Modified Nernst-Planck Equation Research Articles

Numerical model supplemented by thin-layer analysis for diffusiophoresis of a particle incorporating finite ion size effects

The impact of finite-sized ions on the diffusiophoresis of a charged colloid subjected to a concentration gradient of electrolyte solution consisting monovalent or multivalent ionic species, is studied. In diffusiophoresis, the ion concentration is of O(1M). In this non-dilute electrolyte solutions, the ion–ion steric interaction is important. We have adopted the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) model to account for the ion steric interactions and the Batchelor–Green expression for the relative viscosity of suspension. We have solved the standard model numerically considering ions as point charge (PNP-model), the modified Nernst–Planck equations incorporating the ion steric interaction with constant viscosity (MNP-model), and modification of the MNP-model by incorporating the viscosity variation with the ionic volume fraction (MNPV-model). Semi-analytical expressions for mobility based on a linear perturbation technique under a thinner Debye length is presented for PNP- and MNP-models. In the MNP-model, counterion saturation in the Debye layer due to the ion steric interaction enhances the surface potential by attenuating the shielding effect, diminishes the surface conduction, and magnifies the induced electric field. These in combination create a larger mobility at a thinner Debye length compared with the PNP-model. This increment in mobility attenuates when the MNPV-model is considered. The MNPV-model is more appropriate to analyze the finite ion size effects, and it is found to yield the mobility values more close to the experimental data compared with the MNP- and PNP-model. The semi-analytical expressions for mobility based on the PNP- and MNP-models agree with the corresponding exact numerical solutions when the surface potential is in the order of thermal potential. However, a large discrepancy between the simplified expression and the exact numerical results is found for a concentrated electrolyte in which the induced electric field is large.

Energetic variational approach for prediction of thermal electrokinetics in charging and discharging processes of electrical double layer capacitors

This work proposes a new variational, thermodynamically consistent model to predict thermal electrokinetics in electric double layer capacitors (EDLCs) by using an energetic variational approach. The least action principle and maximum dissipation principle from the non-equilibrium thermodynamics are employed to develop modified Nernst–Planck equations for non-isothermal ion transport with temperature inhomogeneity. Laws of thermodynamics are employed to derive a temperature evolution equation with heat sources due to thermal pressure and electrostatic interactions. Numerical simulations successfully predict temperature oscillation in the charging–discharging processes of EDLCs, indicating that the developed model is able to capture reversible and irreversible heat generations. The impact of ionic sizes and scan rate of surface potential on ion transport, heat generation, and charge current is systematically assessed in cyclic voltammetry simulations. It is found that the thermal electrokinetics in EDLCs cannot follow the surface potential with fast scan rates, showing delayed dynamics with hysteresis diagrams. Our work thus provides a useful tool for physics-based prediction of thermal electrokinetics in EDLCs.

Finite ion size and ion permittivity effects on gel electrophoresis of a soft particle

Electrophoresis of a soft particle embedded in a hydrogel medium is studied by considering the hydrated ions as charged dielectric spheres of finite radius. The Nernst-Planck equation for the transport of ions is modified to take into account the ion steric repulsion and the ion-solvent dielectric interactions (MNP-model). The effective polarizability of the hydrated ions lowers the dielectric permittivity of the ionic solution and consequently, the permittivity of the medium varies with local ionic concentration. This spatial variation of the permittivity creates a Born energy difference in ions leading to a Born force experienced by the ions. In addition, ions experience a dielectrophoretic force due to the polarization under the imposed external electric field. The ion steric repulsion are modeled by the Carnahan-Starling equation of state. The convection of ions are governed by the Brinkman extended Navier-Stokes equations. The modified Nernst-Planck equation coupled with the Brinkman Navier-Stokes equations and the equation for electric field are solved numerically through a control volume approach. The ion steric interactions and ion-solvent interactions create a saturation of mobile ions in the polyelectrolyte layer (PEL) of the soft particle, which leads to an attenuation of the counterion condensation of the PEL. This short range steric repulsion and ion-solvent interactions have significance for higher bulk ionic concentration as well as higher range of charge density of the soft particle. These interactions are elucidated by comparing with the mobility obtained by the standard model (PNP-model). The Dukhin number and the effective charge density of the PEL are determined to illustrate the mobile ion saturation. The discrepancy between the present modified model from the standard model magnifies for the case of salts of multivalent counterions.

Electrokinetic actuation of an uncharged polarizable dielectric droplet in charged hydrogel medium.

