Faraday Symp. Chem. SOC.,1983 18 189-197 Rheological Behaviour of Branched Polymer Molecules BY DALE S. PEARSON* Exxon Research and Engineering Company Corporate Research-Science Laboratories Annandale New Jersey 08801 U.S.A. AND EUGENE HELFAND Bell Laboratories Murray Hill New Jersey 07974 USA. Received 18th August 1983 A theory is presented for describing the viscoelastic properties of star-shaped branched polymers in concentrated solutions or melts. We describe the dominant mode of relaxation as diffusion of the chain in a potential field as others have done. Some indication of the form of the potential function is obtained by studying the statistics of random lattice walks entangled with a net of obstacles. The theory is compared with experimental data on well characterized stars covering a wide range of branch-point functionality.We show that it correctly describes the molecular-weight dependence of the zero shear-rate viscosity qo and the steady-state compliance Jt as well as the frequency dependence of the dynamic shear moduli G(o)and G(u>).Some comments are made about the behaviour of branched polymers with a structure more complex than stars. When a sudden strain is applied to a viscoelastic liquid a resisting stress appears which decays away with time. If the sample is a high-molecular-weight polymer the rate of stress relaxation is a measure of the time required for the molecules to disentangle from their oriented configurations and return to an isotropic state. The essential constraint which controls this disengagement is that polymer chains cannot pass through the contours of the neighbouring chains which surround them.In 1971 de Gennesl introduced the concept that the primary mode of relaxation was a simple snake-like diffusion of the polymer along a path approximating its own contour a motion he aptly termed reptation. In 1978 Doi and Edwards2 extended these ideas by developing a theory that predicted the viscoelastic response of entangled liquids. These theories as originally presented apply to Zinear polymer molecules. If the polymer contains long branches it is expected that this simple curvilinear translation will be suppressed. This paper addresses the issue of how branched polymer molecules disentangle. We first provide a description of a model for the entanglement of random lattice walks with a net of fixed obstacles.Certain statistical properties of these walks are obtained and then used to develop a theory for the time-dependent behaviour of star-shaped branched polymers. It is shown that this theory can quite successfully describe the linear viscoelastic properties of model star molecules. We conclude with a review of what is known about the behaviour of more complex branched polymers such as those with a comb-like or H-shaped structure. STATISTICS OF ENTANGLED CHAINS After the initial success of the reptation theory it became clear that further progress would require a deeper understanding of how molecules are entangled with each other. This was evident from work on the early stages of the relaxation p~ocess,~ on the 189 RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS Ilx X X Fig.1. An unrestricted random walk shown by solid lines on a square lattice represented by the open circles. The crosses form a net of obstacles with which the walk can entangle. The walk has several unentangled loops such as the one beginning one down from the upper left corner. Reeling in the walk from its ends produces the primitive path which is traced by the dashed lines. anomalous ‘3.4 law’ of viscosity (r,,xM3.4),4on the behaviour of branched polymer^^-^ and even on the equilibrium properties of rubber network^.^ In the model of Doi and Edwards2 a polymer is assumed to be confined to a tube-like region by the other polymers which surround it.Edwards and Evans studied some of the properties of this tube by using computer simulations of random walks entangled with fixed obstacles.8 In particular they determined the length of a walk which remains after it is reeled in from its ends without intersecting any of the obstacles. In this section we discuss some analytical results we have obtained on a model very similar to that of Edwards and Evans. Polymer chains are depicted by unrestricted random walks on a lattice with a coordination number q. The steps of the walk are meant to represent sections of the molecule and not the real polymer bonds. Furthermore the lattice is permeated by a net of fixed obstacles. This is illustrated in fig. 1 for a planar lattice (q =4) where the obstacles are placed at the centre of each lattice square.The position of the entanglement net for more complicated lattices can be ~pecified.