896
Macromolecules 1989,22, 896-908
(17) Kolinski, A,; Skolnick, J.; Varis, R. J. Chem. Phys. 1987,86, 1567;1987,86,7164; 1987,86,7174. (18) Rouse, P.E.J. Chem. Phys. 1953,21,1272. (19) Brown, W.Macromolecules 1984,17,66. (20) Fujita, H.; Einaga, Y. Polym. J. 1985,17,1131. (21) LBger, L.;Hervet, H.; Rondelez, F. Macromolecules 1981,14, 1732. (22) Kim, H.; Chang, T.; Yohanan, J. M.; Wang, L.; Yu, H. Macromolecules 1986,19,2737. (23) Callaghan, P.T.;Pinder, D. N. Macromolecules 1984,17,431; 1981,14,1334. (24) Wesson, J. A.; Noh, I.; Kitano, T. Yu. H. Macromolecules 1984, 17,782. (25) Fleischer, G.; Straube, E. Polymer 1985,26,241. (26) von Meerwall, E. D.; Amis, E. J.; Ferry, J. D. Macromolecules 1985,18,260. (27) Amis, E.J.; Han, C. C. Polymer 1982,23,1403. (28) de Gennes, P.-G. Macromolecules 1976,9,587,594.
(29) Adam, M.;Delsanti, M. J. Phys. (Les Ulis, Fr.) 1983,44,1185. (30) Brown, W.; RymdBn, R. Macromolecules 1986,19, 2942. (31) Kendall, M. G.; Buckland, W. R. A Dictionary of Statistical Terms; Oliver and Boyd: London, 1960. (32) Brown, W.; Johnsen, R. M.; Stilbs, P. L.; Lindman, B. J. Phys. Chem. 1983,87,4548. (33) Brown, W.;Mortensen, K. Macromolecules 1988,21,420. (34) Cukier, R. I. J . Chem. Phys. 1983,79,3911. (35) Cukier, R. I. Macromolecules 1984,17,252. (36) Altenberger, A. R.; Tirrell, M. J. Chem. Phys. 1984,80,2208. (37) Altenberger, A. R.; Tirrell, M.; Dahler, J. S. J. Chem. Phys. 1986,84,5122. (38) Freed, K.F.; Muthukumar, M. J. Chem. Phys. 1978,68,2088. (39) Muthukumar, M.;Freed, K. F. J. Chem. Phys. 1979,70,5875. (40) Rendell, R. W.; Ngai, K. L.; McKenna, G. B. Macromolecules 1987,20,2250. (41) Doi, M.; Edwards, S. F. J. Chem. SOC.,Faraday Trans. 2 1978, 74, 1789.
Diffusion and Relaxation in Oriented Polymer Media William W. Merrill and Matthew Tirrell* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
Jean-Frangois Tassin and Lucien Monnerie Laboratoire de Physicochimie Structurale et Macromoleculaire, AssociZ au CNRS, Ecole Supgrieure de Physique et de Chimie Industrielles de la Ville de Paris, 10,rue Vauquelin, 75231 Paris Cedex 05,France. Received May 31,1988; Revised Manuscript Received August 8,1988
ABSTRACT: A model for reptation in oriented polymer melts is presented in which local orientational correlations between the segments of the chain molecules are incorporated. The presence of local orientational correlations has been indicated by an array of equilibrium experiments on alkane fluids and on oriented polymer networks. More recent dynamic experiments by Monnerie and co-workers have indicated that the relaxation behavior of polymers can also be affected. Our model to account for the effect of this orientational bias on reptation is a modification of the Doi-Edwards model. The Doi-Edwards aseumption of isotropic chain end segments is replaced by a coupling condition between the chain end orientation and the average orientation of the medium. Since the medium is the melt of chains itself, this is a self-consistent mean-field treatment. The retarded stress relaxation rate and other viscoelastic properties are calculated for monodisperse polymers and polydisperse mixtures. The potential implications of this orientational bias for the dynamic properties of polymers are discussed.
I. Introduction The reptation model of de Gennes,l as adapted by Doi and Edwards2-6for chains under deformation, has been a highly successful theory for describing polymer dynamics. It forms a theoretical bridge between macroscopic viscoelastic relaxation behavior and microscopic molecular diffusional phenomena. In the Doi-Edwards model, the chain moves by wriggling along its axis like a snake because a mean-field, tubelike constraint formed by the neighboring macromolecules surrounding each chain prohibits perpendicular motion. Except for this tube constraint, each chain moves independently in the melt. In particular, the unconstrained chain ends, acting like the head of a snake, can freely choose the direction of motion as the chain moves out of its existing tube into a new tube. Because interchain interactions occur only through the tube constraint, i.e., through "entanglement", the chain end chooses the direction of the new tube path isotropically. An array of equilibrium experiments on alkane fluidsand on oriented polymer networks,%13however, indicate the existence of local orientational correlations which lead to interchain interactions not accounted for by the tube constraint (entanglement couplings). Since these correlations are directional, they also cannot be lumped into a
scalar friction coefficient. These correlations have their origins in packing constraints in the liquid state.l"" When the mean orientation of segments in a polymer melt increases because of the application of a strain, orientational correlation could enhance the anisotropy of the system. Dynamic experiments on polymer melts by Tassin and Monnerie1*J9suggest that these correlations can retard the relaxation behavior of entangled polymers. In this paper, we incorporate the effects of these orientational couplings on the configurational relaxation of entangled polymer melts by modifying the Doi-Edwards postulate of orientational isotropy of new tube segments. We begin by reviewing the aforementioned experimental evidence in section 11. In section IIa, we review the current Doi-Edwards (henceforth DE) model and introduce our notation. We present our model in section IIIb, where we assume that the new tube acquires some residual orientation related to the mean orientation present in the medium during that new tube creation. Specifically, we allow a new tube segment at its creation time t = t' to assume a fraction a of the orientation H(t'). To avoid mathematical complexity, we neglect tube fluctuations5 (a relaxation of the constant contour length postulate) and constraint release" (a relaxation of the strictly axial motion
002~-929~/89/2222-0896$01.50/0 0 1989 American Chemical Society
Macromolecules, Vol. 22, No. 2, 1989 postulate) and focus on the basic model during the terminal relaxation process. We concentrate on the case of uniaxial extension in which the Hermanns' orientation function H ( t ) naturally describes the level of orientation in the medium. We then extend our treatment to general deformations. In section IVa, we provide an analytic solution for the orientation function H ( t ) for the case of linear coupling (constant a)and in section IVb explore the effects of nonlinear coupling which could occur in highly oriented systems. In section IVc, we present the effects of linear coupling on linear viscoelasticity. We find qualitative similarity to the DE model for monodisperse systems although we find quantitative changes in the properties resulting from the retardation of the relaxation process. In the polydisperse case, we find the component contributions to relaxation significantly altered although we still find a linear blending law in the linear coupling case. Finally, we comment on diffusional behavior in section V.