Electrokinetic transport of an uncharged nonconducting microsized liquid droplet in a charged hydrogel medium is studied. Dielectric polarization of the liquid drop under the action of an externally imposed electric field induces a non-homogeneous charge density at the droplet surface. The interactions of the induced surface charge of the droplet with the immobile charges of the hydrogel medium generates an electric force to the droplet, which actuates the drop through the charged hydrogel medium. A numerical study based on the first principle of electrokinetics is adopted. Dependence of the droplet velocity on its dielectric permittivity, bulk ionic concentration, and immobile charge density of the gel is analyzed. The surface conduction is significant in presence of charged gel, which creates a concentration polarization. The impact of the counterion saturation in the Debye layer due to the dielectric decrement of the medium is addressed. The modified Nernst-Planck equation for ion transport and the Poisson equation for the electric field is considered to take into account the dielectric polarization. A quadrupolar vortex around the uncharged droplet is observed when the gel medium is considered to be uncharged, which is similar to the induced charge electroosmosis around an uncharged dielectric colloid in free-solution. We find that the induced charge electrokinetic mechanism creates a strong recirculation of liquid within the droplet and the translational velocity of the droplet strongly depends on its size for the dielectric droplet embedded in a charged gelmedium.

Multiphysics Simulator for the IPMC Actuator: Mathematical Model, Finite Difference Scheme, Fast Numerical Algorithm, and Verification.

The article is devoted to the development and creation of a multiphysics simulator that can, on the one hand, simulate the most significant physical processes in the IPMC actuator, and on the other hand, unlike commercial products such as COMSOL, can use computing resources economically. The developed mathematical model is an adjoint differential equation describing the transport of charged particles and water molecules in the ion-exchange membrane, the electrostatic field inside, and the mechanical deformation of the actuator. The distribution of the electrostatic potential in the interelectrode space is located by means of the solution of the Poisson equation with the Dirichlet boundary conditions, where the charge density is a function of the concentration of cations inside the membrane. The cation distribution was obtained by means of the solution of the equation system, in which the fluxes of ions and water molecules are described by the modified Nernst-Planck equations with boundary conditions of the third kind (the Robin problem). The cantilever beam forced oscillation equation in the presence of resistance (allowing for dissipative processes) with assumptions of elasticity theory was used to describe the actuator motion. A combination of the following computational methods was used as a numerical algorithm for the solution: the Poisson equation was solved by a direct method, the modified Nernst-Planck equations were solved by the Newton-Raphson method, and the mechanical oscillation equation was solved using an explicit scheme. For this model, a difference scheme has been created and an algorithm has been described, which can be implemented in any programming language and allows for fast computational experiments. On the basis of the created algorithm and with the help of the obtained experimental data, a program has been created and the verification of the difference scheme and the algorithm has been performed. Model parameters have been determined, and recommendations on the ranges of applicability of the algorithm and the program have been given.

Impact of finite ion size, Born energy difference and dielectric decrement on the electroosmosis of multivalent ionic mixtures in a nanotube

The electroosmotic flow (EOF) in a nanotube is analyzed by considering the finite ion size and dielectric decrement effects. The ion transport is governed by the modified Nernst-Planck equation, which takes into account the ion steric repulsion, Born energy difference, and dielectrophoretic force. The dielectric permittivity is considered to depend on the ionic concentration and ion size. The governing equations for fluid flow, ion transport, and electric field are solved numerically in their full form. The steric interaction is accounted, based on the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) model, which considers the nonuniform ion size for monovalent as well as multivalent salts. We made a quantitative comparison with the standard PNP model for the lower range of charge density and highlighted the impact of ion size and dielectric decrement effects. For a highly charged surface, the impact due to finite ion size is non-negligible when the Debye length is in the order of the nanotube radius. However, the dielectric decrement is found to have a strong impact on a higher range of surface charge density or bulk ionic concentration. The EOF and the ion selectivity of the mixed electrolyte solution of different counterion size are considered in the present study. Present study provides an in-depth analysis of EOF through a cylindrical nanotube beyond the dilute solution and low charge density limits, which has not been addressed before.

Influence of slip velocity at the core of a diffuse soft particle and ion partition effects on mobility.

