Royal Society Publishing

Electrokinetic self-propulsion by inhomogeneous surface kinetics

Ehud Yariv

Abstract

Electrokinetic self-propulsion was conceptually proven in recent experiments wherein bimetallic nano-rods were observed to migrate when placed in aqueous solutions. We present here a systematic theoretical model of the self-propulsion mechanism, analysing the steady-state transport occurring about an autonomously moving inhomogeneous particle. The non-uniform catalysis on the particle surface is modelled via position-dependent cation redox coefficients. The particle shape is axisymmetric but otherwise arbitrary, as are the distributions of the (possibly discontinuous) kinetic coefficients along its boundary. We formulate the mathematical problem governing this electrokinetic transport. In the thin-Debye-layer limit, the microscale description is transformed into a macroscale one, applying in the electro-neutral bulk. Effective boundary conditions represent asymptotic matching with the Debye-layer fields. A linearized model is derived for weak variation of the kinetic coefficients and is solved for a spherical-particle geometry. With a view towards understanding existing experiments, the macroscale model is used for analysing slender particles. Matched asymptotic expansions provide the particle velocity as a functional of its shape and kinetic-coefficient distributions. The predicted self-propulsion is in the direction observed in nanorod experiments.

1. Introduction

One of the challenges in nanotechnology is the construction of miniaturized motors that can autonomously propel themselves without the need for externally applied fields. Because of size limitations, it is natural to seek propulsion mechanisms that do not require on-board carrying of fuel. Such nanoscale mechanisms abound in biology, where complex protein motors perform mechanical operations by exploiting the chemical energy of the carrying fluid.

It was suggested by Mitchell (1972) that several of these motors propel themselves by generating electric fields. The possibility of such ‘auto-electrophoresis’, wherein a body generates an electric field by asymmetric ion pumping at its boundary, was discussed by Anderson (1989). The fundamental principles underlying such ‘osmotic motors’ were investigated using both theoretical modelling (Lammert et al. 1996; Golestanian et al. 2005; Córdova-Figueroa & Brady 2008; Golestanian 2009; Wei & Jan 2010) and numerical simulations (Rückner & Kapral 2007; Tao & Kapral 2010).

The auto-electrophoresis concept was experimentally proven by Paxton et al. (2004) who demonstrated autonomous non-Brownian motion of platinum–gold bimetallic nanorods in aqueous hydrogen peroxide solutions. The asymmetric ion pumping is naturally achieved by the different chemical redox reactions on the two metals: at the platinum end, peroxide oxidation generates hydrogen cations (as well as electrons and oxygen molecules); the electrons are conducted through the metallic particle towards its golden end, where they combine with the hydrogen cations and peroxide molecules to produce water molecules. Paxton et al. (2004) observed systematic particle motion in the direction pointing from the golden end to the platinum end, at velocities of up to 10 body lengths per second. Auto-rotation of similar synthetic structures was observed by other groups (Catchmark et al. 2005; Fournier-Bidoz et al. 2005). A different particle geometry was considered by Howse et al. (2007); these authors employed micron-size polystyrene spheres whose half-side was coated with platinum in order to verify the theoretical model of Golestanian et al. (2005).

Clearly, the energy required for the self-propulsion process is extracted from the overall transformation of hydrogen peroxide to water and oxygen. The precise physico-chemical mechanism, by which chemical energy is transformed into a mechanical one, was, however, initially unclear (Paxton et al. 2004, 2005). Following voltage measurements at zero-current conditions (Wang et al. 2006) and current measurements between platinum and gold micro-electrodes (Paxton et al. 2006), it became obvious that the dominant mechanism is an electrokinetic one. This hypothesis was corroborated by Wang et al. (2006), who fabricated an insulating nanorod, where the catalysis of peroxide to oxygen and water occurs without any cation exchange. These particles exhibited Brownian motion, with no preferential swimming along their axis.

Following the experimental work and the evidence for the dominance of electrokinetic mechanisms, it is desirable to construct a theoretical model that would describe the self-propulsion phenomenon based upon electrokinetic first principles, namely continuum transport (Saville 1977) and cation redox kinetics (Newman 1973). In relating binary-species models to transport processes in hydrogen peroxide solutions, the two ionic species represent hydrogen cations (H+), which are generated at the platinum surface and consumed at the gold end, and hydroxide anions (OH). A first step in this direction was carried out by Moran et al. (2010), who solved the full electrokinetic equations numerically. Upon modelling the surface kinetics by a prescribed piecewise uniform flux of cations, the simulations were found to agree with the experimental trends.

