Time Dependent and Equilibrium Stress-Strain Behavior of Rubbers in Moderately Large Deformations Citation Gurer, Cigdem (1984) Time Dependent and Equilibrium Stress-Strain Behavior of Rubbers in Moderately Large Deformations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/e0xe-8w83. https://resolver.caltech.edu/CaltechETD:etd-02252004-105813 Abstract In this work, time dependent and equilibrium stress-strain properties of elastomeric networks are investigated for moderately large deformation. A two network potential is proposed, which constitutively describes stress-strain behavior at elastic equilibrium. The potential is applied to time dependent deformations thorough a molecular model. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemical Engineering) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Leal, L. Gary Thesis Committee: Tschoegl, Nicholas W. (chair) Stephanopoulos, Gregory N. Goddard, William A., III Leal, L. Gary Defense Date: 16 December 1983 Funders: Funding Agency Grant Number Department of Energy (DOE) UNSPECIFIED Record Number: CaltechETD:etd-02252004-105813 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-02252004-105813 DOI: 10.7907/e0xe-8w83 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 751 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 27 Feb 2004 Last Modified: 31 Oct 2025 22:45 Thesis Files Preview PDF (gurer_c_1984.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

TIME DEPENDENT AND EQUILIBRIUM STRESS-STRAIN BEHAVIOR OF RUBBERS IN MODERATELY LARGE DEFORMATIONS. by Cigdem Gurer In Partial Fulflilment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, CA. 91125 1984 (Submitted December 16, 1983) -ii- to Omar -ili- ACKNOWLEDGEMENTS I would like to acknowledge the moral, and at times financial, support of my - family. The contribution of my husband Omar Newland to this thesis, both in terms of moral support and in terms of help with the figures, is greatly appreci- ated. I would like to thank all my friends and collegues at Cal Tech for their numerous suggestions and recommendations at times. Among those, I would especially like to mention Bill Moonan for his technical help. The guidance of my advisor, Dr. N. W. Tschoegl is greatly appreciated. 1 would like to extend my sincere thanks to the people of the Chemical Engineering Machine Shop, especially George Griffith, for their technical advise. Miss Linda Liu's help with experiments and data reduction is appreciated. I also would like to acknowledge the financial support of the Department of . Energy, which made this work possible. -i[v~ ABSTRACT In this work, time dependent and equilibrium stress-strain properties of elas- tomeric networks are investigated for moderately large deformation. A two net- work potential is proposed, which constitutively describes stress-strain behavior at elastic equilibrium. The potential is applied to time dependent deformations thorough a molecular model. —Vm TABLE OF CONTENTS INTRODUCTION I. PROPERTIES OF ELASTOMER NETWORKS AT ELASTIC EQUILIBRIUM 1. Introduction 2. Molecular Theories of Rubber Elasticity 3. Phenomenological Equations in Terms of the Invariants of the Deformation Tensor 4. Phenomenological Equations in Terms of the Stretch Ratios 5. The 2NW Potential 6. Constitutiveness in the Narrower Sense 7. Crosslink Density 8. Swelling of Elastomeric Networks 9. Conclusions 10. References 11. Figures Il. DETERMINATION OF THE PARAMETERS OF THE TWO-NETWORK POTENTIAL FROM MEASUREMENTS IN SMALL DEFORMATIONS 1. Introduction 2. Results 3, The Loose End Correction 4, Conclusions 5. References 6. Figures 10 11 14 15 18 20 22 34 34 35 38 40 42 43 -vi- Ill. MOLECULAR DESCRIPTION OF STRESS RELAXATION IN POLYMER NETWORKS 1. Introduction 2. Theory 3. Experimental Procedure 4, Constant Rate of Strain 5. Trapezoidal Excitation 8. Conclusions 7. References B. Figures APPENDIX I 1. Figure APPENDIX II 1. Figure APPENDIX III APPENDIX IV 48 48 48 59 63 65 66 68 69 81 83 84 87 88 93 -l- INTRODUCTION This work addresses some questions relating to nonlinear deformations of crosslinked polymers. In Chapter I, the phenomenological theory of hyperelastic materials is presented in its most general form. The theory is then specialized to moderately large deformations of polymer networks through the introduction of a strain energy density function, W, of the form Gy w= Hope ge+ne—3) + m2 (AM + AR + AM - 3) (1) where Gy is the modulus associated with the chemical crosslinks of the network, and Gy is the modulus contribution arising from the presence of topological constraints. The moduli are linked by the relation Gy+Gy=G, where G is the shear modulus. The parameter m can be taken as 0.34. The constitutiveness of this two network potential is demonstrated on pub- lished data in general biaxial deformation of natural rubber. Some of our own data on natural and styrene-butadiene rubber crosslinked to different degrees are also discussed. The nature of Gy has been studied on published data on several swollen networks. In Chapter II, it is shown that, in moderately large deformations, Gy is pro- portional to GJ, the plateau modulus of a high-molecular-weight uncrosslinked sample of the same polymer. Since the sum of Gy and Gy, the equilibrium modulus, G, can be obtained from stress relaxation measurements on the crosslinked material in the linear region, it is possible to predict non-linear stress-strain behavior in large deformations from linear behavior in small (theoretically infinitesimal) deformations. In Chapter II a molecular theory is proposed which is capable of describing the nonlinear, time dependent properties of polymer networks. The model is -2- derived for a uniaxial deformation applied as a step function of time. It is then generalized to other strain histories via the Boltzmann superposition integral. Although a full tensorial form for the constitutive equation is obtained through the molecular model, the resulting strain function can be determined analyti- cally only for a uniaxial deformation. To avoid the necessity of numerical integrations, an approximation is proposed and its range of applicability is dis- cussed. This thesis was written as a collection of three independent papers. Chapters I and II will be submitted to Macromolecules for publication. Chapter Il] together with Appendix III will be submitted to the Journal of Polymer Sci- ence. For that reason, each chapter is complete in itself. Thus, the equations, the figures and the references are numbered starting from 1 in each chapter. -3- 1. PROPERTIES OF ELASTOMER NETWORKS AT ELASTIC EQUILIBRIUM Introduction The basic laws of motion, conservation of mass, balance of momenta, con- servation of energy, and principle of entropy, valid for all types of continuum, do not form a complete set of equations to describe the response of a material toa mechanical excitation. It is necessary to complement the set with constitutive equations that characterize the material response. Though largely arbitrary in form, constitutive equations are subject to the restrictions of the constitutive theory! To formulate a constitutive equation valid for all types of material is a useless task because, due to the generality of such an equation, it would have to contain too many experimentally determined parameters. It is more expedient to group materials into various classes and find a constitutive equation for each class. Because of their novel characteristics and easy processibility, polymeric materials are being used in increasing amounts instead of conventional materi- als. For design purposes it is necessary to be able to predict the deformation properties of these materials under given conditions. The theories of linear elas- ticity and linear viscoelasticity are exact theories that describe the equilibrium and time dependent properties of polymeric materials when they undergo infinitesimal deformations. In applications, however, the deformations are not infinitesimal, and it is therefore necessary to be able to predict their behavior in finite deformations. Most crosslinked polymers in the rubbery state are capable of large defor- mations vastly exceeding those of other elastic materials. Elongations of 300- 400% are not uncommon with rubber networks. Due to this characteristic, they have been termed Ayperelastic materials. -4- To describe the equilibrium stress-strain behavior of hyperelastic materi- als, two paths are available for the formulation of an elastic potential valid for large deformations. Hither the small-scale response of the polymer chains to a given macroscopic excitation is assumed to be known, and one then describes the material response using the methods of statistical mechanics; or one pro- poses a potential consistent with the axioms of constitutive theory and the available experimental data, and then tries to interpret the molecular nature of the phenomenological coefficients. This is needed to understand the region of validity of the equation and to enable one, eventually, to design a material for a given application. A potential is commonly called constitutive when it does not violate any of the principles of continuum mechanics. Many potentials which are constitutive in this sense, are restricted to certain types of deformation. We call a potential constitutive in the narrower sense when it appears to describe the mechanical properties of the network with the same material parameters in any deforma- tion. This will be discussed separately further below. Molecular Theories of Rubber Elasticity One of the earliest theories to describe the isothermal equilibrium stress- strain behavior of polymeric materials in finite deformation has been formu- lated by modelling a polymer network as a network of freely jointed chains. The displacement length ( end-to-end distance ) of a chain is considered to be Gaus- sian in the undeformed state. The chains are allowed to pass through each other without hindrance, constituting what is commonly known as a phantom net- work. The stress resulting from a given elongation then is a result of a decrease in entropy of the hypothetical network. Imperfections in the network structure, such as dangling chain ends, loops, and entanglements, are neglected. The -5- equilibrium stress resulting from an elongation is then given by & = RTIF(X* —1/d) (1) where @ is the Cauchy stress based on the deformed cross-sectional area, F is the structure factor, A is the stretch ratio in the direction of the pull, R is the gas constant, and T is the absolute temperature. The derivation and the thermo- dynamic implications of eq 1 have been discussed in detail by Flory.” The structure factor is determined by the specific assumptions made about the microscopic movement of the junction points. James and Guth® assumed that the displacements of the mean position of the junction points are affine in the macroscopic strain and that the fluctuations about the junction points are independent of the strain. These assumptions lead to a structure factor that can be expressed as F = vyx(1-2/4) (2) where vy is the number of chains between crosslink points per unit volume, and y is the functionality of the junction points. Flory” made the assumption that the junction points are firmly imbedded in the polymer matrix and therefore do not fluctuate. Their displacements, not the mean positions of the junctions, are affine in the macroscopic strain. Flory's assumption led to a structure factor given simply by F=vy (3) Stress resulting from a given strain in a crosslinked rubber follows Flory's predictions in small deformations, but is consistently overestimated in even moderately large deformations. In simple tension we consider a deformation to be moderately large if the principal stretch ratio remains below the inflection point in a plot of the nominal stress vs the ratio. Our discussion will be confined to moderately large deformations. We will give a more general description further below. More recently, Flory‘, and Flory and Erman® proposed a new statistical mechanical theory which is based on the idea that topological constraints affect the fluctuations of the junction points. According to this theory, in small defor- mations topological constraints restrict the fluctuations about the junction points to a large extent so that their displacements are still affine in the macroscopic strain. In large deformations, however, the chains are further apart; therefore, only the mean positions of the junction points are affine in the macroscopic strain. The fluctuations about the junction points are independent of the strain in this region. To be able to make the transition between the two limiting cases Flory expresses the stress resulting from a given strain as the sum of two additive terms. The first arises from modelling the displacement of the mean positions of the junction points and has the form given by eq 1 with eq 2. The second term accounts for the severity of the constraints. The form of the equation is too lengthy to be reported here, but it is important to point out that the second term is evaluated using statistical mechanical methods and contains one adjustable parameter to characterize the severity of topological constraints. We will refer to this theory as the Flory-Hrman theory. Phenomenological Equations in Terms of the Invariants of the Deformation Tensor One can formulate a constitutive equation phenomenologically for large equilibrium deformations of hyperelastic materials by postulating the existence of a strain energy density function ( reversible work of deformation ), W , from which the stress is derivable. The most general constitutive equation for equili- brium deformations of an initially isotropic, homogeneous, incompressible material has been derived by Rivlin ® and also by Ariano.’