Poisson--Nernst--Planck Systems for Ion Flow with a Local Hard-Sphere Potential for Ion Size Effects

Abstract

In this work, we analyze a one-dimensional steady-state Poisson-Nernst-Planck-type model for ionic flow through a membrane channel with fixed boundary ion concentrations (charges) and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. A local hard-sphere potential that depends pointwise on ion concentrations is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive an approximation of the I-V (current-voltage) relation and identify two critical potentials or voltages for ion size effects. Under electroneutrality (zero net charge) boundary conditions, each of these two critical potentials separates the potential into two regions over which the ion size effects are qualitatively opposite to each other. On the other hand, without electroneutrality boundary conditions, the qualitative effects of ion sizes will depend not only on the critical potentials but also on boundary concentrations. Important scaling laws of I-V relations and critical potentials in boundary concentrations are obtained. Similar results about ion size effects on the flow of matter are also discussed. Under electroneutrality boundary conditions, the results on the first order approximation in ion diameters of solutions, I-V relations, and critical potentials agree with those with a nonlocal hard-sphere potential examined by Ji and Liu [J. Dynam. Differential Equations, 24 (2012), pp. 955-983].

Effects of (Small) Permanent Charge and Channel Geometry on Ionic Flows via Classical Poisson--Nernst--Planck Models

“…A strategy for analyzing this connecting problem of classical PNP models was developed in [14] (in [36] for a general setting), which has been successfully extended to handle PNP with hard-sphere ions in [27,34,38]. The classical PNP system is first reduced to two subsystems: the limiting fast and the limiting slow system.…”

“…Numerical simulations of PNP with approximated models of μ ex k (x) have been conducted for ion channel problems in comparison with experimental data and have shown great successes for properties such as ion permeation and ion selectivity (e.g., [18,19,20,21]). Other important phenomena involving μ ex k (x) such as steric effects, layering, charge inversion, and critical potentials have also been studied [3,16,23,24,25,27,30,31,33,34,38,56].…”

A Local Approximation of Fundamental Measure Theory Incorporated into Three Dimensional Poisson–Nernst–Planck Equations to Account for Hard Sphere Repulsion Among Ions

“…This local expression is much simpler than the nonlocal one in numerical calculation. For one dimensional situation, a local hard sphere potential is also proposed by Liu et al [32] to investigate ion flow through channels and shows great improvements. The final modified Nernst-Planck (NP) equations of the local hard sphere PNP (LHSPNP) can be obtained by replacing the µ ex i in Eq.…”

“…We aim to construct an excess chemical potential in a point to point way like the ideal component for 3D simulations of ionic solutions. Liu et al have derived a LHS excess chemical potential in 1D case and made theoretical analysis on the model problem based on geometric singular perturbation theory [32]. In our work, we concentrate on the more complex 3D case and simplify the integration using an expansion of the integrand under small ionic diameters to obtain the final LHSPNP model.…”

Electroneutral models for dynamic Poisson-Nernst-Planck systems

“…For example, singular perturbation analysis of PNP system has been carried out for narrow ion channels with certain geometric structure [5,6]. Geometric singular perturbation approach has been developed to investigate the existence and uniqueness of solutions in stationary PNP system [7,8] as well as the effects of permanent charge and ion size [9,10]. Recently, Wang et al [11] have tackled the steady state PNP system with arbitrary number of ion species and arbitrary valences, and have successfully reduced the asymptotic solutions to a single scalar transcendental equation.…”