## Abstract

A zero net charge ideally polarizable particle is suspended within an electrolyte solution, nearly in contact with an uncharged non-polarizable wall. This system is exposed to a uniform electric field that is applied parallel to the wall. Assuming a thin Debye thickness, the induced-charge electro-osmotic flow is investigated with the goal of obtaining an approximation for the force experienced by the particle. Singular perturbations in terms of the dimensionless gap width *δ* are used to represent the small-gap singular limit *δ*≪1. The fluid is decomposed into two asymptotic regions: an inner gap region, where the electric field and strain rate are large, and an outer region, where they are moderate. The leading contribution to the force arises from hydrodynamic stresses in the inner region, while contributions from both hydrodynamic stresses at the outer region and Maxwell stresses in both regions appear in higher order correction terms.

## 1. Introduction

There is currently an increasing theoretical and experimental interest in electrokinetic flows about polarizable surfaces (‘induced-charge’ flows), wherein the zeta potential distribution depends upon the applied field. The prototypical configuration entails an ideally polarizable (perfectly conducting) particle in a uniformly applied electric field (Gamayunov *et al*. 1986). This problem has been extensively studied in recent years (Bazant & Squires 2004; Squires & Bazant 2004, 2006; Yariv 2005, 2008; Saintillan *et al*. 2006; Yossifon *et al*. 2007; Saintillan 2008) under the assumption of an unbounded fluid domain.

Understanding of real systems requires modelling the effects of bounding walls. In the context of *non-polarizable* particles, wall effects on electrophoresis (Keh & Anderson 1985; Keh & Chen 1989) are pretty well understood, as they are in other electrokinetic phenomena (Keh & Jan 1996; Chen & Keh 2002, 2005). Similar understanding is desired for induced-charge processes about *polarizable* particles. In a sense, wall effects are even more important in the latter case due to the high symmetry of the associated flow fields. For example, the induced-charge flow about an initially uncharged spherical particle does not result in a net force on it. This symmetry disappears when a wall is introduced. Thus, the presence of walls may actually *engender* particle motion that would have been absent in unbounded fluid domains.

Owing to the unique velocity scaling in induced-charge flows, Maxwell stresses are comparable in magnitude to hydrodynamic stresses (Yariv 2005). Thus, it is generally required to calculate both the electrical force and the hydrodynamic force (an example of electric force calculation was provided by Yariv (2006)). Boundary effects on polarizable particles were initially studied by Zhao & Bau (2007), who considered a cylindrical particle near a plane wall. The case of spherical particles was analysed in a previous publication (Yariv 2009), using reflection techniques that are applicable for remote particle–wall interactions. In this paper, we carry out the comparable analysis of the other extreme limit, where the particle–wall separation is small. We follow Yariv (2009) in considering the simplest model of a particle–wall interaction, entailing an initially uncharged ideally polarizable stationary particle in the vicinity of an uncharged non-polarizable wall. The thin-Debye-layer model is employed.

The present investigation resembles the near-contact electrophoretic study of non-polarizable particles (Yariv & Brenner 2003). As in that study (and in contrast with the large separation investigation of Yariv (2009)), the present investigation is singular and requires use of inner–outer asymptotic expansions (Cole 1968). The inner region comprises the narrow gap between the particle and the wall, while the outer region consists of the remaining fluid domain. The electric potential in the narrow gap has been solved by Solomentsev *et al*. (1997) in the context of an analogous heat conduction problem. The electrokinetic slip animated by the large electric field in the gap drives a strongly sheared flow. Scaling analysis readily reveals that this flow results in a large repulsive hydrodynamic force, which dominates the hydrodynamic contribution from the outer fluid domain. The electrical force on the particle is also subdominant to that contribution.

In principle, it is possible to obtain the dominant gap contribution using a lubrication analysis of the inner flow structure. This procedure, however, proves unnecessary in view of an existing elegant representation (Brenner 1964) of the hydrodynamic force as a surface quadrature of the electrokinetic slip, the corresponding kernel being the traction field associated with pure translation of the particle. The latter translation problem was solved by Cox & Brenner (1967) in the near-contact limit.

