Modeling Swelling of pH-Responsive Microgels: Theory and Simulations

Combining a mean-field swelling model─which incorporates the Poisson–Boltzmann cell model for describing the electrostatics of microgels and a Flory–Rehner-based model for describing the polymer network─with the law of mass action to account for chemical reactions, we present a comprehensive swelling model for weakly charged microgels. This model provides an expression for the microgel osmotic pressure, used to determine the equilibrium swelling and, consequently, the net charge of the microgel as a function of reservoir pH, salt concentration, degree of polymerization, and other suspension and microscopic network properties. The model allows us to relate microscopic microgel features with the equilibrium swelling properties. The weak-field limiting case of the Poisson–Boltzmann theory is analyzed, yielding closed formulas. We validate the model against state-of-the-art coarse-grained simulations of a microgel, utilizing molecular dynamics to explore configurational degrees of freedom and the Monte Carlo grand-reaction method to simulate chemical reactions in equilibrium with a pH and salt reservoir. We test the model predictions for equilibrium ionization, size, and net charge against particle-based simulations and experiments. Our findings show that the model accurately describes microgel swelling and net charge over a wide range of pH levels. Although the accuracy decreases for larger salt concentrations, its overall qualitative accuracy makes it a reliable tool for parameter exploration and data interpretation, aiding in the rational design of microgel suspensions.