Relaxation and self-diffusion of a polymer chain in a melt

Relaxation and self-diffusion of a polymer chain in a melt are discussed on the basis of the results of our recent Monte Carlo simulations of the bond fluctuation model, where only the excluded volume interaction is considered. Polymer chains are located on an L × L × L simple cubic lattice under periodic boundary conditions. Each chain consists of N segments, each of which 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 examined, where L- 128 for N ≤ 256 and L= 192 for N ≥ 384. The longest relaxation time τ is estimated by solving generalized eigenvalue problems for the equilibrium time correlation matrices of the positions of segments of a polymer chain. The self-diffusion constant D is estimated from the mean square displacements of the center of mass of a single polymer chain at the times larger than τ. From the data for N = 256, 384 and 512, the apparent exponents x _r and x_d, which describe the power law dependences of τ and D on N as τ ∝ N^xr and D ∝ N^xd, are estimated to be x_r ≈ 3.5 and x_d ≈ 2.4, respectively. For N = 192,256, 384 and 512, Dτ/〈R_e ²〉 appears to be a constant, where 〈R_e ²〉 denotes the mean square end-to-end distance of a polymer chain.