Forces and pressures in adsorbing partially directed walks

TL;DR

This paper models the entropic forces exerted by adsorbing partially directed lattice walks confined between walls, identifying zero-force points and analyzing how free energy varies with wall separation.

Contribution

It introduces a lattice walk model to analyze forces in confined polymers, providing exact and asymptotic locations of zero-force points.

Findings

Zero-force points are identified in the model.

The free energy as a function of wall separation is characterized.

Locations of zero-force points are determined exactly or asymptotically.

Abstract

Polymers in confined spaces lose conformational entropy. This induces a net repulsive entropic force on the walls of the confining space. A model for this phenomenon is a lattice walk between confining walls, and in this paper a model of an adsorbing partially directed walk is used. The walk is placed in a half square lattice $L^2_+$ with boundary $\partial L^2_+$, and confined between two vertical parallel walls, which are vertical lines in the lattice, a distance $w$ apart. The free energy of the walk is determined, as a function of $w$, for walks with endpoints in the confining walls and adsorbing in $\partial L^2_+$. This gives the entropic force on the confining walls as a function of $w$. It is shown that there are zero force points in this model and the locations of these points are determined, in some cases exactly, and in other cases asymptotically.