Relaxation Mode Analysis of a Single Polymer Chain in a Melt

Relaxation modes and rates of a polymer chain in a melt are studied by Monte Carlo simulations of the bond fluctuation model, where only the excluded volume interaction is taken into account. Polymer chains, each of which consists of N segments, are located on an L × L × L simple cubic lattice under periodic boundary conditions, where each segment occupies 2 × 2 × 2 unit cells. The results for N = 32, 48, 64, 96, 128, 192, 256, 384 and 512 at the volume fraction φ ≃ 0.5 are reported, where L = 128 for N ≤ 256 and L = 192 for N ≥ 384. The relaxation modes and rates are estimated by solving generalized eigenvalue problems for the equilibrium time correlation matrices C_i,j(t) = 1/3 〈R̄_i(t) · R̄_j(0)〉 of the coarse-grained relative positions R̄_i of segments of a polymer chain defined by R̄ _i = 1/n ∑_k=1ⁿ R_(i-1)n+k, where R_k denotes the position of the kth segment relative to the center of mass of the polymer chain. The apparent exponent z which describes the N-dependence of the slowest relaxation rate λ₁ as λ₁ ∝ N^-z increases beyond three as N increases. From the data for N = 256, 384 and 512, the apparent exponent is estimated to be z ≃ 3.5. The behavior of the pth slowest relaxation rate λ_p for a fixed value of N is consistent with the prediction of the reptation theory λ_p ∝ p². The first and second slowest relaxation modes show the Rouse-like behavior.