The modern electrokinetic approach in analysing electrokinetic problems exploits the relative smallness of the Debye thickness in typical systems. While the electrokinetic transport in the present problem is complicated by the cation exchange on the particle boundary, it is still characterized by a conceptual decomposition of the fluid domain into the thin Debye layer surrounding the particle, in quasi-equilibrium, and the electro-neutral ‘bulk’ fluid outside it. This scale disparity naturally lends itself to boundary-layer analysis, which allows us to extract a macroscale model in which the Debye layer is manifested as effective boundary conditions. This has been an effective approach in describing ‘conventional’ electrokinetic phenomena such as electrophoresis (Keh & Anderson 1985) and diffusio-phoresis (Prieve et al. 1984). The goal of the present paper is to provide the first application of this asymptotic methodology to the problem of electrokinetic self-propulsion.

Another innovative feature of the present paper is the systematic modelling of surface cation exchange. In contrast to previous analyses, where this process was modelled by rather ad hoc specification of ionic fluxes or concentrations, it is represented here by a realistic model of reduction–oxidation surface transport, expressed in terms of two kinetic coefficients: dissolution rate constant and kinetic-equilibrium cation concentration. For a given metal, these coefficients can be calculated from two independent current–voltage measurements. One of them, at zero current, is tabulated at available references (Newman 1973).

The proposed paradigm, using systematic electrokinetic and electrochemical description, together with an asymptotic transformation of a microscale model to a macroscale one, is innovative in the analysis of phoretic swimmers. Recently, we described a comparable methodology for analysing field-driven motion of cation-selective particles (Yariv 2010b). In both problems, the highly conducting particles exchange cations with the surrounding electrolyte. The difference between the two problems lies only in the asymmetry driver (external field in one, inhomogeneous catalysis in the other). Thus, the macroscale equations and boundary conditions derived by Yariv (2010b) readily apply to the present self-propulsion problem.

In the purpose of generality, we consider a rather general configuration, comprising a highly conducting axisymmetric particle (of an otherwise arbitrary shape). Cation exchange on the particle surface is represented by redox kinetics; the respective kinetic coefficients may vary along the particle boundary, but are uniform at each cross section, whereby axial symmetry is retained. The exact continuum-level description of the electrokinetic transport consists of: (i) the governing differential equations, (ii) boundary conditions on the particle surface, which, in view of cation exchange, differ from those pertinent to inert surfaces, (iii) far-field conditions representing the absence of any external drivers, and (iv) an integral ‘force-free’ condition, representing the fact that the particle is freely suspended in the electrolyte. In addition, this description comprises a unique integral constraint, postulating zero net cationic flux into the particle.

Using the method of Yariv (2010b), a comparable macroscopic model is constructed in the thin-Debye-layer limit. The effective boundary conditions in that model represent matching with the Debye-scale fields, rather than physical conditions at the literal particle surface. In the macroscale description, the inhomogeneous kinetics are transformed into a boundary condition governing the zeta-potential distribution. This condition illuminates how a non-uniform distribution of the kinetic-equilibrium cation concentration on the particle boundary naturally results in autonomous particle motion.

The macroscale equations are linearized for small variations of the kinetic-equilibrium cation concentration. In the linearized model, ion transport and electrostatics are decoupled from the flow. This model highlights the physical mechanism responsible to the induced particle motion, linking the microscale transport and redox reaction to the macroscale generation of electrokinetic slip. It is demonstrated for the case of a spherical particle. The resulting particle velocity describes a transition from the electrokinetic velocity scale (Saville 1977) at fast kinetics to a modified scale at slow kinetics, the latter velocity being inversely proportional to electrolyte conductivity.

Motivated by the use of a rod-like shape in prevailing experiments (Paxton et al. 2004), we also present an application of the macroscale model to slender geometries. For sufficiently concentrated solution, the Debye thickness is small even compared with the small particle dimension. (In the experiments of Paxton et al. (2004), for example, the bimetallic nanorods were 370 nm in diameter and 2 μm in length.) Such a three-scale disparity has already been employed in other slender-body electrokinetic analyses (Solomentsev & Anderson 1994; Saintillan et al. 2006; Yariv 2008b). Using matched asymptotic expansions, we analyse a slender body of arbitrary axisymmetric shape, obtaining the particle velocity as a functional of its shape and kinetic properties.

