Electrokinetic Phenomena in Concentrated Disperse Systems: General Problem Formulation and Spherical Cell Approach

ce Science xx (2007) xxx–xxx

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CIS-00918; No of Pages 43

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Advances in Colloid and Interfa

Electrokinetic Phenomena in concentrated disperse systems: General problemformulation and Spherical Cell Approach

Emilij K. Zholkovskij a, Jacob H. Masliyah b,⁎, Vladimir N. Shilov a, Subir Bhattacharjee c

a Institute of Bio-Colloid Chemistry of Ukrainian Academy of Sciences, Vernadskogo,42, 03142, Kiev, Ukraineb University of Alberta, Department of Chemical and Materials Engineering, 536Chemical-Mineral Engineering Building, Edmonton, Alberta, Canada T6G 2G6

c University of Alberta, Department of Mechanical Engineering, 4-8C Mechanical Engineering Building, Edmonton, Alberta, Canada T6G 2G6

Abstract

Electrokinetic Phenomena in concentrated disperse and colloid systems have been studied employing Spherical Cell Approach for over threedecades. The critical review of the advances in this area, which is conducted in the present paper, demonstrates a number of contradictions between theresults reported by different authors. These contradictions are largely associated with imposition of boundary conditions at the outer boundary of therepresentative Spherical Cell. In order to establish a correct version of the Spherical Cell Approach, in the present paper, the theory of ElectrokineticPhenomena in concentrated suspensions is revisited by primarily focusing on the boundary conditions employed at the Spherical Cell outer boundary. Tothis end, a general mathematical problem is formulated for addressing the behavior of a planar layer of a macroscopically homogeneous disperse systemunder simultaneous influence of the pressure difference, gravitation and applied electric fields. On the basis of the general problem formulation, wepresent strict definitions of the kinetic coefficients which describe the system behavior. Making use of such definitions, some general relationships arerederived for the kinetic coefficients, namely, the Smoluchowski asymptotic expressions and the Onsager irreversible thermodynamic relationships.

The general problem is reformulated for describing the electric, hydrodynamic and ion concentration fields inside the representative Spherical Cell. Usingan original approach, a complete set of the boundary conditions is derived by employing the only assumption: the average over the disperse systemvolume isequal to the average over a representative Spherical Cell volume. A general method for predicting the kinetic coefficients is developed by employing thesolution of the formulated problem. The developedmethod is combinedwith themethod of small perturbation parameter using the normalized zeta potential.Final expressions for the kinetic coefficients are obtained while accounting for the terms proportional to zeta potential. The predictions are compared withresults of other publications. On this basis, conclusions are made about the validity of different models proposed in the literature.© 2007 Published by Elsevier B.V.

Keywords: Electrophoresis; Electroosmosis; Sedimentation potential; Zeta potential; Volume fraction

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01.1. Macroscopically homogeneous and isotropic medium: representative cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01.2. Spherical Cell Approach in physics: historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01.3. Macrokinetic method for analyzing Electrokinetic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01.4. Spherical Cell Approach for Electrokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

1.4.1. Electrochemical outer conditions: Levine–Neale and Shilov–Zharkikh–Borkovskaya versions . . . . . . . . . . . . . 01.4.2. Hydrodynamic outer conditions: Happel and Kuwabara versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

1.5. Criteria of model validity: Onsager and Smoluchowski principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01.6. Unexpected results and contradictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

1.6.1. Common and different predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01.6.2. Onsager's principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01.6.3. Smoluchowski principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

1.7. Objective and structure of the paper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

⁎ Corresponding author.E-mail address: [email protected] (J.H. Masliyah).

0001-8686/$ - see front matter © 2007 Published by Elsevier B.V.doi:10.1016/j.cis.2007.04.025

Please cite this article as: Zholkovskij EK et al. Electrokinetic Phenomena in concentrated disperse systems: General problem formulation and Spherical CellApproach. Adv Colloid Interface Sci (2007), doi:10.1016/j.cis.2007.04.025

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2 E.K. Zholkovskij et al. / Advances in Colloid and Interface Science xx (2007) xxx–xxx

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2. General problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02.1. System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02.2. Equilibrium state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02.3. Electrokinetic regime: governing equations and interfacial boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . 02.4. Electrokinetic regime: outer boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

2.4.1. External influences and system responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02.4.2. Electroosmosis and electrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02.4.3. Streaming and sedimentation potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

2.5. Discussion on the general problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03. Kinetic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

3.1. External and internal applied field strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.2. Definitions of kinetic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

3.2.1. Disperse system conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.2.2. Two types of electrophoretic mobility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.2.3. Electroosmotic coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.2.4. Electroosmotic pressure coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.2.5. Sedimentation and streaming potential coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.2.6. Hydraulic permeability, filtration and sedimentation velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

3.3. Expressions for kinetic coefficients using volume mean quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.4. Smoluchowski limiting case for kinetic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

3.4.1. Smoluchowski limit for electrophoretic mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03.4.2. Smoluchowski limit for streaming (sedimentation) potential coefficient . . . . . . . . . . . . . . . . . . . . . . . . 0

3.5. Irreversible thermodynamic consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 04. Spherical cell approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

4.1. Governing equations for cell bulk and boundary conditions at the particle surface . . . . . . . . . . . . . . . . . . . . . . . . . 04.2. Derivation of the outer cell boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

4.2.1. Boundary condition for equilibrium potential distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 04.2.2. Boundary condition for a given applied voltage: Shilov–Zharkikh–Borkovskaya condition . . . . . . . . . . . . . . 04.2.3. Two boundary conditions for a given pressure difference: Kuwabara and Ohshima conditions . . . . . . . . . . . . 04.2.4. Boundary condition for a given external flow velocity: the first Happel condition . . . . . . . . . . . . . . . . . . . 04.2.5. Boundary condition for a given electric current density: Generalized Levine–Neale Condition . . . . . . . . . . . . 0

4.3. Expressions for kinetic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 04.3.1. Zero gravitation and pressure differences (δg=0, Δp=0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 04.3.2. Zero gravitation and zero velocity (δg=0, δu∞=0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 04.3.3. Zero electric current regime (δJ∞=0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

5. Description of the Electrokinetic Phenomena at low zeta potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.1. Method of small parameter perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

5.1.1. Obtaining the zero order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.1.2. Obtaining the first order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

5.2. Prediction of the kinetic coefficients for low zeta potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.2.1. Conductivity at low zeta potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.2.2. Electrophoretic mobility at low zeta potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.2.3. Electroosmotic and electroosmotic pressure coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.2.4. Streaming and sedimentation potential coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

5.3. Discussion on predictions from alternative models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.3.1. Why do different models give the same predictions for the zero and first order terms in expansion of conductivity by

powers of zeta potential? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.3.2. When the Levine–Neale boundary condition is valid? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.3.3. Critique of the sedimentation potential predictions based on the Levine–Neale model . . . . . . . . . . . . . . . . . 0

6. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

1. Introduction

Electrokinetic Phenomena comprise a wide class of effectsthat are observed when oppositely charged constituent phases ofa multiphase system move with reference to each other. Such amovement is provided by applying external force fields. For

Please cite this article as: Zholkovskij EK et al. Electrokinetic Phenomena in conApproach. Adv Colloid Interface Sci (2007), doi:10.1016/j.cis.2007.04.025

instance, an externally applied electric field gives rise toelectrophoresis of the charged solid phase (when the liquidphase is kept in rest) or electroosmosis of the liquid phase (whenthe solid phase is kept in rest). Gravitationally driven settling ofcharged particles in a stationary liquid and pressure driven flowof a liquid phase through a stationary porous solid medium

centrated disperse systems: General problem formulation and Spherical Cell


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(diaphragm) generate electric fields (sedimentation and stream-ing potentials, respectively).

For over a century, Electrokinetic Phenomena have beenavidly studied using different theoretical and experimentalapproaches. Theoretical analysis of Electrokinetic Phenomenadeals with a complex boundary problem intended to describe theelectric, hydrodynamic and ion concentration fields inside amultiphase system comprising a solid phase inside a continuousmedium containing solvent and ions. The major difficulties informulating and solving such a problem are connected withcomplexity and uncertainty of the geometry of a multiphasesystem. However, as shown by Smoluchowski [1–4], thesedifficulties do not exist for the case of a suspension or diaphragmhaving vanishingly thin interfacial electric double layers (EDL)formed by the opposite charges of the contacting phases. Ac-cording to Smoluchowski, kinetic coefficients, which describedifferent electrokinetic effects in such systems, are geometryindependent quantities. Consequently, for sufficiently thindouble layers, the kinetic coefficients predicted for the simplestgeometries (single spherical particle, slit channel etc.), can beused for describing a suspension or diaphragm having complexgeometry.

The Smoluchowski approximation becomes invalid when alength-scale parameter characterizing internal structure of themultiphase system (particle radius, mean inter-particle distanceetc.) is comparable with the EDL thickness (or more). For suchsystems, the kinetic coefficients describing Electrokinetic Phe-nomena depend on the internal geometry of multiphase system.For obtaining these coefficients, unavoidably, one should dealwith the complex boundary problem linked to the multiphasesystem geometry. Because of the geometrical difficulties,during a long period of time, theoretical modeling was confinedto analysis of simple geometries: an individual particle [5–19]and straight capillary [12,20–24]. Simultaneously, valuabletheoretical results, which are valid for arbitrary geometry ofmultiphase system, were obtained within the framework of theirreversible thermodynamic approach [25–29]. In these studies,a set of relationships of great generality was established be-tween the kinetic coefficients which describe different Electro-kinetic Phenomena.

For over three decades, substantial attention of researcherswas focused on modeling Electrokinetic Phenomena in con-centrated disperse systems whose behavior, at finite EDLthicknesses, can appreciably deviate from both the Smolu-chowski predictions and the predictions from the single particleand straight capillary models. The present paper is concernedwith critical analysis of the advances in this area.

1.1. Macroscopically homogeneous and isotropic medium:representative cells

As stated above, due to the complex and, strictly speaking,uncertain geometries of multiphase systems, there are manyobstacles in formulating and solving mathematical problemsintended for obtaining spatial distributions of physical (electri-cal, hydrodynamical etc.) quantities inside a multiphase system.However, usually, there is no need to know all the details of the


above mentioned distributions. Conventional experiments withmultiphase systems deal with measuring the responses of theentire system under given external influences. Such responsescan be addressed by averaging the actual distributions ofphysical quantities over the multiphase system volume.

Obtaining the system response due to external influences isdrastically simplified utilizing the concept of a macroscopicallyhomogeneous medium. According to this concept, under certainconditions, a multiphase (or, in general, multiparticle) systemcan be considered as a macroscopically homogeneous mediumwhich, at any given point, is characterized by a set of equivalentparameters called kinetic coefficients. The kinetic coefficientsare determined by averaging the local distributions of physicalquantities over representative part of the disperse system(representative cell), not over the entire volume.

There are two major conditions enabling a multiphase ormultiparticle system to be considered as a macroscopicallyhomogeneous medium:

(i) dimensions of the representative cell should be negligiblecompared to the dimensions of the whole disperse system;

(ii) the averaging conducted over the representative cellvolume leads to the same set of kinetic coefficients irre-spective of the position of the representative cell;

When the second condition is satisfied, an average over thedisperse system volume, ⟨…⟩, is always equal to the averageover the representative cell volume,⟨…⟩cell. Consequently, whena given multiphase system is considered as a macroscopicallyhomogeneous medium, for a local vector or scalar quan-tityðYAðYr Þ or f ðYr ÞÞ, the following equalities are valid

hYAðYr Þi ¼ hYAðYr Þicellh f ðYr Þi ¼ h f ðYr Þicell

ð1:1Þ

Eq. (1.1) gives a basic relationship which is widelyemployed in predicting kinetic coefficients for macroscopicallyhomogeneous systems.

In the general case, kinetic coefficients, which describe theproportionality between the components of the influence andresponse vectors, compose a tensor. In the present paper, we willconsider systems for which, all such tensors degenerate intoscalars (or, equivalently, diagonal tensors with equal diagonalelements). Such systems are referred to as the macroscopicallyhomogeneous and isotropic media.

The Spherical Cell Approach (SCA) is an approximatemethodfor addressing macroscopically homogeneous and isotropicmedia consisting of particles dispersed in a continuous phase.The SCA is based on the following major assumption: Therepresentative cell for the system of spherical particles dispersedin a continuous phase is a sphere which, in its center, contains oneof the particles surrounded by the continuous phase (Fig. 1).

The above assumption enables one to predict the kineticcoefficients by averaging electrical, hydrodynamic and ionconcentration fields over the spherical cell volume. Consequently,the boundary value problem, which should be solved for de-scribing these fields, is formulated inside the spherical cell. The



Fig. 1. Representative spherical cell model.


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latter enables one to avoid the geometrical difficulties originatingfrom the complexity of the real system geometry.

1.2. Spherical Cell Approach in physics: historical overview

In spite of heuristic and approximate nature of the SCA, itbecame a classical theoretical method which, under differentnames, is employed in different fields of physics.

The first examples of using SCA are probably dated to theend of the 19th century. That time, the Spherical Cell Approachhas been utilized for interrelating the polarizability of anindividual constituent molecule and the permittivity of amacroscopically homogeneous and isotropic dielectric. Twoequivalent relationships, which are referred to as Clausius–Mossotti [30,31] and Lorentz–Lorenz [32–34] equations, werederived by considering a spherical cell (usually called the“virtual cave”, in this context) containing one polarized mole-cule. The radius of such a cell was chosen to make equal thepolarization vector and the induced molecular dipole calculatedper unit cell volume. The induced molecular dipole was as-sumed to be proportional to the uniform component of theelectric field strength inside the cave.

In the beginning of the 20th Clausius–Mossotti–Lorentz–Lorenz equation [30–34] was employed byDebye for describingdielectric permittivity for the system of molecules withpermanent dipoles [35,36]. Later, Onsager criticized Debye's“mechanical” use of the Clausius–Mossotti–Lorentz–Lorenzequation and obtained another result based on a more consistentanalysis within the framework on the SCA [37]. The Onsagertheory was extended by Kirkwood [38] who completed the SCAwith analysis of dipole–dipole interaction of a molecule insidethe cell with the nearest neighbors.

There is another classical example of using the SCA thatrelates to the end of 19th century. It is the Maxwell expressionfor the effective dielectric permittivity of a system containingperfect dielectric spherical particles dispersed in a perfectdielectric continuous medium [39]. The Maxwell expressionrepresents the effective dielectric permittivity as a function ofthe dielectric permittivities of the constituent phases and thevolume fraction of the dispersed phase. While deriving the


Maxwell expression, the spherical cell, in its center, containsthe dispersed particle surrounded by the continuous medium(Fig. 1). The cell radius is chosen such that the ratio of theparticle to the cell volume is equal to the dispersed phasevolume fraction, ϕ. Accordingly, the particle and cell radii, aand b, satisfy the equality

a=b ¼ ϕ1=3: ð1:2Þ

TheMaxwell formula is obtained by averaging the preliminarydetermined distributions of the electric displacement and strengthvectors over the cell volume.

Utilizing the similarity between the governing equationdescribing different physical processes, a variety of expressionshaving the same structure as the Maxwell expression [39] butcharacterizing other physical properties have been derived foreffective parameters of multiphase system with the samegeometry. For example, similarity of the continuity equationswritten for the electric displacement and current density (or theheat flux) leads to the Maxwell type expression for the effectiveelectric (or thermal) conductivity. Such an expression, instead ofthe dielectric permittivities, contains electric (thermal) conduc-tivities of the constituent phases. As well, using non-stationaryversion of the continuity equation for the electric current(containing the displacement current term), Wagner [40,41]derived an analogue of the Maxwell formula which describesfrequency dependency (theMaxwell–Wagner dispersion [41,42])of the complex dielectric permittivity of a heterogeneous leakydielectric.

Aversion of the SCA, which is widely used in the Solid StatePhysics, was proposed in the 1930's by Wigner and Seitz[43,44]. The Wigner–Seitz model is employed for describing anatom incorporated into ionic crystal lattice. According to such amodel, the representative cell for the crystal is a spherecontaining an ion in the center. Energy structure of atom isdetermined by solving the Schrödinger equation subject to thespecial boundary conditions, which are set at the cell externalboundary for the wave function and its derivatives [45,46]. Thecell (Wigner–Seitz) radius is determined by equating thematerial mass density and the volume mean mass density insidethe cell.

The next wave of interest in SCA is related to the 50's of thepast century when the SCA was employed for describingmultiphase flows. However, the first work using a version of theSCA for addressing a hydrodynamic problem was publishedmuch earlier-in 1910-by Cunningham [47]. In the 50's of thepast century, three other types of the spherical cell models havebeen proposed by Simha [48], Happel [49,50] and Kuwabara[51]. The cells proposed by Cunningham, Happel and Kuwabarahave radii satisfying Eq. (1.2) whereas Simha proposed adifferent relationship. The models differ from each other due tothe differences in the hydrodynamic boundary conditionsimposed at the cell outer boundary. According to the Cunning-ham [47] and Simha [48] models, the velocity at the cellboundary should be the same as in the flow existing in theabsence of the particles. In Happel's work, the latter outercondition is assumed for the normal velocity only. Additionally,




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Happel proposed to set zero local tangent stresses at the cellboundary. The Kuwabara outer boundary condition amounts tosetting zero vorticity. In the 1970's, an original version of thehydrodynamic SCA was proposed by Kvashnin [52]. TheKvashnin outer boundary condition is that the tangential com-ponent of the velocity reaches a minimum with respect to theradial coordinate.

The Simha and Happel models were employed for addressingeffective viscosity of suspensions. Making use of the Kuwabara,Happel and Kvashnin models, both the Darcy coefficient andsedimentation velocity were predicted for mono-disperseconcentrated diaphragms and suspensions. Among the papersconcerned with the hydrodynamic SCA, one should mention theseries of publications of Smith [53–55] who adopted the Happelcell model for considering settling of poly-disperse systems.Later, Mehta and Morse [56] used a modification of theCunningham cell model for describing hydraulic permeability ofmembranes. Recently, the Cunningham–Mehta–Morse modelwas used by Starov and coworkers [57–59] who addressed aflow through a system of particles covered by porous layers.

Over the three past decades, the SCA has been extensivelyemployed in colloid and interface science. In some of thesestudies, major attention was focused on the analysis ofthermodynamic equilibrium in different colloid and polyelectro-lyte systems [60–73]. In these studies, the representative sphericalcell, whose radius b is given by Eq. (1.2), contains a chargedparticle with radius a surrounded by an electrolyte solution(Fig. 1). Considering the thermodynamic equilibrium state, theions within the cell are assumed to be distributed according to theBoltzmann law. Consequently, the equilibrium electric potentialdistribution is determined from the Poisson–Boltzmann (P–B)equation.At the particle surface, the P–Bequation is subject to theboundary condition of a given interfacial potential, charge densityor adsorption isotherm which describes the interfacial chargeregulations. The outer boundary condition sets a zero electric fieldstrength and thus reflects the overall electroneutrality of therepresentative cell. Solution of the P–B equation subject to theabove boundary conditions yields the electric potential and (usingthe Boltzmann law) ion concentration distributions inside the cell.Such distributions are employed while obtaining electrostaticcontributions into the thermodynamic potentials, osmotic pres-sure, ionic activity coefficients etc.

After the pioneering papers of Levine and Neale [74] andLevine et al. [75], a large number of theoretical studies wereconcerned with applying the SCA for describing ElectrokineticPhenomena [76–112]. Detailed discussion related to thesepublications, which are the focus of the present paper, is givenin the next sections.

1.3. Macrokinetic method for analyzing ElectrokineticPhenomena

Prior to discussing the results of applying the SCA fordescribing Electrokinetic Phenomena, in the present section, wewill give an overview of general theoretical approaches employedin electrokinetic studies. In the next section, we will consider howthe general approaches are specified while using the SCA.


In the present paper, we consider the linear ElectrokineticPhenomena for which a system response is always proportionalto the external influence. Mathematical scheme intended foraddressing the linear Electrokinetic Phenomena amounts to twoboundary value problems to be subsequently solved. The first ofsuch two problems describes the electric and ion concentrationfields in the thermodynamic equilibrium state. The second ofthe problems is formulated to address a weak departure of thedisperse system from the equilibrium state.

Formulation of the first problem includes:

– the P–B equation written in the cell bulk outside thedielectric particle;

– the interfacial boundary condition setting a given equilibri-um interfacial potential;

– the outer boundary condition which imposes zero electricpotential at infinity

As for the second boundary value problem, which describesthe disperse system under non-equilibrium conditions, a varietyof its versions have been proposed in literature. The differencesrelate to both the governing equations and the interfacialboundary conditions. For certainty, in the present paper, we willdiscuss the second problem and conduct all the necessaryderivations in terms of the most consistent and technicallysimple formalism which, more than twenty years ago, has beenindependently developed by two groups of scientists: Ohshima,Healy, White, O'Brian [14,15,17,78–83,86,87] and Shilov,Zharkikh, Borkovskaya [16,18,19,91,92]. Within the frame-work of such a formalism, both the governing equations and theinterfacial boundary conditions are written in terms ofperturbations of the electrochemical potentials of ions, δμk,that are considered as unknown functions. The value of δμk isinterrelated with the local concentration of the kth ion, Ck, andelectric potential, δφ, as

dlk ¼ d½ZkFuþ RT lnðCkÞ� ð1:3Þ

where k=1,2…, N (N number of ionic species), Zk is the kth ioncharge expressed in the Faraday units, F is the Faraday constant,and T is the absolute temperature.

The unknowns δμk are introduced in the papers of Ohshima,Healy, White, O'Brian [14,15,17,78–83,86,87] and in somepapers of Shilov, Zharkikh, Borkovskaya [18,19]. The latterauthors, in their other papers, [16,91–93], employ another set ofN unknowns, δCk

⁎ and δφ⁎ which can be interrelated with δμk ina unique manner by using the following linear relationships.

du⁎ ¼

Xk

mkdlk ðaÞ

dC⁎k ¼ Clk

RT

Xi

miðdlkZi � dliZkÞ ðbÞ

mk ¼ Clk Zk=

Xn

Z2nC

ln ðcÞ

ð1:4Þ

where Ck∞ is the kth ion concentration in the electroneutral equi-

librium solution. Since, according to Eq. (1.4),PN

k¼1 ZkdC⁎k ¼ 0,




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then Eq. (1.4) defines N independent unknowns, namely δφ⁎ andN−1 functions δCk

⁎.It should be noted that, in the general case, the Shilov–

Zharkikh–Borkovskaya unknown functions δCk⁎ and δφ⁎

(which are referred to as the “flow” concentrations andpotential, respectively [16,91–93]) do not coincide with theperturbations of the kth ion concentration and electric potential.The physical meaning of the quantities δCk

⁎ and δφ⁎ is ex-plained by considering an electroneutral electrolyte solutionwhich, being brought in contact with a given point inside theliquid phase of the disperse system, would be in thermodynamicequilibrium with the electrolyte solution in immediate vicinityof this point (Fig. 1). For the first time, the use of such anassumed solution (to which we will refer as the virtual solution)for analysis of transport phenomena was proposed by Kedemand Katchalsky [28].

When the disperse system is in thermodynamic equilibrium,the ion concentrations inside the virtual solution coincide withCk

∞ and are equal for all the points. Under irreversible con-ditions, the concentrations in the virtual solutions deviate fromCk

∞ and become functions of the position of the point withwhich the equilibrium is assumed. Simultaneously, the electricalpotential of the virtual solution deviates from the constant valuecorresponding to the actual equilibrium. According to Shilovet al. [16,91–93], the perturbation δCk

⁎ and δφ⁎ are thedeviations of the ion concentrations and the electric potential inthe virtual solutions from the concentrations and the potentialexisting in equilibrium solution prior an external influence isapplied. Considering variation of electrochemical potential ofthe kth ion in the virtual solution δμk

⁎=ZkFδφ⁎+RTδCk

⁎ /Ck∞

and substituting in the above equation the expressions for δCk⁎

and δφ⁎ given by Eq. (1.4), one obtains δμk⁎=δμk.The latter equality describes thermodynamic equilibrium

and, hence, confirms the above concept.Thus, the second boundary value problem is formulated for

obtaining the small perturbations of the electrochemical poten-tials, δμk (or, alternatively, δCk

⁎ and δφ⁎), velocity, dYu, andpressure δp which describe deviation from thermodynamicequilibrium in each local point of electrolyte being the continuousphase of the disperse system. The problem formulation containsgoverning equations attributed to the electrolyte bulk and relevantboundary conditions, namely:

– continuity equations for each of the individual ion fluxes;– the Stokes equation containing the volumetric electric forceon the right hand side;

– the continuity equations for liquid velocity;– interfacial boundary conditions of zero normal flux set for ofeach of the ions (in the reference system linked to the particles);

– the interfacial condition setting zero velocity of liquid (in thereference system linked to the particles);

– the interfacial condition of zero total force exerted on theparticles;

– the outer boundary conditions.

The above equations and boundary conditions form a closedproblem formulation. In Sections 2 and 4, we rederive all the


above listed equations and boundary conditions for the case of adisperse system of the general type (Section 2) and specify themfor the Spherical Cell Approach (Section 4).

Remarkably, the above outlined second problem formulationdoes not contain the Poisson and Laplace equations for theperturbation of electric potential outside and inside the particleas well as the electrostatic interfacial boundary conditions.Clearly, using the solution of the above problem, both theequations for the electric potential subject to the electrostaticinterfacial boundary conditions can be explicitly written andsolved. However, for addressing linear Electrokinetic Phenom-ena, such a step is not required.

In the above list of governing equations and boundary con-ditions, we did not specify the outer boundary conditions. For thecase of a disperse system surrounded by infinite electrolytesolution, the outer boundary conditions impose uniform electric,hydrodynamic and ion concentration fields at infinity. Forexample, inspecting Eq. (1.3), the boundary condition for δμk,is written at infinity as dlk=ZkFY � d

YE d Yr where d

YE is the

external uniform electric field.

