# An Analytical Understanding of Reentrant Condensation of a Polyelectrolyte in the Presence of an Oppositely Charged Surfactant

## TL;DR

This paper explains how oppositely charged surfactants affect the condensation and phase transitions of polyelectrolytes, offering insights into liquid-liquid phase separation in biological systems.

## Contribution

A novel mean-field theory with analytical solutions explains reentrant condensation and phase transitions in polyelectrolytes with surfactants.

## Key findings

- Strong electrostatic adsorption between ionic monomers and surfactant ions is critical for collapse and re-entry transitions.
- A minimum coupling energy is required for phase transitions, explaining the role of surfactant chain length.
- The theory rationalizes the 'egg shape'-like phase diagram and applies to biological systems involving DNA/RNA and proteins/peptides.

## Abstract

We explore the phase-transition mechanism of the reentrant
condensation
of a polyelectrolyte in the presence of an oppositely charged surfactant,
which is of fundamental importance to the understanding of liquid–liquid
phase separation (LLPS) in soft materials and biological systems.
We focus on the adsorption and attraction effects of surfactants near/on
polymer chains and ignore their own nonessential mixing effects if
surfactant molecules are far away from polymer chains. This novel
approach allows us to construct a simple mean-field equilibrium theory
with closed-form analytical solutions, which can rationalize the essential
features of the emergent “egg shape”-like phase diagram.
The theory addresses that a strong electrostatic adsorption between
the ionic monomers and surfactant ions is critical to understand the
peculiar phenomenon that both the collapse and re-entry transitions
of polyelectrolytes can occur when the concentration of the surfactant
is lower than its bulk critical micelle concentration (CMC). Our theory
also indicates that a minimum coupling energy for the nonlinear hydrophobic-aggregation
effect of the adsorbed surfactant is essential for a phase transition
to occur, which explains why polyelectrolytes show such a phase transition
only if the surfactant chain length is beyond a minimum value. This
work provides insight into the understanding of liquid–liquid
phase separation in biological systems if proteins and/or peptides
bound to DNAs and/or RNAs play an important role.