Interaction between similarly charged surfaces can be attractive at high electrostatic coupling constants Ξ = lBZ2/μGC, where lB is the Bjerrum length, μGC the Gouy–Chapman length, and Z the valency of counterions. While this effect has been studied previously in detail, as a function of surface charge density and valency of the pointlike counterions, much less is known about the effect of counterion size. We apply the Wang–Landau sampling Monte Carlo (MC) simulation method to compute the free energy F as a function of the scaled distance between the plates

${\widetilde{D}}=D/\mu _{\rm GC}$
D̃=D/μ GC for a range of Ξ and scaled counterion radii
${\widetilde{R}}=R/\mu _{\rm GC}$
R̃=R/μ GC ⁠. We find that for large Ξ and small ion radius, there is a global equilibrium distance
$ {\widetilde{D}}= {\widetilde{D}}_{\rm eq} =2(1+{\widetilde{R}})$
D̃=D̃ eq =2(1+R̃)⁠, correctly giving the expected value at the point counterion limit. With increasing
${\widetilde{R}}$
R̃ the global minimum in
$F({\widetilde{D}})$
F(D̃) changes to a metastable state and finally this minimum vanishes when
${\widetilde{R}}$
R̃ reaches a critical value, which depends on Ξ. We present a state diagram indicating approximate boundaries between these three regimes. The Wang–Landau MC method, as it is applied here, offers a possibility to study a wide spectrum of extended problems, which cannot be treated by the use of contact value theorem.

Topics
Electrostatics, Free energy, Elementary particle interactions, Ions and properties, Intermolecular forces, Electrical double layers, Steric effect, Coupling constants, Monte Carlo methods, Statistical mechanics models
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