear. From Mason's measurements the diameter of a snake turn decreases as the rate of shear increases, which change will have an effect on the viscosity. The necessary relationships for this model will now be set up and the Kirkwood-Riseman equation written down. 129 The esaential approximations, aside from those inherent in the Kirkwood theory, are 1. The particles will be treated as eliipses with 2n + 1 hydro- dynamic elements a distance b apart, and each possessing the same friction factor § . Mason's measurements show that this approxima- tion may not be too bad as an average non-linear configuration. 2. The effect of brownian motion will be neglected. Including brownian motion would make an already complicated mathematical problem prohibitively difficult. DNA molecules are anyway quite large, though not as large as rayon fibers, and the diffusion constants are correspondingly quite small in comparison with smaller systems such as TMV or other molecules manifesting rigid-rod characteristics. Consider the following situation 4 _—_—_ x 2 -— The arrows denote the direction of laminar motion of fluid. The fluid velocity is a Sey (4). If a macromolecular The dissipation of energy is 1 of particle is introduced whose center of mass is at the origin of the coordinate system, the dissipation of energy increases to some new 130 value Ne é - if the molecule has 2nt 1 hydrodynamic elements, each with a characteristic friction factor ¢., the increase of energy dis- sipation te given by (4) (1-9) €° “yh vi? q) where F, ia the force which the Pai particle exerts on the fluid, and fa = € (Re - by) ex (2) where Re fe the vector from the center of mass of the particle to element f. Substitution of 2 into | gives (7-7) é* = = é) (F,- 8.) CRY By) (3) fi: Hence Le . zm, Y, CFa- &x)( Ra 8) -£& (4) VE 2e Rz-n where Gis clearly { ane (Fa eC RE &) (s) gen This gives the increase in specific viscosity for one particle, or equivalently the intrinsic viscosity when the concentration is 431 expressed in number of particles per unit volume. Since the concen- tration usually used in intrinsic viscosity is grams solute per 100 grams solvent, ~ {00M Ye (6) [HJ] = Lim Cpo ¢C Yo This is the relationship used in Kirkwood's theory. ifn] isa function of rate of shear, s, then Cy) 2 My) ~ GS) (7) £4) se (o) G Co) This is the relationship of interest in the study of shear dependence. The model is an ellipse whose circumference is L. 3722! Of > > The location of the beads is c*x,” + a’y, 2 = eo“ a" (8) The cigcumference is L-44d [* (-sin ty sin *6)¥ de = 4d E (su) (9) 132 where E(x) is the complete elliptic integral of the second kind, and cose =c/d. dand¢ are thus not independent, but are related by equation 9. Since brownian motion will be neglected, the hydrodynamic torques will be zero if the loop rotates such that the velocity of the particle, O, located at (O,¢,C) is ~ ex — ex 10 Uo “Cad c | + Cow (10) Each element, of course, has the same speed, given by the absolute value of 10. The velocity of other elements is dc (c+ xi + dtye°) % _ SEHR Oy fate” + Oy CoWP Xa [+ our (g*+ Sin ¥ x7) oe The forces F necessary to determine the quantity G are a derived from the equation Foz 6 (M-0,)- 62 Gs Fs “7 S:-n ah The signe differ from Kirkwood's equation because of a slightly dif- ferent convention in defining the F,. The Oseen tensor here is f hs = = = (13) we \Ré — Ro | where “~*~ ~ ~Oo ol] A “Aw Qo aot Ry Rel” (le,-*,) &, + (y,-¥,) ©) (14) ROR, Further [Re - RPI =Vix, - x,)* - (y, - ¥,)° (15) The coordinate Xp {s determined from the condition . =f Xe 7) i ory Rb= (12 sin* psimtede = E (sin (¥)0¢) (16) where E(y,s) is the incomplete elliptic integral of the second kind. The necessary relationships for determining the F, as functions of d have been presented. The solution of equation 12 is feasible but obviously not simple. Itis now necessary to relate the rate of shear to the quantity d. Once this is done, the viscosity as a