2. Formulation

(a) Problem description and dimensionless notation

A symmetric binary electrolyte solution (permittivity d*, ionic valences ±z) is in thermodynamic equilibrium, where the two ionic species possess an identical concentration c*. A conducting axisymmetric particle (length 2L*), whose surface can exchange cations with the electrolyte, is introduced into the solution. The kinetic coefficients describing this exchange may vary along the particle-axis direction. As a result of ion exchange and the ensuing electrokinetic transport, the freely suspended particle acquires a non-zero velocity relative to the solution. Owing to symmetry, this velocity is directed along the particle axis.

In analysing this problem, it is convenient to employ a particle-fixed cylindrical coordinate system. The axial and radial distances are denoted by L*x and L*ϖ, where the x-axis is aligned with the particle axis. With no loss of generality, the particle shape is Embedded Image 2.1 where R is an O(1) function satisfying Embedded Image 2.2 Thus, λ represents the particle slenderness. By definition, the shape function also satisfies Embedded Image 2.3 where the rate at which R vanishes near the ends depends upon its shape: for rounded ends at x=1, for example, R(x)/(1−x) diverges as x→1.

The electrokinetic transport is described using the dimensionless notation of Saville (1977). Thus, in addition to normalizing the position vector x by L*, the ionic concentrations c± are normalized by c* and the electric potential φ by the thermal voltage (about 25 mV in a uni-valence solution) φ*=R*T*/zF*, in which R* is the gas constant, T* is the absolute temperature and F* is Faraday’s constant. Thus, the ionic fluxes, normalized with D*c*/L* (with D* the ionic diffusivity, presumably identical for both species), adopt the form Embedded Image 2.4 The velocity field v and particle velocity Embedded Image are normalized by the electrokinetic scale (Saville 1977) Embedded Image 2.5 wherein μ* is the electrolyte viscosity. Stresses are normalized by μ*v*/L*, and forces by μ*v*L*.

Following Yariv (2010b), we assume that the cation selectivity on the particle boundary is expressed by local redox kinetics, where Embedded Image 2.6 in which the backward deposition term depends upon the cationic concentration c+ adjacent to the boundary. Here, k is an effective rate constant, normalized by D*c*/L*, and γ is a reference cation concentration, normalized by c*. This model was already used to analyse the field-driven motion of a homogenous particle (Yariv 2010b); in the present problem, both k and γ are functions of x, representing non-uniform surface kinetics. For the Embedded Image relation, equation (2.6) implies c+=γ; thus γ reflects a kinetic-equilibrium cation concentration. In equation (2.6), we neglect the effect of Stern-layer potential.

(b) Governing equations

The steady-state differential equations governing the pertinent fields comprise the Nernst–Planck conservation equations, Embedded Image 2.7 describing ionic transport by the combined action of diffusion, electro-migration and convection; Poisson’s equation, Embedded Image 2.8 as well as the continuity and inhomogeneous Stokes equations, Embedded Image 2.9 The dimensionless group appearing in equation (2.7), α=L*v*/D*, constitutes the Péclet number. In view of equation (2.5), Embedded Image 2.10 thus, this number is independent of both particle size L* and equilibrium concentration c*. For typical ionic diffusivities α≈0.5 (Saville 1977). The parameter δ appearing in equation (2.8) denotes the length-scale ratio δ*/L*, in which δ* is the Debye thickness, Embedded Image 2.11

On the particle surface S, the impermeability to anions is expressed by the no-flux condition Embedded Image 2.12 wherein the unit vector Embedded Image, normal to S, is pointing into the fluid. The cationic selectivity kinetics (2.6) is expressed as a condition governing c+, Embedded Image 2.13 Since the particle is conducting, the electric potential possesses a uniform value, say Embedded Image, on its boundary Embedded Image 2.14 Finally, the velocity vector satisfies the mass impermeability and no-slip on its boundary, Embedded Image 2.15

At large distances away from the particle, the ionic concentrations approach the equilibrium unity value, while the electric field vanishes, Embedded Image 2.16 In the co-moving reference frame, the velocity at large distances approaches the uniform value Embedded Image 2.17 reflecting particle motion relative to the otherwise quiescent fluid.

As usual in problems of freely suspended particles, the particle velocity Embedded Image is determined using the constraint of a force-free particle, Embedded Image 2.18 wherein the contribution of both Newtonian and Maxwell stresses is accounted for. Here, I denotes the idemfactor and superscript ‘†’ signifies transposition.

