Modified Poisson-Boltzmann Equation Research Articles

Modeling the Environment-Dependent Kinetics of Oxygen Reduction Reaction – a Continuum Model for Electric Double Layer

For proton-exchange-membrane fuel cells (PEMFCs) to achieve broad commercialization, improved energy-conversion efficiency with minimal Pt-based electrocatalyst is required [1]. Because the sluggish rate of oxygen reduction reaction (ORR) limits the efficiency of PEMFCs [2], the efficiency improvement requires a better understanding of ORR kinetics and mechanism to design better catalyst. To understand the ORR mechanism, theoretical and experimental analyses have been conducted. While previous studies reasonably explained the catalyst-dependent activity on single crystal catalysts in 0.1 M perchloric acid solution [3, 4], the explicit effect of electrolyte and related microenvironments [5, 6] is not thoroughly understood. The change in the electrolyte alters the electric-double-layer (EDL) structure and thus the local microenvironment at the electrode/electrolyte interface. Thus, the structure of the EDL should be carefully analyzed to uncover the electrolyte-dependent reaction kinetics. In this talk, we propose a multiscale continuum model to predict the EDL structure and examine the effect of perchloric acid concentration on ORR activity on Pt (111). The model includes Density Potential Functional Theory (DPFT) for electron density and Modified Poisson Boltzmann equation for species’ density and electric potential [7]. Also, the interaction between adsorbents and electric field is taken into account by minimizing the grand potential. After model validation with experimentally measured double-layer capacity data as a function of applied potential and concentration, the effect of the perchloric acid concentration (0.02 M – 0.2 M) on ORR activity is analyzed and discussed. It is shown that the model reproduces the specific activity obtained in the experiments when assuming the oxygen adsorption is limiting the rate, which can be attributed to the large energetic barrier for solvent reorganization [8]. Then, extension of the model to PEMFC ionomer electrolytes will be introduced. Overall, the model framework and findings provide insights into the ORR mechanism and guidance on how to tailor catalyst materials for increased PEMFC performance. Acknowledgements: This work was partially supported by the Hydrogen and Fuel Cell Technologies Office (HFTO), Office of Energy Efficiency and Renewable Energy, U.S. Department of Energy (DOE) through the Million Mile Fuel Cell Truck (M2FCT) consortia, technology managers G. Kleen and D. Papageorgopoulos, under contract number DE-AC02- 05CH11231, as well as by a CRADA with Toyota Central R&D Labs., Inc.. Reference [1] D. A. Cullen, K. C. Neyerlin, R. K. Ahluwalia, R. Mukundan, K. L. More, R. L. Borup, A. Z. Weber, D. J. Myers and A. Kusoglu, Nature Energy, 6, 462 (2021).[2] I. E. L. Stephens, A. S. Bondarenko, U. Grønbjerg, J. Rossmeisl and I. Chorkendorff, Energy & Environmental Science, 5 (2012).[3] M. Shibata, M. Inaba, K. Shinozaki, K. Kodama and R. Jinnouchi, Journal of The Electrochemical Society, 169 (2022).[4] H. A. Hansen, V. Viswanathan and J. K. Nørskov, The Journal of Physical Chemistry C, 118, 6706 (2014).[5] K. Kodama, K. Motobayashi, A. Shinohara, N. Hasegawa, K. Kudo, R. Jinnouchi, M. Osawa and Y. Morimoto, ACS Catalysis, 8, 694 (2017).[6] M. Luo and M. T. M. Koper, Nature Catalysis, 5, 615 (2022).[7] J. Huang, J Chem Theory Comput, 19, 1003 (2023).[8] D. T. Limmer, A. P. Willard, P. Madden and D. Chandler, Proceedings of the National Academy of Sciences, 110, 4200 (2013).

The nanofluidic capacitor: Differential capacitance in the absence of reservoirs.

Within the framework of the classical, mean-field Poisson-Boltzmann (PB) theory, we carry out direct numerical simulations to determine the differential capacitance of a closed nanochannel of a circular cross section, embedded in a polymeric host with charged walls and sealed at both ends by metal electrodes under an external potential bias. Our approach employs the modified PB equation, which accounts for the finite size of ions and the dependency of the electrolyte's relative permittivity on the local electric field. In view of the absence of reservoirs, the modified PB equation becomes subject to global algebraic constraints, without prior knowledge of a bulk electrolyte concentration. Equilibrium ion distributions and differential capacitance curves are investigated as functions of electrolyte properties and the surface charge density modulation. This modulation leads to asymmetric differential capacitance curves that can be tuned. More generally, our approach provides a transparent numerical framework for accurately simulating confined nanofluidic systems with new physical properties that may be exploited in novel iontronic circuit elements.