11. Motivation and Concept Although it is both natural and obviously correct to model configurational distribution functions of molecules as isotropic in an isotropic medium, it is an assumption to do so in an anisotropic medium. Several different bodies of evidence support the related ideas that (a) there can be local correlations in the orientations of neighboring molecules in fluids such as n-alkanes and (b), if the medium is an oriented fluid of polymer chains, then even small molecules imbedded in the medium will adopt a preferential orientation related to that of the medium as a whole. Existence of orientational correlations in low molecular weight fluids has been observed by depolarized Rayleigh scattering and Raman spectroscopy.68 The correlations result from steric or "excluded-volume" interactions between the molecules. Since these correlations are shortranged, they would cause neither long-range ordering nor alteration of the random walk description of unoriented, equilibrium polymer fluids. According to recent simulations," these interactions cause segmental correlations in stretched polymer networks. Polymer networks afford the interesting opportunity to study configurationally oriented media at equilibrium. Deuterium NMR studiesg-l' of unattached free polymer chains in strained networks show definite orientational coupling at the segmental level between the network and free chains. In other words, the orientation distribution of the segments of the free chains does not relax to isotropy when the equilibrium condition of the surrounding network is one of finite strain. These orientational correlation effects are also operative between strands in networks without any free chains, which has implications for the stress-strain behavior of elastomers, as analyzed previ~usly.'"'~ Essentially, these models attempt to bring the packing of oriented molecules into the computation of the total configurational entropy of the network. Small unattached probe molecules imbedded in strained networks are also found to adopt biased equilibrium orientation distributions, coupled to the effect of the segments of the network. The equilibrium average orientation of three rigid rodlike diphenyl polyalkene probes (of length 1.15, 1.4, and 1.65 nm) in uniaxially deformed cis-1,4polyisoprene networks has been studied by Erman et a1.12 using fluorescence polarization methods. A distinct coupling of the orientation of the probes to the maintained orientation of the network was observed and correlated as (11.1) where H is the measured Hermanns' orientation function
Diffusion and Relaxation in Polymer Media 897 and a is the degree of coupling which, in these data,12 is itself a function of finetwo& (11.2) The linear coupling term, fo, was found to decrease linearly with increasing probe length with values of 0.5,0.3, and C 0.2), was about 0.1. The nonlinear term, f 2 (for Hnetwork 12 for all three probes. These probes are too small to experience entanglement interactions with the network. The network mesh size was approximately an order of magnitude larger than the probe lengths so that probe entrapment and forced orientation with the network were also unlikely. Rather, these results were explained by very short range packing correlations between the probes and the neighboring conformational sequences of the subchains in the surrounding network. A second study by Queslel et al.,13 using fluorophore probes with flexible n-alkyl groups attached, showed that these molecules also couple with the orientation of the deformed network. These flexible probes appear to have coupling coefficients that are independent of the orientation of the network ( f 2 is zero in eq 11.2) and values of fo between 0.4 and 0.8, increasing with molecular weight of the alkyl groups in the probes. This increasing coupling may result from the decreasing local free volume around the probes with probe molecular weight. This same group of workers has also shown that probes attached to dangling chains in networks orient as the network is strained. All of these observations support the supposition that if the deformed network were replaced by a deformed polymer melt, orientational coupling between any given chain and the surrounding chains composing the medium should also occur. The distinction between the melt and the network is that the equilibrium stress in a melt is zero after a step strain deformation. Orientational coupling between segments does not change this; however, the short-ranged correlations in segmental orientations would be expected to retard the rate of relaxation to equilibrium. Recently, several investigatorsl8Y2lhave studied the orientational relaxation of polymer chains in a melt using IR-dichroism and fluorescence depolarization. The experimental results of Tassin and Monnerie'*J9 suggest that such a retarded relaxation does occur in uniaxially high extended (e.g., X >> 4) monodisperse polystyrene melts and that the coupling value a is about 0.6. Kornfield et a1.22 have examined relaxation in bimodal molecular weight mixtures by mechanical, birefringence, and dichroism techniques. They have found that the terminal relaxation of short chains is retarded by increasing the fraction of long chains present, consistent with this orientational coupling hypothesis. In this work, we develop a self-consistent calculation of such orientational coupling within the framework of the Doi-Edwards model. The nature of such coupling is intrinsically linked to the packing characteristics of the particular chemical system. For example, recent wideangle X-ray scattering work on polystyrene and polycarbonate by Windle23suggests local correlations caused by the phenyl groups. Therefore, we do not attempt to present a quantitative theory for the strength of the coupling here but rather heuristically use the empirical form, eq 11.2. We treat the coupling coefficient a as a positive quantity because nematic-like interactions suggest that similar segments like to align in the same direction. Likewise, the DiMarzio packing theory14 implies that the principal values describing the orientation of a free segment or probe are proportional to those of the oriented medium (in the small anisotropy limit). Finally, we consider well-entangled chains whose monomeric friction
Macromolecules, Vol. 22, No. 2, 1989
898 Merrill et al.
coefficients and free volumes do not depend on molecular weight.24 Given these considerations and the short-range extent of the coupling interactions, we shall treat the coupling coefficient a as independent of molecular weight.
111. The Model A. The Current Reptation Model. In reptation theory, the chain is modeled as a threadlike object, following the so-called ”primitive path” of constant equilibrium contour length L, with a three-dimensional random coil ensemble of configurations representing that of the real chain. Large-scale chain motions perpendicular to the primitive path axis are prohibited and large-scale configurational rearrangement is accomplished by translating the thread strictly along its own axis. The leading end is free to move isotropically while the following end is constrained to move along the presently defined path, usually called the tube. Motion is randomly reversed over time thereby exchanging the roles of leading and following ends. Thus, configurational rearrangement is accomplished from the ends inward. To follow the effect of this kind of motion on configurational relaxation, a “snapshot” is taken of the polymer configuration at some arbitrary zero of time. At any time thereafter, the primitive path will be composed of three sections: two configurationally altered and growing end sections and one configurationally persistent and shrinking middle section or, equivalently, two “new tube” and one “initial tube” sections (where the new tube is created by the leading end, while the initial tube is destroyed concomitantly by the following end), up to the time where the initial tube section is completely destroyed, meaning all memory of the zero time configuration is lost. Doi and Edwards (henceforth abbreviated DE) discretize their primitive path into a collection of N entanglement vectors, [rl,..., rNJ,with associated magnitudes {al,...,ahr) and unit direction vectors (ul, ...,ury). At equilibrium, each entanglement vector maintains an isotropic distribution in direction (solid angle) and a constant length a (the so-called “mean length between entanglements”). Conditions on the primitive path contour length and mean(which coincides with the square end-to-end distance (R2) real chain (R2)) lead to Nu = L (111.1)
Na2 = (R2) (111.2) One typically takes “a” as the root-mean-squareend-to-end distance of a polymer with the molecular weight between entanglements Me. Furthermore, one calculates N by dividing the actual molecular weight by Me. A step strain affinely deforms each rrin this equilibrium distribution. On the microscopic level, this causes both stretching of the ai (and thus an increase of L ) and orientation of the ui. On the macroscopic level, this produces an observed stress resisting the deformation. The time response of this stress (normalized by the imposed step strain) is the relaxation modulus, G(t) (which in general can be strain dependent), depicted for an “idealized” polymer melt in Figure 1. DE5 explain the three stages of the relaxation as follows. Unconstrained by the longer range effects of the tube, the quick, “Rouse-like” relaxations, which lower the stress to its rubbery plateau, occur through local rearrangement of the chain within the confines of each entanglement segment. These short-ranged relaxations change neither the a, nor the ui, and we will not discuss these further here. The next relaxation process3 occurs through the recoil of the chain ends into the deformed tube of length L(t), thereby reducing the total contour length to the equilib-
“Rouse.like”
A --
=:I
I
\
\
I
I t-N2
t-N3
Log(t) Figure 1. Idealized sketch of the entangled melt relaxation modulus, G(t). The “Rouse-like”relaxations are molecular weight
independent and lower the modulus into the plateau region. The recoil relaxation accounts for relaxation to the lateau value, G,, and has a characteristic time scaling with The reptationcontrolled terminal region has a characteristic time scaling with
hR
N3.
