where the subscript 0 on ⟨...⟩ 0 indicates that the sampling for the averages and the system evolution between the virtual proton insertion events are performed using the energy of the unperturbed system. The surface Widom insertion method brings a virtual proton from infinity into contact with a randomly selected colloidal active site; if the site is empty (has charge -q), it "reacts" with the virtual proton and its charge changes to 0 and the difference in electrostatic energy between the protonated state and the original deprotonated state, ∆E, is calculated using Eqn. (A9). The average in Eqn. ( ) is obtained using 5,000 uncorrelated insertion events. If the site is already protonated, the virtual proton will overlap with the real proton, resulting in infinite ∆E. The virtual protonation process does not affect the actual state of the site and is used just to probe the chemical potential. FIG. : Spherically replicated simulation cell, forming a macroscopic crystal. Each cell has a net electric dipole moment and a non-zero second moment of the charge density tensor. This results in a dipole layer and surface charge at the crystal boundary, producing an electrostatic potential in the crystal's interior.