Rheological impact on thermofluidic transport characteristics of generalized Maxwell fluids through a soft nanopore

The time-dependent electroosmotic flow (EOF) and heat transfer characteristic of a generalized Maxwell fluid through the polyelectrolyte layer (PEL) grafted nanopore are investigated while considering different permittivity between the PEL and electrolyte solution. The ion partitioning effects arise due to the different permittivity among these regions. Taking the ion partitioning effects, the analytic solution for the induced potential is established within and outside the PEL from the modified Poisson–Boltzmann equation assuming the Debye–Hückel approximation for a low surface charge. The Cauchy momentum equation with a suitable constitutive equation for fractional Maxwell fluids is derived, and the corresponding analytic solution is presented to provide the axial fluid flow distribution in the full domain. The energy fluxes that have major contributions to the energy equation mainly depend on axial conduction, convection due to electrolyte transport, and Joule heating effects for the external electric field. The analytical solutions of the energy equation for hydro-dynamically fully developed flow with constant thermophysical properties are presented to provide the temperature distribution considering constant heat flux at the nanopore wall. The influence of several important factors for characterizing heat transfer behavior is investigated in the present study. The maximum fluid velocity occurs when the permittivity between the PEL and electrolyte region is the same. The increasing values of fluid velocity imply higher convective heat transfer and make the Nusselt number higher. This study makes a conscious effort toward highlighting the modality controlling the heat transfer characteristics for the ion partitioning effects.

Impact of hydrodynamics and rheology of the ion partitioning effect on electrokinetic flow through a soft annulus with a retentive and absorptive wall.

The theoretical analysis for the mass transfer process of an oscillatory electroosmotic flow (EOF) in the fractional Jeffrey fluid model is studied through a polyelectrolyte layer (PEL) coated cylindrical annulus with reversible and irreversible wall reactions. The ion partitioning effect is observed due to the difference in permittivity of the PEL and the electrolyte solution, which is accounted for by the Born energy. Considering ion partitioning effects, analytical solutions for induced potential and axial velocity are presented, respectively in both the PEL and electrolyte region from the modified Poisson-Boltzmann equation and the Cauchy momentum equation with a proper constitutive equation, respectively. The Maxwell fluid and classical viscous Newtonian fluid models can be achieved separately by adjusting the relaxation and retardation time in the constitutive equation of this model. The analytical solution of the convection-diffusion equation for solute transport is established in the full domain. The separation of species is found to be dependent mainly on the Damköhler number, absorption parameter, phase partitioning coefficient, etc. It is observed that the osmotic pressure increases with the thickness and fixed charge density of the PEL. The velocity decreases with an increase in the permittivity difference of these layers. Our results suggest that the separation may be achieved through a difference in absorption kinetics.

Coupling Effects of Ionic Surfactants and Electrolytes on the Stability of Bulk Nanobubbles.

As interest in the extensive application of bulk nanobubbles increases, it is becoming progressively important to understand the key factors affecting their anomalous stability. The scientific intrigue over nanobubbles originates from the discrepancy between the Epstein–Plesset prediction and experimental observations. Herein, the coupling effects of ionic surfactants and electrolytes on the stability of bulk nanobubbles is studied. Experimental results show that ionic surfactants not only reduce the surface tension but also promote the accumulation of net charges, which facilitate the nucleation and stabilization of bulk nanobubbles. The addition of an electrolyte in a surfactant solution further results in a decrease in the zeta potential and the number concentration of nanobubbles due to the ion shielding effect, essentially colloidal stability. An adsorption model for the coexistence of ionic surfactants and electrolytes in solution, that specifically considers the effect of the adsorption layer thickness within the framework of the modified Poisson–Boltzmann equation, is developed. A quantitative agreement between the predicted and experimental surface tension is found in a wide range of bulk concentrations. The spatial distribution of the surface potential, surfactant ions and counterions in the vicinity of the interface of bulk nanobubbles are described. Our study intrinsically paves a route to investigate the stability of bulk nanobubbles.

