Nernst Planck Model Research Articles

Electroosmotic flow driven by oscillating zeta potentials: Comparison of the Poisson–Boltzmann model, the Debye–Hückel model and the Nernst–Planck model

We consider electroosmotic flows (EOF) generated by temporally varying zeta potential which is usually adopted for pumping or mixing of fluids. In this case the dynamics of ions in the electric double layer (EDL) influences the induced electric field and consequently the EOF significantly. Therefore, the appropriate model should be the Nernst–Planck (NP) model for all zeta potentials or the Debye–Hückel (DH) model for low zeta potentials rather than the Poisson–Boltzmann (PB) model which is based on the equilibrium distribution of ions in the EDL. In the present investigation, we compare the predictions from the DH model and the PB model with the exact ones from the NP model for a range of frequency of zeta potential oscillation. It is found that one may adopt the PB model when the frequency is low and the DH model when the zeta potential is low. However, for either high frequency of zeta potential oscillation or large value of zeta potential, one must adopt the NP model to get accurate predictions of EOF.

On the capacitance-boost of ionic polymer metal composites due to electroless plating: Theory and experiments

In this paper, we analyze the effect of electrode surface roughness on ionic polymer metal composite (IPMC) capacitance. We use the linearized Poisson–Nernst–Planck model to describe the charge and electric potential distribution in response to a small voltage applied across the IPMC electrodes. We use perturbation methods to develop a comprehensive understanding of the interplay among the scale of the electrode roughness, the Debye screening length, and the IPMC nominal dimensions on the electrical behavior of IPMCs. We derive a closed-form expression of the IPMC capacitance per unit nominal surface area in terms of the Debye screening length, the IPMC nominal thickness, and physically relevant statistical properties of the rough landscape. We find that IPMC capacitance is largely dictated by the effective electrode surface area when the Debye screening length is considerably smaller than the polymer thickness. In this case, the diffuse charge layers that form at the polymer-electrode interface closely follow the rough electrodes profile. As the Debye screening length increases, diffuse layers do not completely adhere to the electrode profile, and local curvature changes and additional geometric factors contribute to the overall IPMC capacitance. We specialize our findings to different electrode models, including fractal electrodes that have been recently observed in IPMC morphological studies. We corroborate our theoretical findings with experimental data on the capacitance of in-house fabricated IPMCs.

The influence of porosity changes in human epidermal membrane during iontophoresis on the permeability enhancement of a model peptide

Purpose: We tested the hypothesis that the increases in the porosity of the skin during iontophoresis would not significantly increase the transport of peptides due to the small size of electrically induced pores. To investigate this mechanistically, we used human epidermal membrane under constant voltage conditions, applying the Nernst–Planck equation to the transport of a small ionic solute, tetraethylammonium bromide (TEAB), and a model peptide, luteinizing hormone releasing hormone. Methods: Steady-state flux of the drugs was determined under passive conditions and also during iontophoresis using constant DC voltages applied across side-by-side diffusion cells. Electrical conductance measurements were used to monitor the porosity changes that occur during electrical field application. Results: Porosity increases observed in the membrane substantially increased the permeability enhancement of the small ionic solute TEAB. The permeability enhancement was well described by Nernst–Planck model predictions after porosity changes in the membrane were taken into account. Enhancement of luteinizing hormone releasing hormone under identical conditions was much less than TEAB. The porosity increases induced by iontophoresis had little or no effect on the permeability enhancement of the larger molecule. Conclusions: These findings closely parallel those reports that have found electrically induced pores to be significantly smaller than preexisting pores in the human epidermal membrane. The data obtained also support the view that iontophoresis-induced pores, alone, may provide limited benefit for macromolecule transport across the skin.