. It contains no -%- assumptions other than the ones stated above. The Rivlin-Ariano equation has the form Oy = -Pby + 22% p, -2 9M ey, (4) al; ale in indicial notation,where dj, is the Cauchy stress tensor, &, is the Finger defor- mation tensor, (87'), is its inverse, and P is an arbitrary hydrostatic pressure required by the assumption of incompressiblity. /, is the first, and /, is the second invariant of the deformation tensor. Thus, it W is known, one can describe the equilibrium deformations of rubberlike materials using the Rivlin- Ariano equation. W is most commonly expressed in terms of the invariants of the deformation tensor for a material that is homogeneous, initially isotropic, and incompressible. The assumption of incompressibility requires that the third invariant be unity. If W is now expanded around the undeformed state of the material in a Taylor series, one obtains W= Dy Cy (h-S) (I-93) (5) tj=0 where C; are the phenomenological coefficients. The two invariants of the defor- mation tensor are given by Ty HAP+AZ+ AG (6) Ip = 1/ AP + 1/AF HAG (7) in terms of the stretch ratios. The fact that W must be zero for an undeformed material imposes the restriction that Copp = 0. Once a form for W is known, the stress resulting from a given deformation characterized by the three principal stretch ratios A;, Ag, Ag (i.e., the positive square roots of the eigenvalues of the deformation tensor) can be obtained as -B- = ow . a= an -P 4=1,2,3 (8) where @; is the Cauchy stress in the ith principal direction. The assumption of incompressibility further relates the three stretch ratios by the equation AyAgdg = 1 (9) Thus only two of the three stretch ratios can be specified independently. Note that eq B is identical with eq 4. If one keeps only the first term of eq 5, then eqs 6, 7, 8, and 9 would yield By — Gg = VC (AP—-1/A1) (10) for simple tension, where A, is the stretch ratio in the direction of the pull, @, is the corresponding Cauchy stress, and G2 is zero since the boundary normal to the direction of the pull is a free boundary, It is apparent that 2C,) = G, the shear modulus, because of the requirement that the results of linear elasticity should be reproduced as A,~1. It can be seen that eq 10 is identical with eq 1 let- ting 2C,, = RIF. Equation 10 models the behavior of a polymer network constitutively in deformations when the stretch ratio is not much larger than about 1.2. Mooney® proposed an extension of eq 10 by postulating that a plot of a/ (A®-A71) vs. 1/A is linear in simple tension, and Hooke's law is obeyed in simple shear. Mooney's equation, which is commonly referred to as the Mooney-Rivlin equa- tion, has the form for simple tension, where C, and Cz are the phenomenological coefficients. It is required that 2C, + 2Cz,=G. Equation 11 can be derived with C, = Cio and Co = Co;, by retaining the first two terms of eq 5. It shows good agreement with experimental data in simple tension, but it does not do so constitutively. In -9- other words, if one estimates the values of C, and Cz from simple tension data, and predicts the results for a different mode of deformation, the predictions do not fit the data. Phenomenological Equations in Terms of the Stretch Ratios Later workers in the field abandoned the idea that W is most easily formu- lated in terms of the invariants of the Finger deformation tensor, and expressed it in terms of the stretch ratios. ) Carmichael and Holdaway*, Mooney!®,and Valanis and Landel!! concluded, on the basis of certain symmetry arguments that W should be the same func- tion of the three stretch ratios. Thus W should have the form W = w(A,) + w(dg) + w{Ag) (12) for a homogeneous, initially isotropic, incompressible rubber. Valanis and Lan- del!!, and Kawabata et al.!*, have confirmed eq 12 experimentally. Most of the well-known nonlinear strain measures can be subsumed in a strain energy density function of the form W= a (AP + AS +A —3) (13) where n is allowed to take on any integer value between -2 and 2. With n=2, eq 13 reduces to the so-called neohookean potential from which eq 10 follows at once. Eq 13 with integer values was first proposed by Seth.!* It was later general- ized by allowing n to be a material parameter,’* which can take non-integral values. Ogden!® used three terms of this same form to describe the behavior of rubbers in large tensile deformations, up to break. By allowing n to be a material parameter, none of the axioms of the constitutive theory are violated.1® Hence from the point of view of continuum mechanics, eq 13 is a valid constitu- tive equation. Equation 13 has been shown by Blatz, Sharda, and Tschoeg],'* to -10- be constitutive in the narrower sense for a styrene-butadiene rubber (SBR), and for natural rubber (NR), below the upswing region. Henceforth we shall call eq 13 the BST potential for convenience. The 2NW Potential Although the BST potential is a valid constitutive equation, the parameter n appears to be difficult to interpret molecularly. Tschoegl!’ proposed a related elastic potential which is based on the assumptions that: (1) A crosslinked polymer can be modelled as consisting of two networks: the phantom, and the constraint network. (2) The contribution of the two networks to W is additive. (3) The stress-strain behavior of the phantom and the constraint networks can be described by the nechookean and the BST potentials, respectively. The first network characterizes the effect of the change in the mean posi- tions of the chemically crosslinked junction points. This network is called the phantom network. The chains in the phantom network can freely pass through each other and have no physical characteristics such as volume. Junction points ean fluctuate freely about their mean positions. The modulus associated with the phantom network is denoted as Gy.Its stress-strain behavior is neohookean. The second network will be called the constraint network. It models the effects of the physical nature of the polymer chains, which causes the fluctua- tions of the junction points to be different from their phantom counterparts. The constraint network can be modelled as if it were held together by equivalent crosslinks and may be considered to have a modulus Gy associated with it. On the basis of these assumptions the most general form for the strain energy density function W can be written as -il- w= Hope erg—3) + Share arenes) (14) where the A’s are the three principal stretch ratios, and m is the nonlinear strain parameter of the BST potential. We use m=0.34 because it closely approximates the strain function we obtain from a molecular theory which we will discuss elsewhere!®, This reduces the unknown parameters to two, and they occur linearly in the model. This is convenient when treating experimental data. We will name the potential represented by eq 14 the 2NW (or two-network) poten- tial. To investigate the validity of the above assumptions, we will show that eq 14 predicts the stress-strain behavior of rubber networks constitutively, in the nar- rower sense, and is able to predict the swelling behavior correctly. The problem of constitutiveness will be addressed first. Constitutiveness in the Narrower Sense In continuum mechanics, any equation that does not violate the axioms of constitutive theory! is a valid constitutive equation. In a stricter sense, a consti- tutive equation must represent material response to any mode of deformation with the same set of phenomenological coefficients. The Flory-Erman theory as well as the BST and the 2NW potentials discussed earlier appear to meet this requirement. The 2NW potential is attractive, because it has only two linear material parameters. It can be shown!® that, for an initially isotropic, homogeneous, and incompressible material, all possible deformations lie in the wedge-shaped region of the /,,/J2 plane shown in Figure 1. Uniaxial tension and equibiaxial ten- sion form the boundaries of this region, pure shear being the straight line with unit slope. Deformations outside of this wedge-shaped region are not admissible. (For a mathematical definition of admissible deformations, see Appendix 1). The whole region inside the wedge can be covered experimentally by biaxial tension -12- experiments in which two dimensions of a rectangular piece of material are changed independently, leaving the third dimension stress-free. Therefore, it is possible to cover al! deformations that an incompressible material can undergo by performing biaxial tension measurements covering the whole range from simple tension to equibiaxial tension. If a proposed equation can describe such experiments with a single set of material parameters, the equation will very likely be constitutive. We emphasize that it is not possible to prove constitutive- ness in the narrower sense; it is only possible to disprove it. One can, however, gather sufficient evidence from biaxial tension measurements to support the constitutiveness of an equation in the narrower sense. When eq 14 is used in the Riviin-Ariano equation, one obtains the true nor- mal stress difference as 2G, Ba Op = Gx(XB—F) + —* APMP) (15) Kawabata et al. have performed ' general biaxial tension measurements on natural rubber crosslinked with dicumyl peroxide. Their data are plotted in Fig- ure 2. The axes were chosen so that the theoretical predictions, as indicated by eq 15, would lie on the straight line with unit slope. The two moduli, Gy and Gy, were obtained from simple tension data. As can be seen from the figure, agree- ment between experiment and theory is excellent. Some of the general biaxial tension data were left out of the figure for clarity, but the agreement for those is just as good. Using eqs 14 and 4, one obtains oH 1 [A(Ga-De) _ AZ(B,-p) | Bh MA LON MM | 08. 