Nernst-Planck Equation Research Articles

Modeling and Simulation of Wastewater Electrocoagulation Reactor

Electro-Coagulation (EC) method is the most effective electrochemical treatment of the wastewater that contains heavy metals as pollutants such as (lead compounds). It is consumed less energy compared with the other conventional methods. In this work, the Computational Fluid Dynamics (CFD) modeling approach for the simulation of the EC unit was studied. Transport of species, current-voltage distribution, and fluid flow were simulated using the diffusion, Nernst-Planck, and Navier-Stokes equations, respectively. The commercial software (COMSOL 5.4) was used to solve the system of equations that govern this problem. The influence of different parameters such as Hydraulic Retention Time (HRT) (15-45 min), applied voltage (2-10v) rotating speed of electrode (0-150 rpm) on the EC reactor also has been studied. The extracted results from the simulation show how the voltage, current, velocity and concentration of the pollutant and coagulant are distributed in the reactor. Besides, the transient behavior of the lead removal from the water was demonstrated in this work. The obtained results from this study were compared with experimental results. The theoretical results showed a good prediction of lead removal from water by electrocoagulation system and they showed good agreement between predictive and experimental data.

Modelling the adsorption of fluoride onto activated alumina in the presence of other ions

Most of the studies on the adsorption of F$^-$ are experimental, and have been done with synthetic solutions. Such studies rarely mimic the field situation. Therefore, selection of an adsorbent that can remove F$^-$ from any kind of feed requires models that can predict the adsorption behavior for any given set of input conditions. From our observations and as also reported by many authors, the adsorption of F$^-$ is affected by the presence of many ions. When modelling the adsorption of F$^-$, it is usually taken as a single entity getting adsorbed on the adsorbent. As this is not a proper assumption, a model was developed which takes into account all the speciation reactions that take place during adsorption, and all the species like H$^+$, OH$^-$, Na$^+$, Cl$^-$, and NO$_3^-$ present in the solution along with F$^-$. As an electrolyte system is involved, the Nernst-Planck equations were used to obtain the flux of each species. Using the model, the equilibrium constants and rate constants for the reactions were obtained. For one initial concentration of F$^-$, a reasonable fit was obtained to the batch adsorption data, except at short times. Because of an uncertainty in the amount of impurity present in the commercial adsorbent used, there was a significant discrepancy between predictions and data at higher initial concentrations of F$^-$. The present model can be applied to any charged adsorbent.

Visco-hyperelastic swelling and mechanical behavior of tough pH-sensitive hydrogels: Theory development and numerical implementation

Tough pH-sensitive hydrogels with noncovalent bonds have been developed in recent years as a promising structural material. They are involved in many engineering devices and natural phenomena due to their attractive properties. In this paper, we develop an electro-chemo-mechanical theory for swelling and mechanical behavior of tough pH-sensitive hydrogels. Due to its large deformation and time-dependent properties, we treat the hydrogel as a single-phase incompressible visco-hyperelastic material. We employ the Nernst-Planck equation to capture the flux of the ionic species through hydrogel boundaries. In this work, we are focused on the macro-size pH-sensitive hydrogels. Due to the Debye length, we assume the electro-neutrality condition both inside and outside of the hydrogel. Therefore, to find the concentration of co-ions and counter-ions inside the hydrogel, we use the Donnan equilibrium. We utilize the Generalized Maxwell model along with the Yeoh strain energy function to characterize the visco-hyperelastic behavior of the hydrogel. Hydrogel swells mainly due to the osmotic pressure. In this theory, we relate the osmotic pressure to the swelling ratio and apply the swelling ratio as a constraint in the strain energy function. After presenting the theory, we implement the theory in a nonlinear finite element framework for simulation. Our proposed theory can capture the transient dual swelling response of the hydrogel in various pH values as well as the time-dependent and self-healing behavior of the hydrogel.

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.

