Simulation studies of permeation through two-dimensional ideal polymer networks

We study the diffusion process through an ideal polymer network, using numerical methods. Polymers are modeled by random walks on the bonds of a two-dimensional square lattice. Molecules occupy the lattice cells and may jump to the nearest-neighbor cells, with probability determined by the occupation of the bond separating the two cells. Subjected to a concentration gradient across the system, a constant average current flows in the steady state. Its behavior appears to be a non-trivial function of polymer length, mass density and temperature, for which we offer qualitative explanations.Comment: 8 pages, 4 figure

Scaling for the Percolation Backbone

We study the backbone connecting two given sites of a two-dimensional lattice separated by an arbitrary distance $r$ in a system of size $L$. We find a scaling form for the average backbone mass: $\sim L^{d_B}G(r/L)$, where $G$ can be well approximated by a power law for $0\le x\le 1$: $G(x)\sim x^{\psi}$ with $\psi=0.37\pm 0.02$. This result implies that $\sim L^{d_B-\psi}r^{\psi}$ for the entire range $0<r<L$. We also propose a scaling form for the probability distribution $P(M_B)$ of backbone mass for a given $r$. For $r\approx L, P(M_B)$ is peaked around $L^{d_B}$, whereas for $r\ll L, P(M_B)$ decreases as a power law, $M_B^{-\tau_B}$, with $\tau_B\simeq 1.20\pm 0.03$. The exponents $\psi$ and $\tau_B$ satisfy the relation $\psi=d_B(\tau_B-1)$, and $\psi$ is the codimension of the backbone, $\psi=d-d_B$.Comment: 3 pages, 5 postscript figures, Latex/Revtex/multicols/eps