Bending rigidity of stiff polyelectrolyte chains: Single chain and a bundle of multichains

TL;DR

This paper develops a theory to analyze how multivalent counterions and charge correlations influence the bending rigidity of stiff polyelectrolytes, revealing complex dependencies on charge fraction and counterion concentration, with implications for polyelectrolyte bundles.

Contribution

The study introduces a theoretical framework that captures the effects of counterion binding and charge correlations on the bending rigidity of polyelectrolytes and their bundles, highlighting nontrivial scaling behaviors.

Findings

Small multivalent counterions dramatically soften stiff polyelectrolytes.

Bending rigidity varies non-monotonically with charge fraction $f_0$.

Inter-loop coupling further reduces rigidity in polyelectrolyte bundles.

Abstract

We study the bending rigidity of highly charged stiff polyelectrolytes, for both a single chain and many chains forming a bundle. A theory is developed to account for the interplay between competitive binding of counterions and charge correlations in softening the polyelectrolyte (PE) chains. The presence of even a small concentration of multivalent counterions leads to a dramatic reduction in the bending rigidity of the chains that are nominally stiffened by the repulsion between their backbone charges. The variation of the bending rigidity as a function of $f_{0}$, the fraction of charged monomers on the chain, does not exhibits simple scaling behavior; it grows with increasing $f_{0}$ below a critical value of $f_{0}$. Beyond the critical value, however, the chain becomes softer as $f_{0}$ increases. The bending rigidity also exhibits intriguing dependence on the concentration of multivalent counterion $n_{2}$; for highly charged PEs, the bending rigidity decreases as $n_2$ increases from zero, while it increases with increasing $n_{2}$ beyond a certain value of $n_{2}$. When polyelectrolyte chains form a $N$-loop condensate (e.g., a toroidal bundle formed by $N$ turns (winds) of the chain), the inter-loop coupling further softens the condensate, resulting in the bending free energy of the condensate that scales as $N$ for large $N$.