Modeling of a pH–tunable dual–response nanopore sensor

We consider a nanopore sensor model proposed previously (Mádai et al. J. Mol. Liq. 283 (2019) 391–398) by extending our study to the effect of pH that is an experimentally controllable parameter. With pH, we can tune the protonation and deprotonation states of the functional groups on the nanopore's wall, and, thus, the charge pattern that strongly influences the ionic current. Sensing is based on the selective binding of analyte ions in the right-hand-side binding region of the pore. The left-hand-side region is a buffer region that determines the main charge carrier. We show that we can optimize the sensitivity of the sensor with pH by creating a bipolar pore, where the buffer region is positive and the binding region is negative. Anions that carry the current are attracted into the pore by the positive analyte ions. The device is dual response, because the nanopore is rectifying: we can use the relative current (in relation to the current in the absence of analyte) and rectification as device functions. We model the electrolyte with charge hard spheres in implicit water, and the binding potential with the square-well potential. The Nernst-Planck equation coupled to Local Equilibrium Monte Carlo simulations is used to compute the ionic flux.

Modeling transport of charged species in pore networks: Solution of the Nernst–Planck equations coupled with fluid flow and charge conservation equations

A pore network modeling (PNM) framework11Computer code available on public repository https://github.com/PMEAL/OpenPNM. for the simulation of transport of charged species, such as ions, in porous media is presented. It includes the Nernst–Planck (NP) equations for each charged species in the electrolytic solution in addition to a charge conservation equation which relates the species concentration to each other. Moreover, momentum and mass conservation equations are adopted and there solution allows for the calculation of the advective contribution to the transport in the NP equations.The proposed framework is developed by first deriving the numerical model equations (NMEs) corresponding to the partial differential equations (PDEs) based on several different time and space discretization schemes, which are compared to assess solutions accuracy. The derivation also considers various charge conservation scenarios, which also have pros and cons in terms of speed and accuracy. Ion transport problems in arbitrary pore networks were considered and solved using both PNM and finite element method (FEM) solvers. Comparisons showed an average deviation, in terms of ions concentration, between PNM and FEM below 5% with the PNM simulations being over 104 times faster than the FEM ones for a medium including about 104 pores. The improved accuracy is achieved by utilizing more accurate discretization schemes for both the advective and migrative terms, adopted from the CFD literature. The NMEs were implemented within the open-source package OpenPNM based on the iterative Gummel algorithm with relaxation.This work presents a comprehensive approach to modeling charged species transport suitable for a wide range of applications from electrochemical devices to nanoparticle movement in the subsurface.

Poisson–Nernst–Planck equations with high-order steric effects

The Poisson–Nernst–Planck–Lennard-Jones (PNP–LJ) model is a mathematical model for ionic solutions with Lennard-Jones interactions between the ions. Due to the singular nature of the Lennard-Jones interaction kernel, however, the PNP–LJ model gives rise to an intractable analytic problem and a highly demanding computational problem. Previous works tackled this problem by replacing the LJ potential with a leading order approximation of the LJ potential giving rise to the steric PNP model. The steric PNP proved to be a successful model for the transport of ions in biological ionic channels. However, in other parameter regimes related to high concentrations, it is ill-posed. Presumably, since the steric PNP model is derived using only a leading-order approximation of the LJ potential. In this study, we go beyond the leading order approximation of the Lennard-Jones (LJ) interaction kernel and develop a new class of steric PNP equations that rely on a high-order approximation of the LJ potential. Surprisingly, we show that the introduction of high-order terms does not regularize the steric PNP model. Namely, at the limit of vanishing ionic sizes, high-order steric PNP equations, at all orders including the PNP–LJ model, are ill-defined at certain parameters regimes. Further analysis shows that this is an inherent property of PNP equations that account for steric effects and is related to pattern formation in these systems.

Understanding the sulfate attack of Portland cement–based materials exposed to applied electric fields: Mineralogical alteration and migration behavior of ionic species

The magnesium and sodium sulfate attacks on Portland cement paste in the presence of applied electric fields were studied, and the mineralogical alterations were investigated by both experiments and thermodynamic modeling. When an electric current flows out of the cement paste, the electric migration of ions induced sulfate ingress and decalcification. Compared with the specimen exposed to Na2SO4, that exposed to MgSO4 for 28 d proceeded to a later degradation stage, which is characterized by the decomposition of ettringite, portlandite, and AFm phases, and the formation of CaSO4. Thermodynamic modeling indicates a neutralization process induced by the electric migration of OH−, which is potentially responsible for the decomposition of ettringite. When an electric current flows into the cement paste, the Mg2+ and Na+ showed different migration behavior. Mg2+ was incorporated to form brucite and M-S-H–like products in a shallow area (~100 μm) on the surface of the specimen, whilst a part of the Na+ could be bonded to form Na-rich silica gel with the other part penetrating through the specimen. By coupling the pore solution chemistry obtained from thermodynamic modeling with the Nernst-Planck equation, the migration behaviors of the ionic species (SO42-, Mg2+, and Na+) were analyzed.

