Modified Poisson Nernst Planck Model Research Articles

Second-order, positive, and unconditional energy dissipative scheme for modified Poisson–Nernst–Planck equations

First-order energy dissipative schemes in time are available in literature for the Poisson–Nernst–Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

Electric-field-induced oscillations in ionic fluids: a unified formulation of modified Poisson-Nernst-Planck models and its relevance to correlation function analysis.

We theoretically investigate an electric-field-driven system of charged spheres as a primitive model of concentrated electrolytes under an applied electric field. First, we provide a unified formulation for the stochastic charge and density dynamics of the electric-field-driven primitive model using the stochastic density functional theory (DFT). The stochastic DFT integrates the four frameworks (the equilibrium and dynamic DFTs, the liquid state theory and the field-theoretic approach), which allows us to justify in a unified manner various modifications previously made for the Poisson-Nernst-Planck model. Next, we consider stationary density-density and charge-charge correlation functions of the primitive model with a static electric field. We predict an electric-field-induced synchronization between emergences of density and charge oscillations. We are mainly concerned with the emergence of stripe states formed by segregation bands transverse to the external field, thereby demonstrating the following: (i) the electric-field-induced crossover occurs prior to the conventional Kirkwood crossover without an applied electric field, and (ii) the ion concentration dependence of the decay lengths at the onset of oscillations bears a similarity to the underscreening behavior found by recent simulation and theoretical studies on equilibrium electrolytes. Also, the 2D inverse Fourier transform of the correlation function illustrates the existence of stripe states beyond the electric-field-induced Kirkwood crossover.

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

To study ion transport in electrolyte solutions, we propose numerical methods for a modified Poisson–Nernst–Planck model with ionic steric effects (SPNP). Positivity preserving schemes based on harmonic-mean approximations are proposed on a nonuniform mesh for the spatial discretization of the SPNP equations. Both explicit and semi-implicit discretization are considered in time. Numerical analysis shows that explicit forward Euler and semi-implicit trapezoidal discretization lead to schemes that maintain fully discrete solution positivity under a constraint on a mesh ratio, while the semi-implicit backward Euler discretization maintain fully discrete solution positivity unconditionally. We further study the condition numbers of the matrix associated with the semi-implicit backward Euler discretization, and establish an upper bound on condition numbers, indicating that the developed discretization based on harmonic-mean approximations effectively solves the issue of the presence of large condition numbers when using the Slotboom-type variables. Further numerical simulations confirm the analysis results on accuracy, positivity, and bounded condition numbers. The proposed schemes are also applied to study practical applications, such as the impact of self and cross steric interactions on ion distribution and rectifying behavior in a slit-shaped nanopore with surface charges. Possible extensions of the numerical methods to other modified PNP models with ion correlations and steric effects are also discussed.

The effects of electrostatic correlations on the ionic current rectification in conical nanopores.

Ion-ion electrostatic correlations are recognized to play a significant role in the presence of concentrated multivalent electrolytes. To account for their impact on ionic current rectification phenomenon in conical nanopores, we use the modified continuum Poisson-Nernst-Planck (PNP) equations by Bazant et al. Coupled with the Stokes equations, the effects of the EOF are also included. We thoroughly investigate the dependence of the ionic current rectification ratios as a function of the double layer thickness and the electrostatic correlation length. By considering the electrostatic correlations, the modified PNP model successfully captures the ionic current rectification reversal in nanopores filled with lanthanum chloride LaCl3 . This finding qualitatively agrees with the experimental observations that cannot be explained by the standard PNP model, suggesting that ion-ion electrostatic correlations are responsible for this reversal behavior. The modified PNP model not only can be used to explain the experiments, but also go beyond to provide a design tool for nanopore applications involving multivalent electrolytes.

On the Impact of Electrostatic Correlations on the Double-Layer Polarization of a Spherical Particle in an Alternating Current Field.

