Titration in Canonical and Grand-Canonical Ensembles

We discuss problems associated with the notion of pH in heterogeneous systems. For homogeneous systems, standardization protocols lead to a well-defined quantity, which, although different from Sørensen's original idea of pH, is well reproducible and has become accepted as the measure of the "hydrogen potential". On the other hand, for heterogeneous systems, pH defined in terms of the chemical part of the electrochemical activity is thermodynamically inconsistent and runs afoul of the Gibbs-Guggenheim principle that forbids splitting of the electrochemical potential into separate chemical and electrostatic parts, since only the sum of two has any thermodynamic meaning. The problem is particularly relevant for modern simulation methods which involve charge regulation of proteins, polyelectrolytes, nanoparticles, colloidal suspensions, and so forth. In this paper, we show that titration isotherms calculated using semigrand canonical simulations can be very different from the ones obtained using canonical reactive Monte Carlo simulations.