1.4. Spherical Cell Approach for Electrokinetics

In all the papers describing Electrokinetic Phenomena withthe help of the SCA[74–112], the derivations have commongeneral structure. The spherical cell shown in Fig. 1 is con-sidered, and the radii, a and b, are assumed to satisfy Eq. (1.2).For describing the hydrodynamic, electric and ion concentrationfields inside the cell bulk, the authors use two boundary valueproblems discussed in Section 1.3. For the case of the SCA, toformulate each of the above two problems in a closed form, it isnecessary to set boundary conditions at the outer cell boundary.

Imposition of the outer boundary conditions is associated withcertain difficulties since the spherical cell boundary does not existin the real system. However, for the first problem, which, asdiscussed in Section 1.3., describes disperse system in theequilibrium state, the difficulties are resolved by all the authors,who deal with SCA for equilibrium state [60–112], in a commonmanner by setting total electroneutrality of the cell. As for thesecond problem,which, according to Section 1.3., describes weakdeviation from equilibrium, a correct choice of the outer boundaryconditions is the most difficult stage in the formulating theproblem. While, in the literature, there is a consensus regardingthe governing equations, interfacial boundary and the outerboundary conditions for the first, equilibrium, problem, the formsof the outer boundary conditions for the second, non-equilibrium,problem are the focus of an intensive discussion.

While formulating the second problem, two groups of con-ditions are required to be set at the cell outer boundary. The first ofthem comprises the electrochemical outer conditions which areset to interrelate the perturbations of the electrochemical po-tentials, δμk, and the applied uniform electric field strength d

YE .

The second group of outer boundary condition, the hydrodynamicone, should impose a macroscopically uniform hydrodynamicflow through the macroscopically homogeneous and isotropicmultiphase system. Let us now consider different versions ofouter boundary conditions employed in the literature.




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1.4.1. Electrochemical outer conditions: Levine–Neale andShilov–Zharkikh–Borkovskaya versions

Two versions of the electrochemical boundary conditions areused by different authors. The first onewas proposed in the paperof Levine and Neale [74] and Levine et al. [75]. The originalform of the Levine and Neale condition can be written as

Yn dY∇du ¼ � Yn d d

YE

� �at the cell outer boundary ð1:5Þ

where Yn is the outward normal vector.It should be noted that, in Refs. [74,75], the analysis was

based on the approach proposed in the paper of Henry [6] wherethe electrophoretic velocity of an individual particle was pre-dicted using certain assumptions. In Henry's paper, the electricpotential perturbation around the particle is assumed to satisfythe Laplace equation. For the case of perfect dielectric particles,the interfacial boundary conditions assumed by Henry amountsto setting a zero normal perturbation of the electric field strength.Exactly the same assumptions were used by Levine and Neale[74] whose expression for the electrophoretic mobility is ageneralization of Henry's result for the case of concentratedsuspensions.

Strictly speaking, Henry's assumptions are valid for the case ofa particle having zero dielectric permittivity which is a physicalimpossibility. However, paradoxically, the Henry theory gives acorrect prediction for the first order term in the Taylor expansionof the electrophoretic mobility by powers of the interfacial po-tential (Debye approximation). This result of Henry was con-firmed in the above cited Refs. [14–18,78–83,86,87,91–93]using rigorous perturbation methods without the Henry assump-tion. In the above references, while considering the Debyeapproximation for electrophoretic mobility, the perturbation ofthe electrochemical potential, δμk, (not the electric potentialperturbation) satisfies Henry's assumptions. The authors of Refs.[78–83,107–109], who dealt with the SCA and used the rigorousperturbation method, rewrote the original Levine–Neale condi-tion (1.5) by replacing δφ with δμk /ZkF, as

Yn dY∇

dlkFZk

¼ � Yn d dYE

� �at the cell outer boundary ð1:6Þ

Hereafter, following Refs. [78–83,107–109], we will refer toEq. (1.6) as the Levine–Neale boundary condition.

Alternative form of the electrochemical outer boundaryconditions, to which we will refer as the Shilov–Zharkikh–Borkovskaya conditions, was proposed in Refs. [91–93]. For thecase of a suspension being in contact with solutions having equalion concentrations, the original form of the Shilov–Zharkikh–Borkovskaya boundary condition was written in terms of the“flow” concentration and potential, δCk

⁎ and δφ⁎, defined in thepresent paper by Eq. (1.4). These conditions take form

du⁎ ¼ �ðdYE d Yr ÞdC⁎

k ¼ 0 at the cell outer boundary:

ð1:7Þ

Combining Eqs. (1.3) and (1.4), one can show that, atδCk

⁎=0, δφ⁎=δφ and δCk=0. The latter enables one to


represent the Shilov–Zharkikh–Borkovskaya boundary condi-tion as

du ¼ �ðdYE d Yr Þ; dCk ¼ 0 at the cell outer boundary: ð1:8ÞSuch an equivalent form of the Shilov–Zharkikh–Borkovs-

kaya conditions was used by Ding and Keh [106].In the next sections, we will conduct our analysis in terms of

the perturbation of the electrochemical potential, δμk discussed inSection 1.3. For such a case, the Shilov–Zharkikh–Borkovskayaconditions can be represented in equivalent form by combiningEqs. (1.4) and (1.7), as

dlkFZk

¼ �ðdYE d Yr Þ at the cell outer boundary ð1:9Þ

This is the form of the Shilov–Zharkikh–Borkovskayacondition employed in the papers of Carrique et al. [107–109].

When the disperse system is placed between two electrolytesolutions having equal ionic concentrations, all the three formsof the Shilov Zharkikh conditions given by Eqs. (1.7)–(1.9) arecompletely equivalent. In the analysis given in the next sectionswe will use Eq. (1.9).

In the literature, some authors use the Levine –Nealeboundary conditions [74–90,112], as given by Eqs. (1.5) or(1.6), other authors use the Shilov-Zharkikh-Borkovskayaconditions [91–103], as presented by Eqs. (1.7) (1.8) or (1.9).In some publications [99–111], the authors use both conditions tocompare the final results. One might add that that in thecorresponding papers, the Levine–Neale condition, Eqs. (1.5) or(1.6), was proposed without any derivation or clarifyingcomments. As for the Shilov–Zharkikh–Borkovskaya condition,which is given by either of Eqs. (1.7)–(1.9), it was derived by theauthors of Refs. [91–93] using irreversible thermodynamics.

1.4.2. Hydrodynamic outer conditions: Happel and Kuwabaraversions

In the papers concerned with Electrokinetic Phenomena, twotypes of the hydrodynamic boundary conditions are used,namely Kuwabara and Happel boundary conditions.

In the studies of Happel [49,50], where the Darcy coefficienthas been obtained for a system of solid spheres, it was proposedto use the following boundary conditions

Yn d dYu ¼ Yn d dYul at the cell outer boundary ð1:10Þ

ðd r¼ d Yn Þ � Yn ¼ 0 at the cell outer boundary; ð1:11Þ

where dYul is the velocity of a uniform flow infinitely far fromthe disperse system; dYu and d ¯r are the local velocity andviscous stress tensor. Eqs. (1.10) and (1.11) are the first andsecond Happel boundary conditions, respectively. According toHappel's interpretation, both the above conditions reflect thefact that any representative cell does not interact with other cellsmaking up the disperse system. Consequently, the first Happelcondition, Eq. (1.10), means that, in the reference system linkedto the liquid at infinity (i.e., dYul ¼ 0), the cell outer boundaryis impermeable for the liquid. The second boundary condition




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of Happel, Eq. (1.11), sets zero local tangential stresses at thecell outer boundary. Happel interpreted it as a condition of azero friction between the cells.

Alternative boundary condition, which is widely employedin Electrokinetics, has been proposed by Kuwabara [51] whoused this condition for obtaining the Darcy coefficient for therandomly distributed cylinders and spheres. According toKuwabara, the flow vorticity is zero at the cell outer boundary

Y∇� dYu ¼ 0: at the cell outer boundary ð1:12Þ

In his paper, considering system of cylinders, as an example,Kuwabara gave some qualitative arguments in support of (1.12).The essence of these arguments is that a deviation from (1.12)leads to variations of the applied pressure along the externalsides of the disperse system.

It should be noted that Eq. (1.6) becomes equivalent toEq. (1.9), and Eqs. (1.10) and (1.11) do not contradict Eq. (1.12)only for the case when, in the outer part of the cell (i.e. outsidethe particle), the electrochemical potential gradient and liquidvelocity field are uniform. In the presence of dispersed phase,such uniformities never exist. Hence, except for the asymptoticbehavior corresponding to the limiting case ϕ→0, using thecontradictory boundary conditions should lead to differentresults. Thus, an important question arises: Which boundaryconditions lead to correct predictions?

1.5. Criteria of model validity: Onsager and Smoluchowskiprinciples

There are two criteria which could be helpful in making achoice of a correct boundary condition. The first one is asso-ciated with Onsager's principle according to which, for properlychosen thermodynamic forces and fluxes, the matrix of thekinetic coefficients describing electrokinetic effects should besymmetric [113,114]. In particular, Onsager's principle definesa certain relationship between the electrophoretic mobility andsedimentation potential [25–29]. Comparing the relationshipsbetween electrophoretic mobility and sedimentation potentialobtained using a model with that defined by Onsager's prin-ciple, a conclusion can be made about the model validity.

As the second criterion of the model validity, we refer to theSmoluchowski principle. According to the Smoluchowskiprinciple, at a constant zeta potential, when the Debye param-eter κ given by

j2 ¼F2X2

Z2k C

lk

eRTð1:13Þ

infinitely increases, both the electrophoretic (electroosmotic)mobility and the sedimentation (streaming) potential shouldapproach finite values (Smoluchowski limits) independent ofthe disperse system geometry [1–4]. In Eq. (1.13), ε is thedielectric permittivity of the electrolyte. For the case ofelectrophoretic (electroosmotic) mobility and streaming poten-tial, the independency of geometry means the independency ofthe dispersed phase volume fraction, ϕ. For the sedimentation


potential, the independency of geometry means proportionalityto ϕ.

Thus, it should be expected, that, using boundary conditions(1.6) and (1.9) leads to different predictions except for the caseof low volume fractions. The same can be concluded regardingboundary conditions (1.10) (1.11) (1.12). To choose a correctmodel, these different results should be examined on the basisof the Onsager and Smoluchowski principles.

1.6. Unexpected results and contradictions

1.6.1. Common and different predictionsAs stated above, similar predictions from Levine–Neale and

Shilov–Zharkikh–Borkovskaya models, as well as Happel andKuwabara models, can be expected for low volume fractionsonly. Analysis of results obtained in [74–112] shows that, whiledecreasing the volume fraction, the results obtained for a samequantity but using the different models do approach each other.When the volume fractions are not low, substantial differencesin the predictions based on the Levine–Neale and Shilov–Zharkikh–Borkovskaya conditions are reported for the electro-phoretic mobility and sedimentation potential [106–111]. Thesame is true in the case of Happel and Kuwabara conditions.

Surprisingly, for the disperse system conductivity, at low zetapotentials, some authors, who reportedly used different models,obtained similar predictions. Remarkably, such predictionsagree even for rather high volume fractions. For example,Ohshima [81,86] (using the Levine–Neale boundary condi-tions) and Carrique et al. [108,109] (using the Shilov–Zharkikh–Borkovskaya boundary conditions) obtained expres-sions containing the identical zeroth and first order terms in theTaylor expansion of the disperse system conductivity in powersof the normalized zeta potential. Similarly, numerical data ofCarrique et al. [108] confirm that, for sufficiently low zeta-potentials, boundary conditions (1.6) or (1.9) lead to the samevalue of the disperse system conductivity. However, at higherzeta- potentials, the numerical predictions are different whenusing boundary conditions (1.6) and (1.9).

Other authors using different boundary condition (forexample, Ding and Keh [106]), report substantially differentpredictions for conductivity obtained with the help of differentmodels at intermediate volume fractions.

1.6.2. Onsager's principleAccording to Shilov et al. [91], the electrophoretic mobility

and the sedimentation potential coefficient obtained using theLevine–Neale boundary conditions do not satisfy the Onsagerprinciple. Consequently, Shilov et al. [91] gave a convincingreason for using any of boundary conditions (1.7)–(1.9) ratherthan Eqs. (1.5) or (1.6). Their derivation of boundary condi-tion (1.7) was based on the requirement of validity of Onsager'sprinciple. Accordingly, it was shown that, using boundarycondition (1.7), such a requirement is satisfied whereas, usingEq. (1.5), it is not satisfied.

Subsequently, Ohshima published an irreversible thermody-namic analysis [86] intended to prove different concept. Ohshima[86] derived an irreversible thermodynamic relationship which,




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due to a factor depending on the volume fraction, differs from therelationship which interrelates electrophoretic mobility andsedimentation potential for a single particle. According to [86],Ohshima's expressions for the electrophoretic mobility [78,86]and the sedimentation potential [81,86], whichwere derived usingthe Levine–Neale model, satisfy Ohshima's new derived versionof the Onsager relationship.

It should be noted that, by its phenomenological nature,irreversible thermodynamic approach does not deal with internalstructure of a system. In other words, irreversible thermodynamicrelationships should have exactly the same form for one, two ormany particles. The latter requirement is not satisfied in theOhshima analysis [86].

Carrique et al. [107] presented the third concept. Accordingto observation of Carrique et al., the Onsager relationshipbetween sedimentation potential and electrophoretic mobilityhas the same form as for the case of one particle provided thatthe sedimentation potential is given by Ohshima's expressionderived from the Levine–Neale model [80,86] and theelectrophoretic mobility is given by the result of Carrique etal. obtained by using the Shilov–Zharkikh–Borkovskaya model[108]. Note that Carrique et al. [108] and Ohshima [78,86]deduced different expressions for electrophoretic mobility.

1.6.3. Smoluchowski principleAll the reported results obtained using Happel boundary

conditions are in a contradiction with the Smoluchowski prin-ciple. A more complex situation takes place when Kuwabaraboundary conditions are used. Below, we discuss the resultswhich are obtained by combining the Kuwabara-boundary con-dition with either Levine-Neale or Shilov–Zharkikh–Borkovs-kaya conditions.

Electrophoretic mobility obtained by Levine and Neale [74]and Ohshima [78,86] using the Levine–Neale model satisfy theSmoluchowski principle. However, using the same model, thederived expressions for sedimentation potential (Levine et al.[75], Ohshima [80,86]), for a vanishingly thin double layer,deviate from the linear dependency on the volume fraction and,hence, do not satisfy the Smoluchowski principle.

Apparently, the electrophoretic mobility predicted using theShilov–Zharkikh boundary conditions does not satisfy theSmoluchowski principle. The latter can be clearly seen con-sidering the curves presented byDing andKeh [106] andCarriqueet al. [108] who obtained the electrophoretic mobility with thehelp of the Shilov –Zharkikh approach.

Dukhin et al. [96] gave a comprehensive explanation why,for a vanishingly thin double layer, the Shilov–Zharkikh–Borkovskaya model yields an electrophoretic mobility thatdepends on volume fraction. The authors of Ref. [96] suggestedto distinguish between two types of electric field strengthswhich are referred to as the external and average electric fields.Consequently, using two different definitions of the fieldstrength, two different electrophoretic mobilities can beintroduced. Dukhin et al. [96] indicated that only one of thesemobilities should satisfy the Smoluchowski principle. Anothermobility should satisfy a modified version of the Smoluchowskiprinciple [12,115]. Accordingly, the mobility obtained using


Shilov–Zharkikh–Borkovskaya model satisfies such a modifiedSmoluchowski principle.

It should be noted that, in the literature, the Smoluchowskiprinciple is widely used for testing the predictions of theelectrophoretic mobility. At the same time, in the studies dealingwith sedimentation potential [75,80,86,107], relevant tests aremissing except for ref. [75]. Importantly, in Refs. [75,80,86,107]using the Levine–Neale model, all the derived expressions forsedimentation potential can be shown to violate the Smolu-chowski principle. The authors of ref. [75] analyzed theirexpression for sedimentation potential using a “weaker” versionof the Smoluchowski principle. The criterion used in ref [75] isthat the sedimentation potential should approach the Smolu-chowski expression for an individual particle, i.e., when both thedouble layer thickness and volume fraction approach to zero. Thesedimentation potential predicted in [75] satisfy such a “weaker”principle but do not satisfy the general Smoluchowski principleaccording to which, at zero double layer thickness, the sedi-mentation potential is proportional to the volume fraction for allvolume fractions.

1.7. Objective and structure of the paper

The above survey shows complex and contradictory situationsexisting in the literature concerned with theoretical study ofElectrokinetic Phenomena in concentrated disperse systems.Sometimes, unexpectedly, the predictions based on differentmodels give the same results. For each model, some predictionsare reported to contradict the basic principles presented abovewhereas other predictions from the same models satisfy theprinciples. The situation is especially complicated because, as itbecomes clear from the literature analysis, different authors havedifferent understanding of the Smoluchowski and Onsagerprinciples.

It becomes quite pertinent to revisit the theory of Electroki-netic Phenomena in concentrated suspensions by primarilyfocusing on the boundary conditions employed at the cell outerboundary.

In the present paper, we start with the general problemformulation which describes Electrokinetic Phenomena in amacroscopically homogeneous and isotropic disperse system(Section 2). We will formulate all the governing equations andboundary conditions that describe such a system of the generaltype. In Section 3, we present a rigorous definition of the kineticcoefficients and rederive the Onsager and Smoluchowskiprinciples. Using the general formulation given in Section 2,in Section 4, we develop an equivalent formulation in terms ofthe Spherical Cell Approach and present an algorithm forobtaining the kinetic coefficients defined in Section 2. Finally,we consider the particular case of low zeta potential andcompare the results reported in the literature with the newlyderived results (Section 5).

2. General problem formulation

In the present section we formulate the general problemindented for addressing Electrokinetic Phenomena in concentrated




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disperse systems. The proposed analysis is based on theapproaches developed independently by Ohshima, Healy, White,O'Brian [14,15,17,78–83,86,87] and Shilov, Zharkikh, Borkovs-kaya [16,18,19,91–93]. Below, we rederive the governingequations and the interfacial boundary conditions, and specifythe outer boundary conditions for describing concentrated dispersesystems of a general type.

2.1. System description

Let us consider a layer of the thickness H, of a dispersesystem. The layer is built up by spherical solid dielectricparticles of equal radii, a, and equilibrium interfacial potentials,ζ. The layer separates two semi-infinite compartments contain-ing the same electrolyte solutions (Fig. 2). The disperse systemis considered to be the macroscopically homogeneous andisotropic medium, i.e., it satisfies the requirements formulatedin Section 1.1. Consequently, for such a disperse system, onecan single out a representative cell whose dimension, b, is muchless than the layer thickness, H.

We analyze the electrically, gravitationally, and pressuredriven motions of the constituent phases (solid particles andliquid electrolyte solution) for two groups of effects:

(i) solid phase is motionless (electroosmosis, streamingpotential, electro-viscous effect etc);

(ii) liquid phase does not move at infinity (electrophoresis,sedimentation potential, electrically hindered settling etc.)

We refer to a disperse system being in the states (i) or (ii) asthe diaphragm or the suspension, respectively.

Each of the particles forming the disperse systems is acted bythe hydrodynamic and electric forces that are computed byintegrating the viscous and Maxwell stress tensors over theparticle surface. Additionally, the particles can be acted byexternal forces. For example, the particles of settling suspen-sions are driven by gravitational force. In the case of a pressuredriven flow through a diaphragm, an external force is applied tothe particles from the side of the apparatus walls to compensatethe externally applied pressure difference and thus maintain theparticles within the stationary layer.

Fig. 2. Disperse system layer.


Analyzing the effects, which belong to any of the abovementioned two groups, we use a reference system linked to theparticles, thus setting their velocities to be zero. For the diaph-ragm, using such a reference system directly leads to a de-scription of the effects. For the suspension, the final expressionspreviously obtained for the diaphragm should be reconsideredby accounting for the change in the reference system. Such asimple method of describing behavior of suspensions (byreconsidering the results obtained for the diaphragms) is validwhen an influence of the apparatus walls on the suspensionbehavior is negligible. Accounting for the role of walls can leadto the effect of the hydrodynamic dispersion which is widelydiscussed in the literature [116–118]. In the present paper, wewill ignore such effects with the understanding that they cansubstantially affect the behavior of suspensions.

2.2. Equilibrium state

As stated in Section 1.3, prior to considering the problem foraddressing the disperse system under external applied fields, itis necessary to analyze the equilibrium state. We considersystem containing N ions. In thermodynamic equilibrium, thekth-ion concentration, C eq

k ðYr Þðk ¼ 1; 2 N ;NÞ, is distributedaccording to the Boltzmann law

Ceqk ðYr Þ ¼ Cl

k exp �WðYr ÞFZk=RT½ � ð2:1Þ

where Ck∞ is the ionic concentration in the bulk of the

electroneutral solution equilibrated with the disperse systemXk

ZkClk ¼ 0 ð2:2Þ

The equilibrium electric potential, WðYr Þ, is defined withreference to the bulk of equilibrium electrolyte solution whereCk

eq =Ck∞. Distribution of WðYr Þ is determined from the P–B

equation

∇2WðYr Þ ¼ �Fe

Xk

ZkClk exp �WðYr ÞFZk=RT½ � ð2:3Þ

subject to the boundary conditions

WðYr Þ ¼ f at the particle surfaces ð2:4Þ

WðYr ÞY0 at infinity ð2:5ÞIn the general case, the ith particle interfacial electric

potential depends on the positions of the all the particles, onelectrolyte concentrations and on the parameters describinginterfacial charge separation (the constants of interfacial groupdissociation, adsorption etc.). In the present paper, we assumethat all the particles are under identical conditions and,therefore, bear equal interfacial potentials,.

Thus, the P–B equation (2.3) subject to boundary conditions(2.4) and (2.5) gives the first, equilibrium, problem discussed inSection 1.3.




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2.3. Electrokinetic regime: governing equations and interfacialboundary conditions

Now we consider the second problem which, as discussed inSection 1.3, is intended for addressing the linear ElectrokineticPhenomena. It is convenient to formulate this problem in termsof the linear perturbation analysis which amounts to two keysteps: each of the unknown functions (ionic concentrations,electric potential, velocity and pressure) is represented as a sumof the equilibrium value (Section 2.2) and a perturbationdescribing deviation from the equilibrium due to the appliedexternal influences and (ii) all the necessary relationships arelinearized with respect to the perturbations.

Thus, we represent the unknown functions, as

Y∇lk ¼ Y∇ l eqk þ dðY∇lkÞ ¼ Y∇dlk ðaÞ u ¼ ueq þ du ¼ Wþ du ðbÞ

Ck ¼ Ceqk þ dCk ðcÞ p ¼ peq þ dp ðdÞ

Yu ¼ Yu eq þ dYu ¼ dYu ðeÞ Yjk ¼ Y

jeq

k þ dYjk ¼ d

Yjk ðf Þ

ð2:6Þwhere the superscript “eq” signifies a value attributed to theequilibrium state. While writing Eq. (2.6), we took into accountthat

Y∇l eqk u0;Yu

equ0;Yjeq

k ¼ 0 and ueq ¼ W.Let us now consider how substitution (2.6) reduces the

general expressions for the kth ion flux. For an ideal electrolytesolution, expression for the kth ion flux,

Yjk , can be represented as

Yjk ¼ � Dk

RTCk

Y∇lk þ YuCk ; ð2:7Þ

where Dk, is the kth ion diffusion coefficient. On the right handside of Eq. (2.7), the first and second terms describe thecontributions of electro-diffusion and convection, respectively.Combining Eqs. (2.6) and (2.7) and retaining the linearperturbation terms only yield

dYjk ¼ � Dk

RTCeqkY∇dlk þ Ceq

k dYu ð2:8Þ

Note, that in Eq. (2.8), the concentrations Ceqk ðYr Þ are

independently obtained by solving the problem given byEqs. (2.3)–(2.5) and applying condition (2.1).

Using Eq. (1.3), the linear perturbation of the electrochem-ical potential, δμk, is interrelated with variations of the electricpotential, δφ, and kth ionic concentration δCk, as

dlk ¼ ZkFduþ RTdCk

Ceqk

ð2:9Þ

where we retained the linear perturbation terms, only. Eq. (2.9)can be used to derive a convenient expression for the perturbationof the body force acting per unit volume of the electrolytesolution, which appears in the Stokes equation

dð�Y∇p� ρY∇uþ ρL

Yg Þ¼�Y∇dp�ρeqY∇du�dρY∇Wþ ρLd

Yg

ð2:10Þwhere dYg is the variation of the gravity acceleration, Yg;ρL isthe liquid mass density; ρeq and δρ are, respectively, the


equilibrium value and perturbation of the local electric chargedensity

ρeq ¼ FXk

ZkCeqk

dρ ¼ FXk

ZkdCk

ð2:11Þ

Let us now introduce the “effective” pressure variation, δΠ,using the substitution proposed by Shilov et al. [16,91–93]

dp ¼ dPþ ρLdYg d Yr þ

Xk

ðCeqk � Cl

k Þðdlk � ZkFduÞ ð2:12Þ

Combining Eqs. (2.1) (2.2) and (2.9)–(2.12) we arrive at thefollowing expression for the variation of the body force givenby Eq. (2.10)

dð�Y∇p� ρY∇uþ ρm

Yg Þ ¼ �Y∇dP�Xk

ðCeqk � Cl

k ÞY∇dlk

ð2:13ÞThus, Eqs. (2.8) and (2.13) give expressions for perturbations

of the kth ion flux and bulk force, respectively. Below, we useEqs. (2.8) and (2.13) for obtaining governing equations and in-terfacial boundary conditions that describe the spatial distributionsof the perturbations of the electrochemical potentials,dlkðYr Þ,effective pressure, djðYr Þ; and velocity dYuðYr Þ.