Electroosmotic flow in micro/nanochannels with surface potential heterogeneity: An analysis through the Nernst–Planck model with convection effect

A complete two-dimensional model is discussed to analyze the electroosmotic flow near a step jump in surface charge distribution of a micro- or nanochannel. The present model does not require the core neutrality of the fluid and it can handle electrolyte of arbitrary ionic valence as well. The governing equations consists of the flow field, mass transfer and electric potential equations. A pressure correction-based iterative algorithm (SIMPLE) is employed for computation. An analytic solution is obtained near the surface heterogeneity based on the Debye–Hückel approximation. The EOF is investigated both for weak and strong electrolytes at different channel heights. The form of the vortical flow near the potential patch and the dependence of the vortex strength on the patch overpotential is analyzed. A linear pressure drop is observed above the vortex. Average streamwise velocity overshoots the Helmholtz–Smoluchowski velocity. A comparison of the present model with the model based on the equilibrium Boltzmann distribution of ions is made. The results due to the two models differ significantly in the region where the convection effect is strong.

Influence of the convective term in the Nernst-Planck equation on properties of ion transport through a layer of solution or membrane

The one-dimensional boundary-value problem of steady-state ion transport, which takes into account the convective component, is formulated and solved in terms of the Nernst-Planck model. This problem is investigated in connection with the diffusion layer, which is understood in a broad sense. This can be the diffusion layer as it is usually understood, i.e., located adjacent to a hydraulically permeable membrane. In another context it can be regarded as a capillary connecting two reservoirs filled with solutions of different concentration or as an uncharged macropore permeating the membrane and separating two solutions. Finally, the solution to the problem is applied to the membrane itself, which is represented as a quasi-homogeneous gel. In the latter case, a virtual electroneutral solution in local equilibrium with a small volume of membrane is considered. The problem is investigated in dimensionless form as a function of the Peclet number. It is shown that the Peclet number is numerically equal to the absolute value of the dimensionless convection velocity. The limiting current, concentration profiles, distributions of the field strength and potential, and effective transport numbers are analyzed as functions of the convective component.

Electrokinetic flow-induced currents in silica nanofluidic channels

Electrokinetic flow-induced currents inside slit-shaped silica nanochannels are investigated. The unusual features observed experimentally in silica nanochannels are described successfully using a new theoretical framework. First, a simple and reliable physicochemical boundary condition at the interface between the channel surface and the solution is suggested. It accounts for the surface conduction effect through the Stern layer and the dependence of the surface charge on the salt concentration and pH, which were commonly neglected in previous studies. Second, the proposed boundary condition is then incorporated into the traditional Poisson–Boltzmann and Nernst–Planck models to complete the self-consistent model. Model predictions are validated by comparison with experimental data. It is found that the direct numerical predictions of the concentration polarization and the induced potential or pressure field are possible, and these allow us to describe the dependence of currents on the solution properties in the nanofluidic channel more accurately than the models proposed in previous studies.

Numerical study on AC electroosmosis in microfluidic channels

This paper deals with parametrical studies focused on determination of selected characteristics of AC electroosmotic micropumps in non-linear (high-amplitude) regimes in the interdigitated arrangement. The net velocity and current–voltage phase shift are computed. Non-equilibrium Navier–Stokes–Poisson–Nernst–Planck model that describes the transport processes in the entire time and spatial domains including electric double layers (EDLs) was derived and analyzed. The obtained results were qualitatively interpreted by means of a simple relation derived for RC circuit in series. Locations of the high net velocity regions in the parametric space were found. The model thus can contribute to development and optimization of microfluidic devices driven by AC electric fields. Flow reversals were observed in frequency and geometric scale characteristics. We suggest that the flow reversals do not necessarily rely on the electric charge injection or the condensed layer formation. The direction of the net flow probably depends only on an interplay of the electric, viscous and pressure forces.

