Theory of polyelectrolytes in solvents

TL;DR

This paper develops a continuum model for polyelectrolytes in ionic solvents, deriving effective potentials and predicting phase behavior, chain conformations, and shape transitions, with results consistent with known scaling laws and experimentally testable predictions.

Contribution

It introduces a novel continuum framework accounting for ionic fluctuations, deriving effective interactions, and predicting phase transitions and shapes of polyelectrolytes.

Findings

Effective short-range potential with repulsive and attractive regimes.

Calculated Flory exponent of approximately 0.63 for long chains.

Predicted shape transitions including spherical and toroidal condensates.

Abstract

Using a continuum description, we account for fluctuations in the ionic solvent surrounding a Gaussian, charged chain and derive an effective short-ranged potential between the charges on the chain. This potential is repulsive at short separations and attractive at longer distances. The chemical potential can be derived from this potential. When the chemical potential is positive, it leads to a melt-like state. For a vanishingly low concentration of segments, this state exhibits scaling behavior for long chains. The Flory exponent characterizing the radius of gyration for long chains is calculated to be approximately 0.63, close to the classical value obtained for second order phase transitions. For short chains, the radius of gyration varies linearly with $N$, the chain length, and is sensitive to the parameters in the interaction potential. The linear dependence on the chain length $N$ indicates a stiff behavior. The chemical potential associated with this interaction changes sign, when the screening length in the ionic solvent exceeds a critical value. This leads to condensation when the chemical potential is negative. In this state, it is shown using the mean-field approximation that spherical and toroidal condensed shapes can be obtained. The thickness of the toroidal polyelectrolyte is studied as a function of the parameters of the model, such as the ionic screening length. The predictions of this theory should be amenable to experimental verification.