Nonlinear effects on the electrophoresis of a soft particle, consisting of a rigid hydrophobic core coated with a diffuse polymer layer (PEL) suspended in an electrolyte medium, are studied. The impact of the ion partitioning effect arising due to the Born energy difference between the PEL and the electrolyte is approximated based on the equilibrium Boltzmann equation, with which the ion distribution and hence, the charge density is modified. The equations describing the electrokinetic transport comprising the Darcy-Brinkman extended Navier-Stokes equations which includes the ion partitioning effect coupled with the modified Nernst-Planck equations and Poisson equations for electric field are solved numerically. The present numerical model for the soft particle compares well with the existing theoretical solutions and experimental results in the limiting cases. A deviation from existing simplified models based on the Boltzmann distribution of ions occurs when the Debye layer polarization, relaxation and the electroosmosis induced by the PEL immobile charge become significant. The hydrophobicity of the inner core strongly influences the nonlinear electrokinetic effects by modifying the Debye layer, electroosmotic flow in the PEL and surface conduction. The results indicate that the ion partitioning can significantly increase the electrophoretic mobility of the soft particle by attenuating the shielding effect. When the Debye layer is in the order of the particle size the hydrophobicity of the core surface and the ion partitioning effect manifest the surface conduction, which implies that the Boltzmann distribution of ions is no longer valid. The core hydrophobicity and ion partitioning effect have influence on the condensation of the PEL immobile charge, which creates a significant impact on the mobility.

Nernst–Planck analysis of reverse-electrodialysis with the thin-composite pore-filling membranes and its upscaling potential

To properly design reverse electrodialysis (RED) stacks, modeling of ion transport and prediction of power generation on the single RED stack are very important. Currently, the Nernst–Planck equation is widely adopted to simulate ion transport through IEMs. However, applying typical Nernst-Planck equation is not proper to analyze ion transport through the heterogeneous thin-composite pore-filling membrane because of the non-conductive site in the membrane matrix. Herein, we firstly introduced modified Nernst-Planck equation by addressing conductive traveling length (CTL) to simulate the ion transport through the thin-composite pore-filling membranes and the performance of a single RED stack with the same membranes. Also, 100 cell-pairs of RED stacks were assembled to validate modified Nernst-Planck equation according to the flow rate and membrane types. Under the OCV condition, the conductivity of the effluents was measured to validate the modified Nernst-Planck equation, and differences between modeling and experiments were less than 1.5 mS/cm. Theoretical OCV and current density were estimated by using modified Nernst-Planck equation. In particular, hydrophobicity on the surface of the heterogeneous membrane was considered to describe ion transport through the pore-filling membranes. Moreover, power generation from RED stacks was calculated according to the flow rate and the number of cell pairs.

Multiscale modeling of ion diffusion in cement paste: electrical double layer effects

Understanding the mechanism of ion diffusion in hardened cement paste is of great importance for predicting long-term durability of concrete structures. Gel pores in calcium silicate hydrate (CSH) phase forms dominant pathway for transport in cement paste with low w/c ratios where the electrical double layer effects play an important role. Experimental results suggest that the effective diffusivity of chloride ions is similar as that of tritiated water (HTO) and higher than the sodium ions. This difference can be attributed to the electrical double layer near the charged CSH surfaces. In order to understand species transport processes in CSH and to quantify its effective diffusivity, a multiscale modeling technique has been proposed to combine atomic-scale and pore-scale modeling. At the pore scale, the lattice Boltzmann method is used to solve a modified Nernst Planck equation to model transport of ions in gel pores. The modified Nernst Planck equation accounts for steric and ion-ion correlation effects by using correction term for excess chemical potential computed through the results from the grand canonical Monte Carlo scheme at atomic scale and in turn bridges atomic scale model with pore scale model. Quantitative analysis of pore size influence on effective diffusivity carried out by this multiscale model shows that the contribution of the Stern layer to ion transport is not negligible for pores with diameter less than 10 nm. The developed model is able to reproduce qualitatively the trends of the diffusivity of different ions reported in literature.

Nanofiltration as a Pretreatment Step in Seawater Desalination: A Review

One of the main applications of nanofiltration (NF) is in the pretreatment stage of seawater desalination. NF has high rejection rates for divalent ions and could eliminate the scaling species that pose serious fouling problems in seawater desalination. This review comprehensively examines recent advances in NF membrane research in seawater desalination. Significant progress has been made in understanding the mechanism of solute transport through NF membranes and has resulted in the development of predictive models based on the Spiegler–Kedem model and the modified Nernst–Planck equation. The contributions of each type of transport mechanism through NF membranes, i.e., convection, diffusion, and electro-migration, have been reported. A review of recent progress made in the development of integrated NF membrane and seawater desalination processes is included. Work related to membrane fouling, which is a key problem in NF, is also discussed.

