Slip-spring model of entangled polymers: a standard Rouse chain of beads is constrained by discrete virtual springs. The virtual springs have one end attached to anchoring points (black squares) and the other is linked to the chain through slip links. The average distance between slip links is segments, and the strength of the confining potential is measured by , the equivalent number of segments of each virtual spring.
Stress relaxation moduli from the slip-spring model for a chain with , , , and . (a) Real-chain relaxation modulus. Symbols: calculation at equilibrium using . Lines: relaxation after step shear, . (b) Total relaxation modulus. Symbols: calculation at equilibrium using . Lines: relaxation after step shear. . Values of the shear strain are indicated in the legend.
Stress relaxation modulus from the slip-spring model (same parameters as in Fig. 2). Symbols: calculation at equilibrium using Eq. (3). Lines: relaxation after step shear, . Values of the shear strain are indicated in the legend.
[Eq. (4)] as a function of from the slip-spring model, , , and . values are shown in the legend. Line: prediction of the Warner-Edwards model for a chain of infinite length [note that when , , and when , ].
Stress relaxations from the Kremer-Grest model (, ). Total relaxation modulus is indicated as a thick line. Also shown are the autocorrelation functions of the bonded (, thin line) and the nonbonded(, dashed) stresses, as well as their cross correlation (dot-dashed line).
Total relaxation modulus and the bonded/nonbonded contributions from molecular dynamics simulations at different densities using the Kremer-Grest model, . Absolute values are shown whenever a contribution is negative (see labels in the plot). At late times, both contributions are proportional to the total modulus.
Decomposition of the shear modulus into complementary partial contributions as a function of the density: bonded and nonbonded (squares), and intra- and interchain (circles).
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