Positivity preserving finite difference methods for Poisson–Nernst–Planck equations with steric interactions: Application to slit-shaped nanopore conductance

Ding, Jie; Wang, Zhongming; Zhou, Shenggao

doi:10.1016/j.jcp.2019.108864

Cited by 42 publications

(44 citation statements)

References 44 publications

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“…The hard-sphere interactions between ions can be described by the Lennard-Jones (LJ) potential energy, which gives rise to a nonlocal model [26]. To avoid computationally intractable integrodifferential equations, local approximations of nonlocal integrals can be employed to obtain local models [16,25,35]. In such a local model, the excess chemical potential is given by…”

Section: Local Steric Poisson-boltzmann Theoriesmentioning

confidence: 99%

“…The excess chemical potential can also represent hard-sphere interactions between ions using the Lennard-Jones (LJ) potential energy [26]. This leads to a nonlocal model, which can be further approximated using the Fourier analysis to obtain a computationally more tractable local model [16,25,35]. Alternatively, the excess chemical potential can be determined by the Carnahan-Starling (CS) equation of state for hard-sphere liquids of a uniform size or the Boublik-Mansoori-Carnahan-Starling-Leland (BM-CSL) equation of state for unequal sizes [5].…”

Section: Introductionmentioning

confidence: 99%

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Fast Newton Iterative Method for Local Steric Poisson--Boltzmann Theories in Biomolecular Solvation

Chen

Dou

Zhou

2021

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This work proposes a fast iterative method for local steric Poisson-Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the method, we focus on a local steric PB theory derived from a lattice-gas model, as an example. The advantages of the proposed method in efficiency lies on a key idea that ionic concentrations as scalar implicit functions of the electrostatic potential, i.e, generalized Boltzmann distributions, are numerically available. The existence, uniqueness, boundness, and smoothness of such distributions are rigorously established. A Newton iteration method with truncation is proposed to solve a nonlinear system discretized from the generalized PB equations. The existence and uniqueness of the solution to the discretized nonlinear system are established by showing that it is a unique minimizer of a constructed convex energy. Thanks to the boundness of ionic concentrations, truncation bounds for the potential are obtained by using the extremum principle. The truncation step in iterations is shown to be energy, residual, and error decreasing. To further speed-up computations, we propose a novel precomputing-interpolation strategy, which is applicable to other local steric PB theories and makes the proposed methods for solving steric PB theories as efficient as for solving the classical PB theory. Analysis on the Newton iteration method with truncation shows local quadratic convergence for the proposed numerical methods. Applications to realistic biomolecular solvation systems reveal that counterions with steric hindrance stratify in an order prescribed by the parameter of ionic valence-to-volume ratio. Finally, we remark that the proposed iterative methods for local steric PB theories can be readily incorporated in well-known classical PB solvers.

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Section: Local Steric Poisson-boltzmann Theoriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fast Newton Iterative Method for Local Steric Poisson--Boltzmann Theories in Biomolecular Solvation

Chen

Dou

Zhou

2021

Preprint

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“…Especially, we noticed that the modified PNP equations with ionic steric effects (SPNP) were studied by finite difference methods based on harmonic-mean approximations to the exponential coefficients of the reformulated NP equations in [12]. The difference from [12] is that the exponential coefficients are approximated based on the inverse averaging of the integral on the element instead of directly on the entire grid node in this paper. Although the finite difference method has been widely used to solve the PNP equations, e.g.…”

Section: Introductionmentioning

confidence: 99%

An inverse averaging finite element method for solving the size-modified Poisson-Nernst-Planck equations in ion channel simulations