A Poisson–Nernst–Planck Model for Biological Ion Channels—An Asymptotic Analysis in a Three-Dimensional Narrow Funnel

We wish to predict ionic currents that flow through narrow protein channels of biological membranes in response to applied potential and concentration differences across the channel when some features of channel structure are known. We propose to apply singular perturbation analysis to the coupled Poisson–Nernst–Planck equations, which are the basic continuum model of ionic permeation and semiconductor physics. In semiconductor physics the problem is a singular perturbation, because the ratio of the Debye length to the width of the channel is a very small parameter that multiplies the Laplacian term in the Poisson equation. In contrast to semiconductors, the atomic scale geometry of narrow ion channels sometimes makes this ratio a large parameter, which, surprisingly, renders the problem a singular perturbation in a different sense. We construct boundary layers and match them asymptotically across the different regions of the channel to derive good approximations for Fick's and Ohm's laws. Our aim is to extend the asymptotic analysis to a class of nonlinear problems hitherto intractable. Analytical and numerical results for the mass flux and the electric current serve as a tool for molecular biophysicists and physiologists to understand, study, and control protein channels, thereby aiding clinical and technological applications.

Electroosmotic flows in an electric double layer overlapped channel with rectangle-waved surface roughness

To further understand the wall-roughness effect, the present study has performed numerical simulations, by employing the Poisson–Nernst–Planck model, on the two-dimensional electroosmotic flow in a plane channel with dielectric walls of rectangle-waved surface roughness where the two electric double layers (EDLs) are overlapped. Results show that the steady electroosmotic flow and ionic-species transport depend significantly on the shape of the surface roughness such as the amplitude and periodic length of wall wave, but their characteristics are basically different from those in the case where the EDLs are not overlapped at all (Kang and Suh in Microfluid Nanofluid, doi:10.1007/s10404-008-0321-5, 2008). It is found that the fluid flows over the waved wall (or wall roughness) with involving a separation or recirculation of flow in the cavity, which resembles much the traditional pressure-driven flow. In addition, the flow characteristics are determined chiefly by the level of the electric-charge density in the bulk region above the waved wall. As a result, with increasing wall-wave amplitude (0.01 ≤ h/H ≤ 0.2), the flow rate increases due to the enhanced amount of electric charges released from the enlarged wet surface at low amplitudes and then decreases due to the reduced flow-passage area at high amplitudes above a certain critical value. With increasing periodic length (0.2 ≤ L/H ≤ 1.2), on the other hand, the flow rate decreases in a hyperbolic fashion due to the reduced amount of electric charges.

Numerical analysis on electroosmotic flows in a microchannel with rectangle-waved surface roughness using the Poisson–Nernst–Planck model

The present study has numerically investigated two-dimensional electroosmotic flows in a microchannel with dielectric walls of rectangle-waved surface roughness to understand the roughness effect. For the study, numerical simulations are performed by employing the Nernst–Planck equation for the ionic species and the Poisson equation for the electric potential, together with the traditional Navier–Stokes equation. Results show that the steady electroosmotic flow and ionic-species transport in a microscale channel are well predicted by the Poisson–Nernst–Planck model and depend significantly on the shape of surface roughness such as the amplitude and periodic length of wall wave. It is found that the fluid flows along the surface of waved wall without involving any flow separation because of the very strong normal component of EDL (electric double layer) electric field. The flow rate decreases exponentially with the amplitude of wall wave, whereas it increases linearly with the periodic length. It is mainly due to the fact that the external electric-potential distribution plays a crucial role in driving the electroosmotic flow through a microscale channel with surface roughness. Finally, the present results using the Poisson–Nernst–Planck model are compared with those using the traditional Poisson–Boltzmann model which may be valid in these scales.