rium value L. Since this is the first relaxation process controlled by the tube, we will henceforth refer to it as the “first relaxation process”. In a discrete chain model, this relaxation process not only reduces the ai but also alters the ui, because the recoil process essentially shrinks the chain into a middle portion, L long, of the deformed tube of length L ( t )> L , and thereby completely vacates some of the initial segments. DE switch to a continuous chain model to avoid this complication. They replace the ai and the ui by a contour functional describing the unit tangent vector u(s,t)along the primitive path coordinate s, defined over the interval [O,L(t)]. This recoil relaxation occurs over the time T (which scales with M ) and accounts for relaxation wiain the plateau region. In the limit of small strain, this relaxation becomes negligible. The terminalu (or “second”)relaxation occurs through the snakelike motions described above. These motions remove the remaining orientation and corresponding stress that resulted from the step strain deformation. This residual orientation resides in the persistent middle section of the initial, deformed tube. The new sections of the primitive path, created by the reptating chains as they escape from these deformed tube sections, are isotropic in spatial distribution according to the DE model. The time-dependent orientation (and hence the stress) is therefore proportional to the time-decayingfraction, F(t ) , of initial tube segments remaining in the ensemble, since these are the only ones assumed to retain the orientation induced by deformation. The tube (or curvilinear) diffusion coefficient DT (D, in ref 5) controls the rate of this tube escape by fixing the rate of this one-dimensional sliding motion in the tube (reptation). The tube diffusion coefficient, in terms of the experimentally measurable self-diffusion coefficient D,,is DT = 3ND, (111.3) The “reptation” or disengagement time, Td, required for this second relaxation process3 scales as p: Td = L2/r2DT = ( R 2 ) / 3 ~ 2 D , (111.4) For large N , the first and second relaxation processes are substantially separated in time and can be treated separately. In order to capture the essential features of orientational coupling on the relaxation process, we will concentrate on the terminal relaxation behavior and thereby implicitly assume either small deformations or large N .
Macromolecules, Vol. 22, No. 2, 1989
Diffusion and Relaxation in Polymer Media 899
According to the DE model for the terminal relaxation, the extra stress of a polymer due to segmental orientation, T, on the a coordinate plane in the 0coordinate direction is
We note immediately that
u,(s,t) = cos (O(s,t))
(111.1la)
(u2,(s,t))= (cos2 (O(s,t)))
(1II.llb)
To find ux,we note axial symmetry, switch to cylindrical coordinates, and find where u,(s,t) is the a component of the unit vector tangent to the primitive path at s (in the continuous version of this path), 6,, is the Kronecker delta function and p , k, and T are the number density of chains, Boltzmann constant and temperature, respectively. The inverse dependence of p on molecular weight cancels the factor of N , making the term in parenthesis independent of molecular weight. We rename this constant By renaming the normalized integral in eq 111.5 as the ensemble-averaged orientation function S,,(t), we may write the equation as T,&)
=
c-ls,,(t)
where GN is the modulus a t the start of this process (in the limit of zero strain), fst(A) is a strain-dependent function, and p ( t ) is a memory function. In uniaxial extension, we may rewrite eq 111.7 in terms of the Hermanns’ orientation function: 3(COS2
e(t)) - 1
(111.8) 2 where 0 is the angle between the tangent vector u(t) and the stretch direction 2 . The Hermanns function is zero when the segmental distribution function is isotropic and unity when the segments are fully aligned along the stretch axis. (Note this does not necessarily mean the chain is fully extended. We shall return to this point in section V.) The result is 7ext(t) = Gd.I(O)[H(t)/H(O)l (111.9) We show this transformation by first writing out eq 111.7 in detail using eq 111.5: Text(t)
=
T&)
(111.12)
(u2,(s,t))= (u2,(s,t)) = f/z(u2,(s,t))= y2 - 1/(cOsz ( e ( s , t ) ) ) (111.13)
Substituting into eq 111.10 and using the definition of H ( t ) , eq 111.8, we obtain L
H(s,t) ds
Text(t)
= [3pkTNJ
Text(t)
= [3~kTNlH(t)
0
(III.14a)
(111.6)
Equation 111.6 states that the deviatoric stress tensor T (here represented by components a@) is proportional to an orientation tensor S. Since several optical measurements, including dichroism and birefringence, are also proportional to this same orientation tensor, eq 111.6 is equivalent to the experimentally ~ u b s t a n t i a t e d“stress~~ optical law”. The constant C is equivalent to the stressoptical coefficient, although C is larger than it by the (assumed) constant ratio of the average orientation of the entanglement segments to the average orientation of the substituent Kuhn or monomeric segments which are directly responsible for the optical birefringence. (Letting “nnequal the number of such Kuhn segments in a single entanglement, this ratio equals n in the classical theory of rubber elasticity26and equals n(1 - V) in ref 16 where V is the nematic interaction parameter described therein.) We now simplify our discussion by focusing on the case of uniaxial extension in which there is axial symmetry about a single stretch direction, z. We begin with eq 111.5 and instead consider the measured difference in the principal normal stresses, T,, - T ~ The ~ . DE model gives this quantity as a product of separable functions of strain, A, and time, t:
H(t)=
(u2,(s,t))= (sin2 ( e ( s , t ) ) )= 1 - (cos2 ( e ( s , t ) ) )
- T x x ( t )=
( 3 ~ n r k T ) L0 - ~ ~ ~ ( u ~- *(u2,(s,t)) ( s , t ) ) ds (111.10)
rext(t)=
(111.14~)
[~P~TNI[H(O)I[H(~)/H(O)J (III.14d)
The three terms in eq III.14d can be identified term by term respectively with the terms in eq 111.7, namely, fifteen-quarters times the plateau modulus (4/5pkTN), the strain function, and the memory function. For all deformations, DE interpret p ( t ) as F ( t ) , the fraction of initial tube remaining in the ensemble of segments at t. We show this by conceptually partitioning the chain segments in eq IIL14a not according to s (as done in the integral) but rather into two groups according to whether the entanglement segment is an initial segment with orientation H(0) or a new segment with orientation zero. Now H ( t ) clearly becomes H(O)F(t)and thus p ( t ) is F ( t ) . So, with the DE model, the Hermanns’ function relaxes proportionally to the stress, a result closely associated with the experimentally observed stress-optical law5sZ5as previously stated. de G e n n e ~ ’ ,first ~ * ~derived ~ F ( t ) from a simple one-dimensional diffusion argument along the curvilinear tube coordinate axis s. An initial segment a t s’with 0 < s’ < L persists (that is, has not disengaged from the initial tube) at time t’if neither the end a t s = 0 at t = 0 nor the end at s = L a t t = 0 has reached s’during the interval 0 < t’ < t. With the constant contour length constraint, both ends were accounted for by requiring that the end initially at s = 0 has reached neither s‘nor s‘- L. The probability density F(s’,t) of survival of an initial tube segment situated between s‘and s’ ds‘is the solution to the ordinary diffusion equation with a 6 function initial condition at s = 0 and absorbing boundary conditions a t both s’ and s’ - L. Integrating F(s’,t) over s’ and dividing by L (i.e., averaging over s? to obtain the fraction of initial tube surviving (i.e., the survival probability), de Gennes’ found
+
We use this function extensively in what follows. B. The Orientational Coupling Model. We now introduce our orientational coupling hypothesis in the form of the experimental correlation given in eq 11.1. In other words, the orientation function of any given chain corresponding to the lowest free energy in a matrix of orientation H ( t ) is a ( t ) H ( t ) where , a ( t ) is a coupling constant that may, in general, be orientation dependent. By this we mean that if the matrix were a polymer network in equilibrium at some orientation H , then a free chain in that network a t equilibriuq would adopt an orientation aH.