7y=1,2,3 (18) -13- aw __1 GygSp a,—Bp | aie MAE |g A 2 oB.y=1,2,3 (17) where the stress differences are given by eq 15. Figures 3-6 show the contour plots of 6W/@/,, and @W/8l, vs /, and Jz, The general shapes of the curves at large J, and Jz are in agreement with those shown by Kawabata”? et. al. They observe rather complex behavior for these contours at infinitesimal deforma- tions which are not reproduced by our theory. We would like to point out that 8W/dl, and 8W/@lg have meaning only in large deformation theory. Values corresponding to infinitesimal deformations must be obtained from extrapola- tion of large deformation data, and are critically dependent on experimental accuracy in small extensions. Unfortunately, data in that region are subject to the largest experimental error. Therefore, the discrepancy between the contours given by our theory and Kawabata's experimental contours is not critical. Figure 3 shows plots of 8W/@/, vs J, for various constant values of J, for representative values of ?y. Here Vy is the ratio Gy/G. The various values of this parameter represent different materials, ¥y=0 corresponding to a material whose stress-strain behavior is neohookean, and ¥y=1 corresponding to an uncrosslinked rubber. In most phenomenological theories, it is assumed that 8W/al, is constant with J, independent of /,. As can be seen from the figure, aw/al, as given by the 2NW potential varies linearly with /, but it is not independent of J. The variation with Jp is small but it is nonzero for ¥y=0. The dependence of 8W/8/, on Jz is shown in Figure 4. Figure 5 shows the variation of OW/ é/,2 with J, for various constant values of Je. It is usually assumed that this quantity is independent of /, and is con- stant with Jp. It is apparent from the figure that the predicted behavior of 8W/ lz is very complex with respect to the invariants. Clearly, if the 2NW poten- tial describes observed behavior, the Mooney Rivlin equation can not be a valid -~14- constitutive equation. Figure 6 shows the variation of 8W/ 4/2 with /g for vari- ous constant values of /). Crosslink Density Figure 2 strongly supports the constitutiveness, in the narrower sense, of the 2NW potential. We now proceed to probe the nature of the phenomenological parameters. For that purpose natural rubber networks were prepared by crosslinking them to various degrees with dicumyl peroxide. (The composition of the networks and the conditions of crosslinking can be found in Appendix 2.) The crosslinking conditions were chosen on the basis of the work of Lorenz and Parks®!, The networks were tested in simple tension using a floor model Instron Tensile Tester. The specimens prepared from the networks were end-bonded to copper U-tabs to minimize problems with clamping. A large aspect ratio was used to minimize end effects. Applied to simple tension, eq 15 yields Oy = Gy + Gy m(A) (18) where oy is the Mooney stress given by _ 51752 and ¢,,(A) is given by _ 2(Ap Ar?) These equations follow from eq 15 with Gg=0, Ay=A, and Ag=Ag=A7!” 2. Figure 7 shows of the experimental data plotted according to eq 18. The straight lines are the lines of best fit. A value of m=0.34 was used as mentioned earlier. The lines for these samples are almost parallel to each other indicating that Gy is closely the same for these networks. -15- Similar results are shown in figure 8 for SBR networks crosslinked to vari- ous extents with dicumy! peroxide. For composition of the networks, see Appen- dix 1. Once again the lines of best fit to the data, for the three crosslinked sam- ples, are parallel to each other indicating that Gy is again the same for all of the networks. This point will be discussed in detail later. We have produced evidence in support of the constitutiveness of the 2NW potential in the narrower sense. It is capable of explaining the data on networks crosslinked to various extent. The second term of the potential appears to be independent of the degree of crosslinking. This supports the assumption of additivity of the moduli as a plausible one. Swelling of Flastomeric Networks Swelling has been used for a long time to study the structure of polymeric networks. The stress-strain behavior of swollen networks in a uniaxial deforma- tion is frequently analysed using the Mooney-Rivlin equation. The most impor- tant observation then concerns the Mooney constant, 2Cz. This decreases with increasing swelling ratio, eventually becoming zero. Therefore,it has commonly been held that the values of the phantom modulus, 2C), obtained from extrapo- lation of swollen stress-strain data are closer to the modulus calculated from the chemical constitution of these networks than 2C, obtained on the dry net- works. In this section we interpret the stress-strain behavior of swollen networks using the 2NW potential. We discuss in detail the importance of the study of swollen networks, and their topological environment. We demonstrate that the parameters of the 2NW potential required to describe the behavior of dry net- works serve equally to describe the behavior of swollen networks. This implies that the effect of an inert solvent is merely a dilution effect and the nature of -16- the topological constraints are not altered in swelling. In conclusion, we discuss these implications. The strain energy density function, W, for unit volume of unswollen rubber is given by eq 14. Equation 14 applies to swollen networks as well as dry net- works. However, to describe the stress-strain behavior of swollen networks, one needs to alter the reference state from the unswollen, undeformed state to the swollen, undeformed state. To that end, let vg denote the volume fraction of the polymer in the swollen state, and W,, denote the strain energy density function per unit volume of swollen rubber. Then, Wey = UgW (21) and a, =uZ/3 >, 4=1,2,3 (22) Equation 14 then becomes Wow = SE ul (a8 +084 08 —3) + 2% uzim-0V3 (aMtastas'—3) (23) 2 me the a 's being the stretch ratios referred to the swollen, undeformed lengths. Using the Rivlin-Ariano equation, and the condition of incompressibility, one obtains 2Gy B= Gud/§ (af—ajt) +E ugin8”9 (ap aim?) (24) for the true stress @ in simple tension. The symbol @ has been used to denote the quantity @, — Ge, since in simple tension, 92 is zero. It is customary to report the nominal Mooney stress f° when relating the results of stress-strain meas- urements. One can obtain the appropriate expression for f ° from eq 5 by con- verting the swollen to the unswollen area, and dividing by the factor a®-a"!. Thus -17- » fA) UY = —m+2)/3 f aa Gy + Gy uf 73 9,,(a) (25) where A is the unswollen, undeformed cross-sectional area, and f is the force acting on it. Both eqs 24 and 25 describe the behavior of swollen networks in a uniaxial deformation, if the solvent is inert, ie., it does not interact with the polymer chains to change their configuration, but acts merely as a diluent. Figure 9 shows the data of Allen® et al. on natural rubber (NR) networks swollen in n-decane. The solid lines show the predictions of eq 25, The parame- ters of the two-network potential, Gy and Gy, were obtained from stress-strain measurements on the dry network as Gy=0.107 MPa and Gy=0.144 MPa. The value of m=0.34 was used for all predictions. As can be seen from the figure, the predictions of the theory are excellent for the first three volume fractions, ve=1.00, 0.79, and 0.61. For vg=0.42 and vg=0.36, the predictions are fair but not very good. The discrepancy between the theory and the experiments is attri- buted to the solvent interacting strongly with the polymer at those swelling ratios. Such a behavior would be expected from a strong concentration depen- dence of the Flory-Huggins x, factor. For vg=0.24 the agreement becomes very good again. If the above inconsistency is due to a concentration effect, it is ‘not observed at this volume fraction, because the data lie on a curve that is rela- tively flat. Figure 10 shows the data of Flory and Tatara®® on poly(dimethyl siloxane) (PDMS) networks swollen in benzene. It is apparent from the figure that the polymer-solvent interactions play a dominant role in this system. Here the x, factor is highly concentration dependent, implying a marked difference in poly- mer configuration as the concentration is changed. Part of the difficulty arises from the fact that the experiments were performed at constant solvent activity rather than constant composition. Again the two parameters of the potential 18 - were obtained from experiments on the dry network, and a value of m=0,34 was used. The value of the moduli were Gy=0.056 MPa and Gy=0.149 MPa. Figure 11 shows the effect of polymer solvent interaction more clearly. In the figure Mooney plots are shown for a PDMS network swollen in three different diluents to the same volume fraction. The points in Figure 11 were calculated using the values of 2C, and 2C2 reported by Mark". At vg=0.80 the data points for all three diluents lie on the same curve indicating the absence of polymer- solvent interactions. At v2=0.60, differences between the three diluents are apparent. Figure 12 shows the data of Erman and Flory * on poly(ethyl acrylate) (PEA) networks swollen in bis(2-ethoxy ethyl) ether at 25°C. The experiments were made on networks crosslinked to different degrees which were then swollen to equilibrium. The open circles are the experimental points for dry networks, and the closed circles are the experimental points for the swollen networks. The solid lines show the predictions of eq 24 with m=0.34, and the two moduli obtained from the dry network stress-strain data. As can be seen, the agreement is very good, One point is worth mentioning here. Flory and Erman*® have examined the same data sets using their two-network theory. The value of the phantom modulus they obtain from their analysis is always larger than our Gy. Although the two quantities are supposed to represent the phantom network contribution to stress, there apparently is a difference. The reason for this difference and the resulting implications is unknown. Conclusions We have demonstrated the constitutiveness, in the narrower sense, of the eNW potential. Further, we have shown that it represents data on varying -19- crosslink densities adequately, with a Gy that is sensibly independent of crosslink density. We have also examined the stress-strain properties of swollen networks. When the diluent does not interact with the polymer strongly (changing the configuration of the chains in space), parameters obtained from the dry net- work are adequate to describe the stress-strain behavior of swollen networks. This implies that, when a network is swollen, the topological environment around the network chains is not altered significantly. The effect of the neigh- boring chains on a given network chain is diminished due to dilution only and is not altered significantly in character. This observation casts doubt on the existence of trapped entanglements because once swollen, their nature as well as their concentration would change. -20- References 1. 10. 11. 12. 13. 14. 15. 16. 17. 18. AC. Eringen, Mechanics of Continuous Media, John Wiley and Sons Inc; New York, NY (1967), 142 . PJ. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY (1953) _ HM. James, E.J. Guth, J. Chem Phys., 11 (1943), 455 . PJ. Flory, J. Chem. Phys., 66 (1977), 5720 . PJ. Flory, B. Erman, Macromolecules, 15 (1982), 800 _RS. Rivlin, Phil. Trans. Roy. Soc. London (A), 240 (1948), 459 _R. Ariano, Rend. Ist. Lombardo Sci. (2), 63 (1930), 740 . M. Mooney, J. App. Prys., 10 (1940), 485 _ AJ. Carmichael, H.W. Holdaway, J. App. Phys., 32 (1961), 159 M. Mooney, J. App. Phys., 11 (1940),582 K.C. Valanis, R.F. Landel, J. App. Phys., 3B (1967), 2997 S. Kawabata, H. Kawai, Adv. Polym. Sci., Springer Verlag, Berlin, 24 (1977), 89 Seth, in Second Order Effects in Flasticity, Plasticity, and Fluid Dynamics, Reiner and Abir, eds., Mc Millan, New York, NY (1973), 162 P. Blatz, S. Sharda, N.W. Tschoegl, Trans. Soc. Rheol., 18 (1974), 145 R.W. Ogden, Proc. Roy. Soc., A328 (1975), 485 RS. Rivlin, K.N. Sawyers, Trans. Soc. Rheol., 20 (1976), 545 N.W. Tschoegl, Polymer, 20 (1979), 1365 Chapter III of this thesis -21- 19. S. Kawabata, M. Matsuda, K. Tei, H. Kawai, Macromolecules, 14 (1981), 154 20. Y. Obata, S. Kawabata, H. Kawai, J Polym. Sci A-2, B (1970), 903 21. O. Lorenz and C.R. Parks, J. Polym. Sci., 50 (1961), 299 22. G. Allen, MJ. Kirkham, J. Padget, C. Price, Trans. Furaday Soc., 67 (1971), 1278 23. P.J. Flory, YJ. Tatara, J. Polym. Sci. Polym. Phys. ed., 13 (1975), 683 24, C.U. Yu, J.E. Mark, Macromolecules, 7 (1974), 229 25. B, Erman, PJ. Flory, Macromolecules 15 (1982), 806 ~22— Figure 1: 1,—_ Region of admissible deformations for an incompressible material, -23- 8't *TeTJUajod MNZ aya Jo SuOTIOTpaad ay YITA peieduos YN Uo FIeq ( fv -sv )(mstaz)« ( ov2y ) %3 vl ot 90 ZO J : ' Li Lj J Ls 7£°O = Ww 2% Van3tTy apaysg aang r uofTsuay, TeTXeTpqEnby @ 3Ss°0 uojTsuay atdwts a a { 701 2! —_-~ = bn =] @ —— "Te 39 BIeqemey jo Bled uN St ~24- *m Suofiouny Aqpsuap AZzeue uTe1zS 9yy JO aATJeALAap 9y}3 Jo TOT ABYaq rAt Ub OL 6 3 Z Ss % Foot 7 eo § $°0 —en J ———___. ee 5 oo J oo ee — ah JS. poroTpeid N fe) ° 3S (tie /me)(3/t) t€ Bansty *m ‘uopjouny Ajpsuap ABzaua upeiys BY JO SATIBAFASp 947 JO AOFARYSG PeIdTperg ty Van3yzy 2 | SZ Gi OL S T oe T T ro 0°T _ . « P 8°0 420 —_] ee id a oh r ~ —_ 1. ry a q _— 460 = em ~ Q —ae, red ~~” - ¥°O be . ~~, 5 4970 z°O t N t — . Tt= I ray 6= I a! « - i= I culty -2?6- *m ‘uoT390uny Ajysuep AB1aue UfeIIS 94] JO BATIeATISP BYi Jo TOFALYSq powopperg tg ean3tyq zLoOti«anhC(<‘éitaR:C“Cté‘té«~CC‘ 8 Z 9 «6S v € a 7 t 5 J ¥ q qv ' ¥ ooo O°T —_— : 8°0 —— a 9°0 — — 9°0 —_—_ 5 3s0'0 W-—~, 2 = N Ot="1 —ee__ ee o> A —— ~~ >_> “ow = é 4 0z= I = ~ ee ee @ . ®% ew ot="1 5 ~sto g="t -27— *m SuoT JoUN; Ajypsuep AZieue upfeiqys 94} jo saz JeaTsap ayy jo 10fAeYyaq palotpedad $9 o1n3Tg 000 soo | O2 SL OL S z°0 > ee 9°0 = a - i. Pe 8'0 ~ 1 Il='I . oS OT . 