Applicability of equilibrium and kinetic models on the herbicide 4-chloro-2-methyl phenoxyacetic acid adsorption on bituminous shale

Abstract Adsorption characteristics of herbicide 4-chloro-2-methylphenoxyacetic acid (MCPA) have been evaluated using bituminous shale (BS) as a model adsorbent-containing solid organic matter in a mineral matrix. The adsorption of MCPA on BS has been studied with varying concentration, temperature, pH and contact time, using batch technique. Adsorption ability of BS increases with increasing temperature and decreasing pH in the studied concentration range of (0.6–4.0) × 10−4 M. Theoretical curves calculated from Freundlich, Dubinin-Radushkevich (D-R), Langmuir and Temkin isotherm equations show a two-step isotherm shape. The results could be explained by assuming the presence of two-type sites with different affinity on adsorbent surface. Adsorption process is endothermic and entropy controlled at the first stage, and exothermic and enthalpy controlled at the second stage. The mechanism proposed based on surface ionization and complexation model is consistent with the pH dependent experimental results. Kinetic data fit well to both Paterson’s and Nernst Planck model based on homogeneous solid phase diffusion (HSPD). The values of particle diffusion coefficients (D p ) predicted from both models are comparable each other and independent of temperature and concentration.

Comparison of the Nernst–Planck model and the Poisson–Boltzmann model for electroosmotic flows in microchannels

In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson-Boltzmann equation. The Poisson-Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distributions are not affected by fluid flows. Although this is a reasonable assumption for steady electroosmotic flows through straight microchannels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst-Planck equation instead of the Poisson-Boltzmann equation to model the internal electric field. In the present work, the predictions of the Nernst-Planck equation are compared with those of the Poisson-Boltzmann equation for electroosmotic flows in various microchannels where the convective transport of ions is not negligible.

Thermal in vivo skin electroporation pore development and charged macromolecule transdermal delivery: A numerical study of the influence of chemically enhanced lower lipid phase transition temperatures

Electroporation is an approach used to enhance transdermal transport of large molecules in which the skin is exposed to a series of electric pulses. Electroporation temporarily destabilizes the structure of the outer skin layer, the stratum corneum, by creating microscopic pores through which agents, which ordinarily are unable to pass into the skin, are able to pass through this outer barrier. Long duration electroporation pulses can cause localized temperature rises which result in thermotropic phase transitions within the lipid bilayer matrix of the stratum corneum. Chemical agents applied to the skin can reduce the lipid phase transition temperatures. This paper studies the benefits of the combination of the chemical enhancer, terpene d-limonene with low voltage electroporation pulses in order to further aid in electroporation pore development resulting from fluidization of the lipid structures within the stratum corneum. A transient finite volume model is developed in which the thermal and electrical behavior associated with electroporation of in vivo human skin is analyzed and lipid phase transition is represented by a melting process. The Nernst–Planck model is used to represent the electrophoretic-assisted transport of large charged molecules through the skin. The results show that the lower lipid phase transition temperatures associated with the topical application of chemical enhancers to the skin allow for increased solute delivery and solute penetration of the skin reaching radial locations much further than in the untreated case. Solute transport solutions of both cases exhibit local accumulation of concentrations below the stratum corneum – epidermis interface which exceed concentration values initially contained within the applicator gel.

A 3-D Finite Element Poisson–Nernst–Planck Model for the Analysis of Ion Transport Across Ionic Channels

A 3-D finite element steady-state Poisson-Nernst-Planck model is presented and applied to the analysis of the ion transport across ionic channels of the cellular membrane. The model allows us to obtain an accurate description of ion flow across the cell membrane. An example of application to the case of a K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> channel is also illustrated. The resulting current-voltage relationship for the K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> channel shows excellent agreement with experimental measurements