Macromolecules, Vol. 22, No. 2, 1989
900 Merrill et al.
In a polymer melt, the matrix orientation results from the orientation of all the chains. As the chains relax toward equilibrium isotropy, H ( t )will eventually become zero, so that the coupling assumed does not prevent the return to isotropy. It does, however, retard the rate of relaxation. Defining the normalized Hermanns’orientation function (i.e., the orientational memory function) as e(t) = H ( ~ ) / H ( o ) (111.16) we must now adapt eq 111.14 to the coupling case. There is a delicate point here. In classical rubber the stress results from entropic forces acting to restore the oriented chain ensemble to its isotropic equilibrium. The average restoring force on free chains in the ensemble is zero at equilibrium. If we assume the same property for the restoring force acting on the equilibrium ensemble in an anisotropic medium, then we imply that free chains in this equilibrium ensemble do not contribute to the stress. Writing the stress as a function of the existing orientation, it follows that we must subtract from the total orientation H ( t ) the instantaneous equilibrium orientation, i.e., aH(t): T,&) = [3pkTN][H(t) - a ( t ) H ( t ) ] (III.17a) 7,&)
=
[3pkTN(1 - 4O))H(O)I[O(t)(l- a ( t ) ) / ( l - a(O))1 (111.17b) Equation 111.17 states that the stress decreases with increased coupling for a fixed amount of orientation. This is consistent with packing entropy notions1*that orientation implies a smaller loss in the number of available configurations relative to the isotropic equilibrium state and therefore a smaller increase in stress with orientation. The memory function is now (111.18) p ( t ) = O(t)(l - a ( t ) ) / ( l - a ( 0 ) ) This ansatz for calculating the stress contribution appears to us to be consistent with the treatment of packing entropy in stretched rubbers.16 Use of an alternative [e.g., p ( t ) = e(t)]would not fundamentally alter our analysis or conclusions. For linear coupling, a = fo # f ( t ) ,so that we have simply ~ ( t=)e(t)as in eq III.14d. Linear coupling therefore does not modify the structure of the DE model at all, as we have said only the relaxation rates are modified. However, nonlinear coupling destroys the proportionality between orientational and stress relaxation, because the instantaneous equilibrium changes with the additional time-dependent factor a(t). To proceed further, we require an equation for H ( t ) , which we obtain using reptation dynamics. We conceptually divide the primitive path segments in our ensemble according to their time of creation (meaning the time at which they first emerge from the initial tube) and their corresponding orientation. First, there is the fraction F ( t ) of initial segments remaining, each with the initial orientation H(0). The remaining segments are new tube segments, each created at an intermediate time 0 C t ’ C t. We define K(t’1t) as the density of such new tube segments created between t ‘and t ’+ dt ’given that they have not yet experienced a subsequent visit by a chain end. According to our hypothesis, each segment emerging from the old tube at t’assumes, on the average, a fraction a of the mean orientation of the surrounding chains, Le., crH(t?. The total orientation of our ensemble at t , H ( t ) , is then the sum over all of these contributions, hence
H ( t ) = H(O)F(t)+ JtK(t’lt)aH(t? 0 dt’ (111.19)
We now expand
CY
as a series in H(t?:
a(t? = a. a(t? =
a0
+ alH(t? + azEFZ(t?+ ...
(111.20a)
+ [~lH(O)I[H(t?/H +(O)l
a&P(0)[H(t?/H(0)I2 + ... (III.20b)
C Y ( t ?=
+ pe(t? + y e 2 ( t ? + ...
(111.20~)
Substituting eq 111.20~into (111.19) and dividing by H(0) we obtain O(t) = F ( t )
+ SfK(tflt){rO(t?+ p02(t? + y03(t9 + ...)dt’ (111.21)
which is our general equation for O(t), the orientational relaxation function. We must now specify our integral kernel, K(t’Jt).Let us consider the present tube at t. Using the method of de Gennes’ previously described, we label the two ends at s = 0 and s = L, where s is the tube axis coordinate. Now focus on a single segment at s’in the present tube so that 0 C s’ C L. This segment was created at t’, the last time either end visited this coordinate. The time elapsed between this creation and the present time is t - t’. Since we assume the first relaxation process has finished, the microscopic density of the chain segments is at equilibrium along the tube. The overall reptation motion results from the hopping of chain segments along the tube. Hence it follows that this hopping process after the first relaxation process should occur at the equilibrium rate. We now assume that the microscopic hopping process in the tube is indeed time independent (stationary). We assume throughout this work that this process, governing the basic rate of diffusion dynamics, is unaffected by orientation. Another way of saying this is that we are assuming that the segmental friction coefficient is unaffected by orientation. We may therefore use the concept of time reversibility and note that the probability density of the last visit by either end at t’, given t , is equal to the probability density of the first passage by the segment s‘ to either s = 0 or s = L over the same amount of time t - t’. (One simply reverses the direction of time and notes a one-to-one correspondence between the paths of these processes.) The probability that such a first passage has not yet hit s = 0 or s = L is equivalent to the previously discussed F(s’,t). The average of this probability over all possible tube positions s‘ is F ( t ) . The probability density that such a first passage occurred exactly between (t - t? and ( t - t? + dt ’is the negative time derivative of F ( t ) evaluated at t - t’. Therefore, our last visit kernel, K(t’lt), is
Wt’lt) = - = -F‘(t
-
t?
(111.22)
which we may readily obtain from eq 111.15. Substituting this into eq 111.21, we obtain t
~ ( t=) ~ ( t+) J - ~ ’ ( t t?ite(t? + 0
@P(t?+ yB3(t9 + ...)dt’ (111.23)
In the context of this derivation, the meaning of our first term F(t) is clear. It is the fraction of chain segments never yet visited between 0 C t ’ C t , i.e., last visited before t = 0. The solution to eq II1.23,O(t),governs the orientational relaxation in eq 111.16 and likewise gives the memory function p ( t ) in eq 111.18, which governs the stress relaxation. Finally, we point out that O(t) is actually O(tJl(0)) since the nonlinear coupling constants 0,y contain factors
Diffusion and Relaxation in Polymer Media 901
Macromolecules, Vol. 22, No. 2, 1989 of H(0);however, e(t)is independent of the initial orientation H(O),in the case of linear coupling (where 0 = y = ... = 0). Thus far, we have limited our discussion to uniaxial extension, in part because uniaxial deformation experiments at high extension have suggested this orientational coupling effect. Our model is actually applicable to general deformations. We start by restating our hypothesis in terms of the overall time-dependent probability distribution function P(u,t)for the unit tangent vector u of the newly created tube segments at t. P(u,t)solves an equation similar to (111.23):
[P(u,t)- P(u,m)]du = (F(t)[P(u,O)- P ( u , m ) ]
+
Table I Longest Relaxation Times0 ~
0 0.01 0.03 0.05 0.1 0.2 0.3
1.570796 1.564404 1.551 462 1.538304 1.504423 1.432032 1.352522
1.OOOOOO 1.008189 1.025079 1.042690 1.090 184 1.203190 1.348810
0.4 0.5
0.6 0.7 0.8 0.9
1.264404 1.165561 1.052794 0.920787 0.759308 0.542281
1.543364 1.816 225 2.226 141 2.910191 4.279610 8.390564
‘Values of (Td, 0) case is most pronounced in the long time tail.