We confine our analysis to the case of a strong electrolytesolution thereby assuming complete dissociation of solutemolecules. Detailed discussion of the governing equationsdescribing electrolyte of the general type are given in Refs.[119–121]. For the case of strong electrolyte, the continuityequation for the kth ion flux perturbation is written as

Y∇ d dYjk ¼ 0 ð2:14Þ

Substituting Eq. (2.8) into Eq. (2.14) leads to

Y∇ d � Dk

RTCeqkY∇dlk

� �þ dYu d

Y∇Ceqk ¼ 0 ð2:15Þ

Since k=1, 2,…, N, relationship (2.15) gives N governingequations.

Let us now take into account that Eq. (2.13) gives aperturbation of the right hand side of the Stokes equation.Hence, the Stokes equation is represented in the form

gY∇� Y∇� dYu ¼ �Y∇dP�

Xk

ðCeqk � Cl

k ÞY∇dlk ð2:16Þ

where η is the electrolyte solution dynamic viscosity. Eqs. (2.15)and (2.16) are completed by continuity equation for the velocityperturbation

Y∇ d dYu ¼ 0 ð2:17Þ

Thus, relationships (2.15)–(2.17) give N+2 equations todetermine the following N+2 unknown functions dYuðYr Þ;dlkðYr Þ and djðYr Þ, that describe deviations of the correspondingquantities (respectively, velocity, the kth ion electrochemicalpotential and the effective pressure) from their values in the




ARTICLE IN PRESS

thermodynamic equilibrium state. The equilibrium concentrations,Ceqk ðYr Þðk ¼ 1; 2; N ;NÞ, which are represented in Eqs. (2.15)–

(2.17), describe the ion concentration distributions at theequilibrium state and are preliminary obtained by substitutinginto Eq. (2.1) the solution of the P–B equation (2.3) subject toboundary conditions (2.4) and (2.5).

Eqs. (2.15)–(2.17) are subject to the boundary conditions atboth the particle surfaces and the external boundaries of thedisperse system. Restricting the discussion to cases where allthe particles move with equal velocities, it is convenient to linkthe reference system to the particles. Using such a referencesystem, a hydrodynamic boundary condition takes the form:

dYu ¼ 0 at the particle surface: ð2:18ÞFor the case of perfect dielectric particles, their surfaces are

impermeable for ions. Using Eqs. (2.8) and (2.18), the latteryields another group of boundary conditions given by

Yn dY∇dlk ¼ 0 at the particle surfaces; ð2:19Þ

where Yn is the outward normal vector.

2.4. Electrokinetic regime: outer boundary conditions

2.4.1. External influences and system responsesThe boundary conditions set at the disperse system external

boundaries can describe both the external influences and themeasured responses. To define the externally imposed influencesand the system responses, we will consider two hypotheticalparallel planes AA and BB chosen sufficiently close to thedisperse system so that one can ignore the electric and hydraulicresistances of the gaps between the planes and the dispersesystem (Fig. 2). At the same time, similarly to H, the gapthickness, d, should be much bigger than the linear dimension ofthe disperse system representative cell, b. Additionally, the gapthickness is assumed to be mach bigger than the equilibriumdouble layer thickness, κ−1 where κ2 is given by Eq. (1.13).Thus the gap thickness, d, should satisfy the following inequal-ities

b b bd b bHjdN N1

�

We consider disperse systems satisfying conditions of ma-croscopically homogeneous and isotropic medium formulatedin Section 1.1. For such a case, when the above inequalities arevalid, inside the adjacent compartments at distance d (or bigger)from the disperse system, all the distributions becomedependent on the coordinate z, only (Fig. 2). Consequently,each of the planes bears a constant electric potential (φA andφB, respectively) and pressures ( pA and pB, respectively). Also,at each of the planes, the liquid velocity and electric currentdensity are constant vectors normal to the planes.

Each of the ionic species is assumed to have equal con-centrations, Ck

∞, in different compartments separated by the layerof the disperse system, Hence, at each of the planes AA and BB(Fig. 2), the electrochemical potential differs from its value at


equilibrium due to the non-zero value of the electric potential.Thus, setting δCk=0 in Eq. (2.9) leads to the following boundaryconditions

dlkðYr Þ ¼ dlAk ¼ duAZkF at plane AAdlkðYr Þ ¼ dlBk ¼ duBZkF at plane BB

Each of the planes, is also assumed to be at a given pressure.Consequently, using the definition of the “effective” pressurevariation, δΠ, given by Eq. (2.12) and taking into account that,in an electroneutral solution, Ck

eq =Ck∞, the following boundary

conditions are valid

dPðYr Þ ¼ dPA ¼ dpA � ρLdgzA at plane AA;dPðYr Þ ¼ dPB ¼ dpB � ρLdgzB at plane BB;

where zA and zB are z coordinates of the planes AA and BB.All measurable quantities do not depend on the reference

point for the pressure and the electric potential, hence, theyalways depend on the potential (Δφ=δφA− δφB) and/orpressure (Δp−δpA−δpB) differences. Thus, the above condi-tions can be rewritten as

dlAk � dlBk ¼ DuZkF; ð2:20Þand

dPA � dPB ¼ Dpþ ρLHdg; ð2:21Þwhere we took into account that zB− zA=H (Fig. 2).

Employing the assumptions formulated in the beginning ofpresent Section 2.4.1, both the local liquid velocity and currentdensity should be uniform at planes AA and BB. Since boththese vectors satisfy the continuity equation, it is sufficient to setthe velocity, dYul, and the current density, d

YJl, at one of the

planes. Thus, for velocity we obtain

dYuðYr Þ ¼ dulYiz at the planes AA or BB; ð2:22Þ

Similar condition written for the current density gives

FXk

ZkdYjk ¼ dJl

Yiz at the planes AA or BB:

Substituting Eq. (2.8) into Eq. (2.22), taking into accountthat, at the planes, Ck

eq =Ck∞ and using Eq. (2.2) one obtains the

following condition

� FRT

Xk

DkZkClkY∇dlk ¼ dJl

Yiz at the planes AA or BB:

ð2:23ÞIt should be noted that, while addressing different electro-

kinetic effects, some of relationships (2.20)–(2.23) define dif-ferent types of the imposed influences which are considered asgiven quantities. Such relationships are used as boundary con-ditions. Solving Eqs. (2.15)–(2.17) subject to such boundaryconditions one obtains the distributions δμk ( r

→) (k=1, 2,…, N),dYuðYr Þ and djðYr Þ. Substituting these distributions into otherrelationships from Eqs. (2.20)–(2.23) one can determine thesystem responses. Next, considering classical Electrokinetic




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Phenomena (electroosmosis, electrophoresis, sedimentationand streaming potential) we give examples of using some ofEqs. (2.20)–(2.23) as the boundary conditions and as the re-lationships for obtaining the system responses.

2.4.2. Electroosmosis and electrophoresisClassical electroosmosis and electrophoresis are observed

under the conditions of zero pressure difference and gravitation(Δp=0 and δg=0). The applied potential difference, Δφ, is theexternal influence. Hence, electrochemical boundary conditionsfor electroosmosis and electrophoresis are given by Eq. (2.20).The hydrodynamic boundary condition is obtained fromEq. (2.21)to take the form

dPA � dPB ¼ 0 ð2:24ÞChoosing the boundary conditions of zero velocity at the

particle surface, Eq. (2.18), we link the reference system to thesolid phase. The electroosmotic velocity d

Yueo coincides with

dulYiz represented in Eq. (2.22). While considering electropho-

resis, the particle (electrophoretic) velocity, dYueph, is opposite to

dYul.To obtain both the electroosmotic and electrophoretic

velocities (dueo and dYueph) at a given applied potential

difference, Δφ, one should solve Eqs. (2.15)–(2.17) subjectto boundary conditions (2.18)–(2.20). Having solved theproblem one obtains the velocity distribution, dYuðYr Þ, and,hence, the electroosmotic and electrophoretic velocities usingEq. (2.22), as

dYueo ¼ dYul ¼ dYuðYr ÞdYueph ¼ �dYul ¼ �dYuðYr Þ at the planes AA or BB:

ð2:25ÞHaving solved the problem given by Eqs. (2.15)–(2.20), one

can substitute the obtained distribution, dlkðYr Þ, into Eq. (2.23)to derive an expression for the electric current density, δJ∞,passing through the disperse system. Note that, due to totalelectroneutrality, the electric current density δJ∞ is an invariantwith respect to changes in the reference system.

2.4.3. Streaming and sedimentation potentialsStreaming and sedimentation potentials are measured under

the condition of zero electric current density (δJ∞=0). Undersuch a condition, both the streaming and sedimentationpotentials are measured as the electric potential difference Δφbetween the planes AA and BB. The electrochemical boundaryconditions used for obtaining the streaming and sedimentationpotentials are common for both the cases. This group includesEq. (2.20) and a specified version of Eq. (2.23) which takes theform

dJl ¼Xk

DkZkClkY∇dlk ¼ 0 at the planes AA or BB

ð2:26Þ

The hydrodynamic boundary condition takes slightly differentforms for the cases of the streaming and sedimentation potentials.


Classical streaming potential is observed in the absence ofgravity (δg=0) under applied pressure difference, Δp, when anelectrolyte solution flows through a diaphragm. Using Eq. (2.21)we conclude

dPA � dPB ¼ Dp ð2:27ÞThus, Eq. (2.27) gives the required additional boundary

condition for solving the streaming potential problem.Sedimentation potential is observed under the action of both

gravitation and pressure difference. Expression for the pressuredifference can be obtained considering a balance of forcesexerted to a layer of the disperse system and taking into accountthat the total electric force is zero due to electroneutrality.Equating the gravitation force and the force due to the pressuredifference yields

Dp ¼ � ρpϕþ ð1� ϕÞρL�Hdg� ð2:28Þ

where ρp is the particle mass density. Combining Eqs. (2.21)and (2.28) leads to the boundary condition which is required forsolving the sedimentation potential problem

dPA � dPB ¼ �ðρp � ρLÞϕHδg ð2:29Þ

Thus, for a given pressure difference, Δp, obtaining thestreaming potential requires solving equation set (2.15)–(2.17)subject to boundary conditions (2.18–2.20) (2.26) and (2.27).Finally, the electric potential difference Δφ is determined fromthe algebraic equations which results from the boundaryconditions. Using the same derivations with boundary condition(2.29) instead of Eq. (2.27) one obtains the sedimentationpotential. Sedimentation potential can also be calculated whilereplacing Δp by − (ρp−ρL)ϕHδg in the expression for thestreaming potential.

For the zero electric current regime, the liquid filtrationvelocity (δu∞) at an applied pressure difference is determined bysubstituting the solution of the problem given by Eqs. (2.15)–(2.20), (2.26), and (2.27) into Eq. (2.22). Changing thereference system, and replacing Δp by −(ρp−ρL)ϕHδg, oneobtains the expression for the sedimentation velocity, −δu∞.

2.5. Discussion on the general problem formulation

According to the above analysis, which is based on theapproach of Ohshima, Healy, White, O'Brian [14,15,17,78–83,86,87] and Shilov, Zharkikh, Borkovskaya [16,18,19,91–93], to address linear Electrokinetic Phenomena in a concen-trated disperse system, one should subsequently solve twoboundary value problems formulated in Sections 2.1–2.4. Thefirst of them describes a disperse system in the thermodynamicequilibrium state (Section 2.2). The second problem relates tonon-equilibrium, electrokinetic regime and deals with thespatial distributions of small perturbations of the ion electro-chemical potentials, δμk, the effective pressure δΠ and the localliquid velocity, dYu (Sections 2.3 and 2.4).

In the problems formulated in Sections 2.1–2.4, governingequations (2.3) (2.16) (2.17) and (2.18), interfacial boundary




ARTICLE IN PRESS

conditions (2.4) (2.18) and (2.19), and boundary condition forthe equilibrium potential at infinity distances, Eq. (2.5), arevalid for an arbitrary ensemble of dielectric particles inside anelectrolyte solution. By imposing the outer boundary conditionsgiven by Eqs. (2.20) and (2.23), the problem is specified foraddressing the disperse systems that can be considered asmacroscopically homogeneous and isotropic medium. Accord-ing to the condition formulated in Section 1.1., such a dispersesystem is constituted by representative cells. Assuming inSection 2.4 that all the particles move with the same velocity,we confine our analysis to the systems for which the smallestrepresentative cell contains a single particle.

A representative cell containing a single particle is a typicalconstituent element for a variety of regular lattices that areformed by disperse particles. Another example of such arepresentative cell is associated with the approach when, in thethermodynamic equilibrium state, all the distributions around agiven particle are assumed to have the spherical symmetry. Suchspherically symmetric distributions are considered as approx-imations of functions obtained by averaging the actual distribu-tions over time or ensemble. The SCA, which is discussed inSections 4 and 5 of the present paper, deals with this type of therepresentative cells.

While using the problem formulated in Sections 2.1–2.4 foraddressing regular lattices, boundary condition (2.4), whichimposes the uniform interfacial potential ζ, is valid for a speciallimiting adsorption isotherm and/or for a weak electrostaticinteraction between the particles, κb>>1. Considering the SCA,boundary condition (2.4) is always valid. In the latter case, thevalue of ζ can be interrelated with the parameters, which describeadsorption, and the volume fraction (see, for example, [71]).However, in the present paper, we do not discuss how theinterfacial potential depends on the above parameters. Conse-quently, all the distributions are considered as explicit functionsof ζ and the volume fraction ϕ (for example:WðYr; f;ϕÞand dlkðYr; f;ϕÞ) with understanding that an additional depen-dency on ϕ can be embedded in ζ.

In the problems formulated in Sections 2.3 and 2.4 fordescribing the small perturbations, the set of the electrochemicalpotentials, δμk, containsN unknown functions (N is the number ofionic species). Remarkably, even when all the N perturbations ofthe electrochemical potentials are known, one can not determineN+1 perturbations of the ion concentrations, δCk, and the electricpotential, δφ. Thus, solving the problems formulated in sectionsin Sections 2.1–2.4 one obtains an exhaustive description of thelinear Electrokinetic Phenomena whereas the concentration andelectric potential distributions remain unknown.

The above observation reflects an important fact that the ionelectrochemical potentials and effective pressure gradients canbe considered as driving forces in Electrokinetics. Accordingly,the boundary value problem given by Eqs. (2.15)–(2.20) issufficient for addressing Electrokinetic Phenomena sincesolving this problem enables one to determine the distributionsof all the electrochemical potentials and the effective pressurewithin the system.

As for the concentration and electric potential perturbations,they can be computed apart of the general problem formulated


Sections 2.2 and 2.3. The governing equation for δφ is derivedby combining a perturbation version of the Poisson equation,

∇2du ¼ �Fe

Xk

ZkdCk ; ð2:30Þ

with Eq. (2.9). Consequently, one obtains the followinggoverning equation

∇2du� j2duXk

mkZkCeqk =Cl

k ¼ � j2

F

Xk

dlkmkCeqk =Cl

k

ð2:31ÞWhen both the problems formulated in Sections 2.1–2.4

have been solved, the spatial distributions of Ckeq and δμk are

known functions. For such a case, Eq. (2.31) contains oneunknown function, δφ, only. However, to find this function,additionally to Eq. (2.31), it is necessary to use the Laplaceequation for the perturbation of the electric potential inside aperfect dielectric particles, δφp:

D2dup ¼ 0 ð2:32Þ

Eqs. (2.31) and (2.32) are subject to the electrostaticboundary conditions at the interface:

du ¼ dup at the particle surfaces ð2:33Þ

epY∇dup � e

Y∇du ¼ dr at the particle surfaces ð2:34Þwhere εp is the dielectric permittivity of particle and δσ is theperturbation of the interfacial charge density. Changes in theinterfacial charge are defined by the changes in the ion adsorption,dissociation of surface groups etc. Assuming a quasi-equilibriumadsorption, the changes in the interfacial charges are completelydefined by the changes in the ion electrochemical potentials in theimmediate vicinity of the interface. Accordingly, consideringlinear perturbations, Eq. (2.34) can be rewritten as

epY∇dup � e

Y∇du ¼Xk

bkdlk at the particle surfaces

ð2:35Þwhere the coefficients βk are determined using the ion adsorptionisotherms. For the particular case of the constant interfacial chargedensity: βk=0.

To close the problem formulation, one should set the outerboundary condition which imposes the applied electric potentialdifference

duA � duB ¼ Du ð2:36Þwhere δφA and δφB are the perturbations of electric potentialsat the planes AA and BB (Fig. 2), respectively.

Thus, for given functionsCeqk ðYr Þ and dlkðYr Þ, Eqs. (2.31)

and (2.32) subject to boundary conditions ), (2.35) and (2.36form a closed problem for obtaining the electric potentialdistribution, duðYr Þ. Substituting the obtained distributions intoEq. (2.9) yields perturbations of the local concentrations,dCkðYr Þ.


ARTICLE IN PRESS

It should be noted that, due to boundary condition (2.36),both the electric potential and concentration perturbations,duðYr Þ and dCkðYr Þ, depend on the adsorption parameters, αk,and the dielectric permittivity of particles, εp. Remarkably, thedistributions of dlkðYr Þ and, thus, the linear ElectrokineticPhenomena are independent of both αk and εp. The latterparameters are not represented in the equations derived Sections2.1–2.4.

The above discussion gives an explanation why the Henryapproach [6], which is valid for an unrealistic case of εp→0,leads to correct predictions. Since the linear ElectrokineticPhenomena are independent of εp, assuming different values ofthe particle permittivity leads to common results.

It should be stressed that such an independency of both theadsorption parameters and the internal dielectric permittivity isa specific property of the stationary linear ElectrokineticPhenomena. While considering non-stationary and/or non-linear effects, the adsorption parameters and the internaldielectric permittivity are represented in the final expressionsdescribing Electrokinetic Phenomena. An analysis of non-linearElectrokinetic Phenomena can be conducted using a techniquesimilar to that discussed in Sections 2.1–2.4, but accounting fornon-linear perturbation terms. Within the frameworks of suchan approach, obtaining solution of the above formulatedproblem, which is given by Eqs. (2.31–2.33) (2.35) (2.36),becomes a necessary step in the analysis.

3. Kinetic coefficients

Since the classical electrokinetic effects are linear withrespect to small perturbations of the external influence (appliedelectric field, pressure difference etc.), it is convenient tointroduce a set of parameters-kinetic coefficients which are theproportionality coefficients between the imposed externalinfluences and the measured system responses. Due to thelinearity of the problem, the kinetic coefficients should notdepend on external influences and are completely defined by theproperties of the disperse system. In this section, we considerdefinitions of the kinetic coefficients used in the literature. Aswell, we show how each of the kinetic coefficients is obtainedby using solution of the problem formulated in this Sections2.1–2.4.

Definitions of some kinetic coefficients deal with the appliedelectric field strength. The applied electric field strength isrepresented in boundary conditions (1.6) and (1.9) which arewidely discussed in the literature. Therefore, prior to definingthe kinetic coefficient, we justify the definition of the appliedelectric field strength.

3.1. External and internal applied field strengths

According to Dukhin et al. [96], it is necessary to distinguishbetween two methods of setting applied electric field. UsingFig. 2, let us define the internal electric field strength, d

YE int, as

dYE int ¼ Y

izduA � duB

H¼ Y

izDuH

ð3:1Þ


whereYiz is the unit vector along the positive direction of Z axis

(Fig. 2). We assume that the total resistance between the planesAA and BB is due to the disperse system (i.e. the resistances ofthe small gaps are negligible). Hence, the electric potentialdifference,Δφ, is considered to be applied between the externalsides of the disperse system. Thus, the internal electric fieldstrength, d

YE int, defined by Eq. (3.1) is an average strength of

the electric field existing inside the disperse system. UsingEq. (2.30) one can rewrite boundary condition (2.20) as

dlAk � dlBkFZk

¼ HdEint ð3:2Þ

Thus, Eq. (3.2), which is expressed in terms of dYE int, is an

equivalent of boundary condition (2.20).Now we use Fig. 2 to define the external field strength, as

dYEext ¼ dJl

KlYiz ð3:3Þ

where K∞ is the common electric conductivity of the electrolytesolutions in the adjacent compartments (Fig. 2):

Kl ¼ F2

RT

Xk

Z2k DkC

lk ð3:4Þ

Thus, for the system sketched in Fig. 2, dYEext is the electric

field strength measured within the adjacent electrolyte compart-ments i.e. outside the disperse system.

Using Eqs. (3.3) and (3.4), Eq. (2.23) can be rewritten as

� 1

FXk

z2kDkClk

Xk

DkZkClkY∇dlk ¼ d

YEext ð3:5Þ

Thus, Eq. (3.5) is an equivalent form of boundary condition(2.23) represented in terms of the external electric field, d

YEext.

It should be noted that, dYE int and d

YEext always differ except

for the case of zero volume fraction. For the later case, the solidphase is absent, and the system becomes a homogeneouselectrolyte solution with a uniform electric field. Relationbetween d

YE int and d

YEext is considered below.

3.2. Definitions of kinetic coefficients

3.2.1. Disperse system conductivityDefinition of the disperse system conductivity can be

introduced using the thought experiment illustrated in Fig. 2.Ignoring the resistances of the gaps with thicknesses d, theconductivity of the disperse system is defined as:

K ¼ dJl

ðduA � duBÞ=H¼ dJl=dEint; ð3:6Þ

where we used Eq. (3.1). Thus, in the above definition of theconductivity, the internal field strength, d

YE int, is used.

At a given electric potential difference, both the electriccurrent and conductivity depend on the imposed hydrodynamicconditions (zero external pressure difference, Δp=0, or zero




ARTICLE IN PRESS

volumetric flow, δu∞=0). Imposing different hydrodynamiccondition results in different velocity distributions inside elec-trolyte. In the presence of the equilibrium charge in theelectrolyte bulk, different velocity fields produce differentcontributions of the convective electric current. The latter affectsthe conductivity. Hence, hydrodynamic conditions should bespecified while defining the conductivity attributed to theimposed hydrodynamic regime. We will specify the hydrody-namic conditions when it is necessary.

Combining the definitions of the internal and external fieldstrengths given by Eqs. (3.1) and (3.3) with the definition of theconductivity given by Eq. (2.35) one obtains an importantrelationship between the field strengths:

dEint

dEext¼ Kl

K: ð3:7Þ

Thus, the two versions of the applied electric field becomeequal when K∞→K i.e. for infinitely diluted disperse system(ϕ→0). For concentrated disperse systems K∞ and K differand, hence, d

YEext and d

YE int differ as well.

3.2.2. Two types of electrophoretic mobilityElectrophoretic mobility is defined as the electrophoretic

velocity per unit applied electric field strength. FollowingDukhin et al. [96] one can define two types of electrophoreticmobility, χ:

vint ¼ �ðdul=dEintÞDp¼0;dg¼0 ð3:8Þand

vext ¼ �ðdul=dEextÞDp¼0;dg¼0: ð3:9Þ

Combining Eqs. (3.7)–(3.9) one obtains

vint=vext ¼ KDP¼0=Kl ð3:10Þ

Note, Eq. (3.10) contains the conductivity KΔp=0 which isattributed to the hydrodynamic regime of the electrophoresis,Δp=0.

Thus, Eq. (3.10) gives an interrelation between the two types ofelectrophoretic mobilities, χint and χext. According to Eq. (3.10),as long as K/K∞≠1, χint and χext differ from each other. It isacceptable to use any of the introduced mobilities. However, in thecase of concentrated disperse systems, it is necessary to specify thetype of the mobility used.

3.2.3. Electroosmotic coefficientElectroosmosis can be characterized in the same terms as

electrophoresis, i.e., by introducing the electroosmotic mobility asa velocity per unit applied field strength. Similar to the abovediscussed electrophoretic mobility, one should introduce two typesof electroosmotic mobility. However, frequently, electroosmosis isdescribed using electroosmotic coefficient defined as

k ¼ ðdul=dJlÞDp¼0 ¼ �lext=Kl ð3:11Þ
dg¼0

3.2.4. Electroosmotic pressure coefficientIn the presence of electric current through a diaphragm, one

can consider situation of a “closed cell” in the absence of thehydrodynamic flow through the system, i.e., δu∞=0. For such asituation, a pressure difference (electroosmotic pressure differ-ence) is developed between the compartments separated by the“diaphragm”. Hence, one can introduce an electroosmoticpressure coefficient as

a ¼ ðDp=dJlÞdul¼0

dg¼0

ð3:12Þ

3.2.5. Sedimentation and streaming potential coefficientThe streaming potential coefficient, ω, is defined as the

electric potential difference per unit applied pressure, which aremeasured at zero electric current density regime (J∞=0)

x ¼ DuDp

� �dJ

l¼0dg¼0

ð3:13Þ

Using the internal and external field definitions given aboveby Eqs. (2.30) and (2.32) respectively, Eq. (2.41), can berewritten in the following form

x ¼ dEint

Dp=H

� �dEext¼0dg¼0

ð3:14Þ

According to Eq. (3.14), the streaming potential coefficientis defined as the internal electric field strength per unit pressuregradient.

It was shown in Section 2.4.3 that the description of thepressure driven process and the sedimentation becomecompletely identical while replacing the pressure difference,Δp, by − (ρp−ρL)ϕHδg in corresponding boundary conditions(compare Eqs. (2.27) and (2.29)). Consequently, consideringsedimentation of a suspension having the same parameters(volume fraction, zeta potential etc.) as the diaphragm, such areplacement in Eq. (3.14) leads to another expression for thesame coefficient.

x ¼ � dEint

ϕðρp � ρLÞδg

" #dJ

l¼0Dp¼�½ρpϕþρLð1�ϕÞ�H

ð3:15Þ

In the latter expression, δEint is internal field strengthcorresponding to the sedimentation potential.