Electrosorption-induced deformation of a porous electrode with non-convex pore geometry in electrolyte solutions: A theoretical study

Porous carbon is well known as a good candidate for the development of electrochemical double-layer capacitors. Predominantly, many conventional carbons are microporous and often well described by the assumption of slit pore geometry. However, there is a class of carbons that is significantly different from the others, namely templated mesoporous carbons. In this work, we study electrosorption-induced deformation in CMK-3–like mesopores having non-convex geometry. Our mean-field approach is based on the modified Poisson-Boltzmann equation taking into account the excluded volume of the ions within the hard-sphere model. We assume that the deformation is caused by two effects: ion osmotic pressure and electrostatic interactions of the electric double layers on charged rods. We estimated the pore-load modulus of the CMK-3–like material and found an agreement with the previously obtained values by small-angle neutron scattering (SANS) data analysis. Additionally, we studied the differential capacitance in the non-convex pore geometry and found that the behavior of the differential capacitance profiles was similar to that of the profiles obtained for flat electric double layers: the crowding regime at rather high electric potentials and more pronounced profile asymmetry with increasing differences in the ionic sizes.

Effect of ion partitioning on an oscillatory electro-osmotic flow on solute transport process of fractional Jeffrey fluid through polyelectrolyte-coated nanopore with reversible wall reaction

The ion-partitioning effects on solute transport phenomena of time-periodic electro-osmotic flow in fractional Jeffrey fluid are investigated through a polyelectrolyte layer (PEL)-coated conical nanopore within a reactive wall whose ends are connected with two large reservoirs. By considering the ion-partitioning effects, analytical solutions for the induced potential and the axial velocity are presented, respectively, from the modified Poisson–Boltzmann equation and the Cauchy momentum equation with the proper constitutive equation of the fractional Jeffrey fluid model in the exterior and interior of the PEL. The analytic solution of the convection–diffusion for solute transport is established in the entire domain. The influence of the oscillating Reynolds number Rew, permittivity ratio εr between two mediums, relaxation time λ1ω, retardation time λ2ω, phase partitioning coefficient σp, PEL fixed charge density qfix, Debye–Hückel parameter κa, and softness parameter λs are investigated in this study. Asymptotic solution for the axial velocity was also presented for low-oscillating Reynolds numbers and validated. The maximum axial velocity occurs when the permittivity between the PEL and electrolyte is the same for all models. The volumetric flow rate decreases with the increase in the PEL thickness, positive PEL charge density, and softness parameter in our study. The volume flow rate of the Newtonian fluid increased 24.07% for Maxwell fluid (λ1ω=5, α = 1) and 11.56% for Jeffrey fluid (λ1ω=5, λ1ω=1, α = 1, and β=0.5), when κa=25, Rew = 10, qfix = 5, d = 0.2, εr=0.6, and λs=1.0. The mass transport rate increases with relaxation time, tidal displacement, and permittivity ratio between these layers.

Modified Poisson–Boltzmann equations and macroscopic forces in inhomogeneous ionic fluids

We propose a field-theoretical approach based on the thermodynamic perturbation theory and within it derive a grand thermodynamic potential of the inhomogeneous ionic fluid as a functional of electrostatic potential for an arbitrary reference fluid system. We obtain a modified Poisson–Boltzmann (PB) equation as the Euler–Lagrange equation for the obtained functional. Applying Noether’s theorem to this functional, we derive a general mean-field expression for the stress tensor consistent with the respective modified PB equation. We derive a general expression for the macroscopic force acting on the dielectric or conductive body immersed in an ionic fluid. In particular, we derive a general mean-field expression for the disjoining pressure of an ionic fluid in a slit pore. We apply the developed formalism to describe three ionic fluid models of practical importance: nonpolarizable models (including the well-known PB and Poisson–Fermi equations), polarizable models (ions carry nonzero permanent dipole or static polarizability), and models of ion-dipole mixtures (including the well-known PB–Langevin equation). For these models, we obtain modified PB equations and respective stress tensors, which could be valuable for different applications, where it is necessary to estimate the macroscopic forces acting on the dielectric or conductive bodies (electrodes, colloids, membranes, etc) together with the local electrostatic potential (field) and ionic concentrations.