creasing t. Decreasing a1 means an increasing apparent longest relaxation time, Tdr: Td.6
= Td(7r/2a1)2
(IV.3)
We tabulate the ratio of this apparent longest relaxation time, Td,f to the DE value Td in Table I. We define this ratio as the new quantity u-l. As t increases, C1 increases monotonicallyfrom 8/x2 toward unity, while the other C, decrease accordingly so that the sum over all C, remains unity. Hence, the memory function O(t) becomes more strongly single exponential with increasing t. When t equals unity, C1is unity and the other C, are identically zero; furthermore, a1 approaches zero so that e(t) approaches unity for all t. This is consistent with the notion that t = 1 means perfect coupling, hence no loss of orientation and no relaxation with time. Figure 2 shows e ( T ) determined by using expression IV.2 for various values of the linear coupling constant E, where 7 is the dimensionless time t divided by Td. If we compare eq IV.2 with the simple reptation result (111.15), we find qualitatively little difference. A t long times (IV.4a) and O(t-m)
- C1
exp(-t/Td,J = C1 exp(-ur)
-
(IV.4b)
where C1 is between 8/G 0.81 and unity. When one fits experimental data, the factor of Cl/(8/7r2) absorbs natu-
902 Merrill et al.
Macromolecules, Vol. 22, No. 2, 1989
.
-a
\
C
-- (E,y) - (0.3,0.0) ..... (E,Y)
--
.-c
(0.3, 0.4)
0.2 '
6 I
0' 4
'
0.6 I
'
0.8 '
'
'
1 .o
F ( t - 0 ) = 1 - (4/a312)(T)'/2+
o(7)
while eq IV.2 approaches e(t-0) = 1 - ( 4 / a 2 ) ( 1 - ~)(7)'/'(1 + t(7/a)1/2)
(IV.5)
+ O(T) (IV.6)
These short time expansions, which we derive in Appendix B, are rather good. They are valid to two decimal places (error > Tdg)prior to significant long-chain relaxation. In the dilute long-chain case, this implies zero matrix orientation ( ( H ( M , t ? ) 0). The longer chains relax more quickly in the blend, not because they move more quickly out of their initial tubes, but rather because they pick up less orientation from the matrix and hence the new tube segments are less oriented. Whereas the shorter chains can be arbitrarily slowed down by an increasingly slower matrix with relaxation time Td,L, the longer chains cannot be speeded up beyond their intrinsic DE rates due to coupling effects alone; however, this second case suggests that the long chains would no longer be fully entangled and constraint release effects would become important. Thus constraint release, rather than coupling, probably plays the dominant role experimentally in the long-chain relaxation process. Nonetheless, were it possible to have sufficiently long chains for both the “short” and long components so that constraint release did not occur, then one could measure the coupling coefficient directly by comparing to (1 - t)-l, the ratio of the relaxation rates of the long chains in their own pure melt state to that in the short-chain matrix (or even more precisely, use eq IV.3 and IV.2b). Experimentally, it might be more expedient to compare the relative shift in the long-time relaxation curve rather than to compare the derived relaxation times. (Linear coupling would imply a superimposable shift.) Finally, with regard to the relaxation of nondilute shorter chains with nondilute longer chains, the shorter chains will still track the longer chain relaxation after Td,s but the longer chains may themselves be relaxing via constraint release as well as reptation. These polydisperse effects might be more clearly seen during dynamic optical experimentsz2in which each molecular weight fraction’s contribution to the dynamic moduli could be measured independently22(unlike mechanical measurements that can only measure the total moduli of the blend). The dynamic moduli G’(w) and G”(w) are given by the usual transform formulas:24
-
G’(w) = wGN
So- e ( t )sin (ut)dt dt So- O(t)
G”(w) = wGN
COS ( U t )
(IV.24) (IV.25)
These Fourier transforms can, in turn, be obtained directly from the Laplace transform of the relaxation function. The results are algebraically complicated and have been relegated to Appendix B. The important point is the both chain components contribute to both loss peaks in G”(w) of the blend. For example, the shorter chains not only contribute to their “own” peak at w = TdB-lbut also to the peak of the long chains a t w = Td,L-l. This contribution to the long-chain peak results from the tracking of these chains’ relaxation by the shorter chains; hence, the contribution by the shorter chains at w = Td,L-l should be a factor of t smaller than their contribution to the peak a t w = Td -l. The relative ratios of these two peaks (obtained by proting only the short chains) would therefore give the coupling coefficient t. V. Diffusion Phenomena In the previous section we showed how the qualitative relaxation behavior remained unchanged for the case of linear coupling, whereas quantitatively the relaxation occurs at later times. Essentially, our model requires the chain to move through several tubes (on the average (1t)F1 tubes) before completely relaxing and the longest relaxation time spans the duration of all of these tube es-
capes. Nonetheless, the chain still diffuses through each tube at the same rate, because the effective friction per unit length and thus the “tube” diffusion coefficient remain unaltered by the orientation of the medium. We reflect this invariance of the basic reptation motion by rewriting eq 111.4 as
where Td,6is the measured or apparent relaxation time. Equation V.l shows that by using viscoelastic data, one would underestimate the diffusion coefficient by a factor of (1 - t)-l were one to neglect orientational coupling. The diffusion coefficient is a property of the equilibrium system in that it characterizesthe rate of molecular motion at long times in an unperturbed sample. Deforming the sample perturbs the equilibrium configuration of the chains and thus their respective tubes. The resulting anisotropic tube configurations add anisotropic contributions to the self-diffusion coefficient, converting this property into a quantity with direction. For example, in uniaxial extension, more of the tube orients parallel to the stretch direction than orients perpendicular to it. Over any given time interval, the chain constrained in this tube is therefore more likely to move parallel to the stretch direction and thus the diffusion component in this direction (which we denote z ) is larger than in the perpendicular ( x and y ) directions. We do not expect to see measurable differences between the DE results and the results of our model for the anisotropic diffusion components (DS4,Day,and D,?) because we do not expect orientational coupling to induce significant chain stretching. A neutron scattering study by Bou6 et al.27 on samples exhibiting orientational coupling13 showed that coupling is a local phenomenon that does not cause measurable chain stretching. Bou6 and co-workers explained this result using a discrete chain model with local orientation but without long-range order (stretching). In particular, they modeled the chain as a random walk of N segments in which each segment is favorably aligned along the z axis with directions +z and -z of equal probability. This contrasts with a typical model of dipole orientation in an electric field in which the +z direction is favored over the -z direction. This field-induced preference would lead to a bias in the random walk in the +z direction and hence stretching of the chain end-to-end vector in this direction. In the continuous (Gaussian) version of the random walk model, we would model such stretching by a biasing force term. Clearly, this model would be incorrect for orientational coupling. Rather, we should capture this effect by altering the relative variance of the individual segment vector components, ai*, ab,and aiz, while maintaining constant magnitude a. We may achieve this condition by equating the mean-square values of components of a with the principal (diagonalized)values of the orientation tensor ( u u) and then normalizing to a. Since the differences in these principal values are typically small (the large stresses resulting from the large GN),perturbations from a globally isotropic coil state and hence the anisotropic components of the diffusion tensor would be rather small. If, instead of center-of-mass diffusion, we examine semilocal diffusion processes, we can see some interesting effects of this anisotropic diffusion. By semilocal,28we mean diffusion processes that involve portions of the chain that are large compared to a but small compared to L. A good example of this kind of process is the interfacial welding or healing process that occurs when two polymer
Macromolecules, Vol. 22, No. 2, 1989 surfaces are joined. As has been discussed if one imagines the joining plane as'a perfectly flat interface, then welding is accomplished by chain segments from each side protruding across that plane. This clearly involves semilocal processes since a chain that moves entirely to the other side of the plane will be just as useless to welding as it was before it started to move. Smaller chain segments spanning the joining plane produce welding. These equilibrate across the interface on a time scale Td.
