Entropy of chains placed on the square lattice

TL;DR

This paper calculates the entropy of flexible linear chains on a square lattice considering excluded volume effects, using transfer matrix methods and finite-size scaling to provide exact and approximate results for various chain lengths and occupation densities.

Contribution

It introduces an exact transfer matrix approach combined with finite-size scaling to estimate the entropy of self-avoiding chains on a square lattice for different chain lengths and densities, extending previous methods.

Findings

Exact entropy estimates for dimers and fully occupied lattices.

Approximate entropy values for various chain lengths and partial occupancies.

Consistent results with earlier series and mean-field approximations.

Abstract

We obtain the entropy of flexible linear chains composed of M monomers placed on the square lattice using a transfer matrix approach. An excluded volume interaction is included by considering the chains to be self-and mutually avoiding, and a fraction rho of the sites are occupied by monomers. We solve the problem exactly on stripes of increasing width m and then extrapolate our results to the two-dimensional limit to infinity using finite-size scaling. The extrapolated results for several finite values of M and in the polymer limit M to infinity for the cases where all lattice sites are occupied (rho=1) and for the partially filled case rho<1 are compared with earlier results. These results are exact for dimers (M=2) and full occupation (\rho=1) and derived from series expansions, mean-field like approximations, and transfer matrix calculations for some other cases. For small values of M, as well as for the polymer limit M to infinity, rather precise estimates of the entropy are obtained.