Nernst-Planck Equation Research Articles

PH-sensitive hydrogel-based valves: A transient fully-coupled fluid-solid interaction study

pH-sensitive hydrogels are promising materials to be employed in microfluidic devices, especially microvalves. In this paper, a theory of transient swelling of pH-sensitive micro-valve is presented. A transient constitutive model that captures electrical, chemical, and mechanical fields is considered to model the swelling phenomenon. The diffusion of ions into the hydrogel, the electromigration, and convection are described by implementing the Nernst-Planck equation. Assuming Gent model, hydrogel is considered as a compressible hyperelastic material and osmotic pressure is assumed as an external loading. Due to benefits of in-plane valves, a design of the micro-valve is studied. Design simplicity and great sealing are vital factors which can be considered as an advantage of this valve for fabrication. This design and modeling approach has not been used for pH-sensitive hydrogels in earlier works. Thus, we have studied the transient swelling of pH-sensitive hydrogel microvalve, when effects of fluid-structure-interaction are examined on valve performance. It is noted that in most previous studies, equilibrium conditions have been assumed. While considering transient fully-coupled fluid-structure-interaction is necessary to capture a more realistic modeling. The results illustrate that the microvalve blocks the channel much earlier than reaching the equilibrium-state, which implies importance of the transient behavior of hydrogels.

Efficient, Positive, and Energy Stable Schemes for Multi-D Poisson–Nernst–Planck Systems

In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson–Nernst–Planck equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully-discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O(1) time step restriction. The scheme is also shown to preserve the steady-states. For the fully second order (in both time and space) scheme with large time steps, solution positivity is restored by a local scaling limiter, which is shown to maintain the spatial accuracy. These schemes are easy to implement. Several three-dimensional numerical examples verify our theoretical findings and demonstrate the accuracy, efficiency, and robustness of the proposed schemes, as well as the fast approach to steady-states.

Numerical simulation of salinity gradient power generation using reverse electrodialysis

Reverse electrodialysis (RED) is a promising technology that directly converts salinity gradient energy into electrical energy through the directional permeation of ions across the ion exchange membranes (IEMs). Fundamental understanding of the multi-physical RED process requires a reliable description of all the related phenomena involved in the process. In this work, a two-dimensional RED model based on the Nernst-Planck framework was developed. The fluid dynamics and ion transport were modelled in a full-length cell pair domain by employing the continuity, Navier-Stokes and Nernst-Planck equations complemented by the Donnan exclusion theory and local electroneutrality. The experimentally inaccessible IEM diffusion coefficients were analytically determined using the counterion condensation theory incorporating the tortuosity effect. A numerical simulation was carried out using the developed model. The solution velocity, ion concentration and electric fields were obtained and the characteristics of ion transport were analyzed. The effects of solution inlet velocity, IEM fixed charge concentration and cell pair length on RED performance were investigated and the mechanisms governing the variations of performance parameters were revealed based on ion transport. The present model contributes to a clearer understanding of the connection between the ion transport behavior and the stack performance.

Discontinuous bubble immersed finite element method for Poisson-Boltzmann-Nernst-Planck model

We develop a numerical scheme for Poisson-Boltzmann-Nernst-Planck (PBNP) model. We adopt Gummel's method to treat the nonlinearity of PBNP where Poisson-Boltzmann equation and Nernst-Planck equation are iteratively solved, and then the idea of discontinuous bubble (DB) to solve the Poisson-Boltzmann equation is exploited [6]. First, we regularize the solution of Poisson-Boltzmann equation to remove the singularity. Next, we introduce the DB function as in [6] to treat the nonhomogeneous jump conditions of the regularized solution. Then, we discretize the discontinuous bubble and the bilinear form of Poisson-Boltzmann equation and solve the discretized linear problem by the immersed finite element method. Once Poisson-Boltzmann equation is solved, we apply the control volume method to solve Nernst-Planck equation via an upwinding concept. This process is repeated by updating the previous approximation until the total residual of the system decreases below some tolerance. We provide our numerical experiments. We observe optimal convergence rates for the concentration variable in all examples having analytic solutions. We observe that our scheme reflects well without oscillations the effect on the distribution of electrons caused by locating the singular charge close to the interface.

