A Variational Approach to the Structure and Thermodynamics of Linear Polyelectrolytes with Coulomb and Screened Coulomb Interactions

TL;DR

This paper introduces a variational method to efficiently compute free energy and conformational properties of linear polyelectrolytes with Coulomb interactions, validated against Monte Carlo simulations, and extends to screened Coulomb interactions with some limitations.

Contribution

It develops a novel variational approach with an efficient iterative algorithm for analyzing polyelectrolytes, including high and low temperature limits and validation against simulations.

Findings

Good accuracy for unscreened Coulomb interactions.

Screened Coulomb interactions less accurate with shorter ranges.

Efficient Monte Carlo algorithm for large chain sizes.

Abstract

A variational approach is used to calculate free energy and conformational properties in polyelectrolytes. The true bond and Coulomb potentials are approximated by a trial isotropic harmonic energy containing monomer-monomer force constants as variational parameters. By a judicious choice of representation and the use of incremental matrix inversion, an efficient and fast-convergent iterative algorithm is constructed, that optimizes the free energy. The computational demand scales as N^3. The method has the additional advantage that the entropy is easily computed. An analysis of the high and low temperature limits is given. Also, the variational formulation is shown to respect the appropriate virial identities. The accuracy of the approximations introduced are tested against Monte Carlo simulations for problem sizes ranging from N=20 to 1024. Very good performance is obtained for chains with unscreened Coulomb interactions. The addition of salt is described through a screened Coulomb interaction, for which the accuracy in a certain parameter range turns out to be inferior to the unscreened case. The reason is that the isotropic Gaussian Ansatz becomes less efficient with shorter range interactions. As a by-product a very efficient Monte Carlo algorithm was developed for comparisons, providing high statistics data for very large sizes - 2048 monomers.