Both the above given definitions are special cases of themore general expression that follows from Eq. (2.21)

x ¼ dEint

ρLdg þ Dp=H

� �dJ

l¼0

ð3:16Þ

The latter expression can be employed when one considerssolution of the problem formulated in Section 2 with boundarycondition (2.21).



Fig. 3. Illustration to derivation of Eq. (3.22).


ARTICLE IN PRESS

3.2.6. Hydraulic permeability, filtration and sedimentationvelocities

The hydraulic permeability of the disperse systems is definedas

g ¼ dulρLdg þ Dp=H

ð3:17Þ

For a given pressure difference, the velocity at the planes AAand BB, δu∞, depends on the electric regime: the velocitiesmeasured at zero electric current and at zero potential differencediffer from each other. Thus, the mechanic permeability, γ,depends on the electric regime, as well.

When solvent, under an applied pressure difference, flowsthrough the diaphragm, we assume that δg=0. For a given γ,the filtration velocity δuf=δu

∞ is determined as follows

duf ¼ gDpH

ð3:18Þ

In the case of settling suspension, the pressure difference isgiven by Eq. (2.28), and the sedimentation velocity δused=−δu∞ is determined by replacing Δp by − (ρp−ρL)ϕHδg inEq. (3.18). The latter results in:

dused ¼ gϕðρp � ρLÞδg ð3:19Þ

3.3. Expressions for kinetic coefficients using volume meanquantities

Following the approach proposed by Saville [122], in manystudies, (see, for example, [78–85,107–109]) some of thekinetic coefficients are determined by using expressions wherethe kinetic coefficients are considered as proportionalitycoefficients between two average vectors that are obtained byaveraging respective vector fields over the disperse systemvolume. For macroscopically homogeneous and isotropicdisperse systems, such an averaging always results in twoparallel uniform vectors. Accordingly, for such systems, thekinetic coefficients are spatially uniform scalars.

In the present section we demonstrate that the definitions ofthe kinetic coefficients given in Section 3.1 are completelyequivalent to the definitions based on the proportionalitybetween volume mean quantities. To this end, let us consider thefollowing identity

dJl ¼ 1S

ZSAA

dYJ d dSY ð3:20Þ

where the surface integration is conducted over a part, SAA,of the plane AA, which has area S (Fig. 3). Let us consider apart, S′, of another hypothetic plane which has the same area, S,and is drawn inside the disperse system to be parallel to AA andBB. Assume that the linear dimensions of both SAA and S′ aremuch bigger than H. Accordingly, while integrating over theclosed surface formed by SAA, S′ and side surfaces SL and SR(Fig. 3), the contributions of the integrals over SL and SR can be


neglected. For such a case, using the Gauss theorem and takinginto account that the local current density satisfies the continuityequation,

Y∇ d dYJ ¼ 0, the following equality can be deducedZSAA

dYJ d dSY ¼

ZSdYJ d dS

Y ð3:21Þ

Combining Eqs. (3.20) and (3.21) one obtains

dJl ¼ 1HS

Z L

0

ZSdYJ d

Yizds

� �dz ¼ Y

iz d1V

ZVdYJ ðYr ÞdV

ð3:22ÞThe final expression in Eq. (3.22) contains an average over

the system volume V=HS, hence

dJl ¼ Yiz d hdYJ i ð3:23Þ

Eq. (3.23) gives relationships between the mean volumeelectric current density and the current density, δJ∞, withinplanes AA and BB (Fig. 2).

Exactly the same scheme as that proposed above for derivingEq. (3.23) can be used for obtaining a similar relationship forthe velocities

dul ¼ Yiz d hdYu i ð3:24Þ

The similarity between Eqs. (3.23) and (3.24) is amanifestation of the fact that both vectors, dYJ and dYu, satisfythe continuity equation.

Now, we show that the internal electric field strength, δEint,is simply the local electric field strength averaged over thedisperse system volume. To demonstrate this, we consider thefollowing chain of equalities

dYE int ¼ Y

izduA � duB

H¼ � SðduAYnA þ duAYnBÞ

HS¼� 1

V∮Sv

duYds

ð3:25Þ

The first of the above equalities is the definition given byEq. (3.1). The second equality is derived using

Yiz ¼

YnB ¼ �Y

nA(Fig. 3). To obtain the third of the above identities one shouldconsider a volume enveloped by a closed surface SV. For thepresent analysis, Sv is formed by SAA, S′ SL and SR, and S′ coin-cides with SBB (Fig. 3). According to the assumed geometry, the




ARTICLE IN PRESS

contribution of the integrals over SL and SR into the integralover SV is negligible. For such a case, the third equality becomesobvious.

Since electric potential is a continuous function of coordinatesinside the disperse system, it satisfies the gradient theorem

∮SVduYds ¼ ∮V

Y∇dudV ð3:26Þwhere V is a volume enveloped by SV. Combining Eqs. (3.25)and (3.26) yields

dYE int ¼ � 1

V∮V

Y∇dudV ¼ �hY∇dui ð3:27Þ

Thus, Eq. (3.27) represents the internal field strength throughlocal electric field gradient averaged over the disperse systemvolume.

Making use of Eqs. (3.3) and (3.23) one can obtain a usefulexpression for the external electric field strength

dYEext ¼ hdYJ i

Kl ð3:28Þ

Thus, Eq. (3.28) connects the external electric field strengthand the local electric current density averaged over the dispersesystem volume.

Using Eqs. (3.23) (3.24), (3.27) and (3.28) one can rewriteexpressions for some kinetic coefficients in terms of volumemean quantities. In particular, combining (3.6), (3.23) and(3.27), the expression for conductivity can be written in the form

K ¼ � hdYJ i d Yiz

hY∇dui d Yiz

ð3:29Þ

Combining Eqs. (3.8), (3.23) and (3.27), the electrophoreticmobility calculated per unit internal field strength becomes

vint ¼hdYu i d Y

izhY∇dui d Y

iz

!DP¼0;dg¼0

ð3:30Þ

Similarly, combining Eqs. (3.3), (3.9) and (3.23) one obtains

vext ¼ �Kl hdYu i d Yiz

hdYJ dYizi

!Dp¼0;dg¼0

ð3:31Þ

Eq. (3.30) gives expression for electrophoretic mobilitycalculated per unit external field strength.

Finally, making use of Eqs. (3.15), (3.23) and (3.28) thesedimentation potential coefficient is represented in the form

x ¼ hY∇dui d Yizϕðρp � ρLÞδg

" #hdJ i¼0Dp¼�½ρpϕþρLð1�ϕÞ�H

ð3:32Þ

Earlier, Eq. (3.32) was derived by Saville [122]. Thus, usingEqs. (3.23) (3.24), (3.27) and (3.28), one can express any of thekinetic coefficients defined in Section 3.2. through meanvolume vectors of the local velocity, electric current densityand electric potential gradient.


3.4. Smoluchowski limiting case for kinetic coefficients

When the Debye parameter, κ, given by Eq. (1.13) infinitelyincreases, κ→∞, some of the above discussed kineticcoefficients approach a remarkable asymptotic values whichare independent of the disperse system internal geometry. Suchan asymptotic behavior was discovered by Smoluchowski [1–4]and, therefore, the corresponding geometry independentlimiting values are referred to as the Smoluchowski limits.

Smoluchowski limits are known during more than a centuryand serve as the important criteria for verifying the validity oftheoretical predictions (Section 1.5). However, as stated inSection 1.5, in the literature concerned with ElectrokineticPhenomena in concentrated disperse systems, different points ofview have been expressed regarding correct forms of the Smo-luchowski limits for the electrophoretic mobility and thestreaming (sedimentation) potential coefficient [75,80,86,107].In the present section, we analyze the asymptotic behavior of thesolutions of the problems formulated in Sections 2.1–2.4, atκ→∞. On the basis of this analysis we rederive the Smolu-chowski limiting expressions for the electrophoretic mobility andthe streaming (sedimentation) potential coefficient.

Let us consider the limit of κ→∞ for the first problem whichis formulated in Section 2.2 for describing the behavior of theequilibrium potential, Ψ. This potential is determined as thesolution of the P–B equation subject to boundary conditions(2.4) and (2.5). To analyze the case of κ→∞, it is convenient torepresent Eq. (2.3) in the following form

j�2∇2W ¼ �Xk

vkexp½�WðYr ÞZk � ð3:33Þ

where Ψ =ΨF/RT, and the coefficient vk is given by Eq. (1.4)c.At κ→∞, the left hand side of Eq. (3.33) and, thus, its righthand side approach to zero. Since

Pk vk ¼ 0, the equilibrium

potential, Ψ also takes zero value for a such limiting case.The latter is true everywhere in the electrolyte solution except

for the vanishingly thin regions in the immediate vicinity of theinterface. Inside such regions having thickness of order of κ−1,the potential, Ψ, should rapidly change toward the interface inorder to satisfy boundary condition (2.4). Accordingly, within thevanishingly thin layer, the value of κ−2∇2Ψ remains finite inspite of κ→∞. Such a vanishingly thin layer can be consideredas locally flat. Consequently, inside the layer, the Laplaceoperator can be written as ∇2=∂2Ψ/∂ξn2 where we introducedthe local Cartesian coordinate nn ¼ ðYr � YrsÞ d Yn (Fig. 4).

The above consideration leads to the following relationshipsdescribing the asymptotic behavior of the equilibrium potentialΨ at κ→∞:

W ¼ 0 outside the vanishing thin interfacial regions ðaÞeA2W=An2n ¼ �ρeq within the vanishing thin interfacial regions ðbÞ

ð3:34Þ

Now, we consider the second boundary value problemformulated in Sections 2.3 and 2.4. The problem formulationincludes Eqs. (2.15)–(2.17) subject to boundary conditions(2.18)–(2.23). For convenience of the analysis, we substitute



Fig. 4. Illustration to derivation of Smoluchowski limit for sedimentationpotential.


ARTICLE IN PRESS

Eq. (2.1) into Eq. (2.15). After minor transformations, such asubstitution leads to the following equation

�∇2 dlkZkF

� �þ Zk

Y∇W dY∇

dlkZkF

� �þ 1Dk

dYu dY∇W ¼ 0

ð3:35ÞEq. (3.35) will be used in the next analysis of the asymptotic

behavior of the electrophoretic mobility and the streaming(sedimentation) potential coefficient.

3.4.1. Smoluchowski limit for electrophoretic mobilityAssuming that κ→∞, let us consider the asymptotic

behavior of the solution of Eq. (3.35) subject to boundaryconditions (2.19) and (2.20). Taking into account Eq. (3.34)a,and boundary conditions (2.18) and (2.19), which set,respectively, zero velocity and normal ionic flux at the interface,we deduce that both the second and third terms in Eq. (3.35)approach zero when κ→∞. Omitting these terms in Eq. (3.35)and inspecting the obtained Laplace equation subject toboundary conditions (2.19) and (2.20), we conclude that thefunction δμi/Fzi becomes common for all the ions, i.e.,

dlk=Fzk ¼ du⁎

In Section 1.3, we already use the notation δφ⁎ for signifyingthe Shilov–Zharkikh–Borkovskaya “flow” potential given byEq. (1.4)a [16,91–93]. Combining the above relationship withEq. (1.4)a and c, it becomes clear that δμi/Fzi does coincide withthe “flow” potential introduced in Refs. [16,91–93].

Thus, governing equation (3.35) is simplified as

∇2du⁎ ¼ 0 ð3:36ÞNote that Eq. (3.36) is valid for the whole volume of the

electrolyte solution including the interfacial layer which, atκ→∞ is transformed into a mathematical surface. The latterbecomes clear while considering Eq. (3.35) and boundaryconditions (2.4), (2.18) and (2.19). Therefore, the interfacial


boundary condition for Eq. (3.36) can be obtained from theelectrochemical interfacial boundary condition (2.19), as

Y∇du⁎ d Yn ¼ 0 at the interface ð3:37ÞThe form of the electrochemical boundary condition at the

planes AA and BB depends on the type of electrophoreticmobility to be obtained. While obtaining electrophoretic mobilityper unit external field strength (recall Section 3.1 and 3.2.2) it isconvenient to impose the external field strength with the help ofcondition (3.5) which is an equivalent of Eq. (2.23). Taking intoaccount that δφ⁎=δμk /FZk, boundary condition (3.5) takes asimple form

Y∇du⁎ ¼ �dYEext at the palnes AA and BB ð3:38Þ

Thus, Eq. (3.36) subject to boundary conditions (3.37) and(3.38) yields a closed problem which can be solved forobtaining the unknown function δφ⁎.

Now, we consider the hydrodynamic part of the generalproblem. Using Eqs. (2.2) and (2.11) one can rewrite Eq. (2.16)as

gY∇� Y∇� dYu ¼ �Y∇dj� ρeqY∇du⁎ ð3:39ÞOutside the vanishingly thin interfacial layer ρeq =0. Hence,

for the whole electrolyte bulk except for the interfacial region,Eq. (3.39) takes the form

gY∇� Y∇� dYu ¼ �Y∇dj ð3:40ÞEq. (3.40) and continuity equation for the velocity

perturbation (2.17) form a set of governing equations fordetermining unknown functions dYu and dj in the regionoutside the vanishingly thin interfacial layer. These equationsare subject to the boundary conditions at the solid-liquidinterface and at the planes AA and BB (Fig. 2). While writingthe hydrodynamic boundary conditions at the interface, itshould be taken into account that Eq. (3.40) is not valid insidethe vanishingly thin interfacial layer where Ψ≠0. Hence, thecorresponding boundary conditions should be set at the outerboundary of the interfacial layer.

As stated above, at κ→∞, the interfacial layer can beconsidered as flat. The liquid velocity, which is zero at the actualinterface, takes a non-zero value at the outer boundary of thelayer. Such a “jump” of the velocity across an infinitely thincharged layer is given by the classical Smoluchowskiexpression. Derivation of these expression amounts to integrat-ing the flat versions of Eqs. (2.17) and (3.39) which, usingEq. (3.34), are represented in the form

dðYn d dYu Þdnn

þ Y∇s d dYu ¼ 0 ðaÞ

gd2dYu

dn2n¼ Y∇dj� e

A2W

An2n

Y∇du⁎ ðbÞð3:41Þ

whereY∇S ¼ Y∇�Yn ðYn d

Y∇Þ. Note that, in Eq. (3.41)b, thefunction δφ⁎ is a solution of the problem given by Eqs. (3.36)–




ARTICLE IN PRESS

(3.38). For obtaining the required boundary condition,Eqs. (3.41)a and b are integrated over the coordinate ξn (onceand twice, respectively) within the range 0bξnbξn⁎. Whileintegrating, we take into account boundary conditions (2.4),(2.5) and (2.18). Finally, we consider a limiting transition of theobtained integrals by assuming that n⁎nY0 and jn⁎nYl,simultaneously. Such a limiting transition yields the Smolu-chowski slip boundary condition

Yn � dYu � Yn ¼ efg

Yn � ðY∇ duÞ �Yn ðaÞdYu d Yn ¼ 0 ðbÞ

at the interface;

ð3:42ÞConsequently, Eq. (3.42) give the interfacial boundary

condition for Eqs. (3.40) and (2.17).Thus, to derive an expression for the Smoluchowski limit of

the electrophoretic (electroosmotic) velocity, it is necessary tofind the unique solution of the problem given by Eqs. (3.36)–(3.38). The obtained solution of this problem, δφ⁎, is employedin boundary condition (3.42) of the hydrodynamic problemwhich also contains governing equations (2.17) and (3.40) andexternal boundary condition (2.24). Solving the problem givenby Eqs. (2.17), (2.24), (3.40) and (3.42) one obtains the velocitydistribution, dYu ðYr Þ. Substituting dYu ðYr Þinto Eq. (2.25) yieldsthe electrophoretic or electroosmotic velocities, dYueph or dYueo.

Remarkably, the above discussed scheme leads to a geometryindependent prediction for the electrophoretic (electroosmotic) ve-locity, because the unique distributions of the velocity and pressuresatisfying Eqs. (2.17) (2.24) (3.40) and (3.42) take the forms

dYuðYr Þ ¼ efg

Y∇ du⁎ðYr ÞdjðYr Þ ¼ constant

(3.43)

Let us now show that the hydrodynamic field given byEq. (3.43) does satisfy Eqs. (2.17) (2.24) (3.40) and (3.42).Clearly, dYu ðYr Þ and djðYr Þ given by Eq. (3.43) satisfyEqs. (3.24), (3.40) and (3.42)b. As well, substituting Eq. (3.43)into Eqs. (2.17) and (3.42)a leads to Eqs. (3.36) and (3.37),respectively. Hence, when the distribution du⁎ðYr Þ satisfiesEqs. (3.36)–(3.38), the structure of hydrodynamic field isdescribed by Eq. (3.43).

The final expression for the electrophoretic (electroosmotic)velocity is deduced by combining Eqs. (2.25) and (3.43).

dYueph ¼ �d

Yueo ¼

efgdYEext ð3:44Þ

Finally, using definition (3.9) we arrive at the followingexpressions for the Smoluchowski limit of χext (χext

Sm)

vSmext ¼ ef=g ð3:45ÞClearly, the right hand side of Eq. (3.45) does not depend on

the disperse system geometry.Thus, evaluating the electrophoretic velocity per unit exter-

nal electric field strength, the obtained mobility, χext, satisfiesthe Smoluchowski principle: at κ→∞, χext becomes indepen-


dent of the disperse system geometry. At the same time, theelectrophoretic mobility of another type, which corresponds tothe unit internal field strength, χint, depends on the systemgeometry. The latter immediately follows from the analysis ofEq. (3.10). Combining (3.10) and (3.45), Smoluchowski limitfor χint (χint

Sm) can be expressed as

vSmint ¼ efgKDp¼0

Klð3:46Þ

Even for a vanishingly thin double layer, the ratio KΔp=0/K∞,and, hence, χint

Sm depend on the disperse system geometry, inparticular, on the volume fraction.

3.4.2. Smoluchowski limit for streaming (sedimentation)potential coefficient

Now, we rederive the expression for the Smoluchowski limitof the streaming (sedimentation) potential coefficient, ω,introduced in Subsection 3.2.5. by Eq. (3.13). To this end,considering the limiting case of κ→∞, we will determine theelectric potential difference Δφ=δφA−δφB which correspondsto the zero electric current through the system.Using the referencesystem linked to the particles, we consider a perturbation of theelectrochemical potential, dlkðYr Þ, inside the liquid phase, whichflows through a system of particles under a given pressuredifference, Δp.

Similarly to the above given analysis of the asymptoticbehavior of the electrophoretic mobility, we will use Eq. (3.35)as a starting point of the consideration. While obtaining thestreaming potential, Eq. (3.35) is subject to boundary conditions(2.19), (2.20) and (2.26). For a given velocity distribution,dYu ðYr Þ, substituting the general solution of Eq. (3.35) intoboundary conditions (2.19), (2.20) and (2.26) yields algebraicequation set enabling one to determine the streaming (sedi-mentation) potential difference, Δφ=δφA−δφB.

As stated in Section 3.4.1, at κ→∞, both the second and thirdterms in Eq. (3.35) approach to zero, and Eq. (3.35) istransformed into the Laplace equation with respect to δμk.However, in such a case, we arrive at a homogeneous problemformed by the Laplace equation subject to boundary conditions(2.19), (2.20) and (2.26). Such a homogeneous problem has atrivial solution, only, i.e., Δφ=δφA−δφB=0. Consequently, atκ→∞, the streaming potential difference approaches zero. Fromthe analysis given below, we will see that, at the Smoluchowskilimit, Δφ∝κ−2. While obtaining such an asymptotic expres-sion, it is not correct to omit the third term in Eq. (3.35) in spite ofthe fact that it approaches to zero when κ→∞. At the same time,the second term in Eq. (3.35) can be omitted for such a limitingcase since, as it follows from Eqs. (2.19) and (3.34)

limjYl

Y∇W dY∇dlk

� �=∇2dlk

h i¼ 0:

Thus, for analyzing the Smoluchowski asymptotic behavior,we use the following reduced form of Eq. (3.35)

�∇2 dlkZkF

� �þ 1Dk

dYu dY∇W ¼ 0: ð3:47Þ




ARTICLE IN PRESS

Inspecting the problem given by Eq. (3.47) subject toboundary conditions (2.19), (2.20) and (2.26), the problemsolution can be represented as

dlkZkF

¼ dUE þ DDk

dUp ð3:48Þ

where D ¼Pn Ceqn Z2

nDn=P

n Ceqn Z2

n . The above introduced

functions δΦE and δΦP, are common for all the ions and can bedetermined as solutions of the following two problems:

f∇2dUE ¼ 0

Yn dY∇dUE ¼ 0 at the interface ðaÞ

dUEA � dUE

B ¼ duA � duB

f∇2dU p ¼ 1

DdYu d

Y∇W

Yn d Y∇dUp ¼ 0 at the interface ðbÞdUP

A � dUPB ¼ 0

ð3:49Þ

where the subscripts “A” and “B” signify the values of thecorresponding functions at the planes AA and BB, respectively(Fig. 2).

For obtaining the streaming potential difference, Δφ=δφA−δφB, the functions δΦE and δΦP being the solutions ofproblems (3.49)a and b are substituted into (3.48) that yields thefunction dlkðYr;DuÞ. This function is finally substituted intoboundary condition (2.26) which sets zero electric current. Such asubstitution yields an algebraic equation for obtaining Δφ.

To analyze the case of κ→∞, it is convenient to introduce afunction, δΦ, defined as

dU ¼ dUE þ dU p; ð3:50ÞAnalyzing Eqs. (3.34) (3.49) and (3.50), it becomes clear

that, within the electrolyte bulk outside the vanishingly thindouble layer, δΦ satisfies the Laplace equation,

∇2dU ¼ 0; ð3:51Þand the following condition

dUA ¼ dUB ¼ duA � duB: ð3:52ÞCondition (3.52) is used for determining the streaming

potential difference, Δφ=δφA−δφB, when δΦ is a knownfunction. To obtain δΦ, one should impose two boundaryconditions for Eq. (3.51). For determining one of the requiredconditions, we use the continuity equation for the local electriccurrent. Such a condition follows from continuity equation(2.14) written for the individual fluxes and takes the form

Y∇ d dYJ ¼ 0 ð3:53Þwhere the local electric current density dYJ is expressed as

dYJ ¼ FX

ZkdYjk ¼� F

RT

XZkC

eqk Dk

Y∇dlk þ FdYuX

ZkCeqk

k k k


Combining the above equation with Eqs. (2.11) and (3.48),one obtains

dYj ¼ �KeqY∇dUþ ρeqdYu ð3:54Þwhere Keq ¼ F2

Pk Z

2k DkC

eqk =RT .

To obtain the required boundary condition one shouldintegrate both sides of Eq. (3.53) over the coordinate nn ¼ðYr � YrsÞ d Yn (Fig. 4) within the range 0bξnbξn⁎ and consider alimiting transition when ξn⁎→0 and κξn⁎→∞, simultaneously.Prior to integrating, the local electric current, dYJ , is substitutedinto Eq. (3.53) in the form given by Eq. (3.54) where one shouldexpand the velocity distribution, dYu, in the Taylor series bypowers of ξn and retain the leading term only

dYu ¼ nn½ðYn dY∇ÞdYu �nn¼0 þ Oðn2nÞ ð3:55Þ

The above outlined derivation, whose details are given inAppendix A, leads to the following boundary condition

Yn dY∇dUðYr Þ ¼ � ef

Kl

Y∇s d ½ðYn dY∇ÞdYu � at the interface

ð3:56ÞThus, Eq. (3.56) yields the first boundary condition for

Eq. (3.51).Now, we obtain the second boundary condition, which is

necessary to close the problem formulation. The requiredcondition, which is given by Eq. (2.26), sets the zero electriccurrent regime for measuring sedimentation and streamingpotential

dJl ¼ 0 ð3:57ÞMaking use of Eqs. (3.22) and (3.57) leads to the condition

1V

ZVdYJ ðYr Þ ¼ 0 ð3:58Þ

Earlier, for analyzing sedimentation potential, such acondition was proposed by Saville [122].

For obtaining the integral in Eq. (3.58), we substitute ex-pression for the electric current (3.54) into Eq. (3.58). Realizingthat the second (convective) term in Eq. (3.54) takes a non-zerovalue inside the vanishingly thin interfacial layer region adjacentto the interface and using expansion (3.56) for the velocity insidesuch a layer, one obtains

1V

�KlZVliquid

Y∇dudV þ limnnY0jnnYl

ZSp

Z n⁎n

0½nnðYn d Y∇ÞdYu �nn¼0ρ

eqdnndS

264

375 ¼ 0

ð3:59Þ

Since, within the solid phase, the electric current density iszero, the first integral of Eq. (3.59) is taken over the liquid phasevolume, Vliquid, whose boundary contains the particle surfacesand the surfaces formed by SAA, S′ SL and SR (Fig. 3). Similarlyto the derivation of Eq. (3.25), we consider the limiting positionof S′ which coincides with SBB (Fig. 3).




ARTICLE IN PRESS

By applying the gradient theorem given by Eq. (3.26), thefirst integral in Eq. (3.59) can be transformed into an integralover the liquid phase boundary. Using such a transformationand obtaining the second integral in Eq. (3.59) with the help ofEq. (A5) (see Appendix A), we arrive at the relationship

ðYnAdUA þ YnBdUBÞS þZSp

dUYdS ¼ ef

Kl

ZSp

ðYn dY∇ÞdYudS

ð3:60Þwhere S is the common area of each of the surfaces SAA andSBB (Fig. 3). Similarly to derivation of Eq. (3.25), in Eq. (3.60),we neglected by the contributions of the integral over the sidesurfaces, SL and SR.

Thus, Eq. (3.60) yields the second boundary condition forEq. (3.51). Consequently, for a given creeping flow velocity,dYu ðYr Þ, by solving Eq. (3.51) subject to boundary conditions(3.56) and (3.60), one obtains the distribution dUðYr Þ within theregion which is confined by the planes AA and BB (Fig. 3) andby the particle surfaces.