Analytical Solution of Modified Poisson–Boltzmann Equation and Application to Cylindrical Nanopore Supercapacitor Energy Storage

Analytical Solution of Modified Poisson–Boltzmann Equation and Application to Cylindrical Nanopore Supercapacitor Energy Storage

Linear-scaling selected inversion based on hierarchical interpolative factorization for self Green's function for modified Poisson-Boltzmann equation in two dimensions

This paper studies an efficient numerical method for solving modified Poisson-Boltzmann (MPB) equations with the self Green's function as a state equation to describe electrostatic correlations in ionic systems. Previously, the most expensive point of the MPB solver is the evaluation of Green's function. The evaluation of Green's function requires solving high-dimensional partial differential equations, which is the computational bottleneck for solving MPB equations. Numerically, the MPB solver only requires the evaluation of Green's function as the diagonal part of the inverse of the discrete elliptic differential operator of the Debye-Hückel equation. Therefore, we develop a fast algorithm by a coupling of the selected inversion and hierarchical interpolative factorization. By the interpolative factorization, our new selected inverse algorithm achieves linear scaling to compute the diagonal of the inverse of this discrete operator. The accuracy and efficiency of the proposed algorithm will be demonstrated by extensive numerical results for solving MPB equations.

Influence of finite size of ions on thermal transport of a simultaneous electrokinetic-pressure driven flow of power-law fluids in a slit microchannel

At high values of interfacial electrokinetic potentials, the effects of finite ionic sizes are consequential and must be taken in the purview. This paper addresses the non-trivial implications of steric effects on thermal transport of an electroosmotic-pressure driven flow within a microchannel under an imposed heat flux. The ion size is introduced into the modified Poisson-Boltzmann equation by the steric factor, which allows considering the ions as point charges or finite sizes. The non-linear, non-dimensional form of the electrokinetic potential, momentum, and energy equations are numerically solved in a spatial scheme discretized into the non-equally spaced elements. Our main point here is that the intricate physical interplay among zeta potential, pressure gradient, and steric effects has a considerable impact on the temperature distribution. Results indicated that by decreasing the surface cooling, the effect of ion finite size is significantly intensified while other factors become of peripheral importance. Also, unfavourable pressure gradients tend to mask the effect of additional heat generated by increasing the channel aspect ratio. Finally, adverse pressure gradients generated higher velocity gradients at the surface, results in a higher convective cooling and consequently an improved thermal dissipation.

Osmotic Transport at the Aqueous Graphene and hBN Interfaces: Scaling Laws from a Unified, First-Principles Description.

Osmotic transport in nanoconfined aqueous electrolytes provides alternative venues for water desalination and "blue energy" harvesting. The osmotic response of nanofluidic systems is controlled by the interfacial structure of water and electrolyte solutions in the so-called electrical double layer (EDL), but a molecular-level picture of the EDL is to a large extent still lacking. Particularly, the role of the electronic structure has not been considered in the description of electrolyte/surface interactions. Here, we report enhanced sampling simulations based on ab initio molecular dynamics, aiming at unravelling the free energy of prototypical ions adsorbed at the aqueous graphene and hBN interfaces, and its consequences on nanofluidic osmotic transport. Specifically, we predicted the zeta potential, the diffusio-osmotic mobility, and the diffusio-osmotic conductivity for a wide range of salt concentrations from the ab initio water and ion spatial distributions through an analytical framework based on Stokes equation and a modified Poisson-Boltzmann equation. We observed concentration-dependent scaling laws, together with dramatic differences in osmotic transport between the two interfaces, including diffusio-osmotic flow and current reversal on hBN but not on graphene. We could rationalize the results for the three osmotic responses with a simple model based on characteristic length scales for ion and water adsorption at the surface, which are quite different on graphene and on hBN. Our work provides fundamental insights into the structure and osmotic transport of aqueous electrolytes on 2D materials and explores alternative pathways for efficient water desalination and osmotic energy conversion.

