## Memory Effects in Entangled Polymer Melts

A chain reptating through a high molecular weight polymer melt leaves a trace in the form of elastic distortions of an entanglement network. Neighboring chains can partially relax these elastic distortions by following this trace with some accuracy. Thus the memory of chain configuration is reproduced by other chains when the first one leaves the tube after time id;f. The number of diA’usion steps of duration Td’f required to completely lose the memory of chain configuration grows with the molecular weight M of the chain as z " ,hgzq;t — M’ in agreement with experiments, where z " , , " is the relaxation time of the melt. PACS numbers: 61.25.Hq, 83.10.Nn, 83.20.Fk The dynamics of polymer melts has puzzled scientists for decades. The viscosity of linear polymers crosses over from linear molecular weight dependence tl — M for shorter chains to much stronger dependence g — M . for longer chains above some entanglement molecular weight M, [1]. The stronger dependence at higher molecular weights is attributed to the topological restrictions, called entanglements, imposed on a chain by its neighbors [1,2]. A number of attempts have been made to develop a quan-titative model of entanglements [3-5], but none of them was fully satisfactory. A more successful approach [6] was to bypass the question of the detailed molecular characteristic of an en-tanglement, but rather to represent the effect of topologi-cal constraint of surrounding chains by a confining tube. The motion of the linear chain along this confining tube is the central idea of the reptation model [7]. The main predictions of the reptation model compare quite favor-ably with the experimental results on entangled polymer melts [2,8]. For example, the self-diffusion coefficient is predicted to scale with reciprocal second power of the molecular weight D " p — M, in agreement with diAu-sion experiments on polymer melts [9] Dm, lt — M The major criticism of the reptation model was related to its prediction of the longest relaxation time [7] -M . The experimentally measured relaxation time has a stronger molecular weight dependence r,1t — M with similar behavior for the melt viscosity [1]. Numer-ous models [10-16] were developed to explain this discrepancy between the reptation model and the experi-ments in a melt. Though this question is not finally set-tled, it is believed that the higher exponent is the finite size crossover eAect due to the tube length fluctuations [10,16, 17]. These tube length fluctuations were neglected in the original treatments of the reptation model. Later it was realized [10] that as the chain undergoes Brownian motion, its tube length (or, equivalently, the number of entanglements it has with its neighbors) lluctuates around some average one. The typical magnitude of these thermal fluctuations in the tube length L is of the order of the polymer size R since L — (L) — R — JN. The aver-age tube length (L) grows linearly with polymer molecu-lar weight and therefore the relative size (and impor-tance) of tube length lluctuations decreases with chain degree of polymerization N as L — (L) /(L) — 1/JN. The importance of the tube length fluctuations was tested by numerical simulations of single chain dynamics in the array of fixed obstacles. The early computer simu-lations [13] of the dynamics of a single entangled chain seemed to agree with the predictions of the original repta-tion model [7]. But more extensive numerical studies [16,18-20] demonstrated that in a wide range (over two decades) of chain length the numerically calculated dependence of relaxation time on molecular weight is z " " m — M . The analytical treatment [21] of tube length fluctuations is in agreement with these numerical calculations. Thus the tube length fluctuations lead to a very broad crossover with an apparent exponent for relax-ation time of — 3.4. The asymptotic value of 3 for this exponent can be achieved in a single chain problem only for astronomically high molecular weights. There are some indications [20] that the corrections to the relaxa-tion time scaling could be logarithmic — M lnM. The "3. 4 mystery" seemed to be resolved by including the tube length fluctuation modes in the reptation model. Note that reptation with tube length fluctuations is still a single chain model. Many chain (constraint release) modes [22-26] are necessary for quantitative predictions of the stress relaxation function, but they do not significantly aAect the molecular weight dependence of relaxation time [26]. The "single chain in the tube" model usually provides the input modes into the con-straint release models [26]. This apparent agreement be-tween theory and experiments did not last for long. The same computer simulations of a single entangled chain 1856 0031-9007/93/71 (12)/1856 (4)$06.00 1993 The American Physical Society VOLUME 71, NUMBER 12 PH YSICAL REVI EW LETTERS 20 SEPTEMBER 1993 gave a higher exponent for the molecular weight depen-dence of the diffusion coefficient [18-20] D " " — M Within a single chain model it is natural to assume that a linear chain diA’uses distance of order of its size R dur-ing its relaxation time z, so that D — R /z. Since the tube length Auctuations lead to the relaxation time i — M, it is natural to conclude that the diA’usion coeScient should have a stronger eA’ective molecular weight dependence D — M [18]. The main question we answer in the present paper is why the relation D — R /z is not satisfied in polymer melts [27]. The mean square distance AR, m polymer diA’uses during its relaxation time Re m. & Ielta meit — M’ grows faster [27] than its mean square size R . How can a polymer diAuse many times its size without relaxing stress? The memory of the polymer con-figuration has to remain in the region even after the chain leaves this region. Phenomenological collective relaxation models [15] have assumed this memory but did not ex-plain its physical origin. Below we argue that the origin of the memory eAects is in the excluded volume interac-tion between the chains in the melt. The experimental indication to this origin comes from the fact that the discrepancy between diAusion and relax-ation exponents exists in polymer melts, but not in entan-gled polymer solutions, where the corrections to scaling for both diffusion and stress relaxation are similar [28-30] z» — M and D» — M ’ . Thus the single chain reptation model with tube length fluctuations agrees with the experimental results in entangled solu-tions. But both solution results and the single chain simulations differ from the results in polymer melts. Thus the memory eA’ects are the strongest in dense poly-mer melts. Below we focus on the excluded volume interaction be-tween diAusing chains in polymer melts. We argue that a reptating chain leaves a trace in the melt in the form of elastic distortions of an entanglement net. Neighboring chains are attracted to this trace and can relax these elas-tic distortions by partially reproducing the configuration of the tube of the first polymer. This is the origin of the memory of chain configuration after it diAuses out of the region.

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### Citation

*71*.

*Memory Effects in Entangled Polymer Melts*. Vol. 71, 1993.