We will discuss here neither questions about what aspects of the spanning chain configurations are most important for weld nor issues of fracture mec h a n i c ~ . ~ ~ There - ~ S are important, unresolved matters pertaining to polymer welding in each of these areas. We focus here instead on the possible role of our orientational coupling model in welding. In the basic reptation model, newly formed tube segments sample an isotropic orientation distribution regardless of the orientation of the medium, leading to the conclusion that strain and orientation of the sample at the moment of joining would have no effect on welding. However, for our modification, orientation of the medium could have a strong effect. If the chains near the joining surfaces in the sample were strongly oriented, anisotropic diffusion could cause the chains to cross the interface more slowly (if preoriented parallel to the weld surface) or more quickly (if preoriented normal to the weld surface) than one would imagine from measurements of diffusive motion in isotropic media. More significantly, the segmental orientation at the interface itself may alter the weld strength by changing the fracture mechanics, for example, by enhancing (or retarding) craze i n i t i a t i ~ n . ~Perturbed, ~-~~ oriented configurations are to be expected in many situations near the surfaces of polymers to be welded. One important example is melt flow proce~sing,3~ where two flow fronts may join at a weld line. This kind of flow front can produce strong orientation parallel to the weld line. According t o our model, this would retard welding significantly.
VI. Conclusions Our objective in this work has been to understand the effects of weak orientational coupling on the dynamics of partially oriented, amorphous polymer systems. Packing entropy arguments can explain these experimentally observed orientational correlations. Our analysis of the dynamics has been done as an extension of the Doi-Edwards reptation model for configurational and stress relaxation. We have replaced their assumption that segments emerging from oriented tubes adopt random configurations with the hypothesis that segments moving into an oriented medium partially align with that orientation. In the case of uniaxial extension, we have assumed that Hermanns' orientation function of these segments takes on some fraction cy of the total instantaneous orientation of the system. For general deformations, we replaced this assumption with a condition on the segmental distribution function P(u). We have derived analytic expressions in the linear coupling case for the stress relaxation modulus and apparent diffusion behavior after a general step deformation. These expressions show qualitatively similar behavior to the Doi-Edwards results, making our hypothesis consistent with interpretations of the present data for moderate deformations. Quantitatively, the longest relaxation time shifts relative to the characteristic time for diffusion by a factor of approximately (1This factor represents the average number of tubes of length L from which each chain must escape before achieving relaxation (isotropy).
Diffusion and Relaxation in Polymer Media 905 Like the basic DE theory, our model results in a linear blending law albeit with a different interpretation. Slower, higher molecular weight chains will r e l g more quickly in the polydisperse matrix than they would in a matrix of themselves, while faster, lower molecular weight chains will relax more slowly, because these chains do not relax independently of each other. However, these effects exactly compensate and thus maintain this linear blending law. Nonetheless, new relaxation phenomena result when one observes the relaxation of the individual components in the blend. For example, shorter chains will acquire a second, lower frequency peak in G"(w) characteristic of the longer chains. Such effects could be observed through the optical techniques of Kornfield et a1.22 We have also presented numerical calculations for nonlinear coupling and have shown how this nonlinear coupling causes a breakdown in the proportionality between stress and orientation through &) and thus implies a breakdown in the stress-optical law. Furthermore, we have shown how nonlinear coupling destroys the factorization of the relaxation (memory) function, B(t), into a function of strain only, multiplied by a function of time only. Experimentally, such deviations usually occw at high deformations. We would like to point out that the DE model itself fails at very high deformations because the finite maximum extension of real chains eventually results in the failure of the random walk approximation. Thus, one may not observe these nonlinear coupling effects in a regime where the DE model is applicable except for high molecular weight samples (where degradation may become a problem). Finally, we showed how these effects could modify polymer welding phenomena. Chain orientation normal to the interface would retard craze initiation and enhanced chain interpenetration would increase effective entanglements important in the ensuing plastic deformation. Chain orientation parallel to the interface would have reverse (though not proportional) effects. Experimental results are needed to test these hypotheses. One could conceive of other avenues to model these results of retarded relaxation of orientation and anisotropic diffusion. One route might be analyses of the type carried out by Zimm and Lumpkin:l and refined by Slater and N ~ o l a n d ifor , ~ ~reptation in an electric field. In our case, the field would represent the orienting influence of the medium. In order to do the analysis self-consistently for a polymer melt, this field would be a time-varying one, diminishing as the overall orientation relaxed. Our formulation in terms of integral equations (rather than differential diffusion equations) automatically builds in this self-consistency.
Acknowledgment. We are grateful to Professor S. Prager, Dr. H. Watanabe, Dr. J.-L. Viovy, Dr. M. Moon, Professor W. W. Graessley, Professor D. S. Pearson, J. A. Kornfield, Professor G. G. Fuller, and Professor R. S. Stein for helpful discussions that clarified several points and brought relevant data to our attention. Professors D. S. Pearson and M. Doi also kindly communicated their own independent theoretical ideas (Appendix A) on this problem as this manuscript was under preparation. M. Tirrell would like to express his appreciation to Professor Monnerie and his research group at the Ecole Supgrieure de Physique et Chimie Industrielles de la Ville de Paris for the hospitality and stimulating environment in which this research was initiated. Financial support to the authors was provided during this work by the US.National Science Foundation, Presidential Young Investigator Award, by the John Simon Guggenheim Memorial Foun-
