where w is the dielectric constant of the medium (water). To quantitatively explore the difference between the titration isotherms calculated using the standard RR-cpH algorithm, Eq. 3, and the canonical algorithm Eq. 4, we focus on a colloidal particle with Z active surface groups, placed at the center of a cubic simulation cell of volume V = L 3 , where L is the side length of the cell. The cell also contain H 3 O + , Cl -, and Na + ions, ensuring the overall charge neutrality. We execute the reactive Monte Carlo simulation to ascertain the average number of protonated surface groups after equilibrium has been established. We then use the surface Widom insertion method to obtain the pH inside the system. Briefly the idea behind the surface Widom method involves bringing a virtual proton from infinity to a randomly chosen colloidal active site. If the selected site is unoccupied (bearing charge -q), it "reacts" with the virtual proton, causing its charge to transition to 0. We then calculate the change in the electrostatic energy between the initial and final states ∆U . If the site is already occupied, ∆U is infinite. The system's pH is then given by,