Energy and entropy effects of counterions in salt-free colloidal solutions

We use a shell model to study the counterion interactions in a colloidal solution. In this shell model, the counterions are restricted to move inside a spherical region about their host colloidal particle. In particular, we apply Monte Carlo simulations to derive the energy and entropy contributions of the effective colloidal interaction. Our result reveals an attractive electrostatic energy, which is overpowered by the osmotic repulsion among the counterions, as the latter can be well estimated by an ideal-gas approximation. We also provide an optional algorithm that enables counterion mixing between the two counterion clouds even when the clouds do not overlap. The residual mixing entropy of counterions gives a reduction in free energy that is comparable to the thermal fluctuation, suggesting a possible attractive mechanism between the colloidal particles under non-equilibrium condition.