Macromolecules, Vol. 22, No. 2, 1989
906 Merrill et al.
dation, and by the French Centre National de Recherches Scientifiques. Appendix A: T h e Doi-Edwards Diffusion Equation In this Appendix, we cast our problem in terms of the original Doi-Edwards formalism. They write the following equation for S,,(s,t)? (A.la) with initial condition S,(s,O) = Q,,(E),
a constant
(A.lb)
and boundary conditions
sa,(o,t) = Sa&L,t) = 0
(A.lc)
They assert a similar equation for any property Q , which can be approximated locally. Our hypothesis alters the boundary conditions as follows
sa&o,t) = Sa&L,t) = f J L S a & s , t ) ds
(A.2)
This demonstrates the difficulty that our hypothesis introduces to the original solution technique. Subsequent to discussion of our modeling effort, M. Doi and D. S. Pearson solved this differential equation using an eigenvalue expansion. They obtained - 20 -
r 20,s
cos
1
The principal assumptions of eq A.7 are that each component relaxes according to its longest relaxation time only and that these are well separated in time. One very interesting observation made from eq A.7 is that the apparent shortest relaxation time decreases from the DE result to the coupling result as 4 increases from zero to unity, causing the two peaks to move away from each other. This counterintuitive result can be explained by interpreting the first term as the intrinsic relaxation of the shorter chains as modified by self-coupling and the second term as the relaxation resulting from coupling with the long chains. As 4 approaches zero, there is no self-coupling and only the intrinsic relaxation remains. Appendix B: Analytic Linear Solution Details We take the Laplace transform of eq 111.20 using the convolution and derivative theorems for these transforms and obtain
where f(s)
L[-F'(t)]
(B.2a) (B.2b)
where the a, are given by eq IV.2b and s is defined over the interval -L/2 to L f 2 for this equation only. By averaging Sa,&$)over s, they obtain the same result for 8 ( t ) , eq IV.2. One apparent advantage of the differential method is the derivation of the more detailed function S,,(s,t). In fact, we can derive this result from the integral method as well. Rather than obtaining S,,(t) from Sa,&,t), we actually do the opposite in this method. Let us define for each coordinate s an integral equation corresponding to (111.23):
In the general polydisperse case, we obtain (B.3) We invert the Laplace transform as a sum of residues Ri about the poles, si = x + iy, of the analytic (complex) function O(s): O(t) =
CRi(s=si)exp(si,t)
(J3.4)
We now define 0:
We note that a is a power series in S,,(t? and S,(s,O) not a function of s: S,,(s,t)
= F(s,t)S,&O)
+
1-F'(s,t t
-
0
is
t')a[S,&t?] dt' (A.5)
From the known function' F(s,t) Fls.t) .,, = 4 (2n - 1)as .=1(2n - 1)a sin L
[
= -(7/2)'SiTd ai = x
(B.5a)
+ iy
(B.5b)
expand 8(s) in u, and obtain the condition for the poles (before case-by-case reduction): [ ( x cos x sin x + y sinh y cosh y) + i(x sinh y cosh y - y cos x sin x)]/(cosh y + 1 sin x)(cosh y - sin x)(x2 + y2) = - (B.6) €
]
-t(2n - 1)'
Td
]
we may obtain the result, eq A.3, by direct integration. Finally, Doi and Pearson have derived a useful approximation for the short-chain relaxation function, eq IV.22a, for various short chain volume fractions 4:
We consider four cases: (i) x = 0, y = 0; (ii) x = 0, y # 0; (iii) x # 0, y = 0; and (iv) x # 0, y # 0. Case i is a solution and zero is a pole. Cases ii and iv have no solutions. Case iii, after reduction, has solutions meeting the requirement: tan x = x / t 03.7) Since x is real, si must be negative via equation B.5a in this case.
Diffusion and Relaxation in Polymer Media 907
Macromolecules, Vol. 22, No. 2, 1989 All the poles are simple poles except for a = 0 which is not actually a pole unless e = 1. In this case, e(s) is l / s identically and hence B ( t ) equals unity. [We calculated the residues at the simple poles according to standard methods.] We obtained the short-time expansions of F ( t ) and O(t) from analysis of eq B.1 as s m. We note 1.3.5...(2v) v = f/z, y2,... (B.8) = 2v+1/2sv+1 1r1/2
-
L[t”] = n!/s”+l
(B.9)
We write e ( t ) = 1 - Clt1/2- C,(t)
- C3t3/2- Cdt ... (B.lO)
Furthermore, lim e(s) =
(B.lla)
lim e(s) = l / s - l/s(B.11b)
hence, to first order in e, comparison to eq B.8, B.9, and B.10 yields eq IV.5 and IV.6. As t increases, the higher order terms in e in eq B . l l b become more important and hence eq IV.6 decreases in accuracy with (t/Td). Comparison to the numerical solution shows that eq IV.6 is good within 1-2% within the e-dependent approximate range of (t/Td) 5 (0.7 - (e + 0.1)/2) (B.12) Finally, we turn to the evaluation of the dynamic moduli. These can be obtained directly from the Laplace transform, e(s), in the following way: G ’ b ) = Re [ s e ( ~ ) ] l ~ = - ~ ~ (B.13) G”(w) = Im [ s e ( ~ ) ] l ~ = - ~ ~ (B.14)
indicating the real and imaginary parts of sO(s), respectively. Likewise, we can define real and imaginary parts of (f(s)) as (fR(s))and (fI(s)),respectively. Our expressions for the dynamic moduli then become
G”(u) =
+ e(fI(U))(l - (fR(U))) (1 - c ( f R ( U ) ) ) 2 + e 2 ( f I ( W ) ) 2
(1 - e ( f R ( U ) ) ) ( f I ( U ) )
*
where
+
(B.16)
=
sinh (a) sin (a) a[cosh (a) + cos (a)]
(B.17)
fib) =
sinh (a) - sin (a) a[cosh (a) + cos (a)]
(B.18)
fR(U)
and a
E
1r(UTd/2)1/2
(B.19)
Appendix C: Numerical Solution Details We have solved eq IV.7 using a finite element scheme with linear basis functions, where we have chosen the We have divided this dimensionless time interval as [0,10]. domain into 10000 equal subintervals each representing
a At of 0.001 and solved for the values of 6’ at each node. We have used linear basis functions (Le., we linearly interpolate between Bi and Bi+l to obtain intermediate values) because this removes back correlations from Bi+l (at t At) to di (at t). We could thus take advantage of the form of the integral limit at t , solve stepwise in time, and use the simple Newton’s method for the single at each step. Furthermore, we could easily extend the domain with the existing code simply by feeding in the previously calculated O values up to t as initial data. At each step, we treated the nonlinear cubic terms according to a standard approximation where the Oi are simply cubed and their intermediate values t h e n calculated by linear interpolation. A t each step, no more than six (and usually three) iterations were required to obtain Oi+l to within ten-place accuracy. Equation IV.7 has a singular kernel. As t - t’approaches zero, F’(t - t’) approaches infinity as ( t - t’)-ll2. Numerically, we avoid this problem by analytically integrating the approximate form of F’derived from eq IV.5 with the linear basis functions for all t - t’less than 0.3. For large t - t’ (greater than 0.3), we approximate F’ by F: using linear interpolating (basis) functions. With this scheme, we obtained errors in the integrals less than low5.To gain confidence in our results, we compared our numerical results for y = 0 with the analytical results. For e I0.7, the results show errors much less than at all times. For e = 0.9 the error is less than 3 X The error actually decreases with increasing time. We therefore have confidence that our calculations with the given time step and tolerances have yielded values accurate to four places with minor deviations in the fifth place.