Since the solution of Cox & Brenner (1967) also provides the near-contact mobility, we easily calculate the corresponding velocity of a comparable freely suspended particle. In the presence of a gravity field directed into the wall, the particle reaches a stable equilibrium position. Estimation using characteristic experimental values (Levitan *et al*. 2005) reveals that gap separation corresponding to this position is quite small, even for strong electric fields. This naturally suggests a simple experimental verification of the wall effect.

## 2. Problem formulation

The system we consider was described by Yariv (2009). It comprises an electrolyte solution (permittivity *ϵ* and viscosity *η*) that is bounded by a non-polarizable planar wall and a stationary spherical particle of radius *a*, whose centre *O* is positioned at distance *a*(1+*δ*) from the wall. The particle is ideally polarizable (a perfect conductor). The system is exposed to a uniform and constant external electric field ( being a unit vector in the field direction), which is applied parallel to the wall. A schematic of the system is shown in figure 1.

We employ a standard dimensionless notation (Yariv 2009), using *a*, *E*_{∞}, *aE*_{∞}, and as the respective units of length, electric field, electric potential, velocity and stress. In the thin-Debye-layer limit, the pertinent fields are evaluated in the electroneutral bulk domain lying exterior to the Debye layers. Consistently, these layers are represented by the no-flux boundary condition for the electric field, together with the Smoluchowski slip condition for the fluid velocity. These conditions apply at the particle and wall boundaries, respectively denoted by and . The electric potential is therefore governed by Laplace's equation in the fluid domain,(2.1)together with the no-flux condition on both the particle ( being an outward pointing unit normal to the sphere),(2.2)and the wall,(2.3)It is driven by the non-homogeneous far-field condition,(2.4)The above Neumann-type boundary-value problem uniquely defines *φ* up to a physically meaningless integration constant. It is readily verified that the electric potential can be made an odd function of *x* by a proper choice of that constant. Since the particle retains its zero net charge, we consistently choose its uniform potential as zero (Yariv 2009). The induced zeta potential on the particle is then(2.5)The velocity field ** v** and the pressure field

**are governed by (i) the Stokes equations,(2.6)(ii) Smoluchowski's slip condition on the particle,(2.7)(iii) the no-slip condition on the wall,(2.8)and (iv) the condition of velocity decay at large distances from the particle.**

*p*In what follows, it is convenient to employ a Cartesian coordinate system, with the *z*-axis passing through *O* and lying perpendicular to the wall boundary (which is given by *z*=0), and the *x*-axis lying in the applied field direction . In addition, we also employ cylindrical polar coordinates (*ρ*, *ψ*, *z*) having the same origin, *ψ*=0, coinciding with the applied field direction. The imposed field condition (2.4) then implies that *φ*∼−*ρ* cos *ψ* at large distances. From the symmetry of the problem, it is readily verified that the electric potential possesses the form(2.9)

Once the electric and velocity fields are evaluated, it is possible to calculate the force (normalized with ) experienced by the particle. It comprises the hydrodynamic contribution(2.10)in which is the Newtonian stress (I is the idem factor and † denotes the transposition), as well as the electric contribution(2.11)in which(2.12)is the Maxwell stress. In view of (2.2), the latter contribution appears as(2.13)

Using symmetry arguments, it is readily verified (Yariv 2009) that both contributions are directed perpendicular to the wall,(2.14)and that both the hydrodynamic and electric torques vanish. The scalars *F* and can only depend upon *δ*, the single dimensionless parameter in the problem geometry.