At concentrated electrolytes, the ion-ion electrostatic correlation effect is considered an important factor in electrokinetics. In this paper, we compute, in theory and simulation, the dipole moment for a spherical particle (charged, dielectric) under the action of an alternating electric field using the modified continuum Poisson-Nernst-Planck (PNP) model by Bazant et al. [ Double Layer in Ionic Liquids: Overscreening Versus Crowding . Phys. Rev. Lett. 2011 , 106 , 046102 ] We investigate the dependency of the dipole moment in terms of frequency and its variation with such quantities like ζ-potential, electrostatic correlation length, and double-layer thickness. With thin electric double layers, we develop simple models through performing an asymptotic analysis of the modified PNP model. We also present numerical results for an arbitrary Debye screening length and electrostatic correlation length. From the results, we find a complicated impact of electrostatic correlations on the dipole moment. For instance, with increasing the electrostatic correlation length, the dipole moment decreases and reaches a minimum and then it goes up. This is because of initially decreasing of surface conduction and finally increasing due to the impact of ion-ion electrostatic correlations on ion's convection and migration. Also, we show that in contrast to the standard PNP model, the modified PNP model can qualitatively explain the data from the experimental results in multivalent electrolytes.

A modified Poisson–Nernst–Planck model with excluded volume effect: theory and numerical implementation

The Poisson--Nernst--Planck (PNP) equations have been widely applied to describe ionic transport in ion channels, nanofluidic devices, and many electrochemical systems. Despite their wide applications, the PNP equations fail in predicting dynamics and equilibrium states of ionic concentrations in confined environments, due to the ignorance of the excluded volume effect. In this work, a simple but effective modified PNP (MPNP) model with the excluded volume effect is derived, based on a modification of diffusion coefficients of ions. At the steady state, a modified Poisson--Boltzmann (MPB) equation is obtained with the help of the Lambert-W special function. The existence and uniqueness of a weak solution to the MPB equation are established. Further analysis on the limit of weak and strong electrostatic potential leads to two modified Debye screening lengths, respectively. A numerical scheme that conserves total ionic concentration and satisfies energy dissipation is developed for the MPNP model. Numerical analysis is performed to prove that our scheme respects ionic mass conservation and satisfies a corresponding discrete free energy dissipation law. Positivity of numerical solutions is also discussed and numerically investigated. Numerical tests are conducted to demonstrate that the scheme is of second-order accurate in spatial discretization and has expected properties. Extensive numerical simulations reveal that the excluded volume effect has pronounced impacts on the dynamics of ionic concentration and flux. In addition, the effect of volume exclusion on the timescales of charge diffusion is systematically investigated by studying the evolution of free energies and diffuse charges.

Numerical modelling of degradation of cement-based materials under leaching and external sulfate attack

A finite element model is developed in this paper to simulate the degradation of cement-based materials when subjected to aggressive agents, including deionized water and sodium sulfate solutions. The proposed model includes three main modules, i.e. the ionic diffusion module, the chemical reaction module, and the damage quantification module. A modified Poisson–Nernst–Planck model is developed to model the diffusion of multiple ions in the ionic diffusion module. The cracking suction effect is accounted for the suction action caused by newly-formed unsaturated cracks with potential to accelerate the entire diffusion process. In the chemical reaction module, chemical interactions between ions and cement hydration products are solved based on a local equilibrium assumption. Different approaches in dealing with the simulation of the decalcification process of ill-crystallized Calcium Silicate hydrate (C–S–H) are discussed. The entire reactive diffusion mechanism is achieved by using the operator splitting approach to couple the ionic diffusion and the chemical modules. In the damage quantification module, the diffusivities of material are evaluated at the end of each time step based on the change of porosity and propagation of cracking. Two mechanisms of precipitation of hydration products, i.e. through-solution and topochemical reactions, are discussed in the light of their distinct contributions to the cracking propagation. The proposed model is applied to model the experiments reported in the literature and the computed results are also compared with those obtained from other available models. It is found that the results obtained from the proposed model agree very well with those from experiments and generate more accurate predictions than other models.