An Extended Model Based on the Modified Nernst–Planck Equation for Describing Transdermal Iontophoresis of Weak Electrolytes

An extended model for iontophoretic enhancement of transdermal drug permeation under constant voltage is described based on the previously modified Nernst-Planck equation, which included the effect of convective solvent flow. This model resulted in an analytical expression for the enhancement factor as a function of applied voltage, convective flow velocity due to electroosmosis, ratio of lipid to aqueous pathway passive permeability, and weighted average net ionic valence of the permeant in the aqueous epidermis domain. The shift of pH in the epidermis compared to bulk caused by the electrical double layer at the lipid-aqueous domain interface was evaluated using the Poisson-Boltzmann equation. This was solved numerically for representative surface charge densities and yielded pH differences between bulk and epidermal aqueous domain between 0.05 and 0.4 pH units. The developed model was used to analyze the experimental enhancement of an amphoteric weak electrolyte measured in vitro using human cadaver epidermis and a voltage of 250 mV at different pH values. Parameter values characterizing the involved factors were determined that yielded the experimental enhancement factors and passive permeability coefficients at all pH values. The model provided a very good agreement between experimental and calculated enhancement and passive permeability. The deduced parameters showed (i) that the pH shift in the aqueous permeation pathway had a notable effect on the ionic valence and the partitioning of the drug in this domain for a high surface charge density and depending on the pK(a) and pI of the drug in relation to the bulk pH; (ii) the magnitude and the direction of convective transport due to electroosmosis typically reflected the density and sign, respectively, of surface charge of the tissue and its effect on enhancement was substantial for bulk pH values differing from the pI of epidermal tissue; (iii) the aqueous pathway predominantly determined passive permeability of the studied compound despite its measurable lipophilicity and therefore the lipid pathway did not notably affect enhancement. Hence, the proposed model can provide a good quantitative insight into the interplay between different phenomena and permeant properties influencing iontophoresis and can potentially be used as a predictive tool of the process.

Quantitative assessment of tissue retention, lipophilicity, ionic valence and convective transport of permeant as factors affecting iontophoretic enhancement

Iontophoresis of the dipeptide tyrosine-phenylalanine (TyrPhe), the protected amino acid tyrosine- β -naphthylamide (Tyr- β -NA) and the glucose derivative benzyl-2-acetamido-2-deoxy-α-D-glucopyranoside (BAd-α-Glc) used as model compounds was investigated in human epidermis in vitro at pH 3 and pH 4.5 under constant voltage application, the purpose being to delineate the contribution of tissue retention, lipophilicity, ionic valence and convective transport of these compounds to iontophoretic enhancement in a quantitative fashion and ultimately gain an improved mechanistic understanding of iontophoresis. Retention of BAd-α-Glc in epidermal tissue during permeation was considerable and was reduced upon iontophoresis in the presence of extraneous tyrosine (Tyr) and phenylalanine (Phe) added to the donor solution, this evoking an increase of the apparent iontophoretic permeation. This suggests firstly, the occurrence of an interaction between BAd-α-Glc, Tyr and Phe impacting iontophoretic enhancement and secondly, the potential of using amino acids as adjuvants to modulate iontophoresis of selected compounds. This influence of tissue retention on iontophoretic permeation in the presence of Tyr and Phe was not observed for TyrPhe or Tyr-β-NA or when BAd-α-Glc, TyrPhe and Tyr-β-NA were used concomitantly rather than individually in the absence of Tyr and Phe. The effect of lipophilicity, ionic valence in the aqueous permeation pathway and convective transport by electroosmosis of the three permeants on iontophoresis was assessed simultaneously for all permeants by analyzing the experimental data using a theoretically derived model for iontophoretic enhancement that encompassed these factors. This model was based on an extension of the modified Nernst-Planck equation, whereas the difference of ionic valence between the aqueous domain of tissue and the bulk solution was evaluated on the basis of a pH shift due to the electrical double layer at the lipid/aqueous interface in the epidermis using the Poisson-Boltzmann equation. Using deduced parameter values characterizing the involved factors, a very good agreement between model calculated and experimental enhancement data was obtained. At pH 4.5, a very weak convective transport due to electroosmosis from cathode to anode was evident, indicating an isoelectric point of the epidermis slightly above 4.5. At pH 3, electroosmotic transport was approximately 10-fold stronger than at pH 4.5 which reflected a high positive surface charge density of the epidermis at pH 3 and a low one at pH 4.5. The pH in the aqueous permeation pathway, estimated to differ from that of the bulk in accordance with these charge densities, produced an ionic valence in this pathway dependent on pK a that accounted consistently for the flux due to the direct effect of the electric field on the ionic permeants. The ratio of lipid to aqueous pathway passive permeability coefficient was

Treatment of the tannery effluents from a plant near Algiers by nanofiltration (NF): experimental results and modeling

The treatment of a rejection of tannery effluent was carried out on an organic nanofiltration membrane. The rejection was recovered from a leather manufacturing plant in the area of Algiers (Rouiba). The effluent of the bath of tanning and liming charged with chromium ions was the subject of this study. In the case of synthetic solutions, the experimental variations of the retention ranged from 94–99.9% depending on initial feed concentration. For the tanning bath, the results indicated that chromium retentions of 60 and 30% were observed respectively in acid and basic medium. In acid medium, the rejection of the bath of liming has generated a retention of 15%. The use of the model of Nakao et al. [1] based on the extended modified Nernst Planck equation enabled us to compare the theoretical retention of chromium ions with the experimental results and to calculate the membrane pore radius.

