Origin of 1-pK and 2-pK models for ionizable water-solid interfaces

A discrete charge Ising model planar ionizable water-solid interface is developed. This model treats all individual ionizable sites explicitly and is solved with Monte Carlo simulation techniques. It is shown that a mean-field cluster expansion provides a very accurate scheme for the calculation of the titration curves in this system. From the first two iterations of this approximation scheme one recovers the classical 1-pK and 2-pK models. In most cases, the 1-pK model provides a sufficiently accurate description of the titration behavior of the interface, while in special situations the use of the 2-pK model may be required. The quick convergence of this approximation scheme is related to the long-range nature of the interaction potential between ionizable residues, a feature which is characteristic for water-solid interfaces.