Comparison of the thermal and concentration criteria for determining the effective charge of a spherical colloidal particle in a salt-free suspension

The techniques of canonical Monte Carlo simulations are employed in a comparative study of two recently proposed criteria for the determination of the renormalized or effective charge of a spherical macroion in a charge-stabilized colloidal suspension. The effective charge is determined using (i) the thermal criterion, where the effective charge consists of the intrinsic macroion charge reduced by the integrated thermodynamically bound counterion charge, the bound counterions being defined as those counterions that are within and up to the radial distance (from the macroion center) at which the reduced mean electrostatic potential energy equals the thermal energy, and (ii) the concentration criterion, where the bound counterions are those that lie within the distance at which the local concentration of the counterions equals its average value. It is found that the former criterion applies to the entire range of bare macroion charge/macroion radius ratios investigated, whereas the latter criterion is more suited to large values of this ratio. The observation that the effective charge/bare charge ratio (Z(eff)/Z(m)) depends only on the bare charge/macroion radius ratio (Z(m)/r(m)) suggests that all macroions with the same numerical value of the Z(m)/r(m) ratio are equivalent systems and that one can therefore study highly charged colloids by using a more manageable set of parameters in a simulation. The calculations are extended to divalent counterion systems, and comparisons are also made with other criteria for determining the effective macroion charge in the literature.