  Ionic microgel particles, when dispersed in a solvent, swell to equilibrium
sizes that are governed by a balance between electrostatic and elastic forces.
Tuning of particle size by varying external stimuli, such as $p$H, salt
concentration, and temperature, has relevance for drug delivery, microfluidics,
and filtration. To model swelling of ionic microgels, we derive a statistical
mechanical theorem, which proves exact within the cell model, for the
electrostatic contribution to the osmotic pressure inside a permeable colloidal
macroion. Applying the theorem, we demonstrate how the distribution of
counterions within an ionic microgel determines the internal osmotic pressure.
By combining the electrostatic pressure, which we compute via both
Poisson-Boltzmann theory and molecular dynamics simulation, with the elastic
pressure, modeled via the Flory-Rehner theory of swollen polymer networks, we
show how deswelling of ionic microgels with increasing concentration of
particles can result from a redistribution of counterions that reduces
electrostatic pressure. A linearized approximation for the electrostatic
pressure, which proves remarkably accurate, provides physical insight and
greatly eases numerical calculations for practical applications. Comparing with
experiments, we explain why soft particles in deionized suspensions deswell
upon increasing concentration and why this effect may be suppressed at higher
ionic strength. The failure of the uniform ideal-gas approximation to
adequately account for counterion-induced deswelling below close packing of
microgels is attributed to neglect of spatial variation of the counterion
density profile and the electrostatic pressure of incompletely neutralized
macroions.