Correlation equation for evaluating energy consumption and process performance of brackish water desalination by electrodialysis

Electrodialysis (ED) is an electro-driven desalination technology that relies on the selective transport of ions through ion exchange membranes. Though several approaches have been developed to model and evaluate the performance of ED, mechanistic ion-transport models, which rigorously solve the fundamental Nernst-Planck equation, remain some of the most reliable and utilized. However, complexity of the involved transport phenomena prevents analytical solutions of such models, and numerical solutions can be prohibitively intensive. Here, we use an equivalent circuit analogue to derive a simple correlation equation that predicts the energetic performance of ED for brackish water desalination. Specifically, our correlation equation predicts the specific energy consumption of ED for a given productivity, set of desalination parameters (i.e., feed salinity, salt removal, water recovery), and system properties. The correlation equation demonstrates robustness in predicting the specific energy consumption across a wide range of operational parameters, showing excellent agreement with a Nernst-Planck ion-transport model and literature-reported experimental data. Furthermore, we use the developed correlation equation to show the dependence of the specific energy consumption on the productivity, highlighting the tradeoff between the thermodynamic energy efficiency and desalination rate of the ED process. Overall, our developed correlation equation provides a convenient alternative to computationally intensive mechanistic models for performance analysis of the ED process.

Coupled mutual diffusion in aqueous paracetamol + sodium hydroxide solutions

Ternary mutual diffusion coefficients (Dik) measured by the Taylor dispersion method are reported for aqueous paracetamol + NaOH solutions at 25.00 °C and concentrations up to 0.050 mol dm−3. The coupled diffusion of paracetamol (1) and NaOH (2) is significant, as indicated by the large negative cross-diffusion coefficients. In sodium hydroxide solutions, paracetamol, a weak acid, is partially converted to its sodium salt. To help interpret the results, the measured Dik coefficients are compared with predictions based on Nernst and Nernst-Planck equations for mixed electrolyte solutions. The analysis shows that the diffusion of the paracetamol and sodium hydroxide components is strongly coupled by the electric field (diffusion potential) generated by the ionic concentration gradients.

Impact of ion partitioning and double layer polarization on diffusiophoresis of a pH-regulated nanogel

Diffusiophoresis of a charge regulated spherical polyelectrolyte nanogel (PE) due to an externally imposed ionic concentration gradient is considered. The immobile charge density of the nanogel develops through the association/dissociation reactions of their inorganic functional groups. The nanogel is ion and fluid permeable with dielectric permittivity different from that of the surrounding electrolyte medium. This difference in dielectric permittivity creates an ion partitioning due to the difference in self energy of ions. The Nernst–Planck equation for ion transport and the Poisson equation (PNP) for the electric field are modified to take into account the ion partitioning effects. The diffusiophoresis mechanism is governed by the electrophoresis generated by the induced electric field and chemiphoresis develops due to the mitigation of counterions across the double layer of the nanogel. In addition, the convection dominated double layer polarization and the counterion condensation as well as the electroosmotic flow created by the gel immobile charge play a role in the diffusiophoresis. All these effects are incorporated through the modified PNP equations coupled with the Navier–Stokes equations. The governing equations in their full form are solved numerically through a control volume approach. Present computed solutions for the limiting cases are in good agreement with the existing solutions based on the first-order perturbation analysis. In order to illustrate the diffusiophoresis mechanism we have measured the induced electric field and effective charge density of the PE and analyzed its dependence on several electrokinetic parameters. The contribution due to chemiphoresis is low for PE compared to a rigid colloid. For a highly permeable PE the diffusiophoretic velocity increases and approaches a saturation for higher range of the PE fixed charge density. The ion partitioning effect depletes counterions in PE to manifests its diffusiophoretic velocity. The diffusiophoretic velocity of PE for pH > IEP is higher than the case for which pH < IEP.

