  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.