It should be noted that the velocity distribution, dYu ðYr Þ,which is represented in boundary conditions (3.56) and (3.50)satisfies the continuity and Stokes equations (2.17) and (3.40)subject to boundary conditions (2.18) and (2.27). Using thisfact, one can show that, for any disperse system geometry, thesolution of the problem given by Eq. (3.51) subject to boundaryconditions (3.56) and (3.60) is represented in the form

dUðYr Þ ¼ ef

gKl djðYr Þ ð3:61Þ

To verify whether Eq. (3.61) gives such a solution, wesubstitute Eq. (3.61) into Eqs. (3.51), (3.56) and (3.60) andobserve whether the obtained equalities are valid. SubstitutingEq. (3.61) into Eq. (3.51) leads to the Laplace equation for thepressure:

∇2djðYr Þ ¼ 0 ð3:62Þ

which follows from hydrodynamic equation (3.40). Hence,dUðYr Þ given by Eq. (3.61) satisfies Eq. (3.51).

Substituting Eq. (3.61) into boundary condition (3.56) yields

Yn dY∇djðYr Þ ¼ �g

Y∇s d ½ðYn d

Y∇ÞdYu � at the interface

ð3:63Þ

Using the Stokes Eq. (3.40) and taking into account boundaryconditions (2.18), one can prove the validity of Eq. (3.63).Details of this analysis are given in Appendix A. Thus, thepotential distribution given by Eq. (3.61) satisfies boundarycondition (3.56).

Finally, substituting Eq. (3.61) into Eq. (3.60) we arrive atthe following equality

ðYnAdjA þ YnBdjBÞS þZSp

djYdS ¼ gZSp

ðYn dY∇ÞdYudS

ð3:64Þ


It can be demonstrated (see Appendix A) that

ðYn dY∇Þ dYudS ¼ ½Y∇dYu þ ðY∇dYu Þt� d Y

dS at the interface

ð3:65Þwhere the superscript “t” signifies a transposed tensor. Takinginto account Eq. (3.65), Eq. (3.64) can be rewritten asZSPþSAAþSBB

�dj I¼þg½Y∇dYu þ ðY∇dYu Þt�

n odYdSV¼ 0 ð3:66Þ

where I¼is a unity tensor. Meaning of Eq. (3.66) is that that

the external force exerted on the liquid phase takes a zerovalue. Consequently, for a creeping flow satisfying Eq. (3.40),Eq. (3.66) is an identity. Hence, the potential distribution givenby Eq. (3.61) satisfies boundary condition (3.60) as well. Thus,satisfying Eqs. (3.35), (3.56) and (3.60), Eq. (3.61) gives acorrect expression for the function dUðYr Þ, within thecontinuous liquid phase.

Combining Eqs. (2.27), (3.13), (3.52) and (3.61) one obtains

xSm ¼ðduA � duBÞSmdJl¼0

dg¼0

Dp¼ dUA � dUB

djA �djB¼ ef

gKlð3:67Þ

Eq. (3.67) yields the Smoluchowski limit for the streaming(sedimentation) potential coefficient.

Making use of Eqs. (3.45) and (3.67) yields

xSm ¼ vSmext=Kl ð3:68Þ

Eq. (3.68) establishes a relationship between the Smolu-chowski limits of the streaming (sedimentation) potentialcoefficient, ωSm, and the electrophoretic mobility calculatedper unit external field strength, χext

Sm. In the next section, we willsee that relationship (3.68) is valid for an arbitrary value of theDebye parameter, κ, not only for the Smoluchowski case ofκ→∞.

Combining Eqs. (3.67) and (3.68), we present expressionsfor the Smoluchowski limit of the streaming potential ΔφSP

Sm

DuSmSP ¼ ðduA � duBÞSmdJl¼0

dg¼0

¼ vSmextKl Dp ð3:69Þ

As stated in Section 2.4.3, the sedimentation potentialdifference can be calculated while replacing Δp by [− (ρp−ρL)ϕHδg] in the expression for the streaming potential. The resultof such a replacement in Eq. (3.69) can be represented in termsof the sedimentation potential field strength, dYE

Sm

g , which is theinternal field strength attributed to zero electric current regime.Consequently, using Eq. (3.14), the sedimentation potentialfield strength, dYE Sm

g is represented as

dYESm

g ¼ �vSmextϕρp � ρLK∞ δ

→g ð3:70Þ

Eq. (3.70) establishes a relationship between the Smolu-chowski limits of the sedimentation potential electric fieldstrength, d

YE

Smg , and the electrophoretic mobility χext

Sm. Note that,

although dYE

Sm

g is the internal electric field strength, relationship




ARTICLE IN PRESS

(3.70) contains the electrophoretic mobility, χextSm, which is

calculated per unit external field strength.Above, the validity of Eq. (3.70) has been proved for the

Smoluchowski limits of dYEg and χext. At the same time, as itwas shown in Ref. [26], the same relationship connects dYEg andχext for the general case of an arbitrary value of κ. In the nextsection, using irreversible thermodynamics, we will rederiveEq. (3.70) for such a general case.

In the above analysis, the hydrodynamic field within thewhole liquid phase (including the vanishingly thin interfaciallayer) was considered on the basis of Eq. (3.40). While usingEq. (3.40), we ignore a contribution of the electrically drivenflow which exists due to the presence of the streaming potentialelectric field. Such an approach is correct since, at κ→∞, thestreaming potential field strength approach to zero, as stated inthe beginning of the present Subsection. To demonstrate this,recall that the streaming potential difference is inverselyproportional to K∞ which, using Eqs. (1.4)c, (1.13) and (3.4),can be represented in the form

Kl ¼ j2eXk

mkZkDk

A comparison of the above equation with Eq. (3.69)confirms that, at the Smoluchowski limit, Δφ∝κ−2→0, asκ→∞. Consequently, the perturbation of the electric fieldstrength inside the electrolyte solution approaches to zero, aswell. Thus, the contribution of the electrically driven flowbecomes negligible compared to the pressure driven flow.

3.5. Irreversible thermodynamic consideration

Now, using the irreversible thermodynamic approach we willrederive the general relationship between the electrophoreticmobility and the sedimentation potential obtained earlier by deGroot et al. [26]. We present this well known derivation indetails, since, in Refs. [86,107], another relationship is suggestedthan that obtained by de Groot et al.

Let us consider a linear equation set interrelating thethermodynamic forces and fluxes. From a variety of versions,for convenience of the discussion, we choose the equation setemployed by Ohshima [86]:

dYJl ¼ L11d

YE int þ L12dYg ð3:71Þ

dYupϕðρp � ρLÞ ¼ L21δ→E int þ L22δ

→g ð3:72Þ

where dYup is the particle velocity with reference to the stationaryground; Lmn are the kinetic coefficients. In Eqs. (3.71) and(3.72), instead of the volume mean current density ðhdYJ iÞ andelectric field strength ðhdYE iÞ used by Ohshima [86], we use theuniform current density ðdYJ lÞ within planes AA and BB(Fig. 2) and the internal electric field strength ðdYE intÞ. The abovederived Eqs. (3.23) and (3.27) prove the equivalency of such areplacement.

For Eqs. (3.71) and (3.72), it can be shown that the Onsagercross relationship is valid, i.e.,

L12 ¼ L21 ð3:73Þ


According to Onsager [113,114], to check the validity ofEq. (3.73), one should analyze the expression for the rate, W, ofthe energy dissipation during the irreversible process describedby Eqs. (3.71) and (3.72). The dissipation takes place due to thework produced by the electric and gravity external forces.Consequently, the dissipation rate is a sum of the electric (Joule),Wel, and gravity, Wg, powers that are expressed as

Wel ¼ DuSdJl ¼ DuSdYJld

Yiz

and

Wg ¼ NpdgVpðρp � ρLÞdup ¼ HSϕðρp � ρLÞδ→g : δ→up;

respectively. While obtaining the latter expression, we took intoaccount that dYg ¼ Y

izdg and dYup ¼ Y

izdup. For the system shownin Fig. 2, calculating the dissipation rate, w, per unit volume andusing Eq. (3.1) yield

w ¼ WV

¼ Wel þWg

HS¼ dYE int d d

YJl þ dYg d dYupϕðρp � ρLÞ

¼ →X 1 :

→I 1 þ→

X 2 :→I2

Thus, the dissipation rate, w, can be represented as a diagonalbilinear combination of the thermodynamic forces,

YX 1 ¼ d

YE int

andYX 2 ¼ dYg , and fluxes,

YI 1 ¼ dYJ

land

YI2 ¼ dYupϕðρp � ρLÞ,

that have been chosen while writing the irreversible thermody-namic equation set given by Eqs. (3.71) and (3.72). According toOnsager [113,114], the possibility to express the dissipation rate insuch a diagonal form is a necessary and sufficient condition for thevalidity of cross relationship (3.73) between the coefficients, Lmn,represented in Eqs. (3.71) and (3.72). Next, combining crossrelation (3.73) with Eqs. (3.71) and (3.72), we will obtain therelationship between the sedimentation potential field strength andthe electrophoretic mobility.

Making use of Eq. (3.71), an expression for the sedimen-tation potential field strength, dYEg, which is the internal fieldstrength corresponding to the case of δJ∞=0, takes the form

dYEg ¼ dYE intðdJl¼0Þ ¼ � L12L11

dYg ð3:74Þ

Now, we obtain a convenient expression for the ratio L12/L11represented in Eq. (3.74). To this end, by assuming that dYg ¼ 0,we apply Eqs. (3.71) and (3.72) for describing electrophoresis.Consequently, eliminating dYE int, we derive an expression whichinterrelates the electrophoretic velocity and the ratio L21/L11

dYueph ¼ dYupðdg¼0Þ ¼ 1ϕðρp � ρLÞ

L21L11

dYJldg¼0

¼ Kl

ϕðρp � ρLÞL21L11

dYEextðdg¼0Þ: ð3:75Þ

While obtaining the latter expressions, we employed the defi-nition (3.3) of the external electric field strength, dYEext. By com-bining Eq. (3.75) with Eq. (3.9) (definition of χext) and Eq. (3.73),one obtains the following expression for the ratio L12/L11:

L12L11

¼ L21L11

¼ vextϕρp � ρLK∞ ð3:76Þ




ARTICLE IN PRESS

Consequently, substituting Eq. (3.76) into Eq. (3.74), wearrive at the following relationship

dYEg ¼ �vextϕρp � ρLK∞ δ

→g ð3:77Þ

Thus, Eq. (3.77) yields the expression obtained earlier by deGroot et al. [26] for the sedimentation potential field strength,dYEg. Note, that Eq. (3.77) yields exactly the same relationshipbetween dYEg and χext as Eq. (3.70) which was derived for thevanishingly thin double layer. The irreversible thermodynamicanalysis reproduced above shows that such a relationship is ofgreat generality and is valid for any disperse system geometry,double layer thickness and zeta potential.

Combining Eqs. (3.14) (3.15) and (3.77), it becomes clear thatEq. (3.68), which, in Section 3.4, was derived for the Smolu-chowski limit, is valid for the general case, i.e.,

x ¼ vext=Kl ð3:78Þ

Eq. (3.77) represents the sedimentation potential field strengththrough the electrophoretic mobility calculated per unit externalfield strength, χext. Combining Eqs. (3.10) and (3.77) one canexpress dYEg through the electrophoretic mobility, calculated perunit internal field strength, χint

dYEg ¼ �vintϕρp � ρLKΔP¼0

δ→g ð3:79Þ

Above, Eqs. (3.77)–(3.79) were derived using equation set(3.71) and (3.72) and Onsager's relationship (3.73). Consequent-ly, another expression, which is based on Eqs. (3.71)–(3.73), but,simultaneously, contradicts Eqs. (3.77)–(3.79), should be con-sidered as incorrect. In particular, the irreversible thermodynamicrelationship, which was obtained by Ohshima [86] on the basis ofthe same equations (3.71)–(3.73), contradicts to both Eqs. (3.77)and (3.79). The expression proposed in [86] contains theelectrophoretic mobility which is not specified, χint or χext.However, whichever of the two mobilities is used, the expressionobtained in ref. [86] differs from both Eqs. (3.77) and (3.79).

Now, we will obtain a useful relationship for the externalelectric field which corresponds to the zero electric potentialdifference between the external sides of the disperse system(sedimentation current regime). For such a regime, usingEq. (3.1) yields dYE int ¼ 0. Combining the latter condition withEqs. (3.7), (3.71), (3.72) and (3.76) and realizing that, inEq. (3.71), L11=KΔp=0, one obtains

dYEextðdEint¼0Þ ¼ vextKDP¼0ϕ

ρp � ρLðK∞Þ2 δ→g ð3:80Þ

Thus, Eq. (3.80) yields the external electric field for thesedimentation current regime. Combining Eq. (3.8) and (3.80)yields the following expression

dYEextðdEint¼0Þ ¼ vintϕρp � ρLK∞ δ

→g ð3:81Þ

Remarkably, Eq. (3.81) looks similar to Eq. (3.77). However,there are three important differences between Eqs. (3.77) and(3.81): (i) on its left hand side, Eq. (3.81) contains the external


electric field which corresponds to the sedimentation currentregime whereas Eq. (3.77) contains the internal field corres-ponding to the sedimentation potential (zero current) regime;(ii) on the right hand side of Eq. (3.81), χint is represented insteadof χext in Eq. (3.77) and (iii) signs of the right hand sides differ.

In Section 5.3.3, Eq. (3.81) will be used in the discussion onthe different predictions reported in literature regarding thesedimentation potential field strength.

4. Spherical cell approach

In the present section we consider how the general problemformulation, which was presented in Section 2, is specified whileusing the assumption that the disperse system representative cellis a sphere (spherical cell) containing the continuous medium ofthe disperse system and a particle in the center (Fig. 1). Theparticle and cell radii, a and b, satisfy equality (1.2).

To a large extend, derivations of the governing equations andthe interfacial boundary conditions given in Section 4.1, repeatanalysis of Ohshima et al. [17] who derived the Electrokineticequations that are capable of addressing Electrokinetic Phe-nomena in an axially symmetric system. As for the boundaryconditions imposed at the cell outer boundary, in Section 4.2, wegive original derivations for all of them. In Section 4.3, wedevelop the general algorithm for obtaining the kineticcoefficients defined in Section 3.

Considering the spherical cell as a representative part of thedisperse system, one can expect certain symmetry properties ofthe unknown functions WðYr Þ (equilibrium electric potential),dlkðYr Þ; dYu ðYr Þ and djðYr Þ. In particular, for the sphericalcell, the function WðYr Þ, which describes the electric potentialdistribution in the absence of external influences, should bespherically symmetrical, Ψ (r). For the distributions that arecaused by the pressure and electric potential difference appliedto the disperse system layer as shown in Fig. 2, one can expectexactly the same angular symmetry as in the case of a singleparticle in a uniform field. Consequently, using the sphericalcoordinate system shown in Fig. 5, the unknown functions canbe represented as

WðrÞ ¼ fwðrÞ ð4:1Þ

dlkðr; hÞ ¼ zkFMkðrÞcosðhÞ ð4:2Þ

dj ¼ j⁎ðrÞcosðhÞ ð4:3Þ

dYuðr; hÞ ¼ UrðrÞcosðhÞYir þ UhðrÞsinhY

ih ð4:4Þwhere, using continuity Eq. (2.17), Ur(r) and Uθ(r) areexpressed as

UrðrÞ ¼ � 2r2Y ð4:5Þ

UhðrÞ ¼ 1rdYdr

ð4:6Þ

The unknown functions ψ(r), Y(r), Π⁎(r) and Mk(r) aredetermined by solving the problem formulated below.



Fig. 5. Spherical cell and coordinate system.


ARTICLE IN PRESS

4.1. Governing equations for cell bulk and boundary conditionsat the particle surface

Substituting relationships (4.1)–(4.6) into governingEqs. (2.3),(2.15)–(2.17), after some transformations, one obtains

fr2

ddr

r2dwdr

� �¼ � F2

eRT

Xk

ZkClk expð�fwZkÞ ð4:7Þ

where ζ=ζF/RT;

d2Mk

dr2þ 2

r� Zk f

dwdr

� �dMk

dr� 2r2Mk ¼ 2

RTFDk

fdwdr

Yr2

ð4:8Þ

d2

dr2� 2r2

� �2

Y ¼ Ffgdwdr

Xk

Clk Z2

k expð�Zk fwÞMk ð4:9Þ

While deriving Eq. (4.9), the curl operator was applied toboth sides of Eq. (2.16), and the obtained equation wascombined with Eqs. (4.1)–(4.6). Note that Eqs. (4.7)–(4.9) iscompletely equivalent to the Electrokinetic equations whichhave been derived by Ohshima et al. [17].

Thus, Eqs. (4.7)–(4.9) are the governing equations forobtaining the functionsψ(r),Mk(r), Y(r). The onemore unknownfunction, Π⁎(r), is expressed through ψ(r), Mk (r) and Y(r), as

j⁎ðrÞ ¼ �gddr

d2Ydr2

� 2Yr2

� � F

Xk

Clk Zk ½expð�Zk fwÞ � 1�Mk ð4:10Þ

Eq. (4.10) was derived by combining Eqs. (2.16), (4.1),(4.5), and (4.6). In the next section, Eq. (4.10) will be employedfor obtaining boundary conditions at the cell outer boundary.

At the particle surface, Eqs. (4.7)–(4.9) are subject to theboundary conditions that are obtained by substituting Eqs. (4.1)–(4.6) into Eqs. (2.4), (2.18) and (2.19)

wðaÞ ¼ 1 ð4:11Þ

Y ðaÞ ¼ 0 ð4:12ÞdYdr

ðaÞ ¼ 0 ð4:13Þ

dMk

drðaÞ ¼ 0 ð4:14Þ


Thus, Eqs. (4.11)–(4.14) give the first group of the boundarycondition set at the particle surface. The second group of theboundary conditions,which are imposed at the cell outer boundary,will be discussed next.

4.2. Derivation of the outer cell boundary conditions

To close the formulation of the problem given by Eqs. (4.7)–(4.14), it is necessary to set boundary conditions at the cellboundary. For the equilibrium problem, one should set anequivalent of outer boundary condition (2.5). For the problemintended for obtaining perturbations, the outer cell boundaryconditions should interrelate the distributions to be obtained withthe external influences and system responses. Recall that, for thegeneral problem formulation (Section 2), the external influencesand system responses are set by Eqs. (2.20)–(2.23).

4.2.1. Boundary condition for equilibrium potential distributionNow, we will present the cell model versions of boundary

condition (2.5). It should be noted that Eq. (2.5) reflects the factthat the disperse system layer in Fig. 2 is electroneutral. Otherwise,the potential of a plane parallel layer would not decay at infinity.Consequently, the required boundary condition at the cell outerboundary should set the total electroneutrality of the representativecell. For the spherically symmetrical cell, making use of the Gausstheorem, the electroneutrality condition takes the form

dwdr

ðbÞ ¼ 0 ð4:15Þ

Thus, Eq. (4.15), which sets electroneutrality of the wholedisperse system, is the SCA equivalent of boundary condition(2.5). Condition (4.15) was used by all the authors who analyzedboth the equilibrium and electrokinetic properties of dispersesystems on the basis of the SCA [60–112].

4.2.2. Boundary condition for a given applied voltage:Shilov–Zharkikh–Borkovskaya condition

Here we consider a spherical cell model equivalent of theboundary condition (2.20). Using Fig. 3, the left hand side ofEq. (2.20) can be rewritten as

dlAk � dlBk ¼ Yiz d

YizðdlAk � dlBk Þ

¼ �Yiz d

HSðdlAk YnA þ dlBkYnBÞ

HS

¼ �Yiz d

HV∮SV

dlkYds ð4:16Þ

where SV is the limiting case of the surface formed by SAA, S′SL and SR, when S′ coincides with SBB (Fig. 3). In Section 3.3,we already used SV to derive Eq. (3.25). Similarly to derivationof Eq. (3.25), while obtaining the final expression of Eq. (4.16),the contribution of the integrals over SL and SR into the integralover SV was assumed to be negligible.

In any system, the electrochemical potential, δμk, is acontinuous function of coordinates. The latter enables us toapply the same gradient theorem which was employed in




ARTICLE IN PRESS

Section 3.3 with respect to perturbation of the electric potential,Eq. (3.26). Rewriting Eq. (3.26) with respect to the electro-chemical potential one obtains the relationship

∮SVdlk

Yds ¼ZV

Y∇dlkdV ð4:17Þ

which is valid when SV is a closed surface enveloping volume V.Making use of Eqs. (4.16) and (4.17) one can rewrite boundarycondition (2.20) as

�HhY∇dlkðYr Þi ¼ DuZkF ð4:18Þwhere h N i ¼ R

V N dV �

=V signifies average over the dispersesystem volume V. Recalling the definition of the internal electricfield strength, d

YE int, given by Eq. (3.1), Eq. (4.18) takes the form

hY∇dlkðYr Þi ¼ �dYE intZkF ð4:19Þ

Thus, Eq. (4.19), which interrelates the internal electric fieldstrength with the functiondlkðYr Þ, is an equivalent of boundarycondition (2.20).

Now, we use the major assumption of the SCA: the sphericalcell is a representative part of the disperse system. Consequently,instead of averaging over the disperse system volume, we consideraveraging over the cell volume (see Eq. (1.1)). Using such anassumption, the left hand side of Eq. (4.18) is transformed to

hY∇dlkðYr Þi ¼ hY∇dlkðYr Þicell ¼3

4pb3

ZVcell

Y∇dlkðYr Þdv

¼ 34pb3

∮Scelldlk

Yds ð4:20Þ

where the notation ⟨…⟩cell signifies averaging over the cell volume.For obtaining the final expression in Eq. (4.20), we rewrote thegradient theorem, Eq. (4.17), in terms of the cell volume (Vcell) andboundary (Scell).

CombiningEqs. (4.19) and (4.20) and considering z-componentof all the vectors one obtains

34pb3

∮SbdlkðYr ÞðYds d Y

izÞ ¼¼ �dYE int dYizFZk ¼ �dEintFZk

ð4:21ÞUsing Fig. 5, it can be easily shown that

Yds d Yiz ¼ 2pb2ðYi d Y

nbÞsinðhÞdh ¼ 2pb2cosðhÞsinðhÞdh ð4:22ÞCombining Eqs. (4.21) and (4.22) yields

32b

Z p

�pdlkðb; hÞcosðhÞsinðhÞdh ¼ �FZkdEint ð4:23Þ

Now, substituting δμk(r,θ) in the form given by Eq. (4.2) intoEq. (4.23) we obtain

32b

MkðbÞZ p

�pcos2ðhÞsinðhÞ ¼ �dEint ð4:24Þ

that finally yields

M ðbÞ ¼ �bdE ð4:25Þ
k int

Thus, Eq. (4.25) gives a cell model equivalent of boundarycondition (2.20) which sets a given voltage applied to the dispersesystem or, in other terms, a given internal electric field strength(see Eq. (3.1)).

Substituting Eq. (2.3) into Eq. (1.9) (the Shilov–Zharkikh–Borkovskaya boundary condition) leads to Eq. (4.25) providedthat dYE ¼ dYE int. Hence, Eq. (4.25) is the Shilov–Zharkikh–Borkovskaya boundary condition where the applied field strengthdYE is understood as the internal electric field strength.Consequently, the Shilov–Zharkikh–Borkovskaya boundarycondition is the only correct spherical cell equivalent of theboundary condition (2.20) which sets a given applied voltage.Setting the applied voltage by using other boundary conditionsmay lead to correct predictions of the kinetic coefficients if suchboundary condition does not contradict Eq. (4.25). More detaileddiscussion of the results obtained using other boundary conditionsis given in Section 5.

4.2.3. Two boundary conditions for a given pressure difference:Kuwabara and Ohshima conditions

Now, we will derive the spherical cell model equivalent ofboundary condition (2.21) which imposes a given pressuredifference between the external sides of the disperse system. Itshould be noted that, at the cell boundary, instead of a singlecondition (2.21) at the external boundaries AA and BB (Fig. 2),one should set two boundary conditions. Such a situation occurssince, as stated in Section 2, planes AA and BB are placedsufficiently far from disperse system, d/b→∞. Consequently,boundary condition (2.21) is set at “infinity” and corresponds toan “outer” boundary problem with respect to the particles.While dealing with the outer boundary problems, one shouldomit those particular solutions which, at infinity, lead to adiverging velocity and pressure. While writing the boundaryconditions of the constant pressure or velocity at infinity, suchdiverging particular solutions are omitted. While consideringthe problem inside a closed volume, the correspondingparticular solutions should be retained. Hence, for the case ofthe cell, the closed problem formulation requires an additionalboundary condition.

To derive an expression for the first of the conditions, we willuse the fact that the total external force exerted on the dispersesystem layer shown in Fig. 2 should be zero. Due to the dispersesystem electroneutrality, the electric force does not contribute tothe total applied force. The latter conclusion follows from thefact that the external electric force is defined by the uniformfield which is created by the electric charges outside the dispersesystem.

The total external force becomes zero due to the balancebetween the hydrodynamic force ðDpS d

YizÞ applied to the external

sides of the disperse system (planes AA and BB in Fig. 2), thegravitational force ðHSð1� ϕÞρLδ

→g Þ exerted on the liquid phase,

and the external force ðYGÞ applied to each of the individualparticles. For different experiments, the force YG has differentorigins: it can be applied from the side of vessel walls to keep theparticles composing a diaphragm at rest, or it can be thegravitational force exerted on a settling particle. Within theframework of the SCA, such a force,YG, should be equal for all the




ARTICLE IN PRESS

particles. Consequently, the above mentioned force balance can berepresented as

NpYG þ HSð1� ϕÞρLδ→g þ DpS d Yiz ¼ 0 ð4:26Þ

where Np is the number of particles within the disperse systemlayer, S is the area of the external sides of the disperse system.Realizing that Np=3HS/4πb

3 (the number of the particles is equalto the number of the cells) one obtains

YG ¼ � 4

3pb3½DpH Y

iz þ ð1� ϕÞρLδ→g � ð4:27Þ

Thus, Eq. (4.27) gives expression for the force acting fromoutside on each of the particles.