The significance of modeling electrodialysis desalination using multi-component saline water

This paper focuses on electrodialysis (ED) desalination modeling of standard seawater and brackish water as a multi-component and equivalent NaCl solutions. Experimentally validated two-dimensional models are developed based on ions separation through a cell pair of ED. Nernst-Planck equation is integrated with Navier-Stokes equations to carry out modeling using the COMSOL program. The results show that the multi-component model is important for the selected brackish water separation, while it is not significant for the investigated seawater. NaCl model predicts a little higher (~1%) current density than the standard multi-component seawater while that is lower (~14%) for a multi-component brackish water, indicating that the cell pair resistant of multi-component brackish water is different from that of equivalent NaCl solution. Ionic concentration and flux (total, migrative, and diffusive) profiles for the standard seawater (Na+, Cl−, Mg2+, SO42−, K+, Ca2+, F−, Br−, and HCO3−), a multi-component brackish water (Na+, Cl−, Mg2+, SO42−, K+, Ca2+, F−, NO3−, and HCO3−), and equivalent NaCl solutions are represented and compared. This study recommends that the NaCl model could be used for the seawater ED model, while the multi-component ED model is preferred for brackish water. Since ED systems are highly recommended for brackish water, the multi-component ED-model is needed.

A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson–Nernst–Planck equations

The Poisson–Nernst–Planck (PNP) equations is a macroscopic model widely used to describe the dynamics of ion transport in ion channels. In this paper, we introduce a semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserves the steady state). Solvability of the semi-discrete scheme is proved and a simple fixed point iteration is proposed to solve the fully discrete scheme. Numerical examples in both 1D and 2D and for multiple species are presented to demonstrate the convergence and properties of the proposed scheme.

A mathematical approach based on the Nernst-Planck equation for the total electric voltage demanded by the electrocoagulation process: Effects of a time-dependent electrical conductivity

In this work, an innovative mathematical approach for the total electric voltage in the electrocoagulation process (EC) was proposed. Contributions of electrodes activation and ions species transfer were modeled by analytical solving the non-steady Nernst-Planck equation. The electrical conductivity was adequately represented in terms of available positive ions species. A time-dependent equation that resembles a desorption-diffusion process was found to represent the electrode activation voltage. An Al electrode-based EC reactor was used to perform EC tests for assessing each electric phenomenon voltage. The data of electrical conductivity were well fitted by its proposed model, highlighting its effects on the total voltage. The anode activation voltage was responsible for the starting transient stage and modulating the total voltage profile. Finally, it could state that the time-dependent electrical conductivity has influenced the total voltage profile and drove the pollutant removal performance, acting on both activation and electric migration voltages intrinsically.

Asymmetric double-layer charging in a cylindrical nanopore under closed confinement.

This article presents a physical-mathematical treatment and numerical simulations of electric double layer charging in a closed, finite, and cylindrical nanopore of circular cross section, embedded in a polymeric host with charged walls and sealed at both ends by metal electrodes under an external voltage bias. Modified Poisson-Nernst-Planck equations were used to account for finite ion sizes, subject to an electroneutrality condition. The time evolution of the formation and relaxation of the double layers was explored. Moreover, equilibrium ion distributions and differential capacitance curves were investigated as functions of the pore surface charge density, electrolyte concentration, ion sizes, and pore size. Asymmetric properties of the differential capacitance curves reveal that the structure of the double layer near each electrode is controlled by the charge concentration along the pore surface and by charge asymmetry in the electrolyte. These results carry implications for accurately simulating cylindrical capacitors and electroactuators.