function of rate of shear can be calculated from equation 6. The relationship between d and the rate of shear is established by balancing the components of the F per- L pendicular to the elliptical perimeter and the forces arising from the 134 tendency of the ellipse to revert to a circle, which can be calculated by knowing the bending moment per unit length of the molecule. Part If it’s assumed that the DNA molecule has the configuration of a random coil above shear rates of about 0.5 sec’, the data relating shear rate and specific viscosity advanced by Zimm and Crothers (5) can be used along with the viscosity theory of Kirkwood for random coil molecules to relate the average radius of gyration of a DNA molecule to shear rate. According to Forgacs and Mason (7), very long flexible thread- like molecules tend to coil inte knots upon application of shear. This leads to the conclusion that DNA molecules perhaps coil into tight knots on application of shear, with the tightness related to the mag- nitude of the shear. The data of Zimm demonstrate that the viscosity of DNA suspensions decreases with shear rate increase, 30 the above conclusion is at least not unreasonable. According to Kirkwood and Riseman (8), the intrinsic vi scosity for a random coil molecule is given by - VS be F (Ao, 2*) fy] : Z600N.Mo 2 (1) where N = Avogadro's number & = friction constant for a monomeric unit on the coil 135 N_ 2 viscosity of medium M_ = molecular weight of monomeric unit Z = total number of monomeric units # M/M , he Teer) NO oo 6 1 \ Fix 2 oe —_ () ne a Ke 14 SH For large Z, the following relationship holds Jemt ,2* F(A, 24) = |.539 (2) Ao Zt >e The radius of gyration of a random coil molecule is (9) - Fe be” | (3) Combining 1, 2, and 3 gives 4 .589N (673)* |.3 The important thing here is that the intrinsic viscosity is proportional to r, Thus, log [1] should vary with log - the same as with log nfo) into) , which is the quantity given by Zimm and Crothers asa function of shear rate. It is now assumed that rts) = sor (s) (5) in the range of interest, which, according to Zimm and Crothers' 136 data, Hes between shear rates of about 0.5 sec” and some un- specified upper limit greater than 18.3 sec”), The relationship defined clearly cannot hold as the shear rate vanishes, and this is the reason rf) is introduced, which is the radius of gyration at the bend of the viscosity versus shear rate curve, which is presumed to lie at the beginning of the range of validity. A. plot of log n{°) n(e) versus log sis given on the graph. The Une connecting the points is remarkably straight and thus in- dicates that the assumption of the type of dependence of the radius of gyration on shear rate ie valid in this region. The slope of this line is about -1/5. Hence, log [11] = constant + log rs) = constant + 3 log r (3) + 3mlog s (6) or, as just stated, 3m # -1/5 m= «1/15 Therefore, r(s) « gills , (3) (7) & & From the looks of Zimm and Crothers' graph, a good value of r (8) is the radius of gyration at zero shear, which can be deter- mined from viscosity data and the formula of Kirkwood and Riseman. (4 5 Nse'?? 137 ae Ms ee ees ee Se ee 7 F Ms 20 2.5 30 4.9 $0 66 7.0 Ro Shear rate ,im sec”! 138 It might be pointed out that an error was discovered in the work of Kirkwood and Riseman (9). When it was investigated, these authors were unable to immediately predict the magnitude of its correction. Their beat guess was that it would alter somewhat the numerical constants. The original theory agreed excellently with experiments on polystyrene. In any event, changing the numerical constants will not alter equation 7, but may make rs) somewhat ill- defined. References Phys., 19, 1101-1108 (1951). 2. Scheraga, H. A., J. Chem. Phys., 23, 1526-1532 (1955). 3. Kirkwood, J. G., and Plock, J. P., J. Chem. Phys., 24, 665-669 (1956). 4. Rheology, Vol. 1, Edited by &. R. Eirich, Chapter 13. 