Grand-canonical Monte Carlo method for Donnan equilibria

We present a method that enables the direct simulation of Donnan equilibria. The method is based on a grand-canonical Monte Carlo scheme that properly accounts for the unequal partitioning of small ions on the two sides of a semipermeable membrane, and can be used to determine the Donnan electrochemical potential, osmotic pressure, and other system properties. Positive and negative ions are considered separately in the grand-canonical moves. This violates instantaneous charge neutrality, which is usually considered a prerequisite for simulations using the Ewald sum to compute the long-range charge-charge interactions. In this work, we show that if the system is neutral only in an average sense, it is still possible to get reliable results in grand-canonical simulations of electrolytes performed with Ewald summation of electrostatic interactions. We compare our Donnan method with a theory that accounts for differential partitioning of the salt, and find excellent agreement for the electrochemical potential, the osmotic pressure, and the salt concentrations on the two sides. We also compare our method with experimental results for a system of charged colloids confined by a semipermeable membrane and to a constant-NVT simulation method, which does not account for salt partitioning. Our results for the Donnan potential are much closer to the experimental results than the constant-NVT method, highlighting the important effect of salt partitioning on the Donnan potential.