Variational theory for a single polyelectrolyte chain revisited

TL;DR

This paper revisits the electrostatic contribution to the persistence length of a charged polymer using a Gaussian variational method, clarifying previous theoretical discrepancies and confirming the OSF result for stiff chains.

Contribution

It introduces a variational approach that accurately captures the electrostatic persistence length and corrects prior misconceptions about cutoff dependencies.

Findings

Electrostatic persistence length scales as rac{}{2} with screening length.

Crumpling occurs at small scales for weakly charged, flexible chains.

OSF result is validated for highly charged, stiff chains.

Abstract

We reconsider the electrostatic contribution to the persistence length, $\ell_e$, of a single, infinitely long charged polymer in the presence of screening. A Gaussian variational method is employed, taking $\ell_e$ as the only variational parameter. For weakly charged and flexible chains, crumpling occurs at small length scales because conformational fluctuations overcome electrostatic repulsion. The electrostatic persistence length depends on the square of the screening length, $\ell_e\sim\kappa^{-2}$, as first argued by Khokhlov and Khachaturian by applying the Odijk-Skolnick-Fixman (OSF) theory to a string of crumpled blobs. We compare our approach to previous theoretical works (including variational formulations) and show that the result $\ell_e\sim\kappa^{-1}$ found by several authors comes from the improper use of a cutoff at small length scales. For highly charged and stiff chains, crumpling does not occur; here we recover the OSF result and validate the perturbative calculation for slightly bent rods.