A Numerical Investigation on the Electrochemical Behavior of CaO and Al2O3 in the ESR Slags

Field structures including electromagnetic, concentration of ions, and flow fields in an ESR-like process composed of a graphite crucible containing a molten slag, air, and an iron electrode are computed. Both CaF2—(mass pct 2) CaO and CaF2—(mass pct 2) Al2O3 slags are examined. Tertiary current distribution is calculated. Therefore, polarization overpotential and Faradic reactions at metal–slag interface are considered using Tafel law, whereas transport of ions in the bulk of slag is determined through Nernst–Planck equations. The main goal is to shed light on the invisible phenomena such as magnetohydrodynamics caused by transport of ions, electrical conductivity of CaF2-based slag using additives (e.g., CaO or Al2O3), and the role of complexation of ions (e.g., {text{AlO}}_{3}^{3 - } ) in the molten slag applied to the ESR. An explanation is given for the observation of higher oxygen content in the metal using Al2O3 than that using equivalent amount of CaO in the CaF2-based slag of a DC-operated ESR.

Influence factors of thermal driven ion transport in nano-channel for thermoelectricity application

Many attentions have been paid on discussing isothermal transport in electrolyte-filled wall-charged nano-pores or nano-channels, while the non-isothermal transport has been scarcely considered before. In this work, a calculation model based on Poisson–Nernst–Planck equation is established to describe the thermal driven ion transport in nano-channels. The effects of different influence factors on the ion transport is carefully studied, including effects of the temperature difference across nano-channel, the ion concentration, the half-width of nano-channel, and the length of nano-channel. Results turn out that: performances of nano-channel batteries can be classified and simplified by using characteristic parameters and ion concentrations; the ion concentration of about 1 mol•m−3 and the nano-channel half width (h) of about 1 nm is optimal for thermoelectricity applications, and if the optimal nano-channel length of about 85 h and the optimal temperature difference of about 85 K are further applied, the thermally induced power of the system can be maximized. This study is expected to supply some information for designing a nano-channels system with high thermos-electric energy conversion efficiencies.

A Kinetic Model to Account for Selective Trapping of Weak Bases Inside Acidic Intra-Cellular Vesicles

Recent work from our group has characterized selective trapping of nicotinic acetylcholine receptor (nAChR) ligands within neurons in the brain (Govind et al. eLife 2017). The ligands are weak bases that bind with high-affinity to α4β2-type nAChRs (α4β2Rs), such as the anti-smoking drug varenicline (Chantix) and epibatidine. Selective trapping occurs within α4β2s-containing acidic vesicles of cells and neurons. Slow release of trapped varenicline reduces effects of long-term nicotine exposure. Selective trapping is further regulated by nicotine exposure, which increases the number of α4β2s-containing acidic vesicles. Nicotine, also a weak base, is not trapped due to its lower pKa and lower affinity for α4β2Rs. Here, we present a kinetic model that accounts for the biphasic behavior of these weak bases leaving the cell after accumulating in the acidic vesicles and binding to the nAChRs on the surface of the vesicles, within intracellular membranes, and on the exterior of the cell. Intracellular accumulation and release of molecules by diffusion into and out of cytosol and vesicles is calculated using Fick's Law of diffusion (neutral molecule) and the Nernst-Planck equation for ions (Trapp and Horobin 2005). Ligand binding to α4β2Rs is modeled using empirical receptor-ligand kinetics. Solving our coupled differential equations numerically has accurately approximated experimental data for epibatidine accumulation and release in vesicles. The model is mainly limited by inclusion of pKa and binding affinity as the only parameters directly causing the trapping. Our model also suggests that the vesicle membrane receptor concentration is an important factor, a notion that we hope to verify experimentally. Using this model, we have been able to model kinetic trapping of the weak-base α4β2 ligands Nifene and 2-F-A8538, and to explain differences in their kinetics during positron emission tomography.

Superconvergent estimate of a Galerkin finite element method for nonlinear Poisson–Nernst–Planck equations

A Galerkin finite element method (FEM) is developed for nonlinear Poisson–Nernst–Planck (PNP) equations. Based on the combination technique of the element’s interpolation and Ritz projection together with the special mean value approach, the superclose and superconvergence estimates with order O(h2) are derived under weaker regularity requirements of the exact solution. Some numerical results are provided to confirm the theoretical analysis.

Transient swelling response of pH-sensitive hydrogels: A monophasic constitutive model and numerical implementation.