Shen¹,

Zhang²,

Lu³

2021

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In this work, we introduce an inverse averaging finite element method (IAFEM) for solving the size-modified Poisson-Nernst-Planck (SMPNP) equations. Comparing with the classical Poisson-Nernst-Planck (PNP) equations, the SMPNP equations add a nonlinear term to each of the Nernst-Planck (NP) fluxes to describe the steric repulsion which can treat multiple nonuniform particle sizes in simulations. Since the new terms include sums and gradients of ion concentrations, the nonlinear coupling of SMPNP equations is much stronger than that of PNP equations. By introducing a generalized Slotboom transform, each of the size-modified NP equation is transformed into a self-adjoint equation with exponentially behaved coefficient, which has similar simple form to the standard NP equation with the Slotboom transformation. This treatment enables employing our recently developed inverse averaging technique to deal with the exponential coefficients of the reformulated formulations, featured with advantages of numerical stability and flux conservation especially in strong nonlinear and convection-dominated cases. Comparing with previous stabilization methods, the IAFEM proposed in this paper can still possess the numerical stability when dealing with convection-dominated problems. And it is more concise and easier to be numerically implemented. Numerical experiments about a model problem with analytic solutions are presented to verify the accuracy and order of IAFEM for SMPNP equations. Studies about the size-effects of a sphere model and an ion channel system are presented to show that our IAFEM is more effective and robust than the traditional finite element method (FEM) when solving SMPNP equations in simulations of biological systems.

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“…A semi-implicit finite difference scheme that ensures positivity and discrete energy dissipating properties was established in [24]. Based on harmonicmean approximations [43], a finite difference scheme that is proved to respect mass conservation and unconditional positivity preservation was proposed for PNP equations with steric effects [11]. Estimates on the condition number of the coefficient matrix was established as well.…”

Section: Introductionmentioning

confidence: 99%

“…The ionic steric interaction energy is given in the third term, in which the M × M matrix G = (g mn ) is symmetric with non-negative entries. The entry g mn is related to the second-order virial coefficients of hard spheres, depending on the size of the mth and nth ionic species [11,51]. Diagonal entries of G describe self-steric interactions of ions of the same species, and off-diagonal entries correspond to short-range cross steric interactions between ions of different species.…”

Section: Introductionmentioning

confidence: 99%

A Positive and Energy Stable Numerical Scheme for the Poisson-Nernst-Planck-Cahn-Hilliard Equations with Steric Interactions

Qian,

Wang,

Zhou

2020

Preprint

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In this work, we consider numerical methods for the Poisson-Nernst-Planck-Cahn-Hilliard (PNPCH) equations with steric interactions, which correspond to an H −1 gradient flow of a free-energy functional that consists of electrostatic free energies, steric interaction energies of short range, entropic contribution of ions, and concentration gradient energies. We propose a novel energy stable numerical scheme that respects mass conservation and positivity at the discrete level. Existence and uniqueness of the solution to the proposed nonlinear scheme are established by showing that the solution is a unique minimizer of a convex functional over a closed, convex domain. The positivity of numerical solutions is further theoretically justified by the singularity of the entropy terms, which prevents the minimizer from approaching zero concentrations. A further numerical analysis proves discrete free-energy dissipation. Extensive numerical tests are performed to validate that the numerical scheme is first-order accurate in time and second-order accurate in space, and is capable of preserving the desired properties, such as mass conservation, positivity, and free energy dissipation, at the discrete level. Moreover, the PNPCH equations and the proposed scheme are applied to study charge dynamics and self-assembled nanopatterns in highly concentrated electrolytes that are widely used in electrochemical energy devices. Numerical results demonstrate that the PNPCH equations and our numerical scheme are able to capture nanostructures, such as lamellar patterns and labyrinthine patterns in electric double layers and the bulk, and multiple time relaxation with multiple time scales. The multiple time relaxation dynamics with metastability take long time to reach an equilibrium, highlighting the need for robust, energy stable numerical schemes that allow large time stepping. In addition, we numerically characterize the interplay between cross steric interactions of short range and the concentration gradient regularization, and their impact on the development of nanostructures in the equilibrium state.

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Positivity preserving finite difference methods for Poisson–Nernst–Planck equations with steric interactions: Application to slit-shaped nanopore conductance

Cited by 42 publications

References 44 publications

Fast Newton Iterative Method for Local Steric Poisson--Boltzmann Theories in Biomolecular Solvation

Fast Newton Iterative Method for Local Steric Poisson--Boltzmann Theories in Biomolecular Solvation

An inverse averaging finite element method for solving the size-modified Poisson-Nernst-Planck equations in ion channel simulations

A Positive and Energy Stable Numerical Scheme for the Poisson-Nernst-Planck-Cahn-Hilliard Equations with Steric Interactions

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