We study the pH dependence of the internal energy and osmotic pressure for a dilute suspension of ion-penetrable spheres. The charge density within the sphere is determined by the dissociation of the fixed ionizable groups, which in turn depends on the pH and the electrostatic field. When the electric potential is small compared to the thermal motion, a linear charge regulation model is always obtained with pH-dependent coefficients. The two-site dissociation models have been considered. On the basis of the linearized Poisson-Boltzmann equation, the electric potential distribution can be solved by the perturbation method. A two-particle interaction is then obtained. The effect of pH on the interaction energy is twofold. It affects the total charge carried by an isolated sphere and also the ability of charge-regulation. The charge can be regulated by the potential established by the sphere itself and that induced by other surrounding spheres. The three-body interaction has also been derived by studying the interaction free energy of a system with N particles. On the basis of the two-particle interaction energy, the second virial coefficients can then be evaluated.