Transport processes in heavy metal fluoride glasses

Na self-diffusion, Li self-diffusion, Na +–Li + ion exchange, electrical conductivity, and mechanical relaxation have been studied below T g on glasses of the system ZrF 4–BaF 2–LaF 3–AF (A=Na, Li), with A=10, 20, 30 mol%. Compared to the transport mechanism in alkali-containing silicate glasses, the mechanisms in these non-oxide glasses are anomalous. Thus the self-diffusion coefficient of Na decreases with increasing NaF content, whereas that of Li increases with increasing LiF content. Both the electrical conductivity and the Na +–Li + ion exchange reach a minimum at ≈ 20 mol% LiF, and the mechanical relaxation shows one peak for the 20 and 30 mol% LiF-glasses and two peaks for the glass with 10 mol% LiF, evidencing both a contribution of F − and Li + ions to the transport. Moreover, the presence of the three partially interacting mobile species F −, Na +, Li + obviously leads to an anionic–cationic mixed ion effect. Applying the Nernst–Einstein equation to the Li + transport in LiF-containing glasses shows that its mechanism is dissimilar to that in oxide glasses. Calculated short jump distances possibly can be interpreted as an Li + movement via energetically suitable sites near F − ions. Likewise the Nernst–Planck model, successfully applied to the ionic transport in mixed alkali silicate glasses, obviously does also not hold for the present heavy metal fluoride glasses.

Study of EDL effect on 3‐D developing flow in microchannel with Poisson–Boltzmann and Nernst–Planck models

AbstractMicrochannels have been suggested to be a very effective heat transfer device. However, the electrical double layer (EDL) effect in a microchannel is anticipated to be significant. In this paper, two EDL models coupled with Navier–Stokes equations were used to compute a 3‐D developing microchannel flow. The Poisson–Boltzmann model (PBM) has been proven to be a promising tool in studying the EDL effect for developed microchannel flow, with respect to its accuracy and efficiency. However, it was commended that the assumption of Boltzmann distribution in PBM, for electric ion concentration distribution in particular was questionable in the developing flow. Nernst–Planck model (NPM), with its two extra partial differential equations (PDEs) to predict the ion concentration distribution, was reported to be a more appropriate model for developing microchannel flow though increased RAM and CPU were needed as compared to the PBM. The governing equations for both models were discretized for developing rectangular microchannel flows in Cartesians co‐ordinate. An additional source term, related to the electric potential, resulting from the EDL effect was introduced in the conventional z‐axis momentum equation as a body force, thereby modifying the flow characteristics. A finite‐volume scheme was used to solve the PDEs. The discrepancy in the results predicted by the two models was more dominant in the near‐wall region and not in the mainstream region. However, the performance of the microchannel was significantly affected by the EDL effect. An increase in Schmidt number will lead to decrease in friction coefficient. For the effect of aspect ratios (ARs), the discrepancy of f Re and Nu values for different ARs was obvious. Therefore, a 3‐D analysis was concluded to be important and necessary for investigating the EDL effect in microchannels. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Numerical Analysis of EDL Effect on Heat Transfer Characteristic of 3-D Developing Flow in a Microchannel

It has been suggested that microchannels are very effective heat transfer devices. However, the electrical double layer (EDL) effect in microchannels is suspected to be significant. In this article, two EDL models together with Navier-Stokes equations are used to compute 3-D developing microchannel flow. The Poisson-Boltzmann model (PBM) has been shown to be a promising tool in studying the EDL effect for developed microchannel flow, with acceptable accuracy and efficiency. However, it has been reported that the assumption of Boltzmann distribution in the PBM for electric ion concentration distribution is questionable in the developing flow. The Nernst-Planck model (NPM), with its two extra partial differential equations (PDEs), to predict the ion concentration distribution has been suggested to be a more appropriate model for developing microchannel flow, but more RAM and CPU are needed as compared to the PBM. The governing equations for both models are discretized for developing rectangular microchannel flows in Cartesian coordinates. An additional source term, which is related to the electric potential resulting from the EDL effect is introduced in the conventional z-axis momentum equation as a body force, thereby modifying the flow characteristics. A finite-volume scheme is used to solve the PDEs. The results predicted by both EDL models with and without EDL effects are shown. It is concluded that the differences in heat transfer performance of a microchannel predicted using the two models are insignificant. However, the performance of the microchannel is significantly affected by the EDL effect.