Simulations of Cyclic Voltammetry for Electric Double Layers in Asymmetric Electrolytes: A Generalized Modified Poisson–Nernst–Planck Model

This paper presents a generalized modified Poisson–Nernst–Planck (MPNP) model derived from first principles based on excess chemical potential and Langmuir activity coefficient to simulate electric double-layer dynamics in asymmetric electrolytes. The model accounts simultaneously for (1) asymmetric electrolytes with (2) multiple ion species, (3) finite ion sizes, and (4) Stern and diffuse layers along with Ohmic potential drop in the electrode. It was used to simulate cyclic voltammetry (CV) measurements for binary asymmetric electrolytes. The results demonstrated that the current density increased significantly with decreasing ion diameter and/or increasing valency |zi| of either ion species. By contrast, the ion diffusion coefficients affected the CV curves and capacitance only at large scan rates. Dimensional analysis was also performed, and 11 dimensionless numbers were identified to govern the CV measurements of the electric double layer in binary asymmetric electrolytes between two identical planar electrodes of finite thickness. A self-similar behavior was identified for the electric double-layer integral capacitance estimated from CV measurement simulations. Two regimes were identified by comparing the half cycle period τCV and the “RC time scale” τRC corresponding to the characteristic time of ions’ electrodiffusion. For τRC ≪ τCV, quasi-equilibrium conditions prevailed and the capacitance was diffusion-independent while for τRC ≫ τCV, the capacitance was diffusion-limited. The effect of the electrode was captured by the dimensionless electrode electrical conductivity representing the ratio of characteristic times associated with charge transport in the electrolyte and that in the electrode. The model developed here will be useful for simulating and designing various practical electrochemical, colloidal, and biological systems for a wide range of applications.

First-principles thermal modeling of electric double layer capacitors under constant-current cycling

This study aims to develop physical modeling and understanding of the coupled electrodiffusion, heat generation, and thermal transport occurring in electric double layer capacitors (EDLCs) during constant-current cycling. To do so, the governing energy equation was derived from first principles and coupled with the modified Poisson–Nernst–Planck model for transient electrodiffusion in a binary and symmetric electrolyte. In particular, irreversible Joule heating and reversible heat generation rates due to ion diffusion, steric effects, and changes in entropy of mixing in EDLCs were rigorously formulated. Detailed numerical simulations of the temperature rise in the electrolyte were performed for planar electrodes. The results qualitatively reproduced experimental data reported in the literature under various charging/discharging conditions.

The Influence of Dielectric Decrement on Electrokinetics.

We treat the dielectric decrement induced by excess ion polarization as a source of ion specificity and explore its impact on electrokinetics. We employ a modified Poisson-Nernst-Planck (PNP) equations accounting for the dielectric decrement. The dielectric decrement is determined by the excess ion polarization parameter α and when α = 0 the standard PNP model is recovered. Our model shows that ions saturate at large zeta potentials (ζ). Because of ion saturation, a condensed counterion layer forms adjacent to the charged surface, introducing a new length scale, the thickness of the condensed layer (lc ). For the electro-osmotic mobility, the dielectric decrement weakens the electro-osmotic flow owing to the decrease of the dielectric permittivity. At large ζ, when α ≠ 0, the electro-osmotic mobility is found to be proportional to ζ/2, in contrast to ζ predicted by the standard PNP model. This is attributed to ion saturation at large ζ. In terms of the electrophoretic mobility Me , we carry out both an asymptotic analysis in the thin-double-layer limit and solve the full modified PNP model to compute Me . Our analysis reveals that the impact of the dielectric decrement is intriguing. At small and moderate ζ, the dielectric decrement decreases Me with an increasing α. At large ζ, it is well known that the surface conduction becomes significant and plays an important role in determining Me . It is observed that the dielectric decrement effectively reduces the surface conduction. Hence in stark contrast, Me increases as α increases. Our predictions of the contrast dependence of the mobility on α at different zeta potentials qualitatively agree with experimental results on the dependence of the mobility among ions and provide a possible explanation for such ion specificity. Finally, the comparisons between the thin-double-layer asymptotic analysis and the full simulations of the modified PNP model suggest that at large ζ the validity of the thin-double-layer approximation is determined by lc rather than the traditional Debye length.