Now we consider a balance of the external forces exerted onan individual cell. The corresponding balance equation includesthe integral of the viscous stress tensor, d¯r, over the cellboundary, the gravitational force acting on the liquid inside thecell and the force YG. Due to the cell electroneutrality, theexternal electrical force does not contribute to the force balancewhich takes the form

∮Scelld ¯r d

YdS þ 43pb3ð1� ϕÞρLδ

→g þ→

G ¼ 0 ð4:28Þ

where

d ¯r ¼ �dp¯I þ gY∇dYuþ ðY∇dYu Þth i

ð4:29Þ

Combining Eqs. (4.27) and (4.28) leads to the followingcondition

∮Scell

Yiz d d ¯r d YndS ¼ 4

3pb3

DpH

ð4:30Þ

Thus, Eq. (4.30) gives the required interrelation between theapplied pressure and the distributions inside the cell. To obtain amore convenient form of the boundary condition, we considerthe integral in Eq. (4.30). Making use of the substitutions givenby Eqs. (4.2)–(4.6), after some transformations, the integral inEq. (4.30) can be represented as

∮Scell

Yiz d d ¯r d YndS

¼ � 43pb2 P⁎ðbÞ þ ρLbdg þ 2

bg

d2Ydr2

ðbÞ � 2Y ðbÞb2

� � �ð4:31Þ

Detailed derivation of this important identity, Eq. (4.31), isgiven in Appendix B.

Therefore, by substituting Eq. (4.31) into Eq. (4.30) oneobtains

P⁎ðbÞb

þ g2b2

d2Ydr2


� �¼ �Dp

H� ρLdg ð4:32Þ

Eq. (4.32) can be considered as the first of the requiredequivalents of boundary condition (2.21).


To obtain the second equivalent, we consider a chain ofidentities

Dp

H¼ �Y

iz d ðdpA YnA þ dpBYnBÞS

HS

¼ �Yiz d ðdPA

YnA þ dPBYnBÞS

HS� ρLdg

¼ � 1V

Yiz d ∮SV

dPYdS � ρLdg ð4:33Þ

where SV is the limiting case of the surface formed by SAA, S′SL and SR, when S′ coincides with SBB (Fig. 3). In derivation ofEq. (4.33), we used the substitution given by Eq. (2.12) whichwas simplified by taking into account that, at the planes AA andBB, Ck

eq =Ck∞. As well, we neglected by the contribution of the

integrals over SL and SR into the integral over Sv (Fig. 3).Using the gradient theorem, replacing averaging over the

disperse system volume by averaging over the spherical cellvolume and substituting δΠ in the form given by (4.3), aftersome transformations (see Appendix C), we arrive at theequality which is a hydrodynamic analogue of the Shilov–Zharkikh–Borkovskaya boundary condition (4.25).

1V

Yiz d ∮SdP

YdS ¼ P⁎ðbÞ

bð4:34Þ

It should be noted that in Section 4.2.2, the Shilov–Zharkikh–Borkovskaya condition (4.25) was derived assumingthe continuity of the electrochemical potential at the interface.In Appendix C, a similar condition, Eq. (4.34), was obtainedwithout such an assumption. Hence, Eq. (4.25) is also validwhen the chemical potential is not continuous at the interface.

Combining Eqs. (4.33) and (4.34) we arrive at the equation

P⁎ðbÞb

¼ �DpH

� ρLdg ð4:35Þ

which yields the second cell model equivalent of boundarycondition (2.21).

Thus, Eqs. (4.32) and (4.35) give the two required boundaryconditions at the cell outer boundary. Usually, more convenientforms of these conditions are employed. Comparing conditions(4.32) and (4.35), it becomes clear that the conditions differfrom each other due to the presence of the second term on theleft hand side of Eq. (4.32), only. Hence, conditions (4.32) and(4.35) can be simultaneously satisfied if (and only if) this termbecomes zero, i.e., when

d2Ydr2


¼ 0: ð4:36Þ

Substituting Eqs. (4.4)–(4.6) into the Kuwabara boundarycondition (1.12), one obtains an expression which exactlycoincides with Eq. (4.36). Thus, Eq. (4.36) is the Kuwabaraboundary condition which, in the original paper [51], wassuggested on the basis of some convincing qualitative reasons.Above, we presented an original derivation of the Kuwabaracondition.




ARTICLE IN PRESS

When condition (4.36) is satisfied, Eq. (4.32) takes the sameform as Eq. (4.35). Using Eq. (4.10), Eq. (4.35) can be rewrittenin a more convenient form using unknowns Y and Mk

gdðd2Y=dr2 � 2Y=r2Þ

drðbÞ

þFXk

Clk Zk exp½�Zk fwðbÞ� � 1

n oMkðbÞ

¼ DpH

þ ρLdg

� �b ð4:37Þ

We will refer to Eq. (4.37) as the Ohshima boundarycondition. In Ohshima's papers [81–83,86] where such acondition is presented in the form given by Eq. (4.37), it isinterpreted as a condition of zero total force exerted on the cell(or entire disperse system). Since, in his analysis, Ohshima usesthe Kuwabara condition (4.36), such an interpretation is correct.

Thus, Eqs. (4.36) and (4.37) give two boundary conditions(respectively, Kuwabara's and Ohshima's conditions) being aspherical cell equivalent to the boundary condition of a givenpressure difference applied to the disperse system, Eq. (2.21).

4.2.4. Boundary condition for a given external flow velocity:the first Happel condition

Now, we obtain a boundary condition which sets a givenflow velocity through the planes AA and BB, i.e., the sphericalcell model version of Eq. (2.22).

In Section 3.3, we obtained expression (3.24) whichinterrelates a given velocity in the planes AA and BB and thevolume mean local velocity. Making use the assumption givenby Eq. (1.1) (the average over the disperse system volume isequal to the average over the spherical cell volume), Eq. (3.24)can be rewritten as

dul ¼¼ Yiz d hdYu i ¼ Y

iz d hdYu icell ð4:38ÞThus, Eq. (4.38) interrelates the uniform velocity at the

external boundaries of the disperse system, δu∞, and the localvelocity distribution inside the cell,dYu ðYr Þ.

With the representative cell being the spherical shell shownin Fig. 1, we obtain a more convenient form of Eq. (4.38). Tothis end, we represent dYu ðYr Þ in the form given by Eq. (4.4).Making use of Eqs. (4.5) and (4.6), Eq. (4.4) can be rewritten as

dYu ðr; hÞ ¼ UrðrÞcosðhÞYir �

12r

ddr

½r2UrðrÞ�sinhYih ð4:39Þ

Substituting Eq. (4.39) into Eq. (4.38) and averaging over thespherical cell volume yields

hdYu icell dYi z

¼ 2p4pb3=3

Z b

a

Z p

�pUrðrÞcos2ðhÞ þ 1

2rddr

½r2UrðrÞ�sin2h� �

r2

� sinðhÞdhdrð4:40Þ

In Eq. (4.40), while choosing a as the lower limit of theexternal integral, we took into account that, in the reference


system linked to the solid phase, the local velocity is zero insidethe particle, i.e., for rba.

While obtaining the integral in Eq. (4.40), it should be takeninto account that, at the particle surface, dYu d Yr ¼ 0; i.e., Ur (a)=0. Consequently, from Eqs. (4.38) and (4.40), one obtains asimple result

UrðbÞ ¼ dul ð4:41ÞThus, Eq. (4.41) gives the required boundary condition at the

cell outer boundary. Remarkably, substituting Eq. (4.4) into thefirst Happel boundary condition, Eq. (1.10), leads to Eq. (4.41).Thus, Eq. (4.41) gives the first of the Happel boundaryconditions. The above proposed original derivation, providesthe meaning of Happel's condition: it is a spherical cell modelequivalent of condition (2.22) which sets a given velocity, δu∞,at the planes AA and BB.

For practical calculations, it is convenient to rewrite boundarycondition (4.41) in terms of the function Y(r). Consequently,using Eq. (4.5), boundary condition (4.41) becomes

Y ðbÞ ¼ � b2dul2

ð4:42Þ

Thus, Eq. (4.42) is the cell model version of boundarycondition (2.2) that imposes a given velocity, δu∞, at the planesAA and BB..

4.2.5. Boundary condition for a given electric current density:Generalized Levine–Neale Condition

In this subsection we will derive the cell model equivalent ofboundary condition (2.23) which sets a given value of thecurrent density at the planes AA and BB (Fig. 2). Obtainingsuch a boundary condition is quite similar to the abovederivation of Eq. (4.41) which sets a given velocity. Such asimilarity takes place since the local current density, dYJ ,satisfies continuity equation

Y∇ d dYJ ¼ 0 ð4:43ÞAs well, for the axial symmetrical problem under consider-

ation, YJ can be represented in the form similar to Eq. (4.4)

dYJ ðr; hÞ ¼ IrðrÞcosðhÞYir þ IhðrÞsinhY

ih ð4:44ÞConsequently, the derivation is a repeat of all the steps

presented in Section 4.2.4, and leads to a complete analog ofEq. (4.41)

IrðbÞ ¼ dJl ð4:45Þ

Let us now represent Eq. (4.45) in terms of the unknowns Yand Mk. Consequently, using the expression presented for thekth ion flux, Eq. (2.8), the electric current is given by

dYJ ¼Xk

ZkFdYjk ¼ �F

Xk

DkCeqk Zk

RT

Y∇dlk þ dYu F

Xk

ZkCeqk

ð4:46Þ




ARTICLE IN PRESS

Combining Eqs. (4.1), (4.2), (4.4)–(4.6), (4.45), (4.46),boundary condition (4.46) becomes

�FXk

Clk Zk

FZkDk

RTdMk

drðbÞ þ 2Y ðbÞ

b2

� exp½�fwðbÞZk � ¼ dJl

ð4:47Þ

Thus, Eq. (4.47) gives a cell model version of boundarycondition (2.23), which, in the general problem formulation,imposes a given value of the external current density.

Recalling the definition of the external electric field, dYEext,Eq. (3.3), it is possible to present an equivalent form of theboundary condition (4.47). Accordingly, combining Eqs. (3.3)(3.4) and (4.47) yields

� 1PkDkC

lk Z2

k

Xk

Clk Zk DkZk

dMk

drðbÞ þ 2

RTF

Y ðbÞb2

�

�exp½�fwðbÞZk � ¼ dEext ð4:48Þ

Eq. (4.48) gives a convenient boundary condition when theexternal electric field, dYEext, is set instead of the electric current.

When the applied field strength is understood in Eq. (1.6) tobe the external electric field strength, dYEext, as it will be shownin Section 5.3, the original Levine–Neale boundary condition(1.6), can be considered as a particular case of Eq. (4.48).Consequently, we will refer to more general relationship (4.48)as the generalized Levine–Neale boundary condition.

In the above presented analysis, six relationships (4.15), 4(4.25), (4.36), (4.37), (4.42) and (4.47) or (4.48), form a set ofconditions that are required to close the boundary problems foraddressing Electrokinetic Phenomena. While dealing withdifferent electrokinetic effects, two of these relationships (theelectroneutrality and Kuwabara conditions), are alwaysemployed in unaltered forms as given by Eqs. (4.15) and(4.36), respectively. The other four relationships (4.25), (4.37),(4.42) and (4.47) or (4.48) are simplified by accounting forspecial conditions that are imposed while studying a particularElectrokinetic Phenomenon.

Dealing with particular Electrokinetic Phenomena, similarlyto Eqs. (2.20)–(2.23), some of relationships (4.25), (4.37),(4.42) and (4.47) or (4.48) are considered as boundaryconditions for corresponding boundary value problems. Con-sequently, other relationships from Eqs. (4.25), (4.37), (4.42)and (4.47) or (4.48) are employed for predicting systemresponses.

4.3. Expressions for kinetic coefficients

Now, we present algorithms for obtaining kinetic coefficientsthat describe a disperse system behavior in some specific hy-drodynamic and electric regimes. To this end, we consider solu-tion of a closed problem given by Eqs. (4.7)–(4.9) subject toboundary conditions at the particle surface and the outer boundaryof the spherical cell. At the particle surface, Eqs. (4.7)–(4.9) aresubject to boundary conditions (4.11) (4.12) (4.13) (4.14). At


the cell outer boundary, we will use electroneutrality condition(4.15), the Shilov–Zharkikh–Borkovskaya and Kuwabara con-ditions given by Eqs. (4.25) and (4.36), respectively. As well,we use the Ohshima conditions, Eq. (4.37) which, by making useof Eq. (4.25), can be rewritten in more convenient form

gbdðd2Y=dr2 � 2Y=r2Þ

drðbÞ � FdEint

Xk

Clk Zkexp½�Zk fwðbÞ�

¼ �DpH

þ ρLdg ð4:49Þ

Due to the linearity of Eqs. (4.8) and (4.9), and boundaryconditions (4.11) (4.12) (4.13) (4.14) (4.35) (4.36) (4.48), thefunctions Mk(r) and Y(r) can be represented in the form

MkðrÞ ¼ mEk ðrÞdEint þ mp

k ðrÞDpH

þ ρLdg

� �

Y ðrÞ ¼ yEðrÞdEint þ y pðrÞ DpH

þ ρLdg

� �8>><>>: ð4:50Þ

where the functions mkE(r), mk

p(r) yE(r) and yp(r) depend on thedisperse system properties only.

Thus, solving the problem given by Eqs. (4.7)–(4.9) subjectto the boundary conditions (4.11) (4.12) (4.13) (4.14) (4.15),(4.25), (4.36), and (4.49) and comparing the solution withEq. (4.50) one obtains 2N+3 functions (N is the number of ionicspecies), namely, ψ(r), mk

E(r), mkp(r), yE(r) and yp(r). Any of

the kinetic coefficients defined in Section 2.5 can be expressedthrough the values and derivatives of these functions at the cellouter boundary (i.e. for r=b). Corresponding expressions willbe derived next.

4.3.1. Zero gravitation and pressure differences (δg=0, Δp=0)Both classical effects, electroosmosis and electrophoresis,

are observed under conditions of δg=0 and Δp=0. Theconductivity attributed to such a regime is determined bycombining its definition (3.6) with Eq. (4.47)

KDp¼0 ¼ �FXk

Clk Zk

ZkDkFRT

dMk

drðbÞ þ 2Y ðbÞ

b2

� exp½�fwðbÞZk �

dEint

ð4:51Þ

Realizing that, in Eq. (4.51), the functionsMk(r) and Y(r) areattributed to the case of δg=0 and Δp=0, and taking intoaccount Eq. (4.50) we arrive at the following expression for theconductivity

KDp¼0 ¼ �FXk

Clk Zk

ZkDkFRT

dmEk

drðbÞ þ 2yEðbÞ

b2

�

� exp½�fwðbÞZk � ð4:52Þ

Calculating the liquid velocity, δu∞, per unit external orinternal electric field strength and changing the velocity sign(see Eq. (2.25)) one obtains both the electrophoretic mobilities,




ARTICLE IN PRESS

χint or χext, defined by Eqs. (3.8)–(3.9), respectively. Conse-quently, for δg=0 and Δp=0, combining Eqs. (2.25), (3.8),(4.42) and (4.50) yields

vint ¼ 2yEðbÞ=b2 ð4:53Þ

Subsequently, combining Eqs. (3.10), (4.52), and (4.53)yields

vext ¼ �2yEðbÞ

Xk

DkZ2k C

lk

Xk

Clk Zk ZkDkb

2 dmEk

drðbÞ þ 2

RTF

yEðbÞ�

exp½�fwðbÞZk �

ð4:54ÞWe also present expression for the electroosmotic coeffi-

cient, λ. Combining its definition (3.11) with Eq. (4.54)

k ¼ 2yEðbÞFXk

Clk Zk

FZkDk

RTb2

dmEk

drðbÞ þ 2yEðbÞ


ð4:55Þ

Thus, Eqs. (4.52)–(4.55) give the expression for the kineticcoefficients describing the behavior of the disperse systemunder zero gravitation and pressure difference conditions.

4.3.2. Zero gravitation and zero velocity (δg=0, δu∞=0)Regime of δg=0 and δu∞=0 takes place when an electric

current is passed through a diaphragm separating two compart-ments of a “closed” electrochemical cell. For such a regime wepresent expressions for the conductivity Kδu∞=0 and theelectroosmotic pressure coefficient α defined by Eq. (3.12).

Expression for the conductivity, Kδu∞=0, can be obtainedfrom Eq. (4.51) assuming that δu∞=0. According to Eq. (4.42),for such a regime, Y(b)=0. Consequently, one obtains

Kdul¼0 ¼ � F2

RT

Xk

Clk Z2

k DkdMk

drðbÞexp½�fwðbÞZk �

dEint: ð4:56Þ

Realizing that, in Eq. (4.56), the function Mk(r) is attributedto the case Y(b)=0, a simple rearrangement of Eq. (4.50) yields

dMk

drðbÞ ¼ dmE

k

drðbÞ � dmp

k

drðbÞ y

EðbÞypðbÞ

� dEint: ð4:57Þ

Hence, the required form of the expression for conductivityis given by

Kdul¼0 ¼ � F2

RT

Xk

Clk Z2

k DkdmE

k

drðbÞ � dmp

k

drðbÞ y

EðbÞypðbÞ

�

� exp½�fwðbÞZk �: ð4:58Þ


Combining the definition of the electroosmotic pressurecoefficient, α, given by Eq. (3.12) with Eqs. (4.50) and (4.58)and taking into account that Y(b)=0 and δg=0, one obtains

a ¼ DpdJl

� �dg¼0dul¼0

¼ DpKdEint

� �dg¼0dul¼0

¼ � yEðbÞKdul¼0ypðbÞH

ð4:59ÞThus, Eqs. (4.58) and (4.59) give expressions for, respec-

tively, the conductivity and the electroosmotic coefficient thatare measured in a closed cell.

4.3.3. Zero electric current regime (δJ∞=0)Now, we consider how to obtain the streaming (sedimenta-

tion) potential coefficient, ω, and the hydraulic permeability, γ,given by Eqs. (3.16) and (3.17), respectively. To obtain ω, thesolution represented in the form (4.49) should be substitutedinto boundary condition (4.47), specified for the case of δJ∞=0.Consequently, using Eq. (3.16) one obtains

x ¼ �

Xn

Clk Zk

FZkDk

RT

dmpk

drðbÞ þ 2ypðbÞ

b2


Xn

Clk Zk

FZkDk

RTdmE

k

drðbÞ þ 2yEðbÞ

b2


ð4:60ÞFor obtaining the hydraulic permeability attributed to the

zero current case, we combine the definition of γ, Eq. (3.17),the definition of ω, Eq. (3.16), the first Happel boundarycondition, Eq. (4.43), and the form of solution given by (4.50).Consequently, one obtains

gdJl¼0dg¼0

¼ � 2Y ðbÞb2ðρLdg þ Dp=HÞ ¼ �2

yEðbÞxþ ypðbÞb2

ð4:61ÞBy using Eq. (4.53), the expression for the hydraulic

permeability is rewritten as

gdJl¼0dg¼0

¼ �vintx� 2ypðbÞb2

: ð4:62Þ

Thus, Eqs. (4.60)–(4.62) give the expressions for streaming(sedimentation) potential coefficient and the hydraulic perme-ability measured for the case of zero electric current.

5. Description of the Electrokinetic Phenomena at low zetapotentials

In this Section, on the basis of the development presented inSection 4, we will derive approximate expressions for thekinetic coefficients using the method of a small perturbationparameter. As a small perturbation parameter, we will use thenormalized zeta potential, ζ. This parameter is represented inEqs. (4.7)–(4.9) and boundary conditions (4.37) and (4.48).

Using ζ as a small parameter, for any of the kinetic coefficients,one can predict a required number of terms in the Taylor series




ARTICLE IN PRESS

expansion by powers of ζ. Here, we will describe a generalscheme for obtaining the consecutive terms in Taylor expansionof the kinetic coefficients defined in Section 2.5. As well, we willderive the final expressions which account for the zero and firstorder terms in such a series. Finally, we compare the obtainedresults with predictions of other studies reported in the literature.

5.1. Method of small parameter perturbations

Following a standard scheme we represent each of theunknown functions in Eqs. (4.7)–(4.9) subject to the boundarycondition (4.11) (4.12) (4.13) (4.14) (4.15), (4.25), (4.36), and(4.49) as expansions by powers of ζ.

w ¼ wð0Þ þ fwð1Þ þ NMk ¼ M ð0Þ

k þ fM ð1Þk þ N

Y ¼ Y ð0Þ þ fY ð1Þ þ N

8><>: ð5:1Þ

Substituting Eq. (5.1) into Eqs. (4.7)–(4.9), (4.11)–(4.15),(4.25) (4.36), and (4.49) and collecting the terms proportional tothe same power of ζ, the initial problem formulation is decomposedinto a chain of problems which should be solved in turn forobtaining subsequent terms in the expansions given by Eq. (5.1).

Using the above described method, we will derive approxi-mate expressions for the functions ψ, (r, ζ ),Mk (r, ζ ) and Y(r, ζ).Interrelating the obtained expression with (4.50) will give us thefunctions mE

k(r, ζ), mpk(r, ζ) y

E(r, ζ) and y p(r, ζ). Consequently,using the expressions for mE

k(r, ζ), mpk(r, ζ) y

E(r, ζ) and y p(r, ζ),we will determine all the kinetic coefficients, which, in Section4.3, are expressed through the values that the above functions andtheir derivatives take at the cell outer boundary.

5.1.1. Obtaining the zero order termsThe scheme described above leads to the following reduction

of Eqs. (4.7)–(4.9)

1r2

ddr

r2dwð0Þ

dr

!¼ j2wð0Þ ð5:2Þ

d2M ð0Þk

dr2þ 2

r

dM ð0Þk

dr� 2r2M ð0Þ

k ¼ 0 ð5:3Þ

d2

dr2� 2r2

� �2

Y ð0Þ ¼ 0 ð5:4Þ

Boundary conditions (4.11) (4.12) (4.13) (4.14) (4.15),(4.25), (4.36), and (4.49) for zero order terms become

wð0ÞðaÞ ¼ 1 ðaÞ; Y ð0ÞðaÞ ¼ 0 ðbÞ; dY ð0Þ

drðaÞ ¼ 0 ðcÞ;

dM ð0Þk

drðaÞ ¼ 0 ðdÞ; dwð1Þ

drðbÞ ¼ 0 ðeÞ; M ð0Þ

k ðbÞ ¼ �bdEint ðfÞ;d2Y ð0Þ

dr2ðbÞ � 2Y ð0ÞðbÞ

b2¼ 0 ðgÞ;

gdðd2Y ð0Þ=dr2 � 2bY ð0Þ=r2Þ

drðbÞ ¼ Dp

Hþ ρLdg

� �b ðhÞ ð5:5Þ


Thus, Eqs. (5.2)–(5.4) subject to boundary conditions (5.5)yields a closed problem for obtaining the zero order terms, ψ(0),Mk

(0) and Y (0) in the expansion (5.1). Solution of such a problemtakes the form

wð0ÞðrÞ ¼ ardsinh½jðb� rÞ� � jbcosh½jðb� rÞ�sinh½jðb� aÞ� � jbcosh½jðb� aÞ� ð5:6Þ

M ð0Þk ðrÞ ¼ M ð0ÞðrÞ ¼ �dEintr

1þ a3=2r3

1þ ϕ=2ð5:7Þ

Y ð0ÞðrÞ ¼ 19g

DpH

þ ρLdg

� �bZ r

a

r2

x� x2

r

� �x2

b� b2

x

� �dx

ð5:8ÞIn Eqs. (5.6) and (5.7), the volume fraction, ϕ and the Debye

parameter κ are defined by Eqs. (1.2) and (1.13), respectively.Note that, for all the ionic species, the zero order term, Mk

(0)(r)given by Eq. (5.7) is same, M(0)(r).

5.1.2. Obtaining the first order termsCollecting the terms which, after substituting Eq. (5.1) into

Eqs. (4.7)–(4.9), (4.11)–(4.15), (4.25), (4.36), and (4.49),become proportional to the first power of ζ, yields the followingequations

1r2

ddr

r2dwð1Þ

dr

!� j2wð1Þ ¼ �j2ðwð0ÞÞ2

Xk

Z3k C

lk

2Xk

Z2k C

lk

ð5:9Þ

d2M ð1Þk

dr2þ 2

r

dM ð1Þk

dr� 2M ð1Þ

k

r2¼ dwð0Þ

dr2RTFDk

Y ð0Þ

r2þ Zk

dM ð0Þ

dr

� ð5:10Þ

d2

dr2� 2r2

� �2

Y ð1Þ ¼ eRTFg

j2M ð0Þ dwð0Þ

drð5:11Þ

subject to the boundary conditions

wð1ÞðaÞ ¼ 0; Y ð1ÞðaÞ ¼ 0;dY ð1Þ

drðaÞ ¼ 0;

dM ð1Þk

drðaÞ ¼ 0;

dwð1Þ

drðbÞ ¼ 0; M ð1Þ

k ðbÞ ¼ 0;d2Y ð1Þ

dr2ðbÞ � 2Y ð1ÞðbÞ

b2¼ 0;

dðd2Y ð1Þ=dr2 � 2Y ð1Þ=r2Þdr

ðbÞ þ eRTgF

j2wð0ÞðbÞdEintb ¼ 0

dðd2Y ð1Þ=dr2 � 2Y ð1Þ=r2Þdr

ðbÞ þ eRTgF

j2wð0ÞðbÞdEintb ¼ 0

ð5:12Þ




ARTICLE IN PRESS

Solution of the problem given by Eqs. (5.9)–(5.11) subject toboundary conditions (5.12) takes the form

wð1ÞðrÞ ¼ j

Xk

Z3k C

lk

2Xk

Z2k C

lk

1rd

sinh½jðr � aÞ�jbcosh½jðb� aÞ� � sinh½jðb� aÞ�

�

�Z b

an½jbcosh½jðb� nÞ� � sinh½jðb� nÞ��

�½wð0ÞðnÞ�2dn� 1r

Z r

ansinh½jðr � nÞ�½wð0ÞðnÞ�2dng

ð5:13Þ

M ð1Þk ðrÞ ¼ r

3 ½ Z r

a

dwð0Þ

dx1� x3

r3

� �2RTFDk

Y ð0Þ

x2þ Zk

dM ð0Þ

dx

� �dx

� 1þ a3=2r3

1þ ϕ=2

Z b

a

dwð0Þ

dx1� x3

b3

� �

� 2RT

FDk

Y ð0Þ

x2þ Zk

dM ð0Þ

dx

� �dx� ð5:14Þ

Y ð1ÞðrÞ ¼ eRT9gF

j2Z r

a

r2

x� x2

r

� �½ � dEintwðbÞ x2 � b3

x

� �

þZ x

b

x2

n� n2

x

� �M ð0ÞðnÞ dw

dndn�dx ð5:15Þ

Thus Eqs. (5.6)–(5.8), (5.13)–(5.15) give the coefficients atthe zero and first order terms in the expansions of the functionsψ(r, ζ ), Mk(r, ζ) and Y(r, ζ) in the Taylor series by powers of ζ.In the next section, using such approximate expressions we willdetermine the kinetic coefficients.