~ A walk which begins and ends at the same point is called a loop. A loop will be termed unentangled if it retraces its own path (perhaps more than once) in returning to its origin. In the Edwards and Evans reel such unentangled loops will be completely pulled out (cf. fig. 1). The walk which remains is defined as the axis of the tube or the so-called primitive path. We have calculated the probability pc(K,N) that a random lattice walk of N steps has a K step primitive path or equivalently that N-K of the steps are part of unentangled loops. For N large and K =O(N) the function pc(K,N) is given by9 log(q-where IC =K/N.This function is strongly peaked at a value of K given by D. S. PEARSON AND E. HELFAND 191 0'5m 0.1t I .';...- ,/' \. 1 KIN Fig. 2. A comparison of the exact form of pc(K,N) eqn (1) (-) and the Gaussian approximation eqn (3) (--). The value of q is 6 and the limit N + co is assumed or equivalently the plot is of the coefficient of N in the exponent. In this region pc(K,N) can be approximated by the Gaussian function9 A comparison of the exact form of pc(K,N) and eqn (3) is shown in fig. 2. It is of interest to point out that the configuration of the primitive path is the same as that of a no-reversal random walk. A walk of this type containing [(q -2)/q]N steps has the same mean-squared end-to-end distance as an unrestricted random walk of N steps.ll The probability of a chain's end returning unentangled to the origin arises in ionnection with the diffusion of star-shaped polymers.The result we find is The form of eqn (4) was anticipated by de Gennes5 and a very similar expression has been recently obtained by Needs and Edwards.lO They have also provided an experimental test of this result by generating walks on a simple cubic lattice (q = 6) and counting unentangled returns. Essentially quantitative agreement with eqn (4) is found. From a physical point of view unentangled loops along a polymer chain form and disappear. Small-scale fluctuations in their number and size are related to the 'defect diffusion' which transport the chain1 and to the rapid changes in the length of the primitive path which effect viscoelastic proper tie^.^ Large fluctuations provide a mechanism for the relaxation of dangling ends in a rubber network7* l2and the arms of branched polymers.RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS VISCOELASTIC PROPERTIES OF STARS The simplest type of branched polymer is a star-shaped molecule consisting offarms connected to a central branch point. The viscoelastic properties of stars have been studied by a number of research gro~ps.~~-~~ The different time-dependent behaviour of stars when compared with linear polymers was attributed to changes in the mechanism by which they di~entang1e.l~ X 0 w II III 1 1 IIIIIIIIIIIIJ Fig. 3. An arm of a star in two separate positions of escape.The probability of retracting a distance y is proportional to exp [-U(y)/kT]. In the reptation model of de Gennes' a linear molecule is assumed to diffuse in free Brownian motion along the axis of the tube. If the chain is oriented by an applied strain it will return to an isotropic state because the ends of the chain adopt random configurations as they emerge from the tube. Long branches attached to a polymer molecule should obstruct this simple translational motion. However an arm of a star can still renew its configuration by retracting along its axis pushing out unentangled loops into the surrounding matrix. The low probability of adopting such contracted states can be thought of as creating a potential in which the end of the arm is diffusing.The lifetime of the tube is then related to the time required for the end of the arm to diffuse back to the branch point. Let an arm of the star consist of N segments each containing N freely jointed monomers. N is the number of monomers between entanglements and there are N =NN monomers per arm. Following Doi and KUZUU~ we assume that motion takes place in a potential U. U is taken to be a quadratic function of the position of the chain end x which varies from x =0at its equilibrium position to x =L, at the branch point (see fig. 3) kT U(x)=a--x2. L& The quantity a =v'(N,/N,) proportional to the length of the arm is normally a large number. Experimental data to be presented below will be used to estimate the constant D.S. PEARSON AND E. HELFAND v’. The use of a quadratic form for U can be motivated by the results presented earlier [see eqn (2) and (3) and fig. 21. The problem of determining the time required for an arm to retract a distance y in its tube can be obtained by solving the Smoluchowski diffusion equation.la However an alternate derivation suitable for our needs follows from a Kramers-type analysis for motion of a particle in a well with an infinite sink at y.17 Let x be the root-mean-square fluctuation in the position of the chain end. Its value determined from U is x = ~,,/(2a)k (6) The time z = CLtq/2akT cc NZ (7) is a measure of the time required to diffuse to x,; i.e. z = xt/D where D = kT/[and [is the friction factor for the chain. Then for y % x and t % z the Kramers method gives the probability F(y,t) that the end of the chain has not reached y (y < Lea) When eqn (6) is substituted for U we obtain The fraction of undestroyed tube i.e.the part which has not been visited by the chain end is 1 f(t) = -I”’“dy F(y,t). (10) Le xe