+
References and Notes (1) de Gennes, P.-G. J . Chem. Phys. 1971,55,572. (2) Doi, M.; Edwards, S. F. J. Chem. SOC.,Faraday Trans 2 1978, 74, 1789. (3) Doi, M.; Edwards, S. F. J.Chem. SOC.,Faraday Trans 2 1978, 74, 1802. (4) Doi, M.; Edwards, S. F. J.Chem. SOC.,Faraday Trans 2 1978, 74, 1818. (5) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. (6) Patterson, G. D.; Flory, P. J. J. Chem. SOC.,Faraday Trans 2 1972, 68, 1098. (7) Fischer, E. W.; Strobl, G. R.; Dettenmaier, M.; Sta”, M.; Steidle, N. Faraday Discuss. Chem. Soc. 1979, No. 68, 26. (8) Bothorel, P.; Such, C.; Clement, C. J. Chem. Phys. 1972,69, 1453. (9) Deloche, B.; Samulski, E. T. Macromolecules 1981, 14, 575. (10) Samulski, E. T. Polymer 1985,26, 177. (11) Deloche, B.; Dubault, A.; Herz, J.; Lapp, A. Europhys. Lett. 1986, 1, 629. (12) Erman, B.; Jarry, J. P.; Monnerie, L. Polymer 1987, 28, 727. (13) Queslel, J. P.; Erman, B.; Monnerie, L. Polymer 1988,29,1818. (14) DiMarzio, E. A. J . Chem. Phys. 1962, 36, 1563. (15) Tanaka, T.; Allen, G. Macromolecules 1977, 10, 426. (16) Jarry, J. P.; Monnerie, L. Macromolecules 1979, 12, 316. (17) Gao, J.; Weiner, J. H. Macromolecules 1988, 21, 773. (18) Tassin, J.-F. ThBse, Universit6 de Paris VI Pierre et Marie Curie, Paris, 1986; Polym. Bull., in press. (19) Tassin, J.-F.; Monnerie, L. Macromolecules 1988, 21, 2404. (20) Graessley, W. W. Adu. Polym. Sci. 1982, 47, 68. (21) Lee, A.; Wool, R. P. Macromolecules 1986, 19, 1063. (22) Kornfield, J. A.; Fuller, G. G.; Pearson, D. S. Bull. Am. Phys. SOC.1988, 33, 445. (23) Windle, A. H. Pure Appl. Chem. 1985, 57, 1627. (24) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (25) Janeschitz-Kriegl, H. Polymer Melt Rheology and Flow Birefringence; Springer-Verlag: New York, 1983. (26) Treloar, L. R. G. The Physics of Rubber Elasticity; Clarendon: Oxford. 1975. (27) Bou6, F.;Farnoux, B.; Bastide, J.; Lapp, A.; Herz, J.; Picot, C. Europhys. Lett. 1986, 1, 637. (28) de Gennes, P.-G.; Leger, L. Annu. Reu. Phys. Chem. 1982,33, 49.
Macromolecules 1989,22, 908-913
908
(29) de Gennes, P.-G. C. R. Seances Acad. Sci., Ser. B 1980,B291, 219. (30) Prager, S.;Tirrell, M. J. Chem. Phys. 1981,75, 194. (31) Jud, K.; Kausch, H. H.; Williams, J. G. J. Mater. Sci. 1981,16, 204. (32) Wool, R. P.; OConnor, K. M. J. Polym. Sci., Polym. Lett. Ed. 1982.20,7. (33) Kim, Y. H.; Wool, R. P. Macromolecules 1983,16,1115. (34) Adolf, D. B.; Tirrell, M.; Prager, S. J. Polym. Sci., Polym. Phye. Ed. 1985,23,413.
'
(35) Kausch, H. H. Polymer Fracture, 2nd ed.; Springer-Verlag: New York, 1985. (36) Hull, D.; Hoare, L. Plast. Rubber: Mater. Appl. 1976,1,65. (37) Dettenmaier, M. Adv. Polym. Sci. 1983,52,57. (38) Kramer. E. J. Adv. Polvm. Sci. 1983,52. 1. (39) Evans, K. E. J. Po1ym.-Sci., Polym. Phys. Ed. 1987,25,353. (40) Wei, K. H.; Malone, M. F.; Winter, H. H. Polym. Eng. Sci. 1986,26,1012. (41) Lumpkin, 0. J.; Zimm, B. H. Biopolymers 1982,21, 2315. (42) Slater, G. W.; Noolandi, J. Macromolecules 1986, 19, 2356.
Temperature Dependence of Tracer Diffusion of Homopolymers into Nonequilibrium Diblock Copolymer Structures Peter F. Green* Sandia National Laboratories, Albuquerque, New Mexico 87185-5800
Thomas P . Russell IBM Research Division, Almaden Research Center, San Jose, California 95120-6099
Robert J6rbme and Maryse Granville Laboratory of Molecular Chemistry and Organic Catalysis, University of Liege, Liege, Belgium. Received April 2, 1988; Revised Manuscript Received June 27, 1988
ABSTRACT The morphology of polystyrene/poly(methyl methacrylate) (PSIPMMA) diblock copolymers was investigated as a function of temperature by using small-angle X-ray scattering. It was found that the microphase separation was enhanced up to temperaturesin excess of 200 "C. At 175 O C the copolymers remained microphase separated when mixed with PS or PMMA homopolymer at homopolymer concentrationsless than 50%, provided the molecular weight of the homopolymer was less than or equal to the total molecular weight of the copolymer. The temperature dependence of the tracer coefficients,D*Hc, of deuteriated polystyrene (d-PS)chains diffusing into two symmetric PS/PMMA diblock copolymers was studied by using forward recoil spectrometry (FRES). D*Hc/T was found to have the same temperature dependence as D*m/T, where D*HH is the tracer diffusion coefficient of d-PS chains diffusing into PS. A comparison of the temperature dependence of the zero-shear rate viscosity, qo, of PS indicates that D*Hc/T and v0-l have essentially the same temperature dependence. Experiments on the diffusion of deuteriated poly(methy1methacrylate) (d-PMMA) also indicated that D*Hc/T and v0-' have the same temperature dependence.
Introduction In a previous paper we reported the results of the molecular weight dependence of the tracer diffusion of deuteriated polystyrene (d-PSI and deuteriated poly(methy1 methacrylate) (d-PMMA) chains into three symmetric diblock copolymers of polystyrene (PS)/polylmethyl methacrylate) (PMMA).' The copolymer samples were prepared by solution casting using toluene, which is a nonselective solvent for either component. The morphologies of films prepared in this manner were characterized by using small-angle X-ray scattering (SAXS).ll2 The SAXS measurements indicated that the copolymers were microphase separated and the long period characterizing the combined thicknesses of the PS and PMMA rich domains was found to increase with the square root of the number of monomer segments which comprise the copolymer chain. In addition, the interface separating the two domains was large, on the order of 50 A. It was also found that the domains are not highly oriented with respect to the film surface. An important consideration in studying the diffusion of a thin layer of a homopolymer initially on the surface of the copolymer is how much of the copolymer surface is accessible to the homopolymer. This, of course, is controlled by the relative surface coverage of each component on the copolymer surface. The coverage is influenced by, among other factors, the relative surface energy of each of the homopolymers which comprise the copolymer, the solvent used, and the rate at which it is extracted from the 0024-9297/89/2222-0908$01.50/0
~opolymer.~ An X-ray photoelectron spectroscopy analysis of the surface of these films indicates that there are appreciable amounts of both components on the surface.ll3 This is reasonable since in the present case a nonselective solvent was used and the surface energies of the homopolymer components differ by a fraction of a dyne per en ti meter.^ The tracer diffusion Coefficient,' D*HC, of d-PS and d-PMMA chains of degree of polymerization NH which diffused into the copolymer hosts scaled as Nn2 provided that NH was less than -Nc, where N c is the degree of polymerization of the copolymer chain. The diffusion coefficient of the homopolymer chains that diffused into the copolymer hosts was found to be an order of magnitude lower than that of the chains that diffuse into the corresponding homopolymer hosts. This large reduction was attributed to three factors: a purely partitioning effect since only half of the volume is available to the diffusing chain, the tortuosity of the domains, and the domain orientation. The sample processing conditions were shown to have a pronounced influence on D*Hc. For example, when methylene chloride was used as a solvent for casting the copolymer films the diffusion coefficient was reduced by a factor of 2, evidently because the microphase separation of the copolymer was not as well developed as in the toluene case. In addition, the effect of annealing the ascast copolymer films for times on the order of 3 h or exposing them to toluene vapor for similar time periods before the diffusion process was to increase D*HCby ap0 1989 American Chemical Society
