L n 3 ) are the reciprocal lattice vectors for the cubic simulation box of side length L and volume V = L 3 , and κ e is an arbitrary damping parameter. The expression (2) is also valid for systems with an excess charge Q t = i q i and the corresponding uniform neutralizing background . The prime on the sum indicates that the term i = j is excluded from the summation for n n n = 0. We set κ e = 5/L, so that only n n n = 0 needs to be considered for the short range -sum over erfc terms -contribution to the electrostatic energy. For a cell model with only one colloidal particle located at its center, we expect that the specific boundary condition -hard wall or periodic -should not make a significant difference for thermodynamics. In Fig. , we show a comparison of the density profiles of positive and negative ions inside a SHC and PC containing 100 mM of 1:1 or 1:3 electrolyte, for a nanoparticle of Z = 600 non-reacting surface groups, obtained using a standard canonical MC. As can be seen, the density profiles for both cases are almost identical. This is very convenient, allowing us to use SHC to construct the mean-field theory of CR. The spherical symmetry of the SHC, makes solution of the nonlinear PB equation much simpler, since it can be transformed into a one dimensional integral equation. On the other hand, PC would require us to solve the full three dimensional non-linear differential equation, which is significantly more involved.