Charge regulation of nanoparticles in the presence of multivalent electrolytes

We explore the charge regulation (CR) of spherical nanoparticles immersed in an asymmetric electrolyte of a specified pH. Using a recently developed reactive canonical Monte Carlo (MC) simulation method, titration isotherms are obtained for suspensions containing monovalent, divalent, and trivalent coions. A theory based on the modified Poisson-Boltzmann approximation, which incorporates the electrostatic ion solvation free energy and discrete surface charge effects, is used to compare with the simulation results. A remarkably good agreement is found without any fitting parameters, both for the ion distributions and titration curves, suggesting that ionic correlations between coions and hydronium ions at the nanoparticle surface play only a minor role in determining the association equilibrium between hydroniums and the functional sites on the nanoparticle surface. On the other hand, if suspension contains multivalent counterions, we observe a large deviation between theory and simulations, showing that the electrostatic correlations between counterions and hydronium ions at the nanoparticle surface are very significant and must be properly taken into account to correctly describe CR for such solutions.