Interactions between charged spheres in divalent counterion solution

We simulate model systems of charged spherical particles in their counterion solution and measure the thermodynamic pressure and the pair distribution function from which we derive effective potentials of mean force. For a system with only electrostatic and hard core interactions, we investigate the effective potential between two like-charged spheres in divalent counterion solution as a function of concentration. We find a strong attractive interaction for high concentration and a global repulsive effective interaction for dilute systems. The results indicate a first order phase transition in sphere–counterion density as a function of global concentration and the effective sphere–sphere potentials in the dilute (solvated) regime suggest significant density fluctuations due to short range local minima in the effective energy surface. Our results arise from a minimal approach model of several recent experiments on polystyrene latex particles in monovalent counterion solution.