We consider a spherical nanoparticle of radius a, placed at the center of the cell. In the case of a SHC, the cell radius R c is related to the volume fraction, η, of the nanoparticles inside the suspension, R c a = η -1/3 , as sketched in Fig. a. In the case of a PC, η = 4πa 3 /3L 3 . At the start of the simulation a nanoparticle contains Z fully ionized acidic groups, each of acid dissociation constant K a , uniformly distributed over its surface. The cell also contains fully ionized 1:z -electrolyte of concentration c s , resulting in a total of N s anions of valence z -and N + = z -N s monovalent cations. In addition, Z of hydronium ions are added to preserve the overall charge neutrality. All ions are free to move within the cell volume. For simplicity, we assume that all ions are equal-sized, of radius r i = 2 Å. Water is treated implicitly as a uniform dielectric medium of Bjerrum length λ B = q 2 /k B T = 7.2 Å. The equilibrium charge of the nanoparticle is -Z ef f q, where Z ef f is the average number of deprotonated surface groups after the surface reaction H + + A -----HA reaches equilibrium. Note that a priori we do not know either the concentration or activity of free protons inside the system. These can only be calculated after the simulation is fully equilibrated. To change the pH inside the system we can add a base, such as NaOH. Since the dissociation of water is so weak, addition of a base molecule will result in the reaction between a hydronium ion and a base molecule, resulting in formation of a water molecule and release of Na + . In the present implicit water model, the net effect will be the replacement of a hydronium ion by a sodium ion, H 3 O + → Na + . We can then rerun the simulation with Z -1 hydroniums and one extra Na + to obtain the new charge of the nanoparticle. The new equilibrium pH can then be calculated using the algorithm described later on in this paper . Repeating this procedure -i.e. adding more base molecules -we can explore the full pH range, obtaining the complete titration curve for a suspension of nanoparticles.