5 a © = o sto (-1e sme)(3/t) -28- NR 4 Emenee 20°C O.1+ = = 0.36 01 02 O03 O04 05 06 O7 O08 09 #,() Figure 7: Equilibrium data on NR networks with varying crosslink densities compared with the predictions of the 2NW potential. -29— q q q i 5 0.4 — a Sample Gy (Pa) Coy MPa) SBR 6 0.096 0.266 SBR 2 0.080 0.232 = SBR 4 0.006 0.234 SBR 3 0.00 SBR 6 Oy ( MPa) 0.2 sR 2 SBR 4 SBR 3 wpe 0.34 l i N 1 1 0.5 0.6 0.7 0.8 0.9 $n(A) Figure 8: Equilibrium data on SBR networks with varying crosslink densities compared with the predictions of the 2NW potential. =-30— - a43 03 peieduos * “TeFauajod MNz 347370 suof otpead yAomjeu UaTTOMS B JO ToFABYeq UFPIIS SSEIIS +6 aan3ta RvR | No 8'0 90 vo zo if am j i q i LJ iJ ij | . 4~0Lo *{e 3a UaTTy jo BIB ———SaesSp-0 UT WN , . =3aSto =a ° ; eSeket = Pig @ e 7 eet © = 2 e 9z°O 3 @ . cP eee 9670 2? 4020 © z7°0 “ #~p 19°0 2 . 6L°0 q «dl - o°t e fn SAIND . 7Se°0 i j i i l 1 t j ‘TeTJuajod MNZ ay2 Jo suoTqofTpead ayq 03 peredwod ‘y1OMJeU UaTTOMS BR JO JOTARYaq UTeI}S SSeS OT eANnBTyZ ~31- 1-2 30 9°0 yO 20 ’ Lj ¥ F T c T - eaeqey pue AzoTa jo eed auezueg UE SWdd ; 410o1ro a P “ =n ° * 0g = ame 5 eee zE°0 3 = e 0 — >> 9 ° 9”°0 p 7Sto T9°0 3 e e . . ¢8°0 q O°T e 60° 7 A aaing . 0 +0270 1.2 1.1 1.0 0.9 f* (wpa) 0.8 0.7 0.6 0.5 Figure ll: ~32- T T t T T SOLVENT -e © None ~ e Dimethyl Siloxane - & n-Hexadecane J a n-Octyl Acetate ma a2 1.0 be 0.8 0.6 PDMS ' 7 Data of Yu and Mark a ~ i i A. i. = 0.5 0.6 0.7 0.8 0.9 ao! Stress strain behavior of a PDMS network swollen in different solvents. -33~ *TeTjuejod MNZ 243 jo suoTIOFpeid ayy e1e SaUTT PFTOS *uNT~IGETENba oj uaTToms saaiZap JUalasj}TP O02 PayxUZ{Sso1a sy1omjou yd JO AOfFAeYyaq UFeIzsS BSa13S t7T aan3sta Lt t- vo ZO v 80 90 q i ' AioyTqg pue uemsg jo e1eq zouyea( Aye Axoyza-Z) Std UF Vad - 34- Il. DETERMINATION OF THE PARAMETERS OF THE TWO-NETWORK POTENTIAL FROM MEASUREMENTS IN SMALL DEFORMATIONS Introduction In two previous communications!*, we proposed an elastic potential of the form G w= HF peneng-3)+ area enr—s) (1) This potential was shown to be highly successful in describing arbitrary defor- mations ( general biaxial tension ) of rubberlike materials, if the deformations are moderately large, i.e., if they are restricted to the region below the upswing in a plot of the stress vs. the largest principal stretch ratio. The potential contains three material parameters. Of these, the strain parameter m was originally left to be determined by experiment. Elsewhere,” however, we presented arguments which led to the adoption of a "universal" value of 0,34 for m. Of the remaining two parameters, Gy and Gy, the first represents the modulus ascribed to the presence of chemical crosslinks in the network. The second expresses the contribution arising from topological con- straints. The two moduli can be obtained from stress-strain experiments in sim- ple tension for which eq 1 yields Oy = Gyt Gy pm(r) (2) where _ G,- 82 and - 35 - 2(AmM—Ai™”*) Aj= m=0,34. (4) In these equations A is the stretch ratio and @ is the true stress in the direction of the stretch. The two moduli are linked by the additivity relation Gy + Gy =G (5) in which G is the shear modulus of the material. If any two of the moduli G, Gy, and Gy are known, the third can be obtained using eq 5. We now examine the possibility of determining the parameters of the elastic potential for moderately large deformations of rubberlike materials from meas- urements in small (theoretically infinitesimal) deformations. Stress relaxation measurements in such deformations on the crosslinked polymer yield the (equilibrium) shear modulus, G. We have found that Gy is proportional to the plateau modulus, Gf, of a high molecular weight precursor of the network, i.e., that Gyu=§GR. (6) G§ can be obtained from stress relaxation measurements in small deformations on the uncrosslinked polymer. Thus, if eq 6 is valid, and é is known, Gy and Gy can both be obtained from measurements in small deformations. Results We have tested the hypothesis contained in eq 6 on several materials for which the necessary information could be found in the literature. These materi- als were butyl rubber (IIR), natural rubber (NR), styrene-butadiene rubber (SBR), crosslinked cis-1,4-polybutadiene (PBD), poly{ethyl acrylate) (PEA), and poly(dimethyl siloxane) (PDMS). The values of G# for these polymers were obtained from J§ = 1/G# as given by Ferry®, except for PEA which was taken - 36 - from Janacek* et al. The values are listed in Table 1, together with estimates of Gy which were secured in the following manner. For each material, values of Gy and Gy were obtained for several crosslink densities. If our hypothesis is correct, then Gy is independent of the latter. Hence the values of Gy were aver- aged. These averages are shown in Table 1 as Gy. together with their standard deviations (s.d.). For SBR we obtained Gy and Gy from eq 2 using our own data. For the other materials we used published values of the constants 2C, and 2C2 of the Mooney-Rivlin equation Gy =2C,+2Cyrd . (7) References are listed in Table 1. From reference 6, we selected only those sets which represent variations in crosslink density. Table I Gy and G§ for Different Rubbers Rubber Gy (MPa) s.d. N G8 (MPa) Ref PEA 0.0612 0.0165 5 0.160 5 PDMS 0.0764 0.0241 11 0.119 8 liR 0.108 0.0143 3 0.2BB 7 NR 0.185 0.0513 37 0.575 6,8 SBR 0.244 0.0191 3 0.776 1 PBD 0.43B 0.135 11 1.148 6,9 To obtain Gy and Gy we first calculated pairs of values of oy and 1/A in the interval 0.5 < 1/A <0.8 at increments of A (1/A) = 0.05, using eq 7. We then used those pairs to obtain Gy and Gy from eq 2. A plot of oy calculated from eq 7 is not linear in ¢y(A). However, in the interval 0.5<1/A<0.8 the departure from linearity is indistinguishable within the usual experimental error. Details of the calculations are tabulated elsewhere.!° -37- A plot of Gy vs. Gf is shown in Figure 1. The slope of the straight line gives £ as 0.354. The line is the line of best fit to the natural rubber data. The error bars represent limits of +1 standard deviation. The errors result from a number of sources such as differences in sample preparation, vagaries in the determination of the Mooney-Rivlin constants, differences in the times used to establish equilibrium, etc. Figure 1 suggests that, for the other five materials, é is the same as that for natural rubber within +1 standard deviation. Any crosslink density dependence of Gy, if it exists, would be hidden within the error bars in Figure 1. To determine if Gy is indeed independent of crosslink density, a plot of Gy as a function of the latter would be required. Since for most materials, the crosslinking reaction is not stoichiometric, we decided to use Gy. as a measure of crosslink density instead of the amount of crosslinker. It is shown in another publication® that in crosslinked polymers Gy relaxes to its equilibrium value much faster than Gy. If any long-time relaxation processes persist, ie., it is difficult to attain elastic equilibrium, the quantity that has not relaxed to equilibrium would be Gy. Since G=Gy+Gy, the ratio Gy/G can be expected to minimize any lack of equilibrium. Also, as shown in the next section, Gy depends more than Gy/G on network imperfections such as loose chain ends and the presence of a sol fraction. We therefore attempted to reduce the scatter by employing the ratio Vy=Gy/ G, instead of Gy. The quantity Dy may be viewed as the mole fraction of “equivalent crosslinks” modelling the effect of topological constraints.!! Using eqs 5 and 6 it becomes vy = oe = EA " Gxt Gu ~ Gy+ G8 (8) which may be rearranged to —=1+¢f' => Uy ‘ GH - 38 - (9) The latter form is convenient in assesing the effect of crosslink density, as reflected in Gy. The plot according to eq 9 is shown in Figure 2. The solid line is the line of best fit to the natural rubber data, the slope yielding =0.354. A few of the natural rubber data were omitted from the plot for clarity of presentation. Figure 2 strongly supports the validity of eq 6. It also suggests that ¢ is independent of the nature of the polymer. A closer examination of the data reveals, however, that although eq 9 is obeyed within experimental error by all polymers examined, the main premise underlying that equation, namely, Gy=tG§, sometimes appears not to be valid. A plot of Gy vs. Gy for various natural rubber networks studied by Mullins et al.!*, is shown in Figure 3. The data indicate that Gy may be weakly dependent on crosslink density. We found this dependence to be due to the presence of loose chain ends in the network. As we will show in the next section, the crosslink density dependence of Gy disap- pears when this is accounted for. The Loose End Correction According to Flory'S, the number of moles per unit volume of elastically active crosslinks, vy, is given by vy = vy ™ (1 —2Mc/ M,) (10) where vy“*™ is the number of moles of chemical crosslinks per unit volume put in, Mc is the molecular weight between crosslinks, and M, is the number average molecular weight of the polymer before crosslinking. The modulus due to the presence of chemical crosslinks is then given by -39- Gy = SUyRT (11) The volume fraction of loose ends, vz, is given by Mc Merl, “) UL Therefore, if networks having the same vy" (and thus the same My) were prepared with differing M,,, one would be able to determine the effect of loose chain ends on Gy. Mullins et al.’* report stress-strain data on such networks. As shown in a previous publication,’, in a swollen network the true stress referred to the swollen, deformed area may be represented by t= Gy u3/8 (2-1) + ZEN yf-mesya (x™ —\-m72) (13) where vg is the volume fraction of the polymer. For vg close to unity, the dilu- tion effect on Gy is minimal, while Gy varies linearly with the concentration of network chains. Since at equilibrium the loose ends would act like a diluent, Gy corrected for loose ends is given by Gy corr — G, 1-—w, iN (14) In Figure 4, Mullins’s data are plotted according to eqs 10 and 11. The straight lines represent the least squares fit to the data. The intercept yields the quantity G,(M, =o)=SugRT, and the slope divided by Gy(M,==) yields 2Mc. Using eqs 12 and 14, one can calculate the Gy values corrected for loose ends. Mullins’s network designated by B was left out of our analysis because the data on this particular network were badly scattered. Gyo"? vs. Gy plots are presented in Figure 5. As can be seen, there is no clear trend in the data, and Gj?" appears to be indeed independent of crosslink density. - 40 - Conclusions In this paper we have examined the validity of eq 6, which suggests that the modulus characterizing the topological constraints is independent of crosslink density, and that all the information which differentiates the topological environment of a network can be obtained from the uncrosslinked polymer. One important implication of the validity of eq 6 concerns the concept of trapped entanglements. If the nature of Gy were related to trapped entanglements, its crosslink density dependence would be more pronounced. Our analysis implies that the nature of Gy is related to a decrease in the configurations available to the network chains as compared to their phantom counterparts. This is implied by the fact that the topological environment of the uncrosslinked polymer is carried over to the crosslinked counterpart. Although in form the preceding theory appears to be similar to the theory of Langley,'* it is in fact quite different. The latter theory has been discussed at length by Queslel and Mark!