Rather than evaluating the flow field directly, we here employ the method of Brenner (1964), which exploits the Lorentz reciprocal theorem. In the present configuration, this theorem can be written in the form(2.15)in which is any flow field that satisfies the Stokes equations and vanishes on and at large distances, while is the corresponding stress field. Choosing as the flow field due to the translation of the particle with a unit velocity perpendicular to and away from the wall, and using (2.7), we find that *F* is given by the quadrature(2.16)

## 3. Small-gap analysis

The preceding problem was studied by Yariv (2009) for large separations (*δ*≫1). Here, we focus upon the near-contact limit where the thickness *aδ* of the particle–wall gap is small compared with *a*,(3.1)The comparable limit in the case of a non-polarizable particle was studied by Yariv & Brenner (2003). Following that work, we handle the singular limit using an inner–outer asymptotic expansion. In the inner gap region, we employ the stretched coordinates(3.2)which transform the surface of the sphere lying proximate to the wall into(3.3)where(3.4)In the outer region, the leading-order problem entails a sphere in contact with a plane. In principle, this configuration is handled using tangent-sphere coordinates.

### (a) Electric potential

We briefly outline the asymptotic solution of the inner potential problem, which was obtained by Solomentsev *et al*. (1997). Since the inner problem is homogeneous, the asymptotic scaling of *Φ* is provided by a pre-factor *μ*(*δ*), which can only be found by matching with the outer region. Thus,(3.5)where *G*, presumably *O*(1), possesses the expansion(3.6)The asymptotic analysis of Solomentsev *et al*. (1997) shows that *G*_{0} is a function of *R* alone and is of the form(3.7)where *F* is the hypergeometric function and *D* a constant of integration. At large *R*,(3.8)Asymptotic matching to an exact bipolar-coordinate solution yields (Solomentsev *et al*. 1997)(3.9)and . These results were later confirmed using asymptotic matching with the outer-region solution (Yariv & Brenner 2003).

The requisite leading-order approximation for the force (see (4.6)) requires, in addition to *G*_{0}, the next term, *G*_{1}. While this term can be calculated in principle (Yariv & Brenner 2003), this is not necessary: it is only the behaviour of *G*_{1} on *Z*=*H*_{0}(*R*) that is required. Thus, expanding the no-flux condition on readily yields(3.10)

### (b) Scaling analysis

In view of (3.9), the radial electric field in the narrow gap is singular, . The slip condition (2.7) then results in radial velocities. Standard lubrication scaling consequently shows that the pressure field is , resulting in a contribution to the hydrodynamic force. This contribution dominates the *O*(1) outer contribution.

On the other hand, the Maxwell stresses (2.12) are only in the gap. While still singular, these stresses result in an asymptotically small contribution to the electric force that is dominated by the outer contribution. We conclude that the leading-order force is , and it arises from the inner contribution of the hydrodynamic stresses. When focus lies at this term, there is no need to consider the electric and flow fields in the outer region. The proper scaling for the field-induced force on the particle is(3.11)where *F*_{0} is contributed by hydrodynamic stresses at the inner gap region.

It is easily verified that this leading-order force is repulsive. The strong electric field in the narrow gap results in large positive electric potential at the ‘leading’ part of the gap (*x*<0) and large negative electric potential at the ‘trailing’ part (*x*>0); the electric field itself, however, points inward in the leading part and outward in the trailing part. Consequently, the velocity slip (2.7) always points inward, resulting in an inward Couette-type flow. Mass conservation then necessitates an accompanying outward Poiseuille-type flow, which is associated with large pressure build-up in the gap. This mechanism, which is summarized in figure 2, results in hydrodynamic repulsion. Both the Couette and Poiseuille flows vary quickly on the *δ*-scale and slowly on the *O*(*δ*^{1/2}) scale.

Given the representation (2.16), there is no need to explicitly evaluate the inner flow field. Indeed, since the problem of particle translation normal to a wall has already been solved in the near-contact limit (Cox & Brenner 1967), we simply employ (2.16) to evaluate the inner contribution.

