Grand Canonical Ensemble Monte Carlo Simulations of Donnan Potentials, Nonelectroneutrality, Activity Coefficients and Excess Energy in Spherical Charged or Uncharged Pores with Restricted Primitive Model Electrolytes

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Abstract
A great number of Grand Canonical Ensemble simulations have been performed for restricted primitive model electrolytes in spherical hard-wall pores with the same dielectric permittivity in the pores and in the wall. The pores have a continuously distributed surface charge from 0 to 10 elementary charges and radii from 1 to 35 ionic diameters. When a pore is in equilibrium with an outside bulk solution, it acquires in the average a certain deviation from electroneutrality (spontaneous electrification). For any given pore surface charge this electrification is a given function of the radius of the accessible part of the pore scaled by the Debye length without regard to the Debye length without regard to the Bjerrum parameter or the bulk concentration of the electrolyte and without regard to the pore size relative to the ion diameter. If an additional uniform electric potential is applied in the pore, the electrification changes. At a certain potential the pore is electroneutral. This potential is an approximation to the Donnan potential in a “Swiss cheese” membrane model taking into account inteactions of ions inside each pore, but replacing the detailed interactions between ions in different pores with a mere neutralizing collective potential. The deviations from electroneutrality, the mean ion occupation numbers and the average mean ion activity coeffificients in the pores are investigated as a function of the total applied potential (the external potential + the potential from the surface charge). Also, the average single ion activity coefficients in the pores are investigated for different pore sizes and surface charges. Some consequences for the theory of ion exchange membranes are discussed. The results are also compared to a carricture analytical model for minimal pores containing no more then one ion at a time. The excess interaction energy between the ions in the pore, its variance and the average exponential of the configuration energy divided by kT is also sampled in each simulation. From the variance, the excess heat capacity at constant volume with a correction term for number fluctuations is calculated. From the logarithm of the average exponential of the interaction energy, the electrostitic Helmholtz free energy in the bulk limit may be extrapolated (very large pores). These excess quanities per particle are scaled by the reciprocal of the radius of the accessible sphere measured in Debye lengths. In neutral pores, the excess energy per particle is a fraction of the bulk value extrapolating monotonously to the bulk value for large pores.

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