# Osmotic Pressure of Permeable Ionic Microgels: Poisson-Boltzmann Theory and Exact Statistical Mechanical Relations in the Cell Model

## TL;DR

This paper develops and compares two theoretical methods within the Poisson-Boltzmann framework to accurately calculate the electrostatic osmotic pressure of permeable ionic microgels, enhancing understanding of their swelling behavior.

## Contribution

It introduces a statistical mechanical approach alongside PB theory to derive exact relations for ionic microgel osmotic pressure in various geometries.

## Key findings

- PB and statistical mechanical methods agree for surface-charged microgels.
- Volume charge microgels' osmotic pressure is the average of PB profiles.
- Derived contact value theorems for charged colloids.

## Abstract

Ionic microgels are soft colloidal particles, composed of crosslinked polymer networks, that ionize and swell when dispersed in a good solvent. Swelling of these permeable, compressible particles involves a balance of electrostatic, elastic, and mixing contributions to the single-particle osmotic pressure. The electrostatic contribution depends on the distributions of mobile counterions and coions and of fixed charge on the polymers. Within the cell model, we employ two complementary methods to derive the electrostatic osmotic pressure of ionic microgels. In Poisson-Boltzmann (PB) theory, we minimize a free energy functional with respect to the electrostatic potential to obtain the bulk pressure. From the pressure tensor, we extract the electrostatic and gel contributions to the total pressure. In a statistical mechanical approach, we vary the free energy with respect to microgel size to obtain exact relations for the microgel electrostatic osmotic pressure. We present results for planar, cylindrical, and spherical geometries. For models of membranes and microgels with fixed charge uniformly distributed over their surface or volume, we derive analogues of the contact value theorem for charged colloids. We validate these relations by solving the PB equation and computing ion densities and osmotic pressures. When implemented within PB theory, the two methods yield identical electrostatic osmotic pressures for surface-charged microgels. For volume-charged microgels, the exact electrostatic osmotic pressure equals the average of the corresponding PB profile over the gel volume. We demonstrate that swelling of ionic microgels depends on variation of the electrostatic pressure inside the particle and discuss implications for interpreting experiments.