Exact Thermodynamics of a Polymer Confined to a Lattice of Finite Size

TL;DR

This paper develops an exact matrix formalism to analyze the thermodynamics of a confined linear polymer on finite lattices of arbitrary shape and dimension, accommodating complex boundary conditions and site-specific interactions.

Contribution

It introduces a general exact method for calculating polymer thermodynamics in confined geometries with arbitrary boundary shapes and site-specific interactions, extending previous results.

Findings

Exact equations for thermodynamic properties derived

Method applicable to arbitrary lattice dimensions and shapes

Results include solutions for 1D and 2D cases

Abstract

We write exact equations for the thermodynamic properties of a linear polymer molecule confined to walk on a lattice of finite size. The dimension of the space in which the lattice resides can be arbitrary. We also calculate polymer density. The boundary can be of arbitrary shape and the attraction of the monomers for the sites can be an arbitrary function of each site. The formalism is even more general in that each monomer can have its own energy of attraction for each lattice site. Multiple occupation of lattice sites is allowed which means that we have not solved the excluded volume problem. For one dimension we recover results obtained previously. The 2-d solution obtained here also solves the problem of an infinite parallelepiped. The method is easily extended by the methods of a previous paper to treat the problem of polymer stars or of branched polymers confined within a finite volume. This exact matrix formalism results in sparse matrices with approximately zM non-zero matrix elements where z is the lattice coordination number and the linear dimension M of the Matrix is equal to the number of lattice sites.