®. In the Langley theory, one has no way of separat- ing the effect of topological constraints from the phantom network, because the strain dependence of the constraint network is unspecified. Therefore, the con- clusions one would draw from the Langley theory are dependent on the assump- tion made about the relation of the chemical composition of the network to the modulus due to chemical crosslinks. In our theory, no such assumption is neces- sary, because we obtain both moduli from experimental data. We would like to point out that the conclusions we have drawn are not expected to be valid near the gel point. We have also based all of our conclusions on data on randomly crosslinked, tetrafunctional networks. Major differences are not expected for end-linked networks. The value of § would probably depend on network functionality. Most of the available data on networks that are not tetrafunctional are on PDMS. Unfortunately, PDMS has a very low G§ so that the -41- values of Gy would be relatively small. This fact precluded the study of func- tionality dependence of é, -~42- References 1. 2. 3. 10. 11. 12. 13. 14. 15. Chapter I of this thesis Chapter III of this thesis J.D. Ferry, Viscoelastic Properties of Polymers, John Wiley and Sons, New York, N.Y. (1980), 374 . J. Janacek, J. Hrouz, J. Polym. Sci., Polym. Symp., 53 (1975), 203 | LE. Mark, M. Kato, J.H. Ko, J. Polym. Sci., Polym. Symp., 54 (1976), 217 , J.E. Mark, Rubber Chem. Tech., 4B (1975), 495 . M. Kato, J.E. Mark, Rubber Chem. Tech., 49 (1976), 85 _L. Mullins, J. App. Polym. Sci., 2 (1959), 257 . B.N.E. Van der Hoff, E.J. Buckler, J. Macromol. Sci. (A), 1 (1967), 747 Appendix IV of this thesis N.W. Tschoegl, Polymer, 20 (1979), 1365 L. Mullins, J. App, Polym, Sci. (a), 2 (1959), 1 P.J. Flory, Principles oof Polymer Chemistry, Cornell Univ. Press, Ithaca, N.Y. (1953), 463 R.N. Langley, Macromolecules, 1 (1968), 348 J.P. Queslel, J.E. Mark, Submitted to Adu. Polym. Sci. -43- 0.5} e Vv 0.4} rN @ ~ 03+ ¢ S a S 0.2 0.1} PEA PDMS i i 0.2 Figure 1: Cy as a function of Ge for different rubbers. 0.4 0.6 0.8 6, (MPa) 1.0 1.2 *g10qqnd JUadajjip 103 ‘kz}~Ssuep YUT[Ssord Jo uot uN; e& se sYUTTSsoId jo uoT}IeIJ STOW +7 aandTyd Ng /%9 80 90 vO cO q LJ v LJ q Ly ' LU v -44- s UII 4p sndd = vad aS *A}[Tsuap YULTSsoAD Jo uoTz3ounjy eb se Ny ZuyMoys Suz TMW Jo ejleq i¢ 24N8TA (ean) "9 £'0 ___z0 10 v -45- 7 a |. e 0 le wr v ° H 1 = v fs 2 ve ? @ a — v z0 <@ e ® eonm<«s @ SUFTINW JO Bled Vv p.¢ *zosanoaid ay} Jo WYySTemM Ae[NdseTOW ay} uo “Dg Jo aouapusdaq ty sinsTq got" Ai OL g 9 v Z 5 42 —~— I ow 4 Ne) ~t 2 r 17? = om ~ | » a : H @ d e r q v 49 a a P SUTTINW JO BIE v @ : YN WYOMLAN t i 2 s i l i a i i t L *spue eSOT 1JOJ pd}DAIIOD SUTTINW JO eB3eq iG PaNsTy (®dW) “9 vo £0 rA 6) to 00 . tT T T T T Ul 00 H @ od . ad 7 qa Vv a SUTTINW jo B3ed 4) a “ ro a uN v @ = 2 | Nose WUOMIAN 4 = e B ~~” Vv ° 8 e zo vve” 2 ° ¥, @ oe % . L ry oe ¢ 7 t i I 1 i i i £0 - 48 - J0.MOLECULAR DESCRIPTION OF STRESS RELAXATION IN POLYMER NETWORKS Introduction In a previous publication’ we described a two-network potential for moderately large deformations of rubberlike materials at mechanical equili- brium. Using published data, we showed! that the potential is constitutive in the narrower sense, ie., it is capable of describing equilibrium deformations in different stress fields with the same set of material parameters. The potential was derived from simple considerations. In the present paper we develop a theory for the time dependent behavior of crosslinked networks, again in moderately large deformations. Our approach is based on the ideas developed by Doi? and, for ease of reference, we usemuch of the same notation. We first formulate the theory and then proceed to a discussion of it using published data as well as our own obtained on styrene-butadiene rubber (SBR). Theory We use the mean-field approximation to formulate a molecular picture of the dynamics of crosslinked polymers. We consider the properties of a single chain embedded in a continuum formed by its neighbors. The interaction of the chain with its neighbors are modelled as interactions with this continuum. In this way, we reduce a multi-chain problem to that of a single chain. We also assume that the molecular motions of the chains result solely in changes in configurational entropy, i.e. there are no enthalpic interactions between them. These two assumptions are implicit in most theories describing the equilibrium and time-dependent properties of both crosslinked and uncrosslinked polymers in the rubbery region. Consider a network chain terminated in chemical junction points at both of its ends. At any instant in time, the chain is confined to a certain configuration - 49 - by its neighbors. When deformed, it must move in the field so defined. To describe the effects of its neighbors on the chain, we place N slip links along its length as shown in Figure 1. Let the chain segment between two slip links be modelled by N, Rouse (bead-and-spring) units with mean-square separation b*. Let ¢ denote the friction coefficient of the Rouse units. Let the total number of Rouse units between two crosslink points be denoted by No, and let a be the dis- tance between two slip links. The complete sequence of Rouse units is called the primitive chain. It has the same end-to-end distance as the real chain which wraps around the primitive chain. Let its contour length (called the primitive path length) be denoted by L. With the definitions given above, we have N = No/ Ne (1) and L =aN = Nyb?/a (2) The slip links have been introduced merely as a mathematical convenience in modelling the effect of the surrounding chains. We do not necessarily imply physical entanglements, although our model could not differentiate between entanglements and other types of topological interactions which restrict the configurations of the chain in space. To allow us to describe the dynamics of the chain mathematically, let r,(t) be the position of the nth Rouse unit at time t, and let R,(s,();£) be the vector defining the curvature of the primitive chain at point n, where s,(f) denotes the arc length at the same point measured from the first (n=0) segment. To represent the wriggling motion of the chain along its primitive path, denote Tr(t) as where d,,{f) is a vector normal to Ra(s,(t);t) at s,(f). Then, s,(t) and d,{t) -50- represent the wriggling motion in the longitudinal and transverse directions, respectively. The relaxation of d,,(¢) corresponds to the relaxation of the Rouse units between two slip links. The relaxation of s,(t) describes the relaxation of the primitive chain between two crosslink points. Stress Relaxation It will be assumed that when a macroscopic deformation is applied to the polymer network as a step function of time, all junction points and slip links first move affinely with the deformation. They can then move further during the two relaxation processes just discussed. (a) Relaxation of d,(t) The relaxation of d,(t) corresponds to the transverse wriggling motion of the chain. The mathematics of this process for an uncrosslinked polymer has been described in detail by Doi.” In the time scale of this motion, the crosslinked polymer relaxes in exactly the same way as the uncrosslinked polymer, since the length a of the relaxing unit is much smaller than the the length between two crosslinks (Nob). Therefore, the true stress resulting from the wriggling motion is the same as that given by Doi, and we have Bat) = 3e(No/ NekaT[1 + Si exp (-tp?/t4)] <(Fa)q (Fu)e>a-Pbap (4) p=1 where 74 = Né is the largest relaxation time associated with the relaxa- 0? 6r°k_T tion of d,(t), kg and 7 have their usual significance, c is the number of network chains per unit volume, F is the deformation gradient tensor, and u is a unit vector with random orientation, defined by u = (cosd, sind cos¢ , sin’ sing ) (5) The average over uis defined by -51- 6.20 = f" L-) sindd 3 dy (6) (b) Relaxation of s,(¢) Again following Doi, we assume that the relaxation of s,(t), corresponding to the longitudinal wriggling motion of the chain, can be described by the equa~ tion for the free Rouse chain.” We have as, (t) = 2) ¢ at (SkgT/b Bent) ra(t) (7) where f,(t) is a random force assumed to be Gaussian, characterized by the moments <fa(t)>a=9 and <falt)fm(t')>u= 2¢keTd(n—m)d(t -t') (8) Equation 7 is the continuous form of the diffusion equation for the Rouse chain.® The Cauchy stress tensor is then given by Gag(t ) = (8ckgT/ b*) fic Rnalt) ropht) >udn — Pap (9) 0 where P is the hydrostatic pressure required by the underlying assumption of incompressibility. The approach taken here differs from the original theory of Rouse in two important points. In the Rouse theory the boundary conditions at the chain ends are @s,(t)/8n=0 at n=0,No. As discussed by Doi and Edwards‘, a fictitious tensile force F.q must be assumed to act at the chain ends. If this force were absent, the polymer chain would soon break free of the slip links and assume configurations that would allow it to interpenetrate with the surround- ing chains. For an uncrosslinked polymer the average magnitude of this force is Sk_T Freq = —2 = (3kgT/ b*) < Salt) >a n=0,No (10) It follows from eq 10 that Os, (¢ 2 eosnlt) 6 ee ant n=0,No (11) which implies that the primitive path length of the chain is equal to its equili- brium length, £ =INp, after s,(f) has relaxed. In other words, the chain eventu- ally shrinks back to its undistorted length. In the case of a crosslinked polymer, the chain cannot shrink to its undistorted length because the chemical junction points will restrict its movement. The amount it can shrink will be assumed to be proportional to the macroscopic deformation. Therefore, the boundary con- ditions, eq 11, must be replaced by as,,(t - ctl) ae <| Pu lol n=0,No (12) where 2 is a proportionality constant. Equation 7 and the boundary conditions given by eq 12 describe the dynamics of the chain between two chemical junc- tion points with the stress given by eq 9 or (see Appendix 3), equivalently by, Gag(t) = (8ckyT/ ob?) LM (<in(t)>—) <Vnalt) Vag(t)>a dn — Péag (13) where 1,(é) = Salt) (14) and _ OR, (s,(t):t) = ~asn(f) (18) To evaluate the integral in eq 13, one needs to rewrite eqs 7 and 12 in terms of <l,,(t)> _. Note that, by definition, at Of an ant (16) Substituting eq 7 into eq 16 and taking the average, eq 7 becomes - 53 - oc (t)>a = (3kpT/b?) Fe cinlt)>a (17) with the boundary conditions <la(t)>g = z <| Feu [oy n=0,No5 (18) and the initial condition <l,(0)>_ =0 <| F-u |>y=la(F) (19) Equation 19 is the mathematical equivalent of the statement that the slip links and the chain segments between them deform affinely with the macroscopic deformation imposed at f=0. Equation 17, subject to eqs 18 and 19, can be solved by the standard method of separation of variables. The result is <i, (t)>_ =F alF) (1 -2) Soe sine exp et + 2la(F) (20) where the largest relaxation time, Tg, is given by = _ fo? NE al This is twice the longest Rouse relaxation time. Note that eqs 17 or 7 can be solved only for the average of s,(t). This results in the loss of the effects of the random Gaussian force on the stress. Because the fluctuations of s,(f) are con- sequently not taken into account, eq 13 is true to a first approximation only. It is relatively easy to account for