Ion adsorption stabilizes bulk nanobubbles

The mechanism leading to the extraordinary stability of bulk nanobubbles in aqueous solutions remains an outstanding problem in soft matter, modern surface science, and physical chemistry science. In this work, the stability of bulk nanobubbles in electrolyte solutions under different pH levels and ionic strengths is studied. Nanobubbles are generated via the technique of ultrasonic cavitation, and characterized for size, number concentration and zeta potential under ambient conditions. Experimental results show that nanobubbles can survive in both acidic and basic solutions with pH values far away from the isoelectric point. We attribute the enhanced stability with increasing acidity or alkalinity of the aqueous solutions to the effective accumulation of net charges, regardless of their sign. The kinetic stability of the nanobubbles in various aqueous solutions is evaluated within the classic DLVO framework. Further, by combining a modified Poisson-Boltzmann equation with a modified Langmuir adsorption model, we describe a simple model that captures the influence of ion species and bulk concentration and reproduce the dependence of the nanobubble’s surface potential on pH. We also discuss the apparent contradiction between quantitative calculation by ion stabilization model and experimental results. This essentially requires insight into the structure and dynamics of interfacial water on the atomic-scale.

Steric and Slippage Effects on Mass Transport by Using an Oscillatory Electroosmotic Flow of Power-Law Fluids.

In this paper, the combined effect of the fluid rheology, finite-sized ions, and slippage toward augmenting a non-reacting solute’s mass transport due to an oscillatory electroosmotic flow (OEOF) is determined. Bikerman’s model is used to include the finite-sized ions (steric effects) in the original Poisson-Boltzmann (PB) equation. The volume fraction of ions quantifies the steric effects in the modified Poisson-Boltzmann (MPB) equation to predict the electrical potential and the ion concentration close to the charged microchannel walls. The hydrodynamics is affected by slippage, in which the slip length was used as an index for wall hydrophobicity. A conventional finite difference scheme was used to solve the momentum and species transport equations in the lubrication limit together with the MPB equation. The results suggest that the combined slippage and steric effects promote the best conditions to enhance the mass transport of species in about 90% compared with no steric effect with proper choices of the Debye length, Navier length, steric factor, Womersley number, and the tidal displacement.

The surface potential explains ion specific bubble coalescence inhibition

HypothesisSome ions can prevent bubbles from coalescing in water. The Gibbs-Marangoni pressure has been proposed as an explanation of this phenomenon. This repulsive pressure occurs during thin film drainage whenever surface enhanced or surface depleted solutes are present. However, bubble coalescence inhibition is known to depend on which particular combination of ions are present in a peculiar and unexplained way. This dependence can be explained by the electrostatic surface potential created by the distribution of ions at the interface, which will alter the natural surface propensity of the ions and hence the Gibbs-Marangoni pressure. CalculationsA generalised form of the Gibbs-Marangoni pressure is derived for a mixture of solutes and the modified Poisson–Boltzmann equation is used to calculate this pressure for five different electrolyte solutions made up of four different ions. FindingsCombining ions with differing surface propensities, i.e., one enhanced and one depleted, creates a significant electrostatic surface potential which dampens the natural surface propensity of these ions, resulting in a reduced Gibbs-Marangoni pressure, which allows bubble coalescence. This mechanism explains why the ability of electrolytes to inhibit bubble coalescence is correlated with surface tension for pure electrolytes but not for mixed electrolytes.

On the equivalence of self-consistent equations for nonuniform liquids: a unified description of the various modifications

A variety of self-consistent (SC) equations have been proposed for non-uniform states of liquid particles under external fields, including adsorbed states at solid substrates and confined states in pores. External fields represent not only confining geometries but also fixed solutes. We consider SC equations ranging from the modified Poisson–Boltzmann equations for the Coulomb potential to the hydrostatic linear response equation for the equilibrium density distribution of Lennard-Jones fluids. Here, we present a unified equation that explains the apparent diversity of previous forms and proves the equivalence of various SC equations. This unified description of SC equations is obtained from a hybrid method combining the conventional density functional theory and statistical field theory. The Gaussian approximation of density fluctuations around a mean-field distribution is performed based on the developed hybrid framework, allowing us to derive a novel form of the grand-potential density functional that provides the unified SC equation for equilibrium density.

BION-2: Predicting Positions of Non-Specifically Bound Ions on Protein Surface by a Gaussian-Based Treatment of Electrostatics.