Treatment of the tannery effluents from a plant near Algiers by nanofiltration (NF): experimental results and modeling

The treatment of a rejection of tannery effluent was carried out on an organic nanofiltration membrane. The rejection was recovered from a leather manufacturing plant in the area of Algiers (Rouiba). The effluent of the bath of tanning and liming charged with chromium ions was the subject of this study. In the case of synthetic solutions, the experimental variations of the retention ranged from 94–99.9% depending on initial feed concentration. For the tanning bath, the results indicated that chromium retentions of 60 and 30% were observed respectively in acid and basic medium. In acid medium, the rejection of the bath of liming has generated a retention of 15%. The use of the model of Nakao et al. [1] based on the extended modified Nernst Planck equation enabled us to compare the theoretical retention of chromium ions with the experimental results and to calculate the membrane pore radius.

Distributions of Li+, Na+ K+, Rb+, and Cs+ tracer ions in erythrocytes at 38 degrees C in relation to entry rates of these ions into cells at 0 degree C.

Forces that are able to transport Na+ and K+ into two compartments were investigated. A modified Nernst-Planck equation for coupled flows of electric current, water, and ions was integrated. The result shows that if alkali ions in the ion channel of the cell membrane are separated by their electric-current-induced inward flows against an electro-osmotic outward flow of water, the logarithms of the stationary cell/medium distributions of these ions should be proportional to the inverse of their diffusion mobilities. The relationship was tested in human erythrocytes. From inward and outward movements of tracer alkali ions, calculations were made to obtain their stationary distributions at infinite time. The cell/medium distributions determined in this way at 38 degrees C are Li+ = 0.59, 22Na+ = 0.044, 42K+ = 10.0, 86Rb+ = 11.9, and 137Cs+ = 3.07. The entry rates of ions into the cell at 0 degrees C are understood to represent their diffusion mobilities in the pump channel. The entry rates are Li+ = 1.44, 2Na+ = 1, 42K+ = 2.22, 86Rb+ = 2.39, and 137Cs+ = 1.72 relative to that of 22Na+. There is an expected negative correlation between the logarithms of the stationary cell/ medium distributions at 38 degrees C and the inverse of the entry rates into the cell at 0 degrees C for the five ions. It is suggested that the proposed physical forces cause the separation of alkali ions in the channel of Na,K-ATPase.

Interaction of electric field and electro-osmotic effects in determining iontophoretic enhancement of anions and cations

The enhanced transport of cations and anions across porous membranes under an applied electric field is found to be asymmetric. This asymmetry may be due to the interaction between the direct effect of the field and convective solvent flow, which has been shown to be directional. The modified Nernst-Planck equations are used to model cation, anion, and neutral solute transport enhancement under an applied electric field. The mechanism of solvent flow is examined via electro-osmotic velocity equations. Uncharged solutes have been used to determine the solvent velocity during iontophoresis. The direct electric field effects on ion transport enhancement were then separated from the convective solvent flow contribution and the asymmetry in cation/anion flux enhancement was assessed. Results of experiments using model charged and uncharged solutes demonstrate the utility of the model and electro-osmosis may he used to explain the cation/anion asymmetry.

Transport of charged macromolecules across a biological charged membrane

The glomerular capillary wall of the kidney behaves as an electronegatively charged structure consisting of three layers, the lamina densa and the two laminae rarae, which are differently chrrged. Thus, a three layer model is proposed to analyse the transport of charged macromolecules across this wall. A modified Nernst-Planck equation describes the macromolecule flux across the wall and a Donnan equilibrium is assumed at each interface. For a given value of the fixed charge concentration in each layer, the local sieving coefficient of the macromolecule, i.e. the ratio between the concentrations in the filtrate and in the plasma, is calculated. A sieving curve which relates the sieving coefficient to the Einstein-Stokes radius of the macrosolute is obtained. The fixed charge concentrations in each layer are iteratively modified until simultaneous adjustment is achieved between calculated and experimental curves, for positively and negatively charged tracers and their neutral equivalent.