fluctuations in eq 13 but s,(t), rather than its average, can be calculated only as {+~=, Therefore we can assess the effects of the random force on stress only at equilibrium. This problem is addressed and discussed in detail in Appendix 3. It results that the expression for the stress given by eq 13 lacks a second term whose time dependence is not known. We therefore add it to eq 13 in its equilibrium form. This yields - 64 - Gag(t) = (SckpT/b2 _f"(<tq(t)>a)® <Vna VagPu dN 0 + GN<Yna Ynpru — Poag (22) Using eq 20 we obtain Gag(t) = Grlt) Bap + Gen Qag — Poag (23) where Gx(t) = (cNokeT/ No) 2° f(t) = Gy f(t) (24) in which s(t)= § pelt + ew (25) Qap = <YnaYng>u = aa (26) and Bog = 3Qqg0°(F) = 3 <(F-u)q (Fu)s)>5 (27) In obtaining Qa, from eq 22, we have made use of Doi and Edwards's independent alignment assumption, and the fact that <vaqq Vpg>_ has not yet relaxed and hence is constant. The eigenvalues of B, the Finger deformation tensor, are BaF AZ. Hence, the first term in eq 23 can be evaluated. The second term, how- ever, can only be obtained analytically in case of a uniaxial deformation. For uniaxial tension, eq 23 becomes A(t) = G,,(t) — Geet) = Ge(t) (Bii—Be2) + Gew(@11-Qea) (28) where By; —Beg = ®-1/A (29) and - 55- _ asf oa8 [. tenctva=t}] i Q11—-Qee = afer C Vd - i (30) The expression for Q,:—Qez has been adjusted by a numerical constant of 5/3 to yield a modulus of 1 as A~1. We now adduce evidence that eq 23 results from a potential that appears to be constitutive in the narrower sense!. We then generalize it to different strain histories. As tf ~~, eq 23 yields and in uniaxial tension Oy = Gy + Gaypn(r) (32) where G1, -8 we ET (38) and — 811 — Qee en(\) = Se4 (34) In eq 34, the expression for @1,—Qzz is given by eq 30. A plot of equilibrium uniaxial tension data according to eq 32 yields a straight line with slope Gw and intercept Gy. We plotted the equilibrium simple tension data® of Kawabata et al. accord- ing to eq 32 to obtain the two moduli. We then used those values together with the appropriate expressions for Bg, and @g¢ to predict their data for pure shear and equibiaxial tension. The predictions of our theory for equibiaxial tension were obtained using its predictions for compression, since the two deformations are equivalent for an incompressible material. The expression for @,;—Q@az2 in uniaxial compression can be evaluated as - 56 - _ 15{ a3 {tanh vi-as 4) 1 @11— Gaze = B tr | Vian’ 1 7 (35) The averages involved in evaluating Qa, in the case of pure shear were obtained by numerical integration of eq 26 using eq 6. In Figure 2, we have plotted the predictions of eq 31 together with the experimental data of Kawabata et al. In the figure, the ordinate shows the nor- mal stress differences obtained from the data, and the abscissa shows the pred- ictions of the theory. The agreement between the experiments and the theory is excellent, and suggests that eq 31 results from a constitutive potential. An Approximate Strain Function To avoid the need for numerical integration, we suggest a simple expedient. Let the eigenvalues of Q be given by Qa = Ya Ke a=1,2,3 (36) where w = (2/m?) (AT + MF + AS — 3) (37) is the normalized form (W/G) of the BST potential! with n replaced by m. A fit of Qi — Gee = (2/m) (A™—A-?) (38) to eq 30 over the range 1 = <2.5 by a nonlinear least squares procedure yields m.=0.34, Combining eq 37 with the (normalized) nechookean potential w = (1/2) (AP + Ag + A¥ — 3) (39) yields the 2NW potential W = (Gyy/2) (A? + AF + 9 — 3) + (2Gy/m*) (AM + AF + NP — 3) (40) -57 - which we have shown elsewhere! to be constitutive for deformations in elastic equilibrium in a certain range of deformation. In uniaxial tension, the range is 1<’<2.5. This is the range over which the upswing in the o vs A curve is not observed. We have called such deformations moderately large. The expedient we are suggesting is, therefore, applicable only for moderately large deformations. Currie et al. have given® an expression for the potential from which the ten- sor Q can be obtained. It does not satisfy the Valanis-Landel! hypothesis. In other words, the potential wp which, upon differentiation, would yield Q is not of the form Wp = Wy(Ay) + WyCAg) + Wy (Ag) (41) The expedient we suggest does satisfy this hypothesis. It must therefore be con- sidered to be a purely mathematical one, and must be justified by comparing the mathematical predictions for the stress in both cases. We demonstrate the close agreement between the two predictions in the case of uniaxial tension, pure shear, and equibiaxial tension. In uniaxial ten- sion, the two stresses agree to better than 1% over the range 1 <A<2.5. The corresponding range would be 1 <A <1.5 for equibiaxial tension and 1 <A <2.2 for pure shear. The maximum stretch ratios, 1.5 and 2.2, were calculated from the iso-wp lines® on a plot of /, vs. /g, where /, is the first, and J, is the second invariant of the Finger deformation tensor. When the neohookean potential is used for w, these values do not change significantly. The agreement is demonstrated in Figure 3. The parameter Dy is the ratio Gen/ G. For most rubbers encountered in practical applications, Vy falls in the range’ 0.3 < Dy < 0.7 The largest deviation observed in Figure 3 is 10% for ¥y=0.7 in equibiaxial tension. The expedient we have suggested appears, therefore, to be satisfactory for most crosslinked networks. -BB- Since the two expressions agree to within 1% in uniaxial tension, we will dis- cuss the time dependent data in uniaxial tension in terms of eq 40. Equation 20 has been derived for the stress response to a deformation applied as a step function of time. Extension to arbitrary time histories is straightforward. Using eq 3B, eq 23 gives a(t) = Gy(t) (A2-A*) + (2Gy/m) (A™ —\m/2) (42) in uniaxial tension. Figure 4 shows f (t) given by eq 25 as a function of logt for two different values of the relaxation time Tg. It clearly shifts linearly with Tg along the logarithmic time axis. Consequently, eq 42 can be generalized to other strain histories in uniaxial tension by at) = f'Gy(t—w) 2 eu) —v“l(u)] du + (2Gay/m) (A™—-AM™2) (43) Qa Equation 43 is the time dependent form of the 2NW potential applied to uniaxial deformations in arbitrary strain histories. Although our theory does not predict the time dependence of Gy, we make the assumption that it also satisfies the Boltzmann superposition principle. Therefore we replace eq 43 by a(t) = Sct) 2 1n*(u) —A“u)] du + (asm) f'Gylt ee) Ze (Aum Au)? | da (44) This is the most general equation describing the time dependent behavior of rubber networks in moderately large, uniaxial deformations in the framework of our theory. -§9- Experimental Procedure To investigate the time-dependent properties of polymer networks experi- mentally, uniaxial tension measurements were made on a styrene-butadiene rubber sample designated as SBR 6. Its equilibrium stress-strain curve has been reported elsewhere.! The sample and specimen preparation methods are sum- marized in Appendix 2. The experiments were performed on a Floor Model Instron Tensile Tester equipped with an environmental chamber cooled with ‘liquid nitrogen. The data were taken at -37.5°C. The temperature was monitored continually during the experiment by a thermocouple. The maximum deviation observed was 0.2°C. To test the validity of the equations we obtained in the theoretical section, we needed stress relaxation data recorded at different elongations. To obtain these, we would have had to conduct stress relaxation experiments at a given elongation, let the specimen relax for about 30 hours at room temperature, and repeat the procedure at another elongation. We were able to reproduce a given temperature in our environmental chamber only within two degrees. Hence, data taken under these conditions would require time-temperature superposi- tion. Since, it is not clear whether this is permissible in large deformations, we devised a different procedure. We extended a specimen to a specified elongation, let it relax for about 30 minutes, and then increased the deformation as required. This method eliminates the problems associated with time temperature superposition but requires special processing of the data which will now be discussed. Stress relaxation measurements were made as a step function of time. For a number of uniaxial steps superposed on one another, the time- dependent elongation ratio is given by -60— =] where A (w—t;) is the unit step function applied at w=t,, Aj is the jth elongation ratio referred to the rest state, f; the time at which the jth deformation is applied, Ag=1 and ¢,=0. With these definitions eq 45 implies Mu) =1+ DOP —AP) A(u-ty) (46) where 7 will take the values 2,m,-—1,-m/2 as required. When substituted into eq 44, eq 46 yields a(t) = D» S'Galt 0) (OF —AE3) ~ jt AGA] BUH) tu + (27m) J *Gy(t 1) [CJM-AP) — (Az/? -077172)] 6(u-t,) du (47) where 6(u—t;) is the delta function acting at u=t;. Performing the integration and obtaining @,(t)-a,_,(#) from eq 47 we find Belt) — Galt) = Gr(t—ty) [AF —AP) — AT? AG] + (2/m)Gy(t—t,) [AP APA) — OG? — 7?) ] (48) This furnishes the stress in the ith experiment carried out at elongation 4,, from which the residual stress left over from the preceding experiment at the elonga- tion 4;-,; has been subtracted. The response to a series of step elongations is shown in Figure 5. The residual stress is, of course, not known since it has not been recorded. If we could predict it, o,(t) -3,-,(t) where 4 has been applied at t;, would be completely equivalent to @(f) However, we cannot do this at this stage. Instead, if the interval ¢;—#;_, is large enough so that the residual relaxa- tion is negligible, we can write with excellent approximation, -61- Gltette) — T(t te’) = T(t +t") — Tia fe) (49) where ty’ = t—t Ostis (tri-t) (50) The #,"s can, in general, be selected at will. We make them all equal, however, to insure that the stress in each experiment represents the same state of relaxa- tion, i.e. that it is isochronal, Substituting fati=ti, =... =8f (51) into eq 50 and the result into eq 49, we obtain oy Oy {t,+t°) — T(t) oul) = QFN) ~ Ov NA) = Gy(t Wa) + Gxt") (52) where at (NB) — (Ag? AIA) m{(AE AZ) — (AA) (a) = (53) and m=0.34. The symbol oy(t*) is the equivalent of the Mooney stress at the isochronal time t’ in the ith experiment. Equation 52 implies that a plot of oyi(t’) vs ¥(%) yields a straight line with slope Gy(t*) and intercept G;(t”). Data plotted according to eq 52 are shown in Figure 6 for three different iso- chronal times t’. The linearity of the plots indicates that our procedure for data reduction is adequate. The plots were constructed by taking isochronal cuts from Figure 7 which represents o,(t—f,;;) as a function of t in logarithmic coor- dinates. The symbol o,(f -t;:4;) refers to the nominal stress at the indicated time and elongation. Figure 8 compares the time dependence of Gy(t) obtained from these experiments with the time dependence predicted by our theory. The prediction requires knowledge of z? and tg. The parameter z* was obtained from eq 24 which gives -62- G 2-2 eX e* ‘GkaTNo/ Ne (54) But ckpTNo/ N, = pRT/ MZ (55) where p and R have their usual meaning, and Mf is the molecular mass of a seg- ment between slip links in the crosslinked network. In a previous publication,’ we have shown that Gy = EGR (56) where the value of ¢ has been estimated as 0.354 for a tetrafunctional undiluted network. Gf is the plateau modulus of the uncrosslinked polymer which has a value of 0.778 MPa for SBR! Equation 56 together with eq 27 of Appendix 3 imply pRT/ Mz = &GP (57) which, using eq 54, allows us to estimate z?. The value of log Tg = 1.70 was obtained by shifting the theoretical curve onto the experimental points. Equation 21 can be scaled to yield _ (62) (a9Z'/?)? 