Note that eqn (8) and (10) provide a way to generalize our calculation for the case where the potential is not quadratic. A more detailed derivation of eqn (9) is found in ref. (18). If o(t)is the stress at time t after applying a small strain y at t = 0 then the stress relaxation modulus is given by G(t) = a(t>/r. (1 1) The fraction of the initial stress which remains at t should be proportional to the amount of tube which has not been vacated i.e.where Gois the initial modulus usually referred to as the plateau modulus. Eqn (10) and (1 2) provide a means to calculate the zero shear-rate viscosity q019 Because [ L, and a are all proportional to molecular weight of the arm Ma eqn (1 3 b) predicts that (vf2) Me qocc (‘)‘exp 1 (14) * Kramers did not explicitly give this result but it is easily derived. 7 FAR I94 RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS i.e.the viscosity increases exponentially with molecular In a similar manner the steady-state shear compliance is calculated by19 =a/Go a% 1 or equivalently J:G0 = v'-. Ma Me 20.0 I I I I I I I I -I -r-I-1 18.0 16.0 14.0 12.0 8.0 6.0 4.0 2.0 O"b.0 210 4t0 610 8;O 10.0 lh.0 li.0 16.0 li.0 2d.0 2i.O 24.0 .o MdMe Fig.4. Product of the steady-state compliance J and the plateau modulus GoN for the series of star-shaped polymers. The symbols indicate polybutadiene 4-arm (0); polystyrene melts,14 4-arm (A) and 6-arm (V); and polyisoprene 4-arm (0) and 6-arm (0). Eqn (15c) suggests a method to determine the constant v'. In fig. 4 we have plotted the product of JE and the plateau modulus against Ma/Mefor polyisoprene stars,12 polystyrene starsL3 and polybutadiene stars.14 Both 4-arm and 6-arm stars are included. A linear relationship is obtained which by least-squares analysis gives a value v' M 0.6. The connection between the viscosity of stars and the arm molecular weight also involves the constant v'. In fig. 5 we show the logarithm of qo/(Ma/Me)iagainst MJM for polyisoprene polystyrene rneltsl4 and polyisoprene melts.18-20 These data include stars with functionalities from 4 to 12.In agreement with eqn (14) the relationship is linear at Ma/Me% 1. The full curves were calculated with eqn (13 a) using v' = 0.6. D. S. PEARSON AND E. HELFAND eqn (12) and (13). I I I I - 100 02 u -.-n K hE lo-’ 3 W u 02 u 1 n --10-1 100 10’ 102 103 104 OT,, Fig. 6. Dynamic moduli G(o)(A) and G(o)(O) normalized by the plateau modulus GoN for an 8-arm polyisoprene melt.’* The value ofz is 1050 s. The solid curves were calculated with eqn (1 2) and (1 6). RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS The frequency dependent dynamic moduli of linear viscoelasticity can be obtained from G(t) as G*(w) = iw dt exp (-iot) G(t) (164 JOrn = G’(w)+iG”(w).(16b) An additional test of our theory is provided by comparing it with experimental values of G’(w) and G”(w)obtained on an 8-arm star polyisoprene.20 This is done in fig. 6 where the solid curves were calculated using a v’ value of 0.6. A discussion of the functions G(t)and G*(w) including useful approximations for correlating experimental data can be found in ref. (18). REMARKS Star-shaped polymers have a structure such that all sections of the molecule are connected at only one end to a branch point. Although these sections cannot translate freely they can renew their configuration by fluctuations in the manner discussed above.Branched polymer molecules that have the shape of an H,21 a comb22 or a tree contain internal sections that are connected at both ends to a branch point. Even though the mechanism by which these sections relax is not clear it seems certain that they will have longer relaxation times than the arm of a star.’ An indication that this is the case can be found in the study of Graessley and Roovers22 on comb-shaped polystyrene. The frequency-dependent dynamic moduli contain two sets of relaxation times in the terminal zone. The first is associated with the arms of the comb and the other at still lower frequencies with the backbone. Realizing that JZ is dominated by the longest relaxation times they demonstrate that JE OC wbb/&b (17) where n/fobb and (bbb are the molecular weight and the volume fraction of the backbone.Eqn (17) indicates that in the determination of Jt the relaxation times of the arms can be ignored. The structure of a branched polymer may also effect its non-linear (strain- dependent) properties. Non-linear properties of star polymers have been measured and they appear to be similar to those of linear However experiments in the non-linear range on well-characterized branched polymers with internal sections have not been done. We thank L. J. Fetters for providing materials used in this study and for pointing out to us that viscosity data on stars of different functionality could be reduced to a common curve by plotting viscosity against arm molecular weight. ’ P-G. de Gennes J.Chem. Phys. 1971 92 572. * M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978,74 1789 1802 1818. M. Doi J. 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