Novel Positively Charged Metal-Coordinated Nanofiltration Membrane for Lithium Recovery

Nanofiltration (NF) with high water flux and precise separation performance with high Li+/Mg2+ selectivity is ideal for lithium brine recovery. However, conventional polyamide-based commercial NF membranes are ineffective in lithium recovery processes due to their undesired Li+/Mg2+ selectivity. In addition, they are constrained by the water permeance selectivity trade-off, which means that a highly permeable membrane often has lower selectivity. In this study, we developed a novel nonpolyamide NF membrane based on metal-coordinated structure, which exhibits simultaneously improved water permeance and Li+/Mg2+ selectivity. Specifically, the optimized Cu-m-phenylenediamine (MPD) membrane demonstrated a high water permeance of 16.2 ± 2.7 LMH/bar and a high Li+/Mg2+ selectivity of 8.0 ± 1.0, which surpassed the trade-off of permeance selectivity. Meanwhile, the existence of copper in the Cu-MPD membrane further enhanced anti-biofouling property and the metal-coordinated nanofiltration membrane possesses a pH-responsive property. Finally, a transport model based on the Nernst-Planck equations has been developed to fit the water flux and rejection of uncharged solutes to the experiments conducted. The model had a deviation below 2% for all experiments performed and suggested an average pore radius of 1.25 nm with a porosity of 21% for the Cu-MPD membrane. Overall, our study provides an exciting approach for fabricating a nonpolyamide high-performance nanofiltration membrane in the context of lithium recovery.

Pore-Scale Modeling of Electrokinetics in Geomaterials

Pore-scale finite-volume continuum models of electrokinetic processes are used to predict the Debye lengths, velocity, and potential profiles for two-dimensional arrays of circles, ellipses and squares with different orientations. The pore-scale continuum model solves the coupled Navier–Stokes, Poisson, and Nernst–Planck equations to characterize the electro-osmotic pressure and streaming potentials developed on the application of an external voltage and pressure difference, respectively. This model is used to predict the macroscale permeabilities of geomaterials via the widely used Carmen–Kozeny equation and through the electrokinetic coupling coefficients. The permeability results for a two-dimensional X-ray tomography-derived sand microstructure are within the same order of magnitude as the experimentally calculated values. The effect of the particle aspect ratio and orientation on the electrokinetic coupling coefficients and subsequently the electrical and hydraulic tortuosity of the porous media has been determined. These calculations suggest a highly tortuous geomaterial can be efficient for applications like decontamination and desalination.

Design and analysis of an electroosmotic micro-reactor and its application on controlling a chemical reaction

This paper introduces a micro-reactor based on advection-diffusion-reaction in electroosmotic driven flows. The idea is to use species flow rates to control the production rate of the device. In this study, the electroosmotic chemical reaction is simulated numerically based on the Nernst-Planck equation as one of the most accurate approaches for electroosmotic driven flows. Results showed that for high Damkohler numbers, the production rate will be controlled by the species with a lower flow rate, while the species flow rate itself, depends on the flow concentration ratio, and micro-reactor geometry. For the first time, the maximum production rate of the micro-reactor, and the device sensitivity to electrical voltage tolerance are investigated, revealing that among all geometrical and working fluid parameters, the flow concentration ratio is the most dominant parameter to control the upper limit of the reactor productivity. Nevertheless, a higher concentration ratio increases the performance sensitivity to the applied voltage tolerances. This deficiency can be partially cured by increasing the inlet length or longer distances between inlet branches. The findings presented in this study represent a further step towards designing and developing more effective reactors, specifically in micro scales.

Towards explicit regulating-ion-transport: nanochannels with only function-elements at outer-surface

Function elements (FE) are vital components of nanochannel-systems for artificially regulating ion transport. Conventionally, the FE at inner wall (FEIW) of nanochannel−systems are of concern owing to their recognized effect on the compression of ionic passageways. However, their properties are inexplicit or generally presumed from the properties of the FE at outer surface (FEOS), which will bring potential errors. Here, we show that the FEOS independently regulate ion transport in a nanochannel−system without FEIW. The numerical simulations, assigned the measured parameters of FEOS to the Poisson and Nernst-Planck (PNP) equations, are well fitted with the experiments, indicating the generally explicit regulating-ion-transport accomplished by FEOS without FEIW. Meanwhile, the FEOS fulfill the key features of the pervious nanochannel systems on regulating-ion-transport in osmotic energy conversion devices and biosensors, and show advantages to (1) promote power density through concentrating FE at outer surface, bringing increase of ionic selectivity but no obvious change in internal resistance; (2) accommodate probes or targets with size beyond the diameter of nanochannels. Nanochannel-systems with only FEOS of explicit properties provide a quantitative platform for studying substrate transport phenomena through nanoconfined space, including nanopores, nanochannels, nanopipettes, porous membranes and two-dimensional channels.