1B BrtkyT (58) in terms of monomeric parameters. In eq 58 ¢, is the monomeric friction coefficient, Z is the degree of polymerization, and dy is the monomer length. If one calculates Z from the equilibrium value of Gy(t) as Gy =pRT/2Mc, and uses the published values® for ¢, and ag, one obtains Tp=1.78, in excellent agree- ment with the experimental value. Figure 9 shows the experimental time dependence of Gy(t). Although from our theory we predict its existence and its strain dependence, we cannot predict its time dependence. Knowledge of the linear relaxation modulus of the polymer -63- is required before one can predict its nonlinear time-dependent properties. As afurther test of our theory, we examine its predictions when it is applied to different strain histories. We consider two strain histories, namely constant rate of strain, and trapezoidal excitation. The experimental data were taken from the work of Bloch, Chang, and Tschoegl.® Constant Rate of Strain For a deformation in uniaxial tension at a constant rate of strain é, the time-dependent extension ratio is given by Mu)=1t+éu (59) Substituting this into eq 44 we obtain a(t ;é) = 2é f *Gelt —u) [(1+éu) + 0.5(1+éu)*]du +28 f'Gy(t—w) [(1téu)™ + 0.5(1+6u) 2 du (60) 0 where é has been shown as a parameter in the symbol for the stress to distin- guish the deformation from stress relaxation. From the near equilibrium stress relaxation data at different levels of elon- gation reported in reference 9, we obtained the parameters required for our theory as follows. We took an isochronal cut at 10 minutes, and plotted o(t*} as a function of A. We then used eq 52 with i=1, to obtain G,(t*) and Gy(t*), where t*=10 minutes. We found Gy(t°)=0.102 MPa and Gy(t*)=0.193 MPa. SBR allowed to relax at room temperature for 10 minutes is close to elastic equilibrium. Thus, we obtained logtg=—5.63 min. and z#=0.371 from G;y(t*), using eqs 54 and 58. The sum of the two moduli G=Gy+Gy is different from the estimate of the linear shear modulus, given by Bloch et al. This is due to an error in their data - B4- reduction scheme,’° and associated with converting from engineering units to SI units. Our estimate of G fits their data using their theory. Bloch et al. approximate the linear shear modulus of their sample using the equation G GwH=G, + TH(t/7* (61) with G,=0.249 MPa, G=1.356 MPa, 7=6.8x10~? min, and k =0.1848. Once again the values reported here differ from those of reference 9, by a constant factor. In terms of our theory we have with Gy=0.102 MPa and G.y=0.1470 MPa, and we approximate Gy(t) by G 1HE/T) (83) Gu(t) = Ga + Note that we have taken Gy(f) equal to its equilibrium value. As seen from Fig- ure 4, relaxation of Gy(t) is complete when log? is greater than log Tg+1. Experimentally, in relaxation at a constant rate of strain, the relaxation time of order (1/é) is the most heavily weighted one. The highest strain rate used in reference 9, 4.56 min™, correponds to a time of 0.22 min. That is several decades longer than Tg, and therefore the relaxation of G,(f) is not observable in the experiments. We make the same assumption in eq 60 also, and set Gy(t-—w)=G,x. Using eq 63 for Gy(t), we integrated eq 60 for the given strain rates. The trapezoidal rule, with 500 subintervals, was used for the second term. No appreciable difference was observed in the results when the number of subin- tervals was increased to 1000. Figure 10 shows the experimental results of Bloch, Tschoegl and Chang together with our predictions. The agreement is very good. - 65- Trapezoidal Excitation We now consider the following deformation. Impose a constant rate of strain £, from t=0 to t=t,. Let the sample relax at that elongation until f=¢g, and impose another constant rate of strain —ég in the opposite direction. The excitation function thus has the shape of a trapezoid. The time-dependent elon- gation ratio is now given by Au) =1+é,wh(w)—é, (u—t A (uty) — 2 (u-te)h(u—t) (64) Substituting eq 64 into eq 45, we obtain Bt) =a(tié,) Ost<t, (65) where o(t;:é,) is given by eq 60, and Ot) = G(t it1.81) = Gey [AP(t1)-A*'(E1)] +26, f 'Gy(t—u) [(142u)™1 + O.5(1+eu)m2Jdu tt, <t Sty (68) G O(t) = Ot st 1,81) + Gay [A*(£)—-A ME) ARE) +A TN(t,)] + 2g Jf *Gy(t 2) [(ap(u))™-! + 0.5(Ap(u)) tm 2"2]du tg <t (67) In eqs 65 through 67 a(t;é,) is given by eq 68 and Ap(u) = 1 + &¢, — ea(u—ta) (68) To obtain the stress resulting from this deformation, we used eqs 61 through 63 in eqs 65 through 69. Figure 11 shows the data of Bloch et al. and our predictions as stress plotted as a function of time in logarithmic coordi- nates. The value of ég we used for our predictions is 1/10 of the strain rate they report, a difference attributed to typographical error.!® Since the plot in Figure - 66 - 11 is logarithmic, it is not very sensitive to differences between experiment and theory. Therefore, we calculated the elongation ratio and the stress from the strain rate and plotted the results in Figure 12 together with the data of Bloch et al. The agreement between our theory and the experimental data is again excellent even in this more sensitive plot. Conclusions We have developed a molecular theory which describes the time-dependent properties of rubber networks. The theory predicts a two term potential which determines the stress-strain properties of the network. The first term of the potential has a modulus G,y associated with it at equilibrium. This term arises from the change in the average path length of the primitive path. The predicted stress-strain behavior is nechookean for this term. An expression for its time dependence is also obtained. The second term of the potential, which has a modulus Gy associated with it, arises from segment length fluctuations. It has a complicated strain dependence which requires numerical integration in all but uniaxial deformations. We have shown, however, that the simple BST potential furnishes an adequate approximation. The resemblence of the obtained form and the interpretation of its parame- ters bears a striking resemblence to the equilibrium theory of Flory and Erman,'! although the route we have chosen is quite different from theirs. Our molecular picture for crosslinked networks is based on the ideas intro- duced by Doi.” Thus, we would like to compare our predictions with Doi's predic- tions for an uncrosslinked polymer. In the uncrosslinked polymer, the relaxa- tion process corresponding to the relaxation of s,(t) is not. observed in infinitesimal deformations. It becomes activated at large strains, and manifests itself in a hump in a plot of log of{f) vs. log ¢. Furthermore, time-shift invariance - 67 - is not preserved when this relaxation process is activated. All of these predic- tions are in accordance with the stress relaxation data of Nordermeer and Ferry.!” An examination of eqs 23 through 31 reveals that in a crosslinked network,the restrictions that are imposed by the chemical junction points on the movement of the chains results in a picture that is qualitatively different. First, our theory predicts a plot of log a(t) vs. log ¢ which is monotone non- increasing for a deformation applied as a step function of time. Also as A~1, we obtain o(f)=3G(t)e in uniaxial tension, where G(f)=Gy(t)+Gy(t). Conse- quently, one can use the information obtained in linear stress-relaxation experi- ments to predict nonlinear, time-dependent behavior. Second, when the poly- mer is uncrosslinked, the chains can escape from their constraints and attain random orientation after the relaxation of s,(¢#) reaches equilibrium.Such a reptation motion is impossible in a crosslinked network, because of the pres- ence of the chemical crosslinks. The effects of segment length fluctuations are not considered by Doi in the nonlinear deformation region. Graessley'™ points out that such fluctuations would compete with the reptation motion. This is not a problem for a mathemat- ical description of the crosslinked network. Thus, the nonlinear stress-strain behavior of crosslinked networks is predicted to be less complicated than the behavior of the uncrosslinked polymer. We consider it to be the outstanding feature of our theory that it allows one to use the information gleaned from stress-relaxation experiments in infinitesimal deformations to predict nonlinear, time-dependent stress-strain behaviorin moderately large deformations. - 8B- References 10. 11. 12. 13. . Chapter l of this thesis . M. Doi, J. Polym. Sci., Polym. Phys. Ed., 1B (1980) 1005 . M. Doi, private communication . M. Doi, S.F. Edwards, J. Chem. Soc. Faraday Trans. 2, 74 (1978) 1802 . 5. Kawabata, M. Matsuda, K. Tei, H. Kawai, Macromolecules, 14 (1981), 154 . P.K. Currie, J. Non Newtonian Fluid Mech., 11 (1982), 53 . Chapter II of this thesis .d.D. Ferry, Viscoelastic Properties of Polymers, John Wiley, New York, N.Y., (1980) 374 . R. Bloch, W.V. Chang, N.W. Tschoegl, J. Rheal., 1 (1978) 22 R. Bloch, private communication P.J. Flory, B. Erman, Macromolecules 15 (1982) 800 J.W.M. Nordermeer, J.D. Ferry, J. Polym. Sci. Polym. Phys. Ed. 14 (1976) 509 W.W. Graessley, Adv. in Polym. Sci., 47 (1982) 67 -69- slip link n th Rouse bead chemical junction point Figure 1: Schematic of the model of the polymer chain. io -70—- l 6 B me me | me ms mf mm I e s @ —R_ 1.4 F Data of Kawabats et al. 1.2 - @ 1.0 - 0.6 Tr 0.6 Ff 0.4L @ Equibiaxial tension @ Siaple tension 0.2L @ Pure shear (max. stress) 4 Pure shear (intermed. stress) = | a | | | = | A _f Figure 2: Time dependent theory compared to data. 2... = mm ym -\a) + Vy Agn¥g) (2/m) 2 Qa (1-590 wht -71- @ Equibiaxial tension v a Simple tension @ Pure shear 4 | j S| = | 1 2 3 4 5 (1-0,,) (B,-Bg) + Uy (Q-@) Figure 3: Comparison of the full tensorial form of the strain function with the BST potential. -72- *auf] peonpeiz jo uofzoUNZ & Se Xy jo aouapuadap amy2 pazoFpead ayy ty aaindTd v- S- o- Ob 96°0 = .F 4G'} j © “i "9/"9 *suo;JeZuoTS JusaIazZITp Qe BIEP VOTIEXBTA1 sS9198 UfZeIGO 03 pesn ainpadcoad [eIUauTAedxa ay jo JFZewayosS y i¢ sanBTy ~73- ae | ee ee ee oe ae ae ee ty Fo me ee soe ee mem ste ee cue mane ty tom - ~74- SBR 6 -37.5° C 0.0 a2 0.4 0.6 08 1.0 Figure 6: Isochronal cuts from stress relaxation data on appropriate coordinates to obtain the time dependence of Gy and Gye ~75—< *SUOTIBBUOTA JUaIazJTP 2B BIEP UOTJeKETaI SBaIIG = t/ BaNBTZ oe sz oz st ot v0 T T T T T T T T r T net o— o—__.__, 7° ° o_o -@ o—____» oe | ~76§- *auo pazoTpaad ayy 07 paredwoo Xo $o souapuadap oupa Teauempredxq +9 an8qy (%/1)30 oO s0- O'1- i T T “98 @L°T = (2T89)92 B07 "0998 O/°T = (dxay 2) Bol ° = Zz 647E°O Zz vas e Ak pl So -7/- oN 5 jo eouapuadap ow]. [equamfaadxg 6 aainsty (% 3) 80 O'l s‘0 0'0 sO0- O1L- as B/°T = a, 8oT tO D oS°LE- 720 9 wus a 460 ~~ = _ ee e v0 al e.h6©°8 r e = e s, a a et ee ee e 1s0 -738- T T T T TT € 4.536 A e 0.5 F _ @ 0.4536 0.4b 0.02268 . 0.001134 “ <= 03- . — i -) 0.2- - SBR Data of Bloch et al. ort Y/ J L i 4 _L 1 0.2 0.4 0.6 0.8 2.0 x Figure10; Constant rate of strain experiments at different strain rates compared to the theoretical predictions. -~79— *upei3s jo uoTIUN] [Tepfozedelqy B OF YAS jo asuodsay sTtT aansyy (aw) 1 30; ; ; *{e@ Ia YOoTE Jo BIeq was yEIT*O- = 3 9CS*y = 3 -~80— 0.S- SBR Data of Bloch et al. 0.4} 0.3} & (MPa) 0.2} € = -0,1134 0.1 1.25 1.50 A Figure 12: Response of SBR to a trapezoidal function of strain. 