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The hard sphere repulsion among ions can be considered in thePoisson-Nernst-Planck (PNP) equations by combining the fundamental measuretheory (FMT). To reduce the nonlocal computational complexity in 3D simulationof biological systems, a local approximation of FMT is derived, which forms alocal hard sphere PNP (LHSPNP) model. It is interestingly found that theessential part of free energy term of the previous size modified model has avery similar form to one term of the LHS model, but LHSPNP has more additionalterms accounting for size effects. Equation of state for one componenthomogeneous fluid is studied for the local hard sphere approximation of FMT andis proved to be exact for the first two virial coefficients, while the previoussize modified model only presents the first virial coefficient accurately. Toinvestigate the effects of LHS model and the competitions among differentcounterion species, numerical experiments are performed for the traditional PNPmodel, the LHSPNP model, the previous size modified PNP (SMPNP) model and theMonte Carlo simulation. It's observed that in steady state the LHSPNP resultsare quite different from the PNP results, but are close to the SMPNP resultsunder a wide range of boundary conditions. Besides, in both LHSPNP and SMPNPmodels the stratification of one counterion species can be observed undercertain bulk concentrations.

journal of statistical physics

A local approximation of fundamental measure theory incorporated into  three dimensional Poisson-Nernst-Planck equations to account for hard sphere  repulsion among ions