5.2. Prediction of the kinetic coefficients for low zeta potentials

Prior to obtaining the kinetic coefficients, we determine thefunctions mE

k (r), mpk(r) yE(r) and y p(r). Recall that all the

kinetic coefficients defined in Section 3.3 are expressed throughthese four functions. Combining Eqs. (4.50), (5.1), (5.6)–(5.8),(5.13)–(5.15), we arrive at the following expressions

mEk ðrÞ ¼ �r

1þ a3=2r3

1þ ϕ=2

� fZk3ð1þ ϕ=2Þ r

Z r

a

dwð0Þ

dx1� x3

r3

� �1� a3

x3

� �dx

ð5:16Þ

mpk ¼

2fb27Dkg

r½ � Z r

a

dwð0Þ

dx1� x3

r3

� �1

x2

Z x

a

x2

n� n2

x

� �

� b2

n� n2

b

� �dndxþ 1þ a3=2r3

1þ ϕ=2

Z b

a

dwð0Þ

dx1� x3

b3

� �1x2

Z x

a

� x2 � n2� �

b2 � n2� �

dndx�
n x n b ð5:17Þ

yEðrÞ ¼ ef9g

j2bZ r

a

r2

x� x2

r

� �½wð0ÞðbÞ b2

x� x2

b

� �� 11þ ϕ=2

�Z x

b

x2

n� n2

x

� �nb

1þ a3

2n3

� �dwð0Þ

dndn�dx ð5:18Þ

yp ¼ � b9g

Z r

a

r2

x� x2

r

� �b2

x� x2

b

� �dx ð5:19Þ

Thus, Eqs. (5.16)–(5.19) give approximate expressions forall the functions that are required for obtaining kineticcoefficients. Note that, in Eqs. (5.16)–(5.19), we omitted theterms having order O(ζ2). Next, using Eqs. (5.16)–(5.19) andthe general expressions presented in Section 4.3, we will obtainthe kinetic coefficients.

5.2.1. Conductivity at low zeta potentialsConsidering expansion of conductivity by powers of ζ and

accounting for zero and linear terms only, one can observe thatconductivities attributed to zero pressure difference, KΔp=0,given by Eq. (4.52) and to zero velocity, Kδu∞=0, given byEq. (4.58) become equal.

KDp¼0 ¼ Kdul¼0 þ Oðf2Þ ¼ K þ Oðf2Þ ð5:20ÞEq. (5.20) becomes clear while combining Eq. (4.52) (or

Eq. (4.58)) with Eqs. (2.2), (5.1), (5.16)–(5.19) and retaining onlythe zero and first terms in the expansion of the conductivities byζ. Accordingly, the expression for the conductivity, K, which iscommon for both the hydraulic regimes, becomes

K ¼ � F2

RT fXk

Clk Z2

k DkdmEð0Þ

k

drðbÞ þ f

Xk

Clk Z3

k Dk

� wð0ÞðbÞ dmEð0Þk

drðbÞ � dmEð1Þ

k

drðbÞ

" #g ð5:21Þwhere mk

E(0) and mkE(1) are given by, respectively, the first and

second terms on the right hand side of Eq. (5.16). Consequently,combining Eqs. (5.16) and (5.21), after some transformations,yields

K ¼ Kl 1� ϕ1þ ϕ=2f1� f½wð0ÞðbÞ þ ϕLðκa;ϕÞ�

Xn

Z3k C

∞k DkX

n

Z2k C

∞k Dkgð5:22Þ

where K∞ is given by Eq. (3.4) and L(κa,ϕ) takes the form

Lðja;ϕÞ ¼ � 1ð1� ϕÞð1þ ϕ=2Þ

Z b

a

12þ r3

a3

� �1� a3

r3

� �dwð0Þ

drdr

ð5:23ÞThe conductivity given by Eqs. (5.22) and (5.23) differs from

that predicted in Refs. [81,86,108,109], due to the presence ofthe term ψ(0)(b) in the square brackets on the right hand sideof Eq. (5.22). Using (5.6), it can be shown that ψ(0)(b) → 0for κ(b− a)→∞, and ψ(0)(b)→1 for κ(b−a)→0. Under the




ARTICLE IN PRESS

assumption κ(b−a) >>1 employed in [81,86,108,109], theterm ψ(0)(b) can be omitted for being small

wð0ÞðbÞ ¼ Ofexp½�jðb� aÞ�g ð5:24Þ

The curves plotted in Fig. 6 for different volume fractionsillustrate how the ratio of two terms in the square brackets inEq. (5.22) varies with κa. Remarkably, even for relatively highκa=10, at ϕ=0.1, the term omitted in [81,86,108,109] is largerthan the retained term (point A). For ϕ=0.3 (point B) theomitted term substantially exceeds the retained term.

5.2.2. Electrophoretic mobility at low zeta potentialsNow we obtain the electrophoretic mobility calculated per

unit external electric field strength, χext, given by Eq. (4.54).Combining Eqs. (4.54), (5.16)–(5.19) and retaining the terms oforder of O(ζ ), one obtains

vext ¼29efgb

j21þ ϕ=21� ϕ

� ½wð0ÞðbÞZ b

a

b2

x� x2

b

� �2

dx

� 11þ ϕ=2

Z b

a

b2

x� x2

b

� �Z x

b

x2

n� n2

x

� �nb

1þ a3

2n3

� �

� dwð0Þ

dndn� ð5:25Þ

Using Eq. (5.2) and boundary conditions for the functionψ(0)(r) (ψ(0)(a)=1 and dψ(0)(b)/dr=0, see Eq. (5.5)), after sometransformations, we arrive at the following final expression forthe electrophoretic mobility

vext ¼ � 23

efgð1� ϕÞ

Z b

af1� a

r

� �3þ 32

ar

� �5þϕ10

� 1� 10ra

� �3�6

ar

� �5� g dwð0Þ

drdr ð5:26Þ

Remarkably, the right hand side of Eq. (5.26), which wasobtained using the Shilov–Zharkikh–Borkovskaya boundarycondition, Eq. (4.25), exactly coincides with the expression

Fig. 6. Contribution of correcting term into expression for disperse systemconductivity.


derived by Levine and Neale [74], who used the Henry assump-tions [6] and the electrochemical outer boundary conditions in theform given by Eq. (1.5). On the basis of the general approachemployed in the present paper, the same result was rederived byOhshima [78,86] who used the Levine–Neale outer boundarycondition in the form given by Eq. (1.6). In Section 5.3, we willexplain such an agreement between the predictions from thedifferent models.

In Section 5.3.2, it will be shown that the above observedequivalency of the models occurs in obtaining χext within theframework of the linear approximation in terms of ζ. Whileconsidering the non-linear dependency of the electrophoreticmobilities on the zeta potential, the Levine–Neale boundarycondition leads to incorrect predictions of the electrophoreticmobility except for the asymptotic behavior at ϕ→0.

It should be additionally stressed that, above, Eq. (5.26) wasobtained for addressing χext (not χint). Therefore, the resultgiven by Eq. (5.26) differs from the electrophoretic mobilitypredicted by Ding and Keh [106] and Carrique et al [108] whoalso used the Shilov–Zharkikh–Borkovskaya condition. Carri-que et al. [108] determined the electrophoretic mobility as aproportionality coefficient between the drift velocity and theelectric field strength represented in the Shilov–Zharkikh–Borkovskaya condition given by Eq. (1.9). Since the correctunderstanding of the Shilov–Zharkikh–Borkovskaya boundarycondition is that the internal electric field strength is representedin Eq. (1.9), (see the derivation given in Section 4.2.2), then theresult of Carrique et al yields χint.

Combining Eqs. (3.10) and (5.22) and retaining only the firstorder terms one can express χint as

vint ¼1� ϕ

1þ ϕ=2vext ð5:27Þ

A relationship similar to Eq. (5.27)was empirically establishedby Carrique et al [108]. Instead of χint on the left hand side ofEq. (5.27), the relationship of Carrique et al contains theelectrophoretic mobility which was numerically obtained by theauthors of [108] using the Shilov–Zharkikh–Borkovskayaboundary condition. Instead of χext represented in Eq. (5.27),the expression of ref. [108] contains the electrophoretic mobilitypredicted in Refs. [74,78,86] where the authors do not distinguishbetween two types of the electrophoretic mobility discussed inSections 3.1 and 3.2.

As stated above, while using the Shilov–Zharkikh–Borkovskaya condition, Carrique et al [108] computed theelectrophoretic mobility per unit internal field strength, χint.According to the analysis conducted in the present subsection,the expression obtained in [74,78,86] describes the electropho-retic mobility per unit external field strength, χext. Thus, theempirical expression of Carrique et al [108] is completelyequivalent to Eq. (5.27) derived in the present subsection.

Let us now consider the Smoluchowski limit, κ→∞, for χextgiven by Eq.(5.26). Analysis of Eq. (5.6), which describes thefunction ψ(0)(r), shows that, at κ→∞, dψ/dr≠0 when r/a→1,only. Hence, while considering the Smoluchowski limit for theintegral in Eq. (5.26), one can set r/a=1 in the polynomial



d


ARTICLE IN PRESS

expression under the integral. The latter enables the followingreduction of Eq. (5.26)

limjaYl

vext ¼ � 23

efgð1� ϕÞ lim

jYl ½Z b

a

32ð1� ϕÞ dψ

ð0Þ

drdr�

¼ ðεζ=ηÞfΨ ð0ÞðaÞ � limκ→∞

½Ψ ð0ÞðbÞ�g ¼ εζ=η ¼ χSmext

ð5:28ÞWhile obtaining Eq. (5.28), we used boundary condition

(5.5)a. Thus, χext given by Eq. (5.26) satisfies the Smolu-chowski principle. At the same time, according to Eq. (5.27),the limiting expression for χint is

limjYl

vint ¼1� ϕ

1þ ϕ=2efg: ð5:29Þ

Thus, the above analysis yields an explanation of the resultsdiscussed in Refs.[106,108] where the authors observed that, atκa→∞, the limiting value of the electrophoretic mobilitydepends on the volume fraction. The explanation is that, whileusing the Shilov–Zharkikh–Borkovskaya boundary condition,the authors determined χint which should behave according tothe asymptotic expression given by Eq. (5.29).

5.2.3. Electroosmotic and electroosmotic pressure coefficientsNow we present results for two kinetic coefficients that are

easily expressed through the electrophoretic mobility and theconductivity.

Combining the definition of the electroosmotic coefficient,λ, given by Eq. (3.11) with Eq. (5.26), one obtains

k ¼ 23

efgð1� ϕÞK∞

Z b

a

�f1� ar

� �3þ 32

ar

� �5þϕ10

1� 10ra

� �3� a

r

� �5� g dwð0Þ

drdr

ð5:30ÞIt should be noted that exactly the same prediction for λ can be

obtained by combining Eqs. (5.16)–(5.19), (3.55) and retainingthe linear terms with respect to ζ.

For obtaining the electroosmotic pressure coefficient, α,combining Eqs. (4.53), (4.59) and (5.27) gives

a ¼ � b2

2vint

Kdul¼0 ypðbÞH ¼ � b2

2vext

KlypðbÞ ð5:31Þ

Finally, making use of Eqs. (5.19) and (5.26) we arrive at theresult

a ¼ � 3efKlð1� ϕÞ

�

Z b

a1� a

r

� �3þ 32

ar

� �5þϕ10

1� 10ra

� �3�6

ar

� �5� � �dwð0Þ

drdr

bZ b b� x2� �2

dx
a x b2 ð5:32Þ

5.2.4. Streaming and sedimentation potential coefficientsFor obtaining the streaming and sedimentation potential

coefficients, ω, we combine Eqs. (4.60), (5.16)–(5.18). Retain-ing the leading term in the expansions by powers of ζ, only, oneobtains

x ¼ efgð1� ϕÞK∞ j2fð2þ ϕÞ y

pðbÞb2

ψð0ÞðbÞ

�ϕZ b

a

yp

x2dψð0Þ

dxdx� 2

b3

Z b

ayp

dψð0Þ

dxxdxg ð5:33Þ

Substituting Eq. (5.19) into Eq. (5.33) and combining theobtained result with Eqs. (5.2) and (5.5) yield

x ¼ � 2

3

efgð1� ϕÞKl

Z b

af1� a

r

� �2þ 3

2

a

r

� �5þ ϕ

10

� 1� 10ra

� �3�6

ar

� �5� g dwð0Þ

drdr ð5:34Þ

Thus, within the framework of the linear approximation interms of zeta-potential, Eq. (5.34) describes the streaming(sedimentation) potential coefficient.

Comparing Eqs. (5.26) and (5.34) one obtains

x ¼ � 23

efgð1� ϕÞKl

Z b

af1� a

r

� �3þ 32

ar

� �5þϕ10

� 1� 10ra

� �3�6

ar

� �5� g dwð0Þ

drdr ¼ vext=K

l ð5:35ÞThus, the results given by Eqs. (5.26) and (5.34) satisfy

irreversible thermodynamic relationship (3.78) rederived inSection 3.5 using Onsager cross relation (3.73). ComparingEqs. (5.28) and (5.35), one can note that, as κ→∞, the coefficientω approaches a geometry independent value given by

limjaYl

x ¼ ef=gKl ¼ xSm ð5:36Þ

Thus, the streaming (sedimentation) potential coefficient, ωgiven by Eq. (5.34) satisfies both the Smoluchowski and Onsagercriteria ofmodel validity. Accordingly, combining Eqs. (3.15) and(5.35) leads to thermodynamic relationship (3.77) which inter-relates the sedimentation potential electric field strength with χext.

Now, we will discuss the dependency of the sedimentationpotential field strength on different parameters. CombiningEqs. (5.15) and (5.34) one can represent the sedimentation potentialfield strength in a dimensionless form using the followingrelationships

dEg

dE⁎¼ 2

3ϕ

1� ϕ

Z ja=ϕ1=3

jaf1� ja

x

� �3þ 32

jax

� �5þϕ10

� 1� 10xja

� �3�6

jax

� �5� g dwdx

dx ðaÞ

E⁎ ¼ dgðρp � ρLÞef=gKl ðbÞ

wð0ÞðxÞ ¼ kax

� sinh½ðja=ϕ1=3Þ � x� � ðκa=ϕ1=3Þcosh½ðκa=ϕ1=3Þ � x�sinh½jað1� ϕ1=3Þ=ϕ1=3� � ðκ=ϕ1=3Þcosh½κað1� ϕ1=3Þ=ϕ1=3�

ðcÞ

ð5:37Þ




ARTICLE IN PRESS

The normalized sedimentation potential field strength,δE=δEg/δE⁎, is a function of two dimensionless parameters,κa and ϕ, only. The behavior of the function δE(κa,ϕ) wasanalyzed by Masliyah and Bhattacharjee [123] who obtainedthis function by combining Eq. (5.26), which describes theelectrophoretic mobility per unit external field strength, withirreversible thermodynamic relationship (3.78). Below, usingthe curves plotted in Fig. 7(a and b), we discuss the behavior ofthe function δE(κa ,ϕ) in details.

According to Fig. 7a, at a given κa, δEg/δE depends on ϕwith a maximum, (for the curve plotted at κa=0.1, themaximum is out of the displayed field). Existence of such amaximum is explained through the fact that, for both limitingcases ϕ→0 and ϕ→1, the sedimentation potential fieldstrength should approach zero. When ϕ→0, the zerosedimentation potential corresponds to the zero sedimentationvelocity of the particles having vanishingly small radius. As forthe case of ϕ→1, the behavior of sedimentation potential isdefined by three effects: (i) increase in the hydrodynamic dragforce (which diverges for such a limiting case) and, hence,decrease in the sedimentation velocity; (ii) decrease of particlecharge due to the double layer overlap at constant zeta-potentialand (iii) decrease in the system conductivity. The first andsecond effects lead to a decrease in the sedimentation potentialand, for any finite κa, dominate as compared with the thirdeffect which leads to an increase in the sedimentation potential.For the limiting case κa → ∞, the effects exactly compensateeach other and the curves are transformed into a straight linegiven by the equation δEg/δE⁎=ϕ (the Smoluchowski limit).

Remarkably, with increase of κa, the maxima are shiftedtoward higher volume fractions, and increasing parts of thecurves approach the straight line, δEg/δE⁎=ϕ. Consequently,for κa=50, nearly the whole curve merges with the straight lineδEg/δE⁎=ϕ. Approaching the Smoluchowski limit is alsoillustrated by the curves shown in Fig. 7b. With increasingκa, the normalized sedimentation potential field strengthapproaches an asymptotic value which coincides with thevolume fraction corresponding to a given curve.

Fig. 7. Dependency of the normalized sedimentation potential field strength on


5.3. Discussion on predictions from alternative models

In the previous section we derived expressions for the samekinetic coefficients which were considered by other authorswho, instead of the Shilov–Zharkikh–Borkovskaya boundarycondition, Eqs. (1.9) or (3.25), used the Levine–Neale boundarycondition given by Eq. (1.6) which, using substitution (3.2), canbe rewritten as

dMk

drðbÞ ¼ �dE ð5:38Þ

While writing the quantity on the right hand side of Eq. (5.38),the authors of the corresponding papers do not distinguish be-tween the external and internal electric fields.

In the light of the derivations given in Section 4.2, usingEqs. (5.38) or (1.6) should lead to incorrect predictions regardlessof the interpretation of the field δE: Eq. (5.38) clearly contradictsboth conditions (3.25) and (3.48) which contain the appliedfield. However, the analysis given in Section 5.2 on the basis ofEqs. (3.25) and (3.48) leads to the same expressions for theconductivity and the electrophoretic mobility as those derived byothers using (5.38).

In Sections 5.3.1 and 5.3.2, we will consider why thepredictions from different models coincided for the conductivityand the electrophoretic mobility. As well, we will discuss thecase of sedimentation potential when the different models leadto different results (Section 5.3.3).

5.3.1. Why do different models give the same predictions forthe zero and first order terms in expansion of conductivity bypowers of zeta potential?

Let us consider the equation

K ¼ � hdYJ icell dYiz

hY∇duicell dYiz

ð5:39Þ

which follows from Eq. (3.29) if to equate the averages over theentire system and the representative cell volumes. Eq. (5.39) is

a) volume fraction for different κa; b) on κa for different volume fractions.




ARTICLE IN PRESS

widely employed for obtaining the conductivity on the basis ofthe SCA [78,79,106–109].

Combining Eqs. (3.27), (4.19) and (4.25), it can be shownthat, for equal ion concentrations in the compartments separatedby the disperse system layer (Fig. 2)

hY∇duicell dYiz ¼ hY∇dlk

ZkFicell

dYiz ¼ MkðbÞ

b¼ �dEint ð5:40Þ

Thus, using Eq. (5.39) is completely equivalent to thereplacing δEint by Mk(b)/b=M

(0)(b)+O(ζ ) in the definition ofconductivity (3.6), i.e., to using Shilov–Zharkikh–Borkovskayacondition (4.25). Thus, while using Eq. (5.39), one implicitlyuses Shilov–Zharkikh–Borkovskaya condition (4.25) whichev-er boundary condition is employed at further stages of analysis.

One might add that an analysis consistently based on theLevine–Neale condition should be associatedwith using−dMk(b)/dr in the denominator of Eq. (5.39) instead of −Mk(b)/b. How-ever, such a consistent approach would lead to obviously incor-rect conclusions. For example, under certain conditions, using−dMk(b)/dr in the denominator of Eq. (5.39), the conductivities ofa suspension and its continuous medium become equal. Such aparticular case can be obtained specifying the general expressionderived by Ding and Keh [106] for the case of the Levine–Nealecondition.

At further stages of the analysis, there may be an additionalnecessity to use a boundary condition at the cell outer boundary.Prior to using Eq. (5.39), one should determine the distributionsof the local electric field and current density inside the cell.Such distributions are obtained by solving a boundary valueproblem which includes boundary conditions at the cell outerboundary. Generally, using different boundary conditions(Shilov–Zharkikh–Borkovskaya or Levine–Neale) leads todifferent results for the distributions (electric potential, ionicconcentrations etc.) because of the difference in the integrationconstant to be determined. However, for some special cases, theconductivity obtained from Eq. (5.39) does not depend on theintegration constant. Below we show that such an independencyis an outcome of considering only two leading terms in theTaylor expansion by powers of zeta-potential.

Combining Eqs. (2.6), (5.39) and (5.40) and omitting theconvection terms, that are order ofO(ζ2), the following equationcan be obtained

K ¼Xk

hKkY∇dlkicell d

Yiz

hY∇dlkicell dYiz

ð5:41Þ

where Kk ¼ F2P

k Ceqk Z2

k Dk=RT . Distribution of δμk is deter-mined using the substitution (4.2) where Mk(r) satisfies Eq.(4.8). Omitting the convection term in Eq. (4.8) (the right handside) one obtains

d2Mk

dr2þ 2

r� Zkf

dwdr

� �dMk

dr� 2r2Mk ¼ 0 ð5:42Þ

Since electrochemical potential is a continuous function,Eq. (5.42) can be used for describing the distribution δμk(r,θ)within the whole cell i.e. within the range 0≤ r≤b.


Now, wewill discuss some general properties of Eq. (5.42). Asa second order differential equation, Eq. (5.42) has two families ofparticular solutions. The general solution of Eq. (5.42) is formedas an arbitrary linear combination of two particular solutionscontaining two uncertain multiplicative constants. Under generalassumptions regarding the behavior of Kk (r), when r→0, one ofthe particular solutions diverges. The diverging particular solutionshould be eliminated from the general solution by setting to zerothe corresponding multiplicative constant. Accordingly, the finitegeneral solution of Eq. (5.43) is represented as

MkðrÞ ¼ Ak fkðrÞ ð5:43Þ

whereAk is an arbitrary constant and fk (r) is an arbitrary finite (for0≤r≤b) particular solution of Eq. (5.42).

In principle, the integration constant, Ak, can be obtained bymaking use of a boundary condition at the cell outer boundary.However, substituting Eq. (5.43) into (3.2) and, then,substituting the obtained result into Eq. (5.41) we obtain

K ¼Xk

hKkY∇½MkcosðhÞ�icell d

Yizz

hY∇½MkcosðhÞ�icell dYiz

¼Xk

hKkY∇½ fkcosðhÞ�icell d

Yizz

h½Y∇fkcosðhÞ�icell dYiz

ð5:44Þ

Eq. (5.44) shows that the conductivity of the disperse systemdoes not depend on the constant Ak, and, hence, on the form ofthe boundary condition at the cell outer boundary.

Thus, we arrive at the following conclusion. Omitting con-vectional contribution in conductivity (it is of order of O(ζ2)),the obtained conductivity will be the same regardless of theboundary condition set at the cell outer boundary. That is why,within the framework of such an approximation, Levine–Nealeboundary condition used by Ohshima et al [78,79] leads to thesame correct predictions as Shilov–Zharkikh–Borkovskayaboundary condition used by Carrique et al. [108,109].

5.3.2. When the Levine–Neale boundary condition is valid?In Section 4.2 we derived Eq. (4.47) which sets a given

electric current density through the disperse system. Usingthe definition of the external electric field, we rewrote Eq. (4.47)in terms of the external electric field strength and obtainedEq. (4.48). In the analysis presented subsequently, we usedEq. (4.47) for obtaining the electric current density for a giveninternal electric field strength which is imposed by the Shilov-Zharkikh-Borkovskaya boundary condition, Eq. (4.25).

Let us now consider transformation of Eq. (4.48) usingexpansion Eq. (5.1). Combining Eqs. (4.48) and (5.1), andretaining zero order terms lead to

1PkDkCl

k Z2k

Xk

DkClk Z2

k

dM ð0Þk

drðbÞ ¼ �dEext ð5:45Þ

Now, recall that using the Shilov–Zharkikh–Borkovskayaboundary condition leads to Eq. (5.7). According to Eq. (5.7),




ARTICLE IN PRESS

Mk(0)(r)=M (0)(r), i.e., Mk

(0)(r) is the same function for all theions. Consequently, Eq. (5.45) can be rearranged as

dM ð0Þk

drðbÞ ¼ �dEext ð5:46Þ

Comparing Eq. (5.46) with (5.38) we arrive at the followingconclusion. The Levine–Neale boundary condition is correct (anddoes not contradict the Shilov–Zharkikh–Borkovskaya bound-ary condition) provided that: (i) in Eq. (1.6) or, equivalently,Eq. (5.38), the applied electric field strength, δE, is understood asthe external electric field strength, δEext, and (ii) in the seriesexpansion of Mk(r) by powers of ζ, only zero order perturbationterms are taken into account. For all other cases, the Levine–Neale boundary condition leads to incorrect predictions.