(c) Consistency condition

Since the electric potential is defined to within an arbitrary additive constant, one may naively claim that the physics are indifferent to the value of particle potential Embedded Image. This arbitrariness, however, is already implicit in the far-field condition (2.16). Once the additive constant is set in choosing the value of φ at large distances, the value of Embedded Image represents a relative potential, therefore affecting the electrokinetic processes. (The need for an extraneous condition to determine the particle potential is discussed by Ben et al. (2004) in the context of uniform ion exchangers.)

The requisite condition is obtained from a self-consistency requirement at steady state, expressed in terms of the cationic normal flux at the particle boundary. Because of cation redox reactions, the cation flux in the present problem does not necessarily vanish on that boundary; this lies in sharp contrast with the classical situation of an inert surface (impermeable to both ionic species). The boundary integral of that flux, however, must vanish. Otherwise, the total particle charge would evolve in time, in contradiction to the pre-assumed steady state. Thus, we postulate1 Embedded Image 2.19

3. Macroscale description

The preceding equations are tremendously simplified in the thin-Debye-layer limit, δ→0. Poisson’s equation (2.8) implies electro-neutrality, whereby both ionic concentrations are identical, Embedded Image 3.1 It is well known that this limit is singular: the multiplication of a small parameter by the highest derivative in equation (2.8) implies the existence of an O(δ)-thick boundary (Debye) layer. Thus, the fluid domain is naturally decomposed into the electro-neutral ‘bulk’ and the Debye layer in quasi-equilibrium, surrounding the particle, where the (different) ionic concentrations are governed by Boltzmann distributions (Prieve et al. 1984; Rubinstein & Zaltzman 2001). The standard practice in electrokinetic analyses (Yariv 2010a) is to extract an effective bulk description wherein effective boundary conditions represent asymptotic matching of the ‘outer’ bulk fields with the ‘inner’ Debye-scale variables (which satisfy the boundary conditions (2.12)–(2.15) on the literal particle boundary). An important quantity in the Debye-layer structure is the Debye-layer voltage—the ‘zeta potential’ ζ. The effective boundary conditions are expressed in terms of this (generally non-uniform) voltage, whose distribution must be found throughout the solution course.

The requisite extraction of a macroscale model has already been carried out by Yariv (2010b) for the related problem of a spherical ion exchanger with uniform kinetic properties. Owing to the local nature of the asymptotic matching, the extension of that work to the more general case of non-spherical particles with non-uniform properties, pertinent to the present problem, is straightforward. To distinguish between the exact continuum fields and their bulk extrapolations, we denote the latter by capital variables. In the following, we present the leading-order equations governing the bulk fields in the limit δ→0, obtained using the governing equations of §2 in conjunction with electro-neutrality (3.1). Thus, the far-field conditions now read Embedded Image 3.2 The flow equations retain the form (2.9), Embedded Image 3.3 while addition and subtraction of the ionic conservation equations provide the simplified ion- and charge-conservation balances Embedded Image 3.4 The possibility of non-uniform concentration C results in two non-conventional features: the potential Φ is non-harmonic, and the Coulomb body force does not disappear from the Stokes equations despite leading-order electro-neutrality. This body force reflects O(δ2) charge density (which would vanish for harmonic Φ).

The boundary conditions, obtained from asymptotic matching with the Gouy–Chapman solution of the Debye layer, are (cf. Yariv 2010b)

  • — The requirement (2.12) of zero anionic flux adopts the form Embedded Image 3.5 with Embedded Image.

  • — The cation exchange kinetics (2.13), in conjunction with the Boltzmann distribution of cations in the Debye layer, is expressed in the equation Embedded Image 3.6 which provides the zeta potential Embedded Image 3.7 Note that ζ may vary along the particle boundary, and is accordingly a function of x.

  • — The electric potential now satisfies an inhomogeneous Dirichlet condition, Embedded Image 3.8 representing the modification of the original equi-potential Dirichlet condition (2.14) by the zeta-potential distribution (3.7).

  • — The no-slip condition (2.15) is replaced by the Dukhin–Derjaguin slip condition (Prieve et al. 1984), Embedded Image 3.9 wherein Embedded Image 3.10 is the surface gradient operator. While this macroscale condition was originally developed for inert surfaces that do not exchange ions, it universally applies to any solid surface on which the fluid does not slip, regardless of its kinetic model (see Rubinstein & Zaltzman 2001).

