A statistical mechanical model for the calculation of the permittivity of water in hydrated polymer electrolyte membrane pores

An equilibrium statistical mechanical model is derived to compute the spatial variation in the permittivity of water within the hydrated pores of ion-containing polymeric membranes. The fixed anionic groups within the pore are modeled as periodic arrays of point charges. The Helmholtz free energy is calculated from a total Hamiltonian of the pore that includes energy from (1) interactions between the fields generated by the fixed charge groups and the dipoles of the water molecules, (2) "hard core" interactions between the water molecules, and (3) dipole-dipole interactions between the water molecules. The free energy is divided into two parts: (a) a reference free energy associated with five water molecules in a cluster interacting with each other through the hard core potentials and with the fixed charge groups and (b) an excess free energy due to the dipolar interactions between the water molecules in two cluster units. In the present work we calculate the polarization and corresponding permittivity from this reference free energy. We first show that our calculations, even at this level of sophistication, go beyond all the traditional approaches. Furthermore, with our model we compute radial profiles of the permittivity in the pores of the sulfonic acid-based Nafion(R) and 65% sulfonated poly ether ether ketone ketone polymer electrolyte membranes at several different hydration levels. These numerical results and predictions are in agreement with known experimental measurements. (C) 2001 American Institute of Physics.