Self-Diffusion of a Polymer Chain in a Melt

Self-diffusion of a polymer chain in a melt is 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 $L \times L \times L$ simple cubic lattice under periodic boundary conditions, where each segment occupies $2 \times 2 \times 2$ unit cells. The results for $N=32, 48, 64, 96, 128, 192, 256, 384$ and 512 at the volume fraction $\phi \simeq 0.5$ are reported, where $L = 128$ for $N \leq 256$ and L=192 for $N \geq 384$. The $N$-dependence of the self-diffusion constant $D$ is examined. Here, $D$ is estimated from the mean square displacements of the center of mass of a single polymer chain at the times larger than the longest relaxation time. From the data for $N = 256$, 384 and 512, the apparent exponent $x_{\rm d}$, which describes the apparent power law dependence of $D$ on $N$ as $D \propto N^{- x_{\rm d}}$, is estimated as $x_{\rm d} \simeq 2.4$. The ratio $D \tau / < R_{\rm e}^{2} >$ seems to be a constant for $N = 192, 256, 384$ and 512, where $\tau$ and $<R_{\rm e}^{2}>$ denote the longest relaxation time and the mean square end-to-end distance, respectively.