Reply to comments on “Intrinsic limitations of impedance measurements in determining electric double layer capacitances” by H. Wang, L. Pilon [Electrochimica Acta 63 (2012) 55

In the commentary to our paper [Electrochimica Acta 63 (2012) 55], Roling and Drüschler raised a very important issue regarding the measurements and comparison of the differential and integral capacitances retrieved using electrochemical impedance spectroscopy (EIS). They clearly explained that EIS measures differential capacitance rather than integral capacitance. The present letter aims to correct our previous study. It also clarifies the fact that cyclic voltammetry (CV) and galvanostatic methods can measure both differential and integral capacitances. Similar confusion exists in the literature on electrical energy storage devices and may explain discrepancies reported when measuring the capacitances of supercapacitors using EIS, CV, or galvanostatic methods. Finally, our original paper, for the first time, solved a modified Poisson–Nernst–Planck model with a Stern layer for simulating EIS. It also presented an interpretation of “capacitance dispersion” and a scaling analysis of electric double layers in EIS simulations. The model, scaling analysis, and the associated results were not affected by the confusion pointed out by Roling and Drüschler.

Intrinsic limitations of impedance measurements in determining electric double layer capacitances

This paper aims to clarify the intrinsic limitations of electrochemical impedance spectroscopy (EIS) in measuring electric double layer (EDL) capacitance. For more than two decades, capacitances measured using EIS at low frequencies have been reported to be significantly smaller than those measured using other techniques without any definitive explanations. In this paper, EIS measurements were numerically reproduced for electric double layers formed near a planar electrode in aqueous electrolyte solutions. The transient double layer dynamics was simulated for low and large electrolyte concentrations using the classical Poisson–Nernst–Planck (PNP) model with or without a Stern layer, and a modified PNP model with a Stern layer, respectively. A characteristic time for ion diffusion τm was identified as λm2/D where λm is the Debye length based on the maximum ion concentration. For a given concentration, the predicted capacitance and the phase shift of surface charge density plotted versus dimensionless frequency τmf for various values of diffusion coefficient overlapped on a single line. This was true for all models considered with or without Stern layer. The simulated EIS measurements systematically overestimated the EDL capacitance for dilute electrolyte solutions while they underestimated it for concentrated electrolyte solutions subject to large electric potential. This discrepancy can be attributed to the fact that the RC circuit used in EIS to model electric double layers is not valid. This study established that the EIS measurements have intrinsic limitations and are inadequate for accurately determining EDL capacitances for practical applications with large potentials such as electrochemical capacitors.

On the Influence of Ion Excluded Volume (Steric) Effects on the Double-Layer Polarization of a Nonconducting Spherical Particle in an AC Field

The dipole moment of a charged, dielectric, spherical particle under the influence of a uniform alternating electric field is computed by solving the modified Poisson−Nernst−Planck (PNP) equations (Bikerman’s mean-field model) (Philos. Mag. 1942, 33, 384−397) accounting for excluded volume effects of the finite ion size as a function of the double-layer thickness, the electric field frequency, the particle’s surface charge, and the volume fraction of the ions in the bulk characterizing the excluded volume repulsion. In the limit of thin electric double layers, we carry out an asymptotic analysis to develop simple models calculating dipole moments which are in favorable agreement with the modified PNP model. Our results reveal that excluded volume effects, imposing a maximum on the counterion concentration, reduce the dipole moment at high frequencies and possibly enhance the dipole moment at low frequencies, assuming that the particle bears the same zeta potential. Excluded volume effects often become sig...