Fixed-charge four-site model

Cation, anion, and water transport in ion-exchange membranes have been described by several phenomenological solution-diffusion models and electrokinetic pore-flow theories. Phenomenological models based on irreversible thermodynamics have been applied to cation-exchange membranes, including DuPont s Nafion perfluorosulfonic acid membranes [147, 148]. These models view the membrane as a black box and membrane properties such as ionic fluxes, water transport, and electric potential are related to one another without specifying the membrane structure and molecular-level mechanism for ion and solvent permeation. For a four-component system (one mobile cation, one mobile anion, water, and membrane fixed-charge sites), there are three independent flux equations (for cations, anions, and solvent species) of the form... 

The model of Sprik and Klein has a set of four fluctuating charges distributed on a tetrahedron located at the M site of a TIP4P water. These charges fluctuate such that the net charge is fixed. This scheme is described by an extended Lagrangian so that the equations of motion determine the fluctuations in the effective dipole moment. A thermostat was not employed in this study, so it should be viewed as a test case only. Liquid state water properties were adequately reproduced. 

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