## 4. Force evaluation using Brenner's method

### (a) The solution to the pure translation problem

We briefly outline here the solution to the translation problem (Cox & Brenner 1967). Owing to its axial symmetry, the velocity field adopts the form , where both the radial, , and axial, , velocity components are independent of the azimuth angle, *ψ* (and then so must be the pressure ). In the gap region, the hydrodynamic variables possess the scaling(4.1)where each of the *O*(1)-rescaled fields *U*, *W* and *P* possess an asymptotic expansion of the form (3.6). A leading-order lubrication analysis yields a Poiseuille-type radial velocity profile(4.2)Imposing mass conservation yields an ordinary differential equation for *P*_{0} whose solution is(4.3)In preparation for use of (2.16), we note that the radial and axial components of are respectively given by(4.4a)and(4.4b)Cox & Brenner (1967) used (4.4*b*) to obtain the hydrodynamic resistance to near-contact translation. Normalizing with *ηa*, they found the approximation(4.5)

### (b) Performing the quadrature

The leading-order hydrodynamic force is obtained using (2.16) and (4.4) in conjunction with (2.9) and (3.5). Invoking the asymptotic expansion (3.11) and performing the integration over *ψ* yields(4.6)On the approximated particle boundary (3.4)(4.7)thus, using (3.10) readily shows that the two last terms in (4.6) cancel out at *Z*=*H*_{0}. Substitution of (4.2)–(4.3) yields(4.8)where(4.9)Since the contribution to the leading-order forces arises solely from the inner region, one simply integrates from 0 to ∞. Integrating by parts and using (3.8) yields(4.10)Substitution of the explicit expression (3.7), followed by numerical integration, yields(4.11)

## 5. Concluding remarks

Using lubrication methods, we have obtained an asymptotic approximation for the induced-charge electrokinetic force acting on a stationary conducting particle, which is suspended in close proximity to a non-polarizable wall. This approximation is associated with the hydrodynamic stresses in the narrow gap region separating the particle from the wall. The contribution of the remaining hydrodynamic tractions appears at higher order correction terms, as does the contribution of Maxwell stresses.

If the particle is freely suspended, the velocity it acquires is obtained by the ratio of the total force that the particle experiences to the hydrodynamic resistance (4.5). Thus, the velocity of a neutrally buoyant particle is given by(5.1)and is directed away from the wall.

In the presence of a gravity field that is directed towards the wall, the particle reaches an equilibrium position. Since the electrokinetic force decreases with *δ*, this is a stable equilibrium. Balancing the approximate force (4.8) with gravity yields(5.2)wherein Δ*ρ* is the difference between the particle and electrolyte mass densities. For characteristic numbers in real systems (Levitan *et al*. 2005), one finds that this expression is consistent with the thin-gap analysis. Consider, for example, a platinum sphere of 100 μm radius, which is suspended in an aqueous solution (*ϵ*≈7×10^{−10} C V^{−1} m, Δ*ρ*≈20.4 g cm^{−3}); it is then readily seen that *δ* is extremely small, even at a strong electric field of, for example, 100 V cm^{−1}, *δ*≈5×10^{−4}. This corresponds to gap separations of approximately 50 nm (still large compared with typical Debye thicknesses). The scale disparity confirms *a posteriori* the underlying assumptions of the theoretical model.

In view of the dominant role of the inner gap region and the available near-contact solutions of Cox & Brenner (1967) and Solomentsev *et al*. (1997), obtaining the leading-order approximation for the force was quite straightforward. Further improvement of that approximation seems, however, intractable in view of the complicated expressions for the outer electric field, which appear in the form of Hankel transforms (Yariv & Brenner 2003). We are therefore unable to calculate the leading-order correction term in (3.11). In view of the scaling analysis, this term is expected to be of *O*(*δ*) relative magnitude (although a slightly larger asymptotic term is also possible, see Cox & Brenner (1967)). Agreement of (4.8) with numerical solutions of the particle–wall interaction at arbitrary separations (currently being developed by Saintillan (in preparation)) will therefore require their convergence at small numerical values of *δ*.

## Acknowledgments

This research was supported by the N. Haar and R. Zinn research fund.

## Footnotes

- Received December 24, 2008.
- Accepted March 5, 2009.

- © 2009 The Royal Society