The free energy of compressed lattice knots

TL;DR

This paper models the free energy of knotted lattice polymers confined in a cube, using sampling algorithms and Flory-Huggins theory to analyze how knot type and concentration affect free energy.

Contribution

It introduces a computational approach to estimate free energy of knotted lattice polymers and applies Flory-Huggins theory to model these energies based on concentration.

Findings

Free energy depends on knot type at low concentrations.

Flory-Huggins theory effectively models free energy in this context.

Estimated Flory interaction parameter for knotted lattice polygons.

Abstract

A compressed knotted ring polymer in a confining cavity is modelled by a knotted lattice polygon confined in a cube in ${\mathbb Z}^3$. The GAS algorithm [17] is used to sample lattice polygons of fixed knot type in a confining cube and to estimate the free energy of confined lattice knots. Lattice polygons of knot types the unknot, the trefoil knot, and the figure eight knot, are sampled and the free energies are estimated as functions of the concentration of monomers in the confining cube. The data show that the free energy is a function of knot type at low concentrations, and (mean-field) Flory-Huggins theory [12,15] is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of knotted lattice polygons in ${\mathbb Z}^3$ is also estimated.