1.75 -§1- Appendix I This appendix gives a mathematical description of admissible deformations, and the related deformation gradient tensors. A schematic defining the various vectors and tensors mentioned here, is supplied in Figure A-1. Let the material points of a continuum occupy a region 2 before deforma- tion. The position of a material point P in this region can be specified by a vec- tor P with the cartesian coordinates X;, Xp, X3. Let B be mapped into region 6b, and P into p, as a result of the deformation. Denote the position vector of point p with p, and let it have the cartesian coordinates 2), %2, z3. Mathematically, using indicial notation, this mapping can be described as Ze = Ye (X1,X2.Xo) k=1,2,3 (A-1) or the inverse mapping by Xx = Yx(x 1.2.03) K=123 (A-2) The deformation is admissible if the mapping is one-to-one with a nonvan- ishing Jacobian. Yx and y, are then inverses of each other. Note that the condi- tions of admissibility express the fact that matter is indistructible,; i.e. no region with a finite volume is mapped into one with infinite or zero volume, and that it is impenetrable; i.e. the deformation carries curves into curves,surfaces into surfaces, and a region into another region. Equations A- 1 and A-2 imply dt, =YeaxGXe , GXe= Yep dt (A-3) where a comma indicates partial differentiation. If one now defines the differential position vectors in both systems, one obtains -82- dp=dm%i, , dP=dXx lx (A-4) and (ds)? =dpdp = 6, dz, dz, = dz, dz, - (A-5) (aS)? = dP-dP = dg, dX dX, = dX dXx (A-6) where ds and dS are the line elements in the two systems, and i and Ix are the unit vectors of the coordinate systems. Substituting eq A-3 into eq A-4 one obtains dp=qdm , dP=CydXx (A-7) where %=Yeele . Cr=texk (A-8) Substituting eq A-7 into eqs A-5 and A-8, one obtains (ds)® = Cy, dXx dX, , (dS)* =cy, dz dz, (A-9) ” where Cs =O, | Cx = CxCy, {A-9) and cy; and Cy, are the Cauchy and the Green deformation tensors. Their inverses are the Finger and the Piola deformation tensors respectively. ~83~ *d qupod our g qupod pue q uoT3aq1 OUT gq UOTZeI SaTAIeD Jey UOT JeUIOJap STAFPSSTUpe uy :qT-y aansyyq ~84— Appendix II This appendix summarizes the sample and specimen preparation methods. The styrene-butadiene rubber networks had the following composition: SBR 1502 100 parts antioxidant 1 part crosslinker variable The following amounts of crosslinker were added: SAMPLE CROSSLINKER (phr) SBR 6 1.02 SBR 2 0.67 SBR 4 0.5 SBR 3 0.3 N-phenyl-2-napthyl amine was used as antioxidant.The crosslinker was Her- cules Inc. Di-Cup 40, which is dicumyl peroxide supported on calcium carbonate. The weight fraction of dicumyl peroxide was 40%. This crosslinker was used instead of straight dicumyl peroxide to facilitate dispersion. The ingredients were milled on a two-roll laboratory mill cooled with water. The SBR 1502 was milled for 15 minutes until an even spread was obtained on the rolls. The antioxidant was then added and the sheet was milled for an addi- tional 20 minutes. During this process, the roll separation was reduced and again increased, and the sheet was folded several times both lengthwise and widthwise to obtain better dispersion. The Di-Cup 40 was added, and the sheet was milled for 10 more minutes, following the same procedure as in the case of the antioxidant. Finally, the sheet was allowed to recover at room temperature for at least 48 hours. -55- After the recovery was complete, the required amount of material was placed in a 15.2 by 15.2 by 0.2 cm. mold and cured at 157°C and 175 MPa for 45 minutes. The press, the mold and the rubber were then cooled to room tempera- ture overnight. This molding procedure was necessary to avoid the formation of bubbles in the sheet. The natural rubber (NR) was obtained through the generosity of the Goo- dyear Tire and Rubber Company in the form of cis-pelyisoprene (NATSYN 2200). The samples were prepared for us by the courtesy of Phillips Petroleum Com- pany. The networks were prepared by adding the required amount of dicumyl peroxide to the rubber. The sheets were then molded at 150°C for two hours. No antioxidant was added. The amount of dicumyl peroxide used for the various samples are listed below: SAMPLE CROSSLINKER (phr) B 3.44 C 2.31 D 1.13 E 0.77 Specimen Preparation The same specimen preparation technique was used for both natural rubber and for SBR. Strips were cut from the molded sheets by a knife-edged mill blade. The sheets were bonded to the surface of a polyethylene sheet, using double-sided adhesive tape to facilitate the cutting. The strips had dimensions of 12 x 0.5 x 0.2 em. Phosphor-bronze U-tabs were first bonded to the specimen ends with epoxy resin. After the resin had set, the tabs were removed and were then reglued using Zip-Grip 10. This procedure was necessary, because the Zip- Grip did not form strong enough bonds between the U-tabs and the strips ~36— without the epoxy. To minimize end effects, the tab ends were made to extend over the specimen for no more than 2 mm. Before testing, the specimens were allowed to relax at room temperature. Measurement of Mechanical Properties A Floor Model Instron Tensile Tester was used in our stress relaxation experiments. The temperature was controlled through a Missimer's Environ- mental Chamber, liquid nitrogen was used if cooling was needed. The force measuring system of the Instron uses load cells with an accuracy of + 0.25%.The output of the load cells was observed to be nonlinear and they were therefore calibrated with several weights before and after an experiment. A typical calibration curve is shown in Figure A-2. In the figure, the symbols show the calibration points for the three scales of the load measuring system. The solid line is the curve fitted to the experimental points. In some instances the crosshead would creep, i.e. after the initial deforma- tion, during stress relaxation, the crosshead would move up, changing the initial elongation. For this reason, the elongation was measured before and after every stress relaxation experiment. Data sets during which the elongation changed were discarded. The forces were recorded on a Hewlett-Packard recorder connected to the load weighing system. -87- a t q 250 . Scale A @ 2 1 B 5 0.4 200+ A 10 0.2 = uo oO UJ Ax force( gr ) 100} SOF A. l 50 100 150 mm ef chart paper Figure A-2: A typical calibration curve. Appendix II In this Appendix we derive the contribution of path length fluctuations to stress at equilibrium. For completeness, parts of reference 2 have been repro- duced. Our eqs A-1 through A-5 correspond to eqs 6.1, 6.2, 6.3, 6.5, 6.6 of refer- ence 2. Our eq A-7 corresponds to eq 8.19, our eqs A-8 and A-10 correspond to eqs 6.20 and 6.21, respectively. Consider a Rouse chain consisting of N, + 1 Rouse segments, fo ,F}........ ry with its ends n =0 andn=N, fixed in space at the origin and at R, respectively. Calculate the stress defined by Bap(t) = (SekpT/b2) fs Srna) ornglt) >, dn — Péap (A-1) [+] subject to the initial conditions r,(0)=Fr2 , R=F-R (A-2) where the superscript 0 denotes the quantities before deformation. The equa- tion describing the motion of r, is Or, (t) _ ) dr, (t) (-= (SkgT/ b* a at 5 + fp (t) (A-3) Since r,(f) = 0 at n=0, and r,(t) = R at n=N,, we introduce the normal coordi- nate x, as _n 2 v2 __, pmM _ Tn(t) Ne R+ (WO Dxp(t sin A (A-4) where the time dependence of x, is exponential and decays to zero as f=. Using eq A-4 in eq A-1 the integration can be carried out and yields Gap(t) = (Sekg T/b*)[ <Rolp>a “Ne -39— + Liens Ne)" <Ipalt) Zpelt) >a ]- Pbag (A-5) This describes the relaxation process characterized by T,. At its completion the second term on the right hand side has decayed to zero. Therefore, the stress due to a length of chain between slip links is SekpT og => “N,b2 <Rokp>a (A-6) Next we show that for ¢>r, the stress is given by eq 13 of the text. Note that a chain between two crosslink points has N slip-links, Rp,R,,......Ry denoting their positions. Then it follows from eq A-6 that the stress due to the complete chain is BekgT Cop = "3 »» <( Rea e-1)0)( Rep e-1)p)> 0 (A-7) e = - since R now is R, —Rx-1. We set <( Fea~Re-1)a)t Rep—F &-1)8)> a = <VnaYngraNe <iney>e (A-8) for N,(k—-1)sn<N,k because VnaYng . gives the direction of (Re —Fe-sJa( Ae —Re-i)g and Ne<iy(t)>_ is the average length of (Ry, —Re-1). Therefore k <( Rea -1)0)( Rep—Re-1)8) >a = Ne hy taste (A-9) re — since v, and l, are independent of n provided N,(k-1)<n<N,k. Substituting eq A-9 into eq A-7 we obtain SckyT og = = f " <vpaVnp>a<ln(t)>2 dn. (A-10) which is the stress due to the change in the average path length. This is eq 13 of the text. -90- As discussed in the text, we cannot write the boundary conditions and the initial condition in terms of s,. We can only estimate the magnitude of fluctua- tions on the stress at equilibrium. To calculate the contribution of the fluctua- tions to the stress, we need to consider the boundary conditions at the chain ends at equilibrium. By eq 7 of the text, the motion of S,(t)} is described by é*s, on = (BkgT/ b*) att Salt) (A-11) oe The boundary conditions at equilibrium result from the requirement that the chain ends be fixed at the crosslink points. Thus, we write the boundary condi- tions as Sa(t)ineg =O > Snit)inewy = NozialF) (A-12) In eq A-i2 the random force satisfies <falt)>u=O0 and = <falt)fm(t')>a = 2tkpTd(n—m)d(t—t') (A-18) The following form satisfies eq A-12 s,(t) = n2la(F) + (2/ No)'/*})z,(t) sin (p mM/ No) (A-14) PB Substituting eq A-14 into eq A-12, we obtain elt) —Aptp(t) +fp(t) (A-15) where _ SkpT pFné Ap = Be Ne (A-16) and f,{t) is defined by Sfp(t) = (2/.No)*? (170) f'°sin Ey Salt) an (A-17) 0 Using eq A-13 it can be shown that the average of f, is zero and -9l- Sp (t1) Sp(te)>a = “reer 6(t\- te) (A-18) Equation A-15 can be solved to yield p(t) =ap(O)exp(-Apt) + fTexp[-Ag(t—tn] fo(tiddti (A198) which when used in eq A-14, gives s,(t) = nzloa(F) + (2/No)'/?)isin Ey exp(—Apt) (zp (0) P + f ‘exp(Apt) Sn (t) dt, ] (A-20) From eq A-20 the contribution of fluctuations to the arc length at equilibrium can be calculated as b? Ne yi 2 sin? BA (A-21) <(s,(t)- <sy(t)>_)*>a = P p*n* Nog The square of the primitive path length of the chain is given by L? = [7 + <(L-L)* >, - (A-22) where Z is the average of Z, and f'<(sq(t)—<sq(t)>)?2>a dn <(L-L)*>, = — = Nob! (A-23) f “ndn ° Per chain segment between two slip links, the square of the path length is UB =1/N (<(L-L)*>,) (A-24) where the subscript F has been used to denote fluctuations. Thus the curvature of the chain due to fluctuations results as -92-— <( Rea ge-t)a)( Rep—Re-1)p)>& = <VnaVnp> ul? (A-25) where now the superscript F on the components of R denote fluctuations. In eq (A-25), <vpaVng> is the direction, and Uf is the square of the length due to fluctuations, of R. Substituting eq A-24 and A-25 into eq A-7 gives Bes = chpTNo/ Ne <YnaVnp>a (A-26) which is the contribution to stress at equilibrium due to fluctuations. We set Gen = ckpTNo/ N, = pRT/ MS (A-27) to obtain eq 22 of the text. ~93- Appendix IV This appendix lists the values of C, and C2 from which Gy and Gy were obtained, in section 2. The references are given in Table 1 of section 2. Table entries are grouped in fives for ease of reference Table 1 Poly(Ethyi Acrylate)-(PEA) 2C, (MPa) 2C2 (MPa) Gy (MPa) Gy (MPa) 0.0814 0.0335 0.074 0.038 0.0462 0.0585 0.033 0.067 0.0357 0.0578 0.022 0.066 0.0365 0.0463 0.0260 0.053 0.0181 0.0718 0.001 0.082 Table 2 Butyl Rubber-(IIR) 2C, (MPa) 2Ce (MPa) Gy (MPa) Gy (MPa) 0.0772 0.100 0.054 0.114 0.0995 0.112 0.073 0.128 0.104 0.728 0.087 0.83 —-I4— Table 3 Poly{Dimethyi Siloxane)-(PDMS) 2C, (MPa) 2Ce2 (MPa) Gy (MPa) Gy (MPa) 0.061 0.054 0.048 0.061 0.065 - 0.051 0.054 0.058 0,069 0.039 0.060 0.045 0.074 0.083 0,054 0.095 0.080 | 0.049 0.068 0.058 0.098 0.105 0.0793 0.120 0.100 ; 0.078 0.082 0.089 0.120 0.109 0.095 0.124 0.036 0.062 0,022 0.071 0.071 0.057 0.058 0.065 0.079 0.049 0.068 0.058 -95- Table 4 Natural Rubber-(NR) 2C, (MPa) 2Cz (MPa) Gy (MPa) Gy (MPa) 0.196 0.196 0.150 0.224 0.230 0.196 0.184 0.224 0.280 0.202 0.233 0.230 0.340 0.206 0.292 0.235 0.344 0.206 0.296 0.235 0.498 0.204 0.450 0.233 0.604 0.280 0.557 0.228 0.031 0.088 0.011 0.101 0.048 0.123 0.020 0.140 0.066 0.127 0.037 0.145 0.076 0.128 0.046 0.146 0.092 0.133 0.061 0.152 0.107 0.129 0.077 0.147 0.129 0.145 0.095 0.165 0.150 0.162 0.115 0.173 — 0.088 0.199 0.056 0.159 0.149 0.137 0.117 0.156 0.188 0.157 0.151 0.179 0.196 0.147 0.162 0.168 0.196 0.196 0,150 0.224 0.122 0.112 0.096 0,128 0.140 0.138 0.108 0.155 0.198 0.2B6 0.300 0.358 0.304 0.340 . 0.230 0.188 0.147 0,265 0.142 0.158 0.166 0.176 0.119 0.110 0.196 0,157 0.112 0.108 -96-. 0.165 0.249 0.261 0.317 0.276 0.314 0.184 0.151 0.121 0.240 0.162 0.180 0.189 0.201 0.186 0.125 0.224 0,179 0.128 0.123 -97- Table 5 cis-1,4-Poly-Butadiene Rubber-(PBD) 2C, (MPa) 2C2 (MPa) Gy (MPa) Gy (MPa) 0.14 0.42 0.042 0.479 0.15 0.18 0.108 0.205 0.16 0.30 0.090 0.342 0.18 0.93 0.056 0.8624 0.30 0.34 0.221 0.388 0.33 0.53 0.208 0.604 0.38 0.54 0.254 0.616 0.41 0.45 0.305 0.513 0.085 0.206 0.037 0.235 0.241 0.343 0.161 0.391 0.384 0.376 0.296 0.429 0.192 0.298 0.123 0.340