Conditions (3.5)–(3.9) apply at the outer edge of the Debye layer (as opposed to the literal surface on which equations(2.12)–(2.15) hold). In the limit δ→0, this edge coincides with the particle boundary. Thus, we hereafter reinterpret the surface S as the effective particle boundary.

Since the Stokes equations (2.9) are equivalent to the statement of divergence-free total stress, the integral appearing in equation (2.18) can be taken over any surface enclosing the particle boundary. Accordingly, equation (2.18) applies with the new interpretation of the surface S, Embedded Image 3.11

Finally, it is required to rewrite the consistency condition (2.19). In a Debye layer, the ionic fluxes are O(1) and are uniform in a direction normal to the boundary (Yariv 2010a,b). Thus, the normal flux at the literal particle boundary, Embedded Image, can be replaced by the corresponding flux at the outer edge of the Debye layer. We therefore obtain Embedded Image 3.12

In the macroscopic description, it is evident that self-propulsion is animated by non-uniformity in γ. Indeed, it is easily verified that for uniform γ, the governing equations posses the equilibrium reference solution Embedded Image 3.13 with a uniform zeta potential Embedded Image 3.14 and a vanishing particle velocity, Embedded Image. This equilibrium state holds regardless of the distribution of k, illustrating that the driver of particle motion is the non-uniformity of γ. Indeed, when γ is non-uniform, no equilibrium solution exists; the system reaches a different steady state, wherein electrokinetic transport in conjunction with (3.11) generally leads to particle motion.

4. Linearized problem

Because the Debye layer has been effectively removed in the macroscale description, its analysis is much simpler than that of the original multi-scale problem. Nonetheless, the highly coupled and nonlinear system (3.3)–(3.12) is still not amenable to analytical treatment.

In the following, we simplify further. We write the distribution of kinetic-equilibrium concentration in the form Embedded Image 4.1 where Embedded Image is the average of γ over the interval [−1,1], Embedded Image 4.2 With no loss of generality, we assume that γ′ is O(1), whereby ϵ represents the relative magnitude of deviations from that average. The reason for using the representation (4.1) lies in the simplicity of the case ϵ=0, where γ is uniform, and the governing equations satisfy the reference solution (3.13) and (3.14) with a vanishing particle velocity.

Following the standard electrokinetic practice of a weak-field analysis (O’Brien & White 1978), we focus upon the case of small deviations from the average, ϵ≪1, resulting in the following linearization about the reference state (3.13): Embedded Image 4.3 whereby Embedded Image 4.4

(a) Ionic concentration and electric field

The linearized conservation equations governing the ion-concentration perturbation and electric field are simply Embedded Image 4.5 while the linearized boundary conditions are Embedded Image 4.6 and Embedded Image 4.7 together with the attenuation of both Φ′ and C′, Embedded Image 4.8 The perturbation ζ′ appearing in equation (4.7) is provided by linearization of equation (3.7), Embedded Image 4.9 Finally, the consistency condition (3.12) yields Embedded Image 4.10

The linearized transport is decoupled from the flow. As a matter of fact, with no loss of generality equations (4.5), (4.6) and (4.8) imply the equality Embedded Image 4.11 thus, equations (4.6)–(4.9) provide the mixed-type condition for Φ′, Embedded Image 4.12 and the consistency condition (4.10) becomes Embedded Image 4.13

(b) Flow

Once Φ′ is determined, the velocity field can be obtained. It is governed by the homogeneous Stokes equations, Embedded Image 4.14 the slip condition, obtained via linearization of equation (3.9) in conjunction with (4.11), Embedded Image 4.15 and the far-field condition Embedded Image 4.16

The particle velocity is determined using the force-free condition Embedded Image 4.17 Here, Embedded Image 4.18 is the hydrodynamic force on the particle, with Embedded Image 4.19 being the Newtonian stress tensor associated with the linearized flow. The Maxwell stress, which is quadratic in the electric field, does not affect the linearized problem.