5, Zimm, B. H., and Grothers, D. M., Proc. Nat. Acad. Sci., 48, 905-911 (1962). é. Arlov, A. P., Forgacs, ©. L., and Mason, S. G., Svensk Papperstidning, 61, 61-67 (1958). 7. Forgacs, O. L., and Mason, 5. G., Journal of Collotd Science, 14, 473-491 (1959). 8. Kirkwood, J. G., and Riseman, J.; J. Chem. Phys., 16, 565~ 573 (1948). 139 1626-1627 (1954). 140 PROPOSITION IV A method of calculating fluid velocity perturbations due to a smell sphere moving perpendicularly between two infinite plane horizontal walle is presented. The sphere is approximated by a point force. Conaider a pair of parallel planes extending infinitely in the x and y directions and separated by a distance 2b. Define the s axis perpendicular to the planes, with the origin exactly halfway between them. The space between the planes is filled with a viscous fluid of viscoaity |, and the fluid is at rest. Zt +b oo — ro © z © @b Introduce at the points =2, x 2 y 20 a point force F operat~ ing in the +s direction. zg 2 +b + o—— r+ - a3 8 eb 141 The components of induced velocity will now be derived. The solution for the case of one plane was given by Lorentz (1), by noticing that if the quantities v! 1) vO), v,! 1) satisfy the hydro- dynamic equations, so do the quantities bv (1) v 4) = ov). 2(z + x) — (2) (1) Vv eey * ‘so Aa ta) y y oy ay 8) v (2), v Q), Uz +0) Zz oZ oy . ememenentmaimesnal +(s + a)? 7 v4) (la) + (n +0)" vy? vO (1b) + (2 +0) vv (1) (ic) In the case of two planes, itis possible to build up a series of images, in the spirit of electrostatic potential theory, to satiafy the boundary condition v20 atg=ztbandz« -b, However, as might be expected, since the functions to be added onto the original infinite medium solution are vector functions, things are considerably more complicated than in electrostatic potential theory. Itis clear that the image forces are to be installed as follows: etc § to+-+ g« 8b - f $ ot-+ 6éb+2 $ +-+ 6b - f $ =+ 4b +4 : + 2b -2 a e- ~(2b + £) $+- «(4b - 2) b -+- ~(6b + £) $ +-+- (8b - £) etc 142 The question now is how to transform the infinite medium solutions for these {mages in order to satisfy the boundary conditions. The images are located at the points (0,0, s,), where the 2, are given in the column to the right of the image diagram. The + and + notation by the arrows refers to which image is under consideration. For example, -refers to the image of the real force across the nega- tive, or 2 = -b, plane. ¢ carries the analogous meaning across the positive, or 2 2 +b, plane. ++ means similarly the image of the image - across the positive plane. The same rules apply for the images +-+, -t-; -tet, +-$-; etc. The components of velocity which are induced by a point force in the +z direction ecting at the origin of an infinite medium are _F KZ FO JF Vy= wm y3 (2b) FL f+ =) Uz = 7™ (< t v3 (2e) These may also be written Uy = a (- Px oz x) v (3a) Cc Vy= gam (~ Son) (3b) 143 —<— (ae R)> Vz = 9m lax* PY? (3c) where reaxt ty tere ee’. In the particular case of a point force ats #=%, these become £2 )y, Ux = 97” (-Sx32 ° (4a) 2 vy =e (- 5) ie (4b) y ~ 99 a Ff fa 22) Vz = Pan \ ax™ Toy Ye (4c) where ro Vez ls oa). It becomes particularly advantageous to work with the undif- ferentiated quantity r, since the modifications of the infinite field solutions for the images can be done with this parameter, leaving the differential operators unchanged. It is now convenient to define the matrix R(-a) ~\+ (e+a)* g? O -2(a+0) R(-o) “. oO —[+(ata)*v* -2 (z+0) 55 > > oO os] (t2 Ztoan HZ re) q ) (5) 144 and the column matrix ( 32 \ - 9X2 We (- sha) - ay92 (6) ne Pie ao ox2z + ? 