Due to their ability to swell/deswell under alternating pH conditions, pH-sensitive hydrogels present themselves as a potential candidate for controlled drug delivery. In this paper, a coupled electro-chemo-mechanical transient large deformation homogeneous and inhomogeneous swelling theory is developed for pH-sensitive hydrogels. The hydrogel is treated as a single-phase compressible isotropic hyperelastic material. The Nernst-Planck equation is used for the flux of the ionic species inside the hydrogel through its boundaries. The main reason which causes the hydrogel to swell is the osmotic pressure. The osmotic pressure is treated as external stress to the hydrogel. As soon as the concentration of the ions inside the hydrogel is known one may compute the osmotic pressure. Finally, the mechanical equilibrium equations give the deformation of the hydrogel. After presenting the swelling theory, the theory is implemented into the finite element method, and the experimental data is used in order to validate the theory. The simulation matches closely the experimental data which shows the accuracy of the proposed theory. The free swelling and constrained swelling of the pH-sensitive hydrogel are also simulated to show the capabilities of the proposed formulation and well as its numerical counterpart.

Quasi-One-Dimensional Generator-Collector Electrochemistry in Nanochannels.

Mass transport in fluidic channels under conditions of pressure-driven flow is controlled by a combination of convection and diffusion. For electrochemical measurements the height of a channel is typically of the same order of magnitude as the electrode dimensions, resulting in complex two- or three- dimensional concentration distributions. Electrochemical nanofluidic devices, however, can have such a low height-to-length ratio that they can effectively be considered as one-dimensional. This greatly simplifies the modeling and quantitative interpretation of analytical measurements. Here we study mass transport in nanochannels using electrodes in a generator-collector configuration. The flux of redox molecules is monitored amperometrically. We observe the transition from diffusion-dominated to convection-dominated transport by varying both the flow velocity and the distance between the electrodes. These results are described quantitatively by the one-dimensional Nernst–Planck equation for mass transport over the full range of experimentally accessible parameters.

Electrochemical Flow Reactor for Selective Extraction of Lithium Chloride from Natural Brines

An electrochemical flow reactor for the extraction of lithium chloride from natural brines has been designed and tested. It comprises two 3D porous packed bed electrodes and an anion exchange membrane. The electrodes were filled with conducting petroleum coke particles covered with LiMn2O4 (anode) in contact with dilute LiCl recovery solution and partly de-lithiated Li1−xMn2O4 (cathode) in contact with natural brine. The electrolytes were circulated at constant flow rate through the porous packed bed electrodes and a constant current was passed at the lithium deficient Li1−xMn2O4 (cathode) with Li+ ion intercalation and de-intercalation at the LiMn2O4 (anode), while a flow of chloride ions operated at the anion selective membrane to compensate charge. In a second step, the electrolytes were exchanged and LiCl was recovered from the lithiated Li1−xMn2O4, now acting as anode. A mathematical model for the flow reactor with the diffusion and migration described by the Nernst-Planck equation, the convective flow of electrolyte and the LiMn2O4 battery intercalation model has been developed using the finite element method under the COMSOL environment.

Modeling multicomponent ion transport to investigate selective ion removal in electrodialysis

A 2-dimensional multicomponent ion transport model based on Nernst-Planck (NP) equation and electroneutrality assumption is developed for an electrodialysis (ED) cell operated in the ohmic regime. The flow in channels are assumed incompressible, isothermal, and laminar. Donnan equilibrium and flux continuity are considered at ion-exchange membrane (IEM)-solution interfaces. To account for tortuosity effects inside membranes, effective ionic diffusion coefficients are calculated using membranes water volume fractions. The developed multicomponent model is used to elucidate the effects of feed solution properties, cell properties, system hydrodynamics, operational conditions, and membrane properties on selective divalent ion removal in the cell.The results indicate that the selective removal of divalent ions improves with decreasing the cell length, imposed potential, and ionic strength of feed water. Enhanced mixing in spacer-filled cell also promotes selective divalent ion removal. Higher concentrations of fixed charges on the membranes results in greater selectivity toward divalent ions at short cell length and low imposed potentials. With equal concentrations of fixed charges, membranes with high water content are less favorable for selective divalent ion removal. The developed framework enables the optimum selection of cell design, IEMs, spacer design, and operational conditions to selectively remove ions from multicomponent solutions.