It is convenient to decompose the flow field into two parts, Embedded Image 4.20 with a comparable decomposition for P′ and S′. Flow field ‘I’ satisfies the far-field condition (4.16) and a no-slip condition on the particle boundary; flow field ‘II’ satisfies the slip condition (4.15) and vanishes at infinity. Owing to symmetry, the hydrodynamic forces associated with these flows apply in the x-direction, Embedded Image 4.21

Since the equations governing the axisymmetric flow field ‘I’ are linear and homogeneous in Embedded Image, the associated hydrodynamic force is of the form Embedded Image 4.22 where Embedded Image is a shape-dependent drag coefficient. For similar reasons, the hydrodynamic force associated with flow field ‘II’ is a linear functional of the slip distribution. This force is easily calculated using a method owing to Brenner (1964). Since flow field ‘II’ satisfies the Stokes equations and decays at infinity, it satisfies the Lorentz reciprocal theorem (Happel & Brenner 1965), Embedded Image 4.23 in which Embedded Image is any flow field that satisfies the Stokes equations and vanishes at infinity, and Embedded Image is the corresponding stress field. Choosing Embedded Image as the velocity field that corresponds to a fictitious translation of the particle with velocity Embedded Image and recalling (4.21) yields Embedded Image 4.24 Substitution of equation (4.15) in conjunction with equations (4.17) and (4.22) then furnishes the expression Embedded Image 4.25

The first and second terms in the slip condition (3.9), respectively, represent electro-osmotic slip, directed along (against) the surface gradient of Φ for positive (negative) ζ and diffusio-osmotic slip, directed against the surface gradients of c. To understand the linearized mechanism, consider with no loss of generality a particle where, on average, γ′ is positive for x>0 (and whence, by definition, is negative for x<0). In view of equations (4.7)–(4.9), the perturbation Φ′ (and then also C′) possesses a similar polarity.

For Embedded Image, the reference zeta potential is negative (see (3.14)), and the electro-osmotic slip is pointing on average in the negative x-direction. This is also the direction of the diffusio-osmotic slip. It then follows that the freely suspended particle moves in the positive x-direction. For Embedded Image, where Embedded Image, the electro-osmotic slip direction is reversed, while the diffusio-osmotic slip remains unaltered. In that case, electro-osmosis dominates diffusio-osmosis (see (4.15)), and the direction of motion reverses.

(c) Spherical particle

In principle, the linearized problem can be solved for various geometries using analytical and numerical methods. For λ=O(1), the simplest shape is spherical. Chemical self-propulsion of ‘Janus’-type spheric particles, made out of two homogenous semi-spheres, was investigated in numerous theoretical analyses and in at least one experimental study (Howse et al. 2007).

The spherical geometry is naturally analysed in polar spherical coordinates, r being the radial distance measured from the particle centre and θ the latitudinal angle measured from the x-axis. With Embedded Image and Embedded Image (Happel & Brenner 1965), equation (4.25) becomes Embedded Image 4.26 Since Φ′ satisfies Laplace’s equation and decays at infinity, it can be expanded into spherical harmonics, Embedded Image 4.27 with Pm the Legendre polynomial of degree m. Substitution into equation (4.26) in conjunction with the orthogonality properties of the Legendre polynomials readily yields Embedded Image 4.28

The coefficients {ϕm} depend, through (4.12), upon the axisymmetric distribution γ′, a function of θ. For uniform k, these coefficients are easily obtained via expansion of γ′ into surface harmonics, Embedded Image, and use of equation (4.12) with ∂/∂n=∂/∂r. Substitution into equation (4.28) then provides the velocity Embedded Image 4.29 For example, in a Janus-type particle (as in the experiments of Howse et al. 2007), we can write, with no loss of generality, γ′=sgn(x); then, Γ1=3/2.

5. Slender particles

With a view towards understanding self-propulsion of rod-like particles (as observed by Paxton et al. (2004)), we consider here slender particles, where λ≪1.

(a) Electrostatics

The electrostatic problem is solved using inner–outer expansions. The inner region, at the O(λ) cross-sectional scale, is resolved using the stretched radial coordinate Embedded Image 5.1 In terms of the inner coordinates, Laplace’s equation governing Φ′ is Embedded Image 5.2 Since ∂/∂nλ−1∂/∂ρ, condition (4.12) becomes Embedded Image 5.3 Its structure suggests the inner expansion Embedded Image 5.4 Substitution into equations (5.2) and (5.3) provides the leading-order differential equation Embedded Image 5.5 and boundary condition Embedded Image 5.6 the potential Embedded Image, in turn, is obtained from substitution of equation (5.6) into equation (4.13), Embedded Image 5.7 The solution of equations (5.5) and (5.6) is Embedded Image 5.8 with Embedded Image 5.9 Evaluation of B(x) requires matching with the outer solution.