57 \ / so that equations 1 become —> (.) FL GeV. am RC™ eK Y (7) In terms of the quantities R and K » the velocities induced by the images ¢ and - are tt =~ am R Cb) a Y} = - Fy R(-b). KY (8a) G4 (8b) The requirements on these, namely that nad exactly cancel the effects from the real force at z = +b, and that vo exactly cancel the effects from the real force ats = -b, are clearly satisfied. The velocities arising from the images +- and -+ are derived by double use of the matrix operator R. The results are ft = ae R tb) - R(-38) WY vy. (9a) FO al i+ =a R RCA). R (34) - D4 V4 (9b) 145 The non-obvious part here is that the argument of the first modification operator R is ¢3b. That this {s indeed the case can be most easily demonstrated by direct substitution and verifying that ." exactly cancels v ats -b, and that v* exactly cancels v" ate © +b. Similarly, Gt-+ = “any B Rb) R36). R (se). Ky. -, (20a) rte = 7 7 & RCb)- R (-3b) RCs»): IY +. (100) The s components of some of the images are, less the factor rer] + ZA) f 2 . z + -4 7, + (2-2b+8) 2 (2-bvlb-) (1-3 (2-2b+0)")} V5 3 ~ Y+? Ye be (12) r= 1 Shy eer bee* 2 (arbi b tt) (2+rb+Q) ) Lz {Fr + “vy 3 |-3 == {+ + z+eb-h)" _ 4h (2+2b-2) (1 Otte” ) yz- Ve Ve- y7- Nah (4, 5 Bete -4) -) j Y7- 7 146 In general, the total velocity is given by oo ML, L ) ({t Rs tacbei)| Re (p+ (2-tpbtyt)* Prt ere a (it Reabtaeb -1)]. LE (Pevsner ¢z0 ™ References I. Lorentz, H. A., Abhandlungen liber Theoretische Phy sik I, page 40, Teubner, Leipzig, 1907. 147 PROPOSITION V Criticisms of a paper (1) concerning the concentration de~ pendence of friction factors of random coil molecules are presented. These comments should be of importance since concentration-dependent friction factors for macromolecules are not well understood theoretic~« ally (2). Specifically, it-will be shown that although Yamakawa's resulta regarding flexible molecules are probably valid, the basic theory which led to the relationship used to calculate these results is highly question- able. A more satisfactory basis for the results is presented. The friction factor which Yamakawa presents is that one relative to the macroscopic average velocity of the fluid with respect to the laboratory. Of course, for such a case no infinities arise, but there is no interest in this friction constant unless the average velocity with respect to the laboratory ie known. The Stokes theory which Yamakawa uses will predict infinity for this velocity; however, since such is not the case physically, Yamakawa implicitly assumes that this velocity is zero. Such an assumption is not very enlightening physically. To see why Yamakawa's equation 14 n » AS , ~ g KF ash -5 Pere O46) D Kersty - rs) 6"? jz ~ a joi is valid if u ie defined as the velocity of the center of mazes of the macromolecule with respect to the laboratory, rather than the fluid, it 148 ia only necessary to develop the ideas presented in the text of the thesis concerning concentration-dependent phenomena. There it was found that the effects with respect to the laboratory at the site of one molecule due to all the other molecules was of the form volume allowed where T was some appropriate Oseen tensor (not the infinite medium tensor used by Yamakawa). For both types of Cseen tensor considered (rigid wall and hypothetical concentric sphere), this integral has the properties a > » > cjT-Fdv=c(TFdv +e rE dv x0 e xeluded volume yolum © alfowed The Osean tensor actually used by Yamakawa is (equation 2)) Cg(R) - 1) T (co) where g(R) is the radial distribution function and T(oo) is the infinite medium Oseen tensor. Now if a(R) is unity except in the excluded volume where itis zero, we get > (3¢4)-1) Tt0ee)-F dy =. « {ne = dv Tor al excluded VolLUME , volome 149 where in both integrals infinite medium Cseen tensors are used. In the general case when g(R) is a more realistic radial distribution func- tion, we have fir atly > > Ox (1? dy re fab TF dv - cf t5h-13 TF dv excluded ‘allowed allowed ~ Volume volume volume Now in the region where f g(R) -l} is3 nonzero, Ts T(a@), 30 that ” asd ~ cf ac) LF dv Ze (gch-] te) -F gv-c[z.F du x fcab-7r9) Edo woe else wen eliee which is Yamakawa's result. References h. Yamakawa, H., J. Chem. Phys., 36, 2995-3001 (1962). 2. Schachman, H. K., Ultracentrifugation in Biochemistry, page 90, Academic Press, New York, 1959.