Under the above discussed conditions, the Levine–Nealemodel does not contradict the Shilov–Zharkikh–Borkovskayamodel and can be employed for obtaining quantities for which it isrequired to know the zero order term in the expansion ofMk(r) bypowers of ζ(as it was done in original work of Levine and Neale[74]). The electrophoretic mobility at low zeta potentials analyzedin Section 5.2.2 belongs to such class of quantities. That is whythe predictions from the different models coincide for this case.

5.3.3. Critique of the sedimentation potential predictions basedon the Levine–Neale model

Clearly, the results obtained for the sedimentation potentialusing the Levine–Neale model (Levine et al [75], Ohshima[80,86] and Carrique [107]) are always contradictory toEq. (5.34) obtained using the Shilov–Zharkikh–Borkovskayacondition. Using Eqs. (3.15) and (5.34), the sedimentationpotential field strength is expressed, as

dYEg ¼ 23ϕδ→g

ðρp � ρLÞεζηð1� ϕÞK∞

Z b

af1� a

r

� �3þ 32

ar

� �5þϕ10

� 1� 10ra

� �3�6

ar

� �5� g dwdr

dr ð5:47ÞThe alternative prediction given in Refs. [75,80,86,107], for

convenience of the discussion, can be represented in the form

dYEaltg ¼ 2

3ϕδ

→g

ðρp � ρLÞεζηð1þ ϕ=2ÞK∞

Z b

af1� a

r

� �3þ 32

ar

� �5þϕ10

� 1� 10ra

� �3�6

ar

� �5� g dψdr

dr ð5:48Þ

While comparing Eqs. (5.47) and (5.48), one can note that theresults differ due to the difference in the volume fraction dependentterms in the denominators of the expressions before the integrals. InEq. (5.47), it is (1−ϕ) whereas, in Eq. (5.48), it is (1+ϕ /2).Consequently, the result of using Levine–Neale ðdYE alt

g Þ andShilov–Zharkikh–Borkovskaya ðdYEgÞ models are interrelated as

dYEaltg ¼ 1� ϕ

1þ ϕ=2dYEg ð5:49Þ

As it was discussed in Section 5.2.3, when κa→∞, thesedimentation potential coefficient, ω=−δEint/ϕ(ρp−ρL)δg, de-


termined from Eq. (5.47) approaches the geometry independentSmoluchowski limit (see Eqs. (3.67) and (5.36)). As well, δEg, isconnected with χext by equality (5.37) and thus satisfies theOnsager principle (see Section 3.5, Eq. (3.77)). Consequently, dueto the presence of the factor (1−ϕ) / (1+ϕ / 2) in Eq. (5.49), thealternative result, dYE

altg , given by Eq. (5.48) satisfies neither the

Smoluchowski principle nor the correct version of Onsager'srelationship given by Eq. (3.77). Hence, Eq. (5.49) gives anincorrect expression for the sedimentation potential field strength.

It is interesting, that, the authors using the Levine–Nealeboundary conditions for obtaining sedimentation potential,inadvertently, determined another quantity. To understand themeaning of the alternative result, we will consider in details theboundary condition employed in [80,86,107]. In terms of thepresent paper, instead of Eq. (4.25) (the Shilov–Zharkikh–Borkovskaya boundary condition), the boundary condition usedin [80,86,107] can be represented as

MkðbÞ ¼ 0 ð5:50ÞFrom the analysis given in Section 4.2.2, it is clear that the

latter condition is equivalent to δEint(b)=0 and, consequently, toΔφ=δφB−δφA=0 (Fig. 2). Thus, Refs. [80,86,107] deal with aboundary condition that sets zero potential difference across thelayer of the settling particles. Such a condition corresponds to thesedimentation current regime (Fig. 8).

In Refs. [80,86,107], while obtaining the final expression forthe sedimentation potential electric field strength, the authorsuse an expression which, in terms of Mk, can be represented inthe following equivalent form

dEaltg ¼ dM ð0Þ

drðbÞ þ OðfÞ ð5:51Þ

whereM(0)(r)=Mk(0)(r) is the zero order term in the expansion of

Mk by powers of ζ. The function M k(0)(r) turns out to be the

same for all ions, M(0)(r). As it would be expected, M(0)(r)obtained in [80,86,107] differs from that given by Eq. (5.7).

Recalling the discussion presented in Section 5.3.2, wearrive at the conclusion that Eq. (5.51) yields the externalelectric field but determined with the opposite sign (compareEqs. (5.46) and (5.51)). Thus, in [80,86,107], while attemptingto obtain the sedimentation potential field strength (i.e., the in-ternal field strength at zero current d

YE intðdJl¼0Þ), the authors obtain

a vector dYE

alt

g , exactly opposite to the external field strength,ðdYEextÞdEint¼0, which exists in the adjacent electrolyte solutionswhen, between the external sides of the settling disperse system, thepotential difference is kept zero (sedimentation current regime):

dYE

alt

g ¼ �dYEextðdEint¼0Þ ð5:52Þ

Two sketches given in Fig. 8 illustrate the difference betweenthe vectors d

YEg and d

YE

altg . Accordingly, d

YEg is the internal

electric field strength attributed to the sedimentation potentialregime (zero electric current density through the disperse systemlayer) whereas d

YE

altg is the vector which is opposite to the external

field strength attributed to the sedimentation current regime (zeroelectric potential difference across the disperse system layer).



Fig. 8. Disperse system layer settling under the sedimentation potential and current conditions.


ARTICLE IN PRESS

Let us now recall the irreversible thermodynamic analysisgiven in Section 3.5 There, in addition to Eq. (3.77) interrelatingthe sedimentation potential ðdYE intÞðdJl¼0Þ and the electropho-retic mobility (χext), we derived Eq. (3.81) which interrelates theexternal field strength attributed the sedimentation currentregime, ðdYEextÞdEint¼0, and another type of the electrophoreticmobility, χint. Using Eq. (5.52), Eq. (3.81) can be rewrittenin terms of the vector dYE

alt

g which is determined in [80,86,107]:

dYEaltg ¼ �vintϕ

ρp � ρLK∞ δ

→g ð5:53Þ

Although the expressions on the right hand sides of Eqs. (5.53)and (3.77) look very similar, they are not equivalent. There isan important difference between these expressions: Eq. (5.53)contains χint instead of χext represented in Eq. (3.77).

It can be shown that Eq. (5.53) is completely equivalent to therelationship proposed by Carrique et al. [107]. The relationshipof Carrique et al connects the sedimentation potential fieldstrength predicted using conditions (5.50) and (5.51) (d

YE

altg ,

according to our nomenclature) and the electrophoretic mobilitywhich was calculated in [107] using the Shilov–Zharkikh–Borkovskaya boundary condition. According to the discussionof Section 5.2, while using the Shilov–Zharkikh–Borkovskayaboundary condition, Carrique et al. calculated the mobility perunit internal electric field strength, χint. It is the type of mobilitythat is represented in Eq. (5.53). Consequently Eq. (5.53) and,thus, the observation of Carrique et al. are correct. However, itshould be stressed that the left hand side of the correctrelationship given by Eq. (5.53) is not the sedimentationpotential field strength. As for the sedimentation potential fieldstrength, it is described by correct irreversible thermodynamicrelationship (3.77) whose right hand side contains χext, not χint.

6. Conclusion

In the present communication, we developed a systematicapproach for addressing the behavior of a plane parallel layer of


a concentrated disperse system under simultaneous influence ofa pressure difference, gravitation and electric fields.

The proposed analysis is based on the general mathematicalproblem formulated in Section 2 for describing the disperse systemwhich can be considered as a macroscopically homogeneous andisotropic medium. The corresponding requirements are formulatedin Section 1.1. The general problem formulation includes govern-ing equations (2.3) and (3.15) (3.16) (3.17) subject to boundaryconditions (2.4), (2.5), (3.18), (3.19), at the particle surfaces,Eqs. (2.5), (2.20)–(2.23), at the external sides of the disperse systemlayer.

Solution of the formulated problem enables one to determine aset of kinetic coefficients that have been defined in Section 3 ofthe present paper for describing a variety of linear ElectrokineticPhenomena. General properties of the kinetic coefficients wereanalyzed in the paper. In particular, the Smoluchowski asymptoticexpression andOnsager irreversible thermodynamic relationshipshave been rederived for the kinetic coefficients (Section 3).

The formulated general problem has been specified usingthe SCA (Section 4). The SCA version of the problem includesEqs. (4.7)–(4.9) subject to boundary condition (4.11) (4.12) (4.13)(4.14), at the particle surface, (4.15), (4.25), (4.36) and (4.49),at the cell outer boundary. All the outer boundary conditionswere derived in Section 4.3 using an original approach basedon one assumption, only: the average over the disperse systemvolume is equal to the average over a representative sphericalcell volume.

General expressions for the kinetic coefficients have been de-rived in Section 4. The kinetic coefficients are represented throughthe 2N+3 functions (for N ionic species), ψ(r,ζ ), mk

E(r,ζ ),mk

p(r,ζ ), yE(r,ζ ) and yp(r,ζ ). Obtaining the above functionsamounts to solving the boundary problem given by Eqs. (4.7)–(4.9), (4.11)–(4.15), (4.25), (4.36), and (4.49). Representing theobtained problem solution in the form given by Eq. (4.50) givesexpressions for mk

E (r,ζ ), mkp (r,ζ ) yE (r,ζ ) and yp (r,ζ ).

The developed general approach has been combined withthe method of small parameter (normalized zeta potential)




ARTICLE IN PRESS

perturbation (Section 5). Final expressions for the kineticcoefficients have been derived while accounting for the termsproportional to zeta potential. The obtained results have beenshown to satisfy the Smoluchowski and Onsager criteria ofmodel validity.

On the basis of the conducted analysis, we explained all thecontradictions which exist in the literature dealing with thetheoretical study of Electrokinetic Phenomena by using theSpherical Cell Approach.

NomenclatureLatin lettersa Radius of a disperse particle;b Radius of the cell;Ck∞ Concentration of the kth ion in adjacent solution;

Ck Local concentration of the kth ion;Ckeq Equilibrium concentration of the kth ion;

Dk The kth ion diffusion coefficientd Thickness of the gap between the disperse system layer

and planes AA and BB (Fig. 2)YE Electric field strengthYE int Internal electric field strengthYEext External electric field strengthYEg Sedimentation potential field strengthF Faraday numberYg Gravity accelerationH Thickness (height) of disperse systemYJ Electric current densityYJl Electric current density at the planes AA and BB (Fig. 2)Yjk The kth ion fluxK∞ Conductivity of adjacent electrolyte solutionKk The kth ion partial conductivityKk

∞ The kth ion partial conductivity in adjacent solutionK Conductivity of disperse systemp local pressureΔp Pressure difference between external sides of disperse

systemR Gas constantr Radial coordinateS AreaT Absolute temperatureYu Local velocity of liquidueo Electroosmotic velocityueph Electrophoretic velocityV Disperse system volumeY(r) Radial part of the streaming functionZk Electric charge of the kth in Faraday unitsz Axial coordinate

Greek lettersα Electroosmotic pressure coefficientγ Hydraulic permeabilityδ Departure from the equilibrium valueχint Electrophoretic mobility per unit internal field strengthχext Electrophoretic mobility per unit external field strengthvk ¼ Cl

k Zk=P

n Z2nC

ln Dimensionless coefficient

η Viscosity


ε Dielectric permittivityκ Debye parameterμk Perturbation of kth ion electrochemical potentialMk(r) Function describing radial dependency of perturbation

of kth ion electrochemical potentialΠ Effective pressureΠ⁎(r) Function describing radial dependency of perturbation

of effective pressureρ Electric charge densityρL Liquid mass densityρp Particle mass densityξn Local Cartesian coordinate normal to the interfaceϕ Volume fraction of the dispersed phaseΨ Equilibrium electric potentialψ Equilibrium electric potential normalized by zeta

potentialω Streaming (sedimentation) potential coefficientφ Electric potentialζ Electric potential at the interface in equilibrium state

(zeta potential)

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the Natural Sciences and Engineering Research Council ofCanada.

Appendix A. Supplementary derivations for obtainingSmoluchowski limit for sedimentation potential

A.1. Derivation of boundary condition (3.56)

Integrating Eq. (3.53) over the coordinate ξn within the range0bξnbξn⁎ and considering limit ξn⁎→0 yield

limn⁎nY0

Z n⁎n

0

Y∇ d dYJ dnn ¼ 0 ðA1Þ

The left hand side of Eq. (A1) can be decomposed in twoterms

limn⁎nY0

Z n⁎n

0

Y∇ d dYJ dnn ¼ lim

n⁎nY0

Z n⁎n

0½YnðYn d

Y∇Þ� d dYJ dnn

þ limn⁎nY0

Z n⁎n

0

Y∇s d d

YJ dnn ðA2Þ

whereY∇s ¼ Y

∇�YnðY∇ dY∇Þ. On the right hand side of Eq. (A2),

the expression under the first integral is transformed to

½YnðYn dY∇Þ� d dYJ dnn ¼ ðdnnÞnk Y

eknmA

AxmdJp

Yep

¼ ðdnnÞnpnmAdJpAxm

¼ Yn d ½ðdYr d Y∇Þ d dYJ � ¼ Yn d dðdYJ ÞðA3Þ

where Yek are the unit vector of an arbitrary Cartesian coordinatesystem; the subscripts, “p”, “k” and “m” signify the Cartesiancoordinates of the corresponding vectors.




ARTICLE IN PRESS

Now, we substitute Eqs. (3.54) and (3.55) into the secondintegral on the right hand side of Eq. (A2). After such asubstitution, while approaching the limit ξn⁎→0, the contribu-tion of the migration term (the first term on the right hand side ofEq. (3.54)), becomes vanishingly small. Since we assume thatξn⁎→0 and κξn⁎→∞, simultaneously, the second integral on theright hand side of Eq. (3.50) is transformed as

limn⁎nY0n⁎nYl

Z n⁎n

0

Y∇s d dYJ dnV¼ Y∇s d ½ðYn d

Y∇dYu Þ�nn¼0

� limn⁎nY0n⁎nYl

Z n⁎n

0nVρeqdnVn ¼ �ef

Y∇s d ½ðYn dY∇ÞdYu �nn¼0

ðA4ÞTo obtain the final expression in Eq. (A4), we took into

account Eq. (3.34) which enabled us to use the locally flat versionof the Poisson equation while considering the limit transitionunder the simultaneous conditions of ξn⁎→0 and κξn⁎→∞:

limn⁎nY0n⁎nYl

Z n⁎n

0nnρ

eqdnn ¼ �e limn⁎nY0n⁎nYl

Z n⁎n

0nn

A2WAnn

dnn

¼ �e limn⁎nY0n⁎nYl

n⁎nAWAnn

ðn⁎nÞ �Wðn⁎nÞ þWð0Þ�

¼ �ef

ðA5Þ

In Eq. (A5), in spite of the condition ξn⁎→0, the potentialdifference is finite, Ψ (0)−Ψ (ξn)=ζ. Such a situation occurssince κξn⁎→∞, i.e., for a vanishingly thin double layer, theinterfacial potential difference, which is assumed to be constant,is localized within a mathematical surface at the interface.

Combining Eqs. (A1)–(A4) yields

Yn d ðdYJ � dYJ sÞ ¼ efY∇s d ½ðYn d Y∇ÞdYu �n¼0 ðA6Þ

where dYJ s is the electric current at the true interface; dYJ s is the

electric current at the interface but outside the vanishingly thindouble layer. The conservation law for the electric charge yields

Yn d dYJ s ¼ 0 ðA7ÞUsing Eqs. (3.4), (3.34), (3.54), the current outside the

double layer is expressed as

dYJ ¼ �KlY∇dUðYr Þ ðA8ÞConsequently, combining Eqs. (A6), (A7), (A8) we arrive at

boundary condition (3.56).

A.2. Derivation of Eq. (3.63)

We consider creeping flow whose velocity and pressuresatisfy Eqs. (2.17) and (3.40) subject to boundary condition(2.18). Let us introduce an orthogonal coordinate systemformed by one axis, which is normal to the interface, two otheraxis which are tangent to the interface and perpendicular to eachother. Accordingly, we consider the unit vectors Yn;

Ys1 and

Ys2,

and the orthogonal coordinates ξn, ξ1 and ξ2. Using such a


coordinate system, the gradient operator and the vectors ofinterest are represented as follows

dYu ¼ Yndun þ Ys1du1 þ Y

s2du2 ðA9ÞYr ¼ Ynnn þ Y

s1n1 þ Ys2n2 ðA10Þ

Y∇ ¼ YnhnA

Annþ Y

s1h1A

An1þ Y

s2h2A

An2ðA11Þ

where hn, h1 and h2 are the metric coefficients attributed to thenormal and tangent axes respectively.

Using Eq. (A11) one can obtain the following identity

Yn dY∇� Y∇� dYu

h1h2

¼ A

An1

hnh1h2

A

Ann

du1h2

� �� A

An1

dunhn

� ��

þ A

An2

hnh2h1

A

Ann

du2h2

� �� A

An2

dunhn

� �� ðA12Þ

Using Eq. (2.18), it is obvious that

dun ¼ du1 ¼ du2 ¼ 0 at the interface ðA13Þand, hence,

Adun=An1;2 ¼ Adu1=An1;2 ¼ Adu2=An1;2 ¼ 0 at the interface

ðA14ÞCombining Eqs. (2.18), (A11), (A13) and (A14) it can be

shown that

Adun=Ann ¼ 0 at the interface ðA15ÞSpecifying identity (A12) for the interface and taking into

account Eqs. (A17) and (A14), Eq. (A12) can be reduced as

Yn dY∇� Y∇� dYu ¼ h1h2f A

An1

hnh2

Adu1Ann

� ��

�þ A

An2

hnh1

Adu2Ann

� �� g at the interface

ðA16ÞLet us now consider a transformation of the expression

Y∇s d ½ðYn dY∇ÞdYu � at the interface.Using Eq. (A11) one can obtain

Y∇s ¼ Y∇�Yn ðYn dY∇Þ ¼ Y

s1h1A

An1þ Y

s2h2A

An2ðA17Þ

Combining Eqs. (A9) and (A11) yields

YnðYn dY∇ÞdYu ¼ fhn

Ys1

Adu1Ann

� h1dunA

Ann

1hn

� ��

þYs2

Adu2Ann

� h2dunA

An2

1hn

� ��

þYn ½Adu2Annþ du1h1

A

An1

1hn

� �

þdu1h2A

An2

1hn

� ��g ðA18Þ




ARTICLE IN PRESS

Combining Eqs. (A17) and (A18) and taking into accountEqs. (A13)–(A15) yield expression containing only four non-zero terms

Y∇s d ½ðYn dY∇ÞdYu � ¼ h1

A

An1hn

Adu1Ann

� �

þ h1h2hnA

An2

1h1

� �Adu2Ann

þ h2A

An2hn

Adu2Ann

� �at the interface

þ h1h2hnA

An2

1h2

� �Adu1Ann

¼ h1h2f A

An1

hnh2

Adu1Ann

� ��

þ A

An2

hnh1

Adu2Ann

� �� gðA19Þ

Comparing Eqs. (A16) and (A19) we arrive at the equality

Yn dY∇� Y∇� dYu ¼ Y∇s d ½ðYn d

Y∇ÞdYu � at the interface ðA20ÞCombining Eqs. (3.40) and (A20) one obtains Eq. (3.63)

A.3. Derivation of Eq. (3.65)

Considering a tensor ¯T in the chosen coordinate system, oneobtains

d¯T dYdS ¼ d¯T d YndS ¼ ðYs1dT1n þ Ys2dT2n þ YnTnnÞdS ðA21Þ

For the case when

d¯T ¼ Y∇dYu þ ðY∇dYu Þt ðA22Þusing Eqs. (A9), (A11), (A13)–(A15) we obtain the followingexpressions for the tensor components which are represented inEq. (A21)

dT1n ¼ h1AdunAn1

þ hnAdu1Ann

¼ hnAdu1Ann

ðA23Þ

dT2n ¼ h2AdunAn2

þ hnAdu2Ann

¼ hnAdu2Ann

ðA24Þ

3 dr b

dTnm ¼ 0 ðA25ÞCombining Eqs. (A21)–(A25) yields

Y∇dYu þ ðY∇dYu Þth i

dYdS ¼ hn

Ys1

Adu1Ann

þ Ys2

AdU2

Ann

� �dS ðA26Þ

At the same time, using Eqs. (A13)–(A15) and (A18) oneobtains

YnðYn dY∇ÞdYudS ¼ hn

Ys1

Adu1Ann

þ Ys2

Adu2Ann

� �dS ðA27Þ

Comparing Eqs. (A26) and (A27) we arrive at Eq. (3.65).


Appendix B. Derivation of Eq. (4.31)

Using Eq. (4.29), let us transform the expression underintegral Eq. (4.30)

Yiz d d ¯r d Yn ¼ �Y

iz dYndp

þ g 2Yiz d

Y∇dYu d Yn � Yiz d ½Y∇dYu � ðY∇dYu Þ⁎� d Yn

n oðB1Þ

The first term in the brackets on the right hand side ofEq. (B1) can be transformed with the help of the followingequality

2Y∇dYu d Yn ¼ �2

Yim

AdukAxm

nk

¼ 2r

Yim

AðxkdukÞAxm

� dukAxkAxm

� �

¼ 2r½Y∇ðdYu d Yr Þ � dYu � ðB2Þ

Consequently, combining Eqs. (4.4)–(4.6) and (B2), oneobtains

Yiz d 2

Y∇dYu d Yn ¼ 2½1� 3cos2ðhÞ� d

drYr2

� �ðB3Þ

The second term in the brackets in Eq. (B1) is transformedusing the identity

½Y∇dYu � ðY∇dYu Þ⁎� d Yn ¼ Yn � Y∇� dYu ðB4Þ

Combining Eqs. (4.4)–(4.6) and (B4) yields

Yiz d ½Y∇dYu � ðY∇Yu Þ⁎� d Yn ¼ 1

rd2Ydr2

� 2Yr2

� �sin2ðhÞ ðB5Þ

Thus, the first and second terms in the brackets of Eq. (B1)are given, respectively, by Eqs. (B3) and (B5). Consequently,combining Eqs. (4.30)–(4.35) yields

∮Scell

Yiz d ¯d d YndS ¼ �∮Scell

dpcosðhÞdSþ 2g

ddr

Yr2

� �∮Scell

½1� 3cos2ðhÞ�dS

� g1r

d2Ydr2

� 2Yr2

� �∮Scell

sin2ðhÞdS ðB6Þ

Realizing that the second integral on the right hand side ofEq. (B6) is zero and taking the third integral, Eq. (B6) isrewritten as

∮Scell

Yiz d d ¯r d YndS ¼ �∮Scell

dpcosðhÞdS� 8

pbgd2Y

2ðbÞ � 2Y

2

� ðB7Þ




ARTICLE IN PRESS

Using the definition of the “effective” pressure”, δΠ, givenby Eq. (2.12), the first integral on the right hand side of Eq. (B7)becomes

∮ScelldpcosðhÞdS ¼ ∮Scell

dPcosðhÞdS þ ∮Scellρld

Yg d YrcosðhÞdS

þXk

½Ceqk ðbÞ � Cl

k �ZScell

ðdlk � ZkFduÞ

�cosðhÞdS ðB8ÞUsing the transformation given in (4.22)Z

Scell

ðdlk � ZkFduÞcosðhÞdS

¼ 4pb3

Yiz d ðhY∇ d dlki � ZkFhY∇ d duiÞ ¼ 0 ðB9Þ

Hence, the first integral on the right hand side of Eq. (B7) isexpressed as

∮ScelldpcosðhÞdS ¼ 4

3pb2½P⁎ðbÞ þ gρLb� ðB10Þ

Combining Eqs. (B7) and (B10) yields Eq. (4.31).

Appendix C. Derivation of Eq. (4.34)

Taking into account that, at the interface, the effectivepressure perturbation, δΠ, is not necessary continuous, let usconsider the following identical transformations based on thegradient theorem

1V

Yiz d ∮SdP

YdS

¼ Yiz d ½∮V�Vp

Y∇dPLdV þ ∮Vp

Y∇dPpdV

þN∮SpðdPL � dPpÞYdS� 1V

¼ hY∇dPi þ 1V=N

∮SpðdPL � dPpÞYdSp

ðC1Þ

where δΠp and δΠL are the perturbations of the effectivepressure inside and outside the particles, respectively; Vp and Spare the particle volume and surface. The value of δΠp isassumed to be the same for all the particles that make up thedisperse system. Simultaneously, it is admitted that, at theinterface, δΠL does not necessary coincide with δΠp.

Equating the averages over the disperse system and cellvolumes in the final expression in Eq. (C1) and realizing that V/Np=4πb

3/3 (the cell volume), Eq. (C1) can be rewritten as

1V

Yiz d ∮S dP

YdS ¼ Yizd hY∇dPicell þ

34pb3

∮SpðdPL�dPpÞYdSp

� ðC2Þ

Applying the gradient theorem to the cell yields

Yiz d hY∇dPicell

¼ Yiz d

34pb3

∮Vcell�Vp

Y∇dPLdV þ ∮Vp

Y∇dPpdV

h i

¼ Yiz d

34pb3

∮ScelldPYdS � ∮Sp

ðdPL � dPÞYdSph i

ðC3Þ


Consequently, combining Eqs. (C1)–(C3) one obtains

1

VYiz d ∮SdP

YdS ¼ Yiz d

3

4pb3∮Scell

dPYdS ðC4Þ

Substituting Eq. (4.3) into the integral on the right hand sideof Eq. (B4) we arrive at Eq. (4.34)

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Electrokinetic Phenomena in Concentrated Disperse Systems: General Problem Formulation and Spherical Cell Approach

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