The outer region, at O(1) length scales, is defined by the limit λ→0 with ϖ=O(1). Here, at leading order, the particle appears like a zero-width line segment. In view of equation (5.4), we postulate the comparable expansion Embedded Image 5.10 Since Embedded Image is harmonic, it is represented by a distribution of sources, Embedded Image 5.11 At small ϖ, Embedded Image diverges logarithmically (Batchelor 1970), Embedded Image 5.12 Asymptotic matching with the inner potential yields Embedded Image 5.13 and Embedded Image 5.14 In addition, it also necessitates modifying equation (5.4) to the form Embedded Image 5.15 wherein Embedded Image 5.16

(b) Hydrodynamics

The particle velocity is provided by the surface integral (4.25); its calculation, therefore, requires the preliminary evaluation of the surface gradient of Φ′ on the particle boundary, using the inner expansion (5.15). For λ≪1, the unit vector normal to the particle boundary (2.1) is Embedded Image 5.17 In terms of the inner coordinates, the surface gradient is given by Embedded Image 5.18 with a relative error that is algebraically small in terms of λ. Substitution of equations (5.8), (5.9) and (5.13)–(5.16) yields, at ρ=R(x), Embedded Image 5.19 with an algebraically small relative error. Here, Embedded Image 5.20 and Embedded Image 5.21

For a slender body, moreover, the stress field owing to translation is already available (Cox 1970). The corresponding traction is Embedded Image 5.22 with an algebraically small relative error. Here, Embedded Image is the axial hydrodynamic force, per unit length, which would be exerted upon the particle if it were to translate with a unit velocity in the x-direction. It was evaluated by Cox (1970) as an asymptotic series in Embedded Image, Embedded Image 5.23

Substitution of equations (5.19)–(5.23) in conjunction with axial symmetry yields Embedded Image 5.24 wherein Embedded Image 5.25 and Embedded Image 5.26

Consider now the integrals of H0 and H1 from x=−1 to x=1. In view of equation (2.3), the integrals of both H0 and the third term in equation (5.26) vanish, while that of the fourth term is given by Embedded Image 5.27 Also, integration by parts of the first term in equation (5.26), in conjunction with equation (2.3), yields a term that cancels out with the integral of the second term in equation (5.26). Finally, integration by parts of the fifth term in equation (5.26) in conjunction with equation (2.3) yields Embedded Image 5.28 It is easily verified that the preceding results are obtained for both rounded and conical ends.

Substitution into equation (5.24) of the remaining terms (5.27) and (5.28) yields Embedded Image 5.29 with an O(1/In λ) relative error. Substitution into equation (4.25), together with the slender-body approximation for the drag coefficient (obtained from (5.23)) Embedded Image 5.30 eventually furnishes the particle velocity as a functional of k(x), R(x) and γ′(x), Embedded Image 5.31

In the present description, a fore–aft symmetric bimetallic nanorod is represented by an even shape distribution R(x) and a piecewise constant γ′(x). For that configuration, (5.7) implies that Embedded Image. In view of the representation (4.1), we can write, with no loss of generality, γ′(x)=sgn(x). For uniform k, we then find, using equations (2.2) and (2.3), Embedded Image 5.32 For Embedded Image (negative reference zeta), the particle propels itself towards the side of higher γ, in agreement with the physical mechanism described in §4. In the bimetallic platinum–gold nanorods experiment of Paxton et al. (2004), the measured zeta potential is indeed negative (Paxton et al. 2005); in these experiments, cations (protons) are generated at the platinum end and consumed at the gold end. The redox kinetics (2.6), approximately expressed by (5.6), imply that the value of γ for platinum is greater than that for gold. Thus, the present model predicts motion towards the platinum end, in agreement with the experimental observation.

The dependence of equation (5.32) upon k suggests a monotonic increase of particle velocity with a reaction rate. Note, however, that the entire slender-body analysis fails for large k, where equation (5.3) suggests the existence of a ‘kinetic’ boundary layer at distances ρO(1/k).

6. Discussion

A microscale steady-state model describing electrokinetic self-propulsion of axisymmetric particles driven by non-uniform surface catalysis was formulated. Cation exchange on the particle boundary is described by a reduction–oxidation surface reaction. This kinetic condition signifies a major qualitative difference with the classical description of inert surfaces, where a no-flux condition applies for both ionic species. Another unique feature of the present electrokinetic formulation is an integral constraint of zero net cation flux into the particle, reflecting the presumed steady state. This condition serves to determine the particle electric potential, which is not a priori specified.

