# Two Methods Based on Integral Equation Approaches in Analyzing Polyelectrolyte Solutions: Macrophase Separation

## TL;DR

This paper introduces two analytical methods to study phase behavior in polyelectrolyte solutions, focusing on complex coacervation and phase segregation.

## Contribution

The paper presents two novel integral equation approaches for modeling polyelectrolyte solutions and their phase behaviors.

## Key findings

- The two methods produce pressures consistent with molecular dynamics simulations for polyanion and counterion solutions.
- Both methods effectively calculate two-phase equilibrium and phase segregation strength in the system.
- Scaling exponents near the critical point are analyzed to understand phase behavior.

## Abstract

To understand the phase behaviors of polyelectrolyte solutions, we provide two analytical methods to formulate a molecular equation of state for a system of fully charged polyanions (PAs) and polycations (PCs) in a monomeric neutral component, based on integral equation theories. The mixture is treated in a primitive and restricted manner. The first method utilizes Blum’s approach to charged hard spheres, incorporating the chain connectivity contribution by charged spheres via Stell’s cavity function method. The second method employs Wertheim’s multi-density Ornstein–Zernike treatment of charged hard spheres with Baxter’s adhesive potential. The pressures derived from these methods are compared to available molecular dynamics simulations data for a solution of PAs and monomeric counterions as a limiting case. Two-phase equilibrium for the system is calculated using both methods to evaluate the relative strength of phase segregation that leads to complex coacervation. Additionally, the scaling exponents for a selected solution near its critical point are examined.