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

TL;DR

This paper introduces a local approximation of fundamental measure theory into the 3D Poisson-Nernst-Planck equations to better account for hard sphere repulsion among ions, simplifying computations while maintaining accuracy.

Contribution

A novel local hard sphere PNP (LHSPNP) model is derived, providing a more accurate and computationally efficient way to include ion size effects in 3D biological system simulations.

Findings

LHSPNP matches the size-modified model in free energy terms.

The equation of state is exact for the first two virial coefficients.

LHSPNP results differ from traditional PNP but align with previous size-modified models.

Abstract

The hard sphere repulsion among ions can be considered in the Poisson-Nernst-Planck (PNP) equations by combining the fundamental measure theory (FMT). To reduce the nonlocal computational complexity in 3D simulation of biological systems, a local approximation of FMT is derived, which forms a local hard sphere PNP (LHSPNP) model. It is interestingly found that the essential part of free energy term of the previous size modified model has a very similar form to one term of the LHS model, but LHSPNP has more additional terms accounting for size effects. Equation of state for one component homogeneous fluid is studied for the local hard sphere approximation of FMT and is proved to be exact for the first two virial coefficients, while the previous size modified model only presents the first virial coefficient accurately. To investigate the effects of LHS model and the competitions among different counterion species, numerical experiments are performed for the traditional PNP model, the LHSPNP model, the previous size modified PNP (SMPNP) model and the Monte Carlo simulation. It's observed that in steady state the LHSPNP results are quite different from the PNP results, but are close to the SMPNP results under a wide range of boundary conditions. Besides, in both LHSPNP and SMPNP models the stratification of one counterion species can be observed under certain bulk concentrations.