A macroscale model is extracted in the thin-Debye-layer limit. Unsurprisingly, it significantly differs from familiar thin-Debye-layer models of field-driven particle motion, such as electrophoresis and diffusio-phoresis. Thus, cation selectivity at the particle boundary results in the establishment of concentration polarization in the electro-neutral bulk, despite the absence of any imposed concentration gradients. This polarization leads to inherent nonlinearities in the ion-concentration balance equation (through the advection term), the local momentum balance (through a Coulomb body force), and the charge-conservation equation (through the conductivity dependence upon concentration).

The boundary conditions governing the present macroscale model differ from those familiar from thin-Debye-layer analyses of flows about inert particles. Thus, the uniform electric potential at the literal particle boundary is represented by a non-uniform Dirichlet condition on the effective boundary. The zeta potential is determined by the cation-exchange rate, and, therefore, depends upon the electric field and ion concentration at the effective boundary. The fluid velocity is driven by Coulomb body forces, as well as concentration gradients and electric field on the particle boundary (through diffusio-osmotic and electro-osmotic slip terms). Since both effects are of comparable magnitude, the phrase ‘auto-electrophoresis’ may not constitute an appropriate terminology for macroscopically describing the phoretic self-propulsion mechanism.

The underlying driver of particle motion is the non-uniformity of the kinetic-equilibrium cation concentration on the boundary. For small variations of that concentration, the governing equations can be linearized about a reference state that corresponds to the mean kinetic-equilibrium concentration. In the linear approximation, the (dimensionless) salt concentration and electric potential coincide. The electric field is governed by a mixed-type boundary condition. Once calculated, the ensuing flow is set by a slip condition that reflects the combined action of electro-osmosis and diffusio-osmosis. The linearized problem is demonstrated for a spherical particle, where the calculation is straightforward. The resulting particle velocity (4.29) represents a transition from O(1) velocities at fast kinetics to O(k) velocities at slow kinetics. In dimensional terms, this reflects a transition from particle velocities that scale as v* (see (2.5)) to slower velocities that scale as (L*k*/D*c*)v*. The dependence upon c* appears to be consistent with the experimental results of Paxton et al. (2006), who reported a swimming velocity that scales inversely with solution conductivity.

For slender particles, we employ standard matched asymptotic expansions (Hinch 1991) for calculating the electric field. Use of Lorentz’s reciprocal theorem then provides the particle velocity as the functional (5.31) of its shape and surface distribution of kinetic coefficients. This functional can handle discontinuous distributions, as would be the case when modelling bimetallic nanorods (Paxton et al. 2004). Because of the particle slenderness and the associated dominance of transverse transport, the dependence upon the kinetics rate appears at leading order (see (4.12)), whereby the dimensional velocities again scale as (L*k*/D*c*)v* (but now with an O(λ) dimensionless coefficient).

The derivation of the macroscale model relies upon a sequence of approximations. The most critical one, allowing linearization about an equilibrium reference state, is reasonable if the cationic kinetic-equilibrium concentration on the reactive particle surface does not vary significantly. Two other additional approximations, the thin-Debye-layer limit and the slender-body modelling, naturally exploit the three separate length scales of the problem. Most of the remaining approximations are quite standard. Thus, as is common in the electrochemical literature of ion-exchange membranes (Rubinstein & Zaltzman 2001) and particles (Ben & Chang 2002), we analyse here a binary electrolyte model. The assumption of identical ionic diffusivities is also quite common; as a matter of fact, that assumption is not necessary for deriving the major results of the present analysis (e.g. (5.31)), since ion convection does not appear in the linearized problem.

Additional approximations appear in the particle model; thus, the possible presence of the Stern layer is neglected, and the potential is assumed uniform, despite the possible effect of internal contact potential for bimetallic particles. The assumption of an axisymetric particle shape and chemistry is motivated by existing experiments, although the description of particle shape by a continuous radial distribution excludes such shapes as finite cylinders.

While the accuracy of the derived model is hampered by the preceding approximations, it nevertheless constitutes the first systematic thin-Debye-layer macroscale description of electrokinetic self-propulsion driven by non-uniform cation exchange. It is anticipated that formulae of the forms (4.29) and (5.31), or modifications thereof, will be employed in initial design considerations of future artificial swimmers.

Acknowledgements

This work was supported by the Israel Science Foundation under grant number 2011446.

Footnotes

  • 1 An integral condition is also required to determine the particle potential value in the comparable problem of an ideally polarizable particle whose boundary is inert. In that case, the condition is of different nature: it represents the invariance of total particle charge during the unsteady charging process, see Yariv (2008a).

  • Received September 28, 2010.
  • Accepted November 26, 2010.

References

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