Ions play significant roles in biological processes—they may specifically bind to a protein site or bind non-specifically on its surface. Although the role of specifically bound ions ranges from actively providing structural compactness via coordination of charge–charge interactions to numerous enzymatic activities, non-specifically surface-bound ions are also crucial to maintaining a protein’s stability, responding to pH and ion concentration changes, and contributing to other biological processes. However, the experimental determination of the positions of non-specifically bound ions is not trivial, since they may have a low residential time and experience significant thermal fluctuation of their positions. Here, we report a new release of a computational method, the BION-2 method, that predicts the positions of non-specifically surface-bound ions. The BION-2 utilizes the Gaussian-based treatment of ions within the framework of the modified Poisson–Boltzmann equation, which does not require a sharp boundary between the protein and water phase. Thus, the predictions are done by the balance of the energy of interaction between the protein charges and the corresponding ions and the de-solvation penalty of the ions as they approach the protein. The BION-2 is tested against experimentally determined ion’s positions and it is demonstrated that it outperforms the old BION and other available tools.

Multi-ionic effects on the equilibrium and dynamic properties of electric double layers based on the Bikerman correction

Abstract Multi-ion electrolytes are used in many electrochemical applications; therefore, systematic analyses of the effect of multi-ion electrolytes on the capacitance and charging rate are required. In this study, the electrochemical behaviors of the electrical double layer of multi-ion electrolytes confined between parallel electrodes are predicted using a continuum approach. To study the characteristics of multi-ion electrolytes, modified Poisson–Boltzmann equations and modified Poisson–Nernst–Planck equations with the Bikerman correction term were used. Multi-valence and multi-size ion electrolytes with various compositions were selected, and the ion distribution, capacitance, and charging rates were obtained. We divided fluxes into thermal, electrical, and Bikerman-type fluxes for further investigation. In multi-valence cases, a relatively small number of high-valence ions might significantly enhance the equilibrium capacitance. However, the small number of the ions restricts the system from reaching equilibrium. In multi-size cases, smaller ions enhance the equilibrium capacitance, and the curves for the capacitance and charging rates gradually change as the fraction of the smaller ions increases. The results of this study should provide strategies for modifying the capacitance and charging rates using multi-ion electrolytes.

Electronic structure calculations in electrolyte solutions: Methods for neutralization of extended charged interfaces.

Density functional theory (DFT) is often used for simulating extended materials such as infinite crystals or surfaces, under periodic boundary conditions (PBCs). In such calculations, when the simulation cell has non-zero charge, electrical neutrality has to be imposed, and this is often done via a uniform background charge of opposite sign ("jellium"). This artificial neutralization does not occur in reality, where a different mechanism is followed as in the example of a charged electrode in electrolyte solution, where the surrounding electrolyte screens the local charge at the interface. The neutralizing effect of the surrounding electrolyte can be incorporated within a hybrid quantum-continuum model based on a modified Poisson-Boltzmann equation, where the concentrations of electrolyte ions are modified to achieve electroneutrality. Among the infinite possible ways of modifying the electrolyte charge, we propose here a physically optimal solution, which minimizes the deviation of concentrations of electrolyte ions from those in open boundary conditions (OBCs). This principle of correspondence of PBCs with OBCs leads to the correct concentration profiles of electrolyte ions, and electroneutrality within the simulation cell and in the bulk electrolyte is maintained simultaneously, as observed in experiments. This approach, which we call the Neutralization by Electrolyte Concentration Shift (NECS), is implemented in our electrolyte model in the Order-N Electronic Total Energy Package (ONETEP) linear-scaling DFT code, which makes use of a bespoke highly parallel Poisson-Boltzmann solver, DL_MG. We further propose another neutralization scheme ("accessible jellium"), which is a simplification of NECS. We demonstrate and compare the different neutralization schemes on several examples.

Impact of ion-steric and ion-partitioning effects on electrophoresis of soft particles.

A theoretical study on the electrophoresis of a soft particle is made by taking into account the ion steric interactions and ion partitioning effects under a thin Debye layer consideration with negligible surface conduction. Objective of this study is to provide a simple expression for the mobility of a soft particle which accounts for the finite-ion-size effect and the ion partitioning arise due to the Born energy difference between two media. The Donnan potential in the soft layer is determined by considering the ion steric interactions and the ion partitioning effect. The volume exclusion due to the finite ion size is considered by the Carnahan-Starling equation and the ion partitioning is accounted through the difference in Born energy. The modified Poisson-Boltzmann equation coupled with Stokes-Darcy-Brinkman equations are considered to determine the mobility. A closed-form expression for the electrophoretic mobility is obtained, which reduces to several existing expressions for mobility under various limiting cases.