Performance prediction of flat sheet commercial nanofiltration membrane using Donnan-Steric Pore Model

The rejection of sodium chloride (NaCl) and calcium chloride (CaCl2) single salt solutions were carried out for commercial nanofiltration NFDK membrane. Results showed that the NFDK membrane had a negative surface charge and had a higher observed rejection of 93.65% for calcium (Ca2+) ion and 78.27% for sodium (Na+) ions. Prediction of rejection for aqueous solutions of both salts was made using Donnan Steric Pore Model based on Extended Nernst-Planck Equation in addition to concentration polarization film theory. A MATLAB program was developed to execute the model calculations. Absolute Average Relative Error (% AARE) was found below 5% for real rejection of the NFDK membrane. This research could be used successfully to assess the membrane characterization parameter using a proposed procedure which can reduce the number of experiments.

Radiation and joule heating effects on electroosmosis-modulated peristaltic flow of Prandtl nanofluid via tapered channel

In this paper, a generalized Newtonian nanofluid in a bio-microfluidics channel due to combined effects of peristalsis and external applied electric field in the presence of thermal radiation and Joule heating is considered. The Poisson and Nernst-Planck equations are taken into account. The obtained non-linear coupled ordinary differential equations are simulated numerically. The impacts of several parameters under the applied electric field on velocity, temperature, concentration, trapping isothermal lines, Nusselt, and Sherwood numbers are presented graphically. It is observed that the number of isothermal lines increases with the electro-osmotic parameter thermophoresis parameter and Brownian motion parameter while perter out with thermal radiation. The results are helpful to design the micro pumps/chips used in medical engineering and improved the thermal efficiency and the durability of microchip cooling devices.

Runge-Kutta Numerical Method Followed by Richardson's Extrapolation for Efficient Ion Rejection Reassessment of a Novel Defect-Free Synthesized Nanofiltration Membrane.

A defect-free, loose, and strong layer consisting of zirconium (Zr) nanoparticles (NPs) has been successfully established on a polyacrylonitrile (PAN) ultrafiltration substrate by an in-situ formation process. The resulting organic–inorganic nanofiltration (NF) membrane, NF-PANZr, has been accurately characterized not only with regard to its properties but also its structure by the atomic force microscopy, field emission scanning electron microscopy, and energy dispersive spectroscopy. A sophisticated computing model consisting of the Runge–Kutta method followed by Richardson extrapolation was applied in this investigation to solve the extended Nernst–Planck equations, which govern the solute particles’ transport across the active layer of NF-PANZr. A smart, adaptive step-size routine is chosen for this simple and robust method, also known as RK4 (fourth-order Runge–Kutta). The NF-PANZr membrane was less performant toward monovalent ions, and its rejection rate for multivalent ions reached 99.3%. The water flux of the NF-PANZr membrane was as high as 58 L · m−2 · h−1. Richardson’s extrapolation was then used to get a better approximation of Cl− and Mg2+ rejection, the relative errors were, respectively, 0.09% and 0.01% for Cl− and Mg2+. While waiting for the rise and expansion of machine learning in the prediction of rejection performance, we strongly recommend the development of better NF models and further validation of existing ones.

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.

Euler’s Numerical Method for Ions Rejection Reassessment of a Defect-Free Synthesized Nanofiltration Membrane with Ultrathin Titania Film as the Selective Layer