Relationship between molar volume and rejection of arsenic species in groundwater by low-pressure nanofiltration process

Rejection of arsenite, dimethyl arsinic acid (DMAA), arsenate, chloride, nitrate and sulfate ion by low-pressure nanofiltration was evaluated by the extended Nernst–Planck model coupled with steric hindrance and Donnan equilibrium. The partition coefficient of K i for each solute was introduced to Donnan equilibrium. They could be obtained from the curve fitting with experimental rejection of solutes. In the case of anionic solutes such as H 2AsO 3 −, H 2AsO 4 −, HAsO 4 −, (CH 3) 2AsO 2 −, NO 3 − and SO 4 2−, the partition coefficient K i was correlated with the molar volume. The empirical equation was obtained from the regression analysis between molar volume ( V A) and partition coefficient ( K i ) of anionic solutes for ES10 ( K i =3100 V A 0.64), NTR729HF ( K i =930 V A 0.44) and NTR7250 ( K i =1800 V A 0.63). The effective range of molar volume for the equation was from 0.5×10 −4 to 1.5×10 −4 m 3/mol. The gradient value of the regression line of K i with molar volume in log scale was closed to 2/3 for all membranes used. These results suggest that the interaction between anionic solute and membrane surface was affected mainly by the surface charge density corresponding to molar volume of ionic solute, although another factors need to be considered in order to account for a deviation from the 2/3.

Characterization and identification of rejection mechanisms in nanofiltration membranes using extended Nernst–Planck model

Nanofiltration (NF) membranes have found increasing applications especially in water and wastewater treatment. In order to predict and optimize the performance of NF membranes, the membranes need to be characterized properly in terms of their structural parameters such as pore radius (r p) and effective thickness (Δx/A k) as well as the effective charge density (X d). In this work, three cellulose acetate NF membranes were fabricated to different properties and were characterized using a model based on the extended Nernst–Planck equation. The results show that the variations of r p, Δx/A k and X d are quite distinct between the three membranes. The membranes were found to have r p>1.2 nm and high X d. Thus the rejection mechanism depends very much on the Donnan effect rather than the steric effect. The high rejection of several negatively charged organics and salts confirmed this finding. Apart from being effective, this method of characterization offers a cheaper alternative in understanding NF membranes.

Iontophoretic transport of oligonucleotides across human epidermal membrane: A study of the Nernst–Planck model

The objective of this study was to investigate the transport behavior of a series of oligonucleotides with human epidermal membrane (HEM) and to examine the applicability of the modified NERNST–PLANCK model to transdermal iontophoresis of these macromolecules. Iontophoretic transport experiments were first carried out in a synthetic model membrane system (Nuclepore membranes) with a four‐electrode potentiostat to examine the baseline modified NERNST–PLANCK model. The modified NERNST–PLANCK model derived from the Einstein relation and the Stokes–Einstein equation taken from previous work did not hold for the oligonucleotides. Results obtained in the Nuclepore studies were, however, consistent with predictions of the modified NERNST–PLANCK model using the experimentally determined electromobilities and diffusion coefficients. The electromobilities of the oligonucleotides (determined by capillary electrophoresis) were found to be more than a factor of two smaller than expected from the Einstein relation between electromobilities and diffusion coefficients (the latter determined in diffusion cell experiments). A correlation between these electromobilities and the theoretical electromobilities estimated by considering the effects of counterion binding and the effects of mobility reduction according to colloid theory was also observed. These results suggest that the modified NERNST–PLANCK model predictions are satisfactory only when the electromobilities and the effective molecular size of the oligonucleotides are known and are used directly to predict the iontophoretically enhanced transport. Results with the HEM experiments generally agreed with model predictions based on the experimental electromobilities. The oligonucleotide HEM flux data also suggest the existence of pores with effective pore radii greater than the effective radii estimated in previous studies with small molecular weight model permeants. © 2001 Wiley‐Liss, Inc. and the American Pharmaceutical Association J Pharm Sci 90:915–931, 2001