Titanium (Ti) nanoparticles (NPs) were successfully seeded on the platform of a polyacrylonitrile (PAN) ultrafiltration (UF) membrane previously coated with bio-glue (a co-deposition of dopamine hydrochloric bicarbonate buffer having undergone pyrocatechol deprotonation). The tools in vogue, especially field emission scanning electron microscopy (FESEM), energy dispersive spectroscopy (EDS), and atomic force microscopy (AFM), have made it possible to fully characterize the structure of the new organic-inorganic nanofiltration (NF) membrane, namely NF_PAN_Ti. A soft computing model has been applied to make commonplace the complex and implicit extended Nernst–Planck equations that govern the transport of ions through NF membranes. Euler’s numerical method was applied with a small step-size and the results obtained were very interesting. The filtration velocity approach of GUEROUT-ELFORD-FERRY helped to estimate the average pore size of NF_PAN_Ti to rp = 0.538 nm. A six-day test carried out on NF_PAN_Ti demonstrated its long-term stability and showed a steady-rejection rate of 89.3% of MgCl2 salt and permeate flux of 56 L·m−2·h−1. The Euler’ numerical method corroborated perfectly the experimental findings since the relative error was found to be very low at 0.33% for Cl− and 0.09% for Mg2+ (RE &lt;&lt; 0.1). These practical prediction tools may henceforth help in the choice and calibration of next-generation NF membranes’ synthesis.

Electrochemical deposition induced continuum damage-healing framework for the cementitious composite

The electrochemical deposition method is a promising approach to repair the deteriorated concrete in the aqueous environment. In this paper, a continuum damage-healing framework is presented for the electrochemical deposition method based on the multi-field coupling growth process of the electrochemical deposition products. The ion transportation and the electrode reactions are characterized by employing the Nernst-Planck equation and the current conservation equation. The level set method is adopted to capture the growth of the deposition products. Based on the deposition process, a new empirical healing law is presented, with which a new continuum damage-healing framework is presented for electrochemical deposition method. Numerical examples are conducted by applying the presented framework to the damaged cementitious composite under the tensile loadings. The presented framework is compared with the classic continuum damage-healing theory and the experimental data. The results show that the presented models can describe the electrochemical deposition method induced damage-healing for the cementitious composite. Furthermore, the effects of the healing time, the solution concentration and the external voltage on the damage-healing behaviors are investigated based on our proposed framework.

Sustainable Electrochemical Extraction of Lithium from Natural Brine: Part II. Flow Reactor

An electrochemical flow reactor for the extraction of lithium chloride from natural brine has been designed and experimentally tested. The reactor comprises two three-dimen-sional porous packed bed electrodes and a porous separator immersed in electrolyte. The packed bed electrodes were filled with conducting petroleum coke particles covered respectively with LiMn2O4 selective to lithium ion and polypyrrole selective to anions. The reactor operates in two steps: In the first step the porous electrodes and separator were filled with natural brine to extract lithium and chloride by intercalation and adsorption respectively. After rinsing with water, in the second step the reactor was filled with a dilute LiCl recovery solution and by reversing the electrical current LiCl is recovered in the electrolyte. A two dimensional mathematical model which describes the diffusion and migration of different ions in the electrolyte with the Nernst-Planck equation, the convective flow of electrolyte and the lithium ion intercalation in LiMn2O4 has been developed using a finite element method under the COMSOL environment. The model captures the effect of forced convection on the efficiency of lithium extraction due to diffusion gradients in the porous LiMn2O4 cathode and predicts the best operation parameters.

Global existence of large solutions for the generalized Poisson–Nernst–Planck equations

In this paper, we study the Cauchy problem of the generalized Poisson–Nernst–Planck equations modeling the anomalous (normal) diffusion and transport of charged particles. Making full use of the algebraical structure of the equations and the Fourier localization technique, we obtain a new global well-posedness result without requiring the summation of initial densities of a negatively and positively charged species are small enough.

Simulation of a Solid Phase Cathodic Electro-Fenton Process as an Alternative for Improving the Quality of Treated Water

In electrochemistry, advanced oxidation processes based on electro-Fenton is an advantageous ecological technique due to its excellent efficiency, high availability, and environmental compatibility. In this work, a combined electrolytic process is simulated using a solid phase cathodic electro-Fenton combined with an advanced anodic oxidation, expecting to adapt the whole process to a wastewater treatment plant. The decomposition of methylene blue (MB) was used as a simulation model. An iterative method was used to solve the Nernst-Planck equation in an electrolytic cell whose reaction products are H2O2 and •OH, using COMSOL Multiphysics®. The distance between the electrodes decreases from 1.9 to 0.2 cm. The concentration of the products increases, according to H2O2 from 0.02 to 0.21 and •OH from 0.01 to 0.28 mM. The model also predict the whole MB conversion.