Strong Deformation of the Thick Electric Double Layer around a

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Strong deformation of the thick electric double layer around a charged particle during sedimentation or electrophoresis Aditya S. Khair Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01897 • Publication Date (Web): 31 Jul 2017 Downloaded from http://pubs.acs.org on August 7, 2017

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Strong deformation of the thick electric double layer around a charged particle during sedimentation or electrophoresis Aditya S. Khair∗ Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA E-mail: [email protected]

Abstract

The deformation of the electric double layer around a charged colloidal particle during sedimentation or electrophoresis in a binary, symmetric electrolyte is studied. The surface potential of the particle is assumed to be small compared to the thermal voltage scale. Additionally, the Debye length is assumed to be large compared to the particle size. These assumptions enable a linearization of the electrokinetic equations. The particle appears as a point charge in this thick-double-layer limit; the distribution of charge in the diffuse cloud surrounding it is determined by a balance of advection due to the particle motion; Brownian diffusion of ions; and electrostatic screening of the particle by the cloud. The ability of advection to deform the charge cloud from its equilibrium state is parameterized by a Péclet number, Pe. For weak advection (Pe 1) the cloud is only slightly deformed. In contrast, the cloud can be completely stripped from the particle at Pe 1: consequently, ∗ To

whom correspondence should be addressed

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electrokinetic effects on the particle motion vanish in this regime. Therefore, in sedimentation the drag limits to Stokes’ law for an uncharged particle as Pe → ∞. Likewise, the particle velocity for electrophoresis approaches Huckel’s result. The strongly deformed cloud at large Pe is predicted to generate a concomitant increase in the sedimentation field in a dilute settling suspension.

1

Introduction

The objective of this article is to determine the sedimentation and electrophoretic velocity of a charged colloidal particle under conditions where: (i) the Debye length is large compared to the particle size; and (ii) the electric double layer is strongly deformed by the particle motion. Electrokinetic effects arise due to the action of fluid flows or electric fields on the ionic charge clouds that screen surfaces in electrolytes. Together, the surface charge and the charge in the cloud are known as the electric double layer, which has zero net charge. The linear dimension of the charge cloud in a dilute solution is characterized by the Debye length, κ −1 . The Debye length is around ten nanometers for a milli-molar solution of monovalent ions in water. Much larger values of κ −1 , on the order of hundreds of nanometers, occur in nonpolar fluids to which surfactant molecules are added as charge-stabilization agents. 1 In this paper, we consider two prototypical electrokinetic phenomena: sedimentation and electrophoresis of a charged colloidal particle. In sedimentation a particle is subject to an external force (e.g., gravity) that distorts the charge cloud around it, leading to an electrokinetic resistance to the particle motion, such that the velocity of the particle is smaller than that of an uncharged particle of equal size. Alternatively, if the charged particle is held in a uniform stream of flow, this additional resistance would manifest as an increase in the external force required to keep the particle fixed. Booth 2 utilized the standard electrokinetic equations for dilute electrolytes 3 to derive an analytical expression for the sedimentation velocity of a solid, uniformly charged


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spherical particle of radius a in an unbounded solution, under the condition that the surface, or zeta, potential of the particle, ζ , is small compared to the thermal voltage scale kB T /e, where kB is Boltzmann’s constant, T is absolute temperature, and e is the charge on a proton. For instance, kB T /e ≈ 25mV at T = 298K. Here, we do not account for the presence of an immobile (Stern) layer of adsorbed ions at the particle surface; hence, ζ is equal to the electric potential at the particle surface. Booth 2 considered arbitrary values of κa, thereby spanning ‘thin’ (κa 1) to ‘thick’ (κa 1) charge clouds. The restriction of a small zeta potential was lifted by the numerical calculations of Ohshima et al.

4

These two studies also calculated the

macroscopic electric (‘sedimentation’) field arising from the settling of a dilute suspension of identical particles. Both works also assumed that the particle motion only slightly distorts the charge cloud from its (spherically symmetric) equilibrium state. This assumption is valid if the Péclet number for ion motion, Pe = U/κD, is much less than unity, where U is the characteristic particle speed, and D is the diffusion coefficient of the ions. As we shall discuss in detail later, Pe is indeed small for sedimentation of colloidal particles in aqueous electrolytes under a gravitational field. However, Saville 3 states that there are occasions for which Pe could be O(1) or even larger: a specific example being motion driven by a centrifugal field, as suggested by Dukhin. 5 The cloud can be significantly deformed for Pe ≥ O(1): numerical calculations by Keller et al.

6

and Bhattacharyya and Gopmandal 7 indicate that the cloud adopts a boundary

layer-wake structure at large Pe, which is reminiscent of the temperature distribution around a heated particle in a relatively cold uniform stream. 8 Keller et al.

6

also demonstrated that in-

creasing Pe leads to a decrease in the electrokinetic resistance, such that the particle velocity increases. Saville 3 claimed that a solution via singular perturbation methods at large Pe can be derived for thin clouds, although details of this solution were not presented. Dukhin 5 also discusses the structure of the charge cloud at large Pe and κa 1. Here, we examine the opposite limit of a thick charge cloud, κa 1, at small surface potentials. We derive an analytical expression for the leading order, O(κa), electrokinetic contribution to the particle velocity, which decreases monotonically with increasing Pe, in agreement with Keller et al.


6

However, elec-

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trokinetic effects do not vanish at large Pe; we will demonstrate that the sedimentation field in a dilute suspension increases in proportion to Pe, due to the increasing deformation of the charge cloud. The second problem we consider is electrophoresis of a solid, uniformly charged spherical particle in a steady, uniform electric field, E∞ . Henry 9 derived an analytical expression for the electrophoretic velocity U at small surface potentials and arbitrary κa. Henry’s expression at small κa limits to U = QE∞ /6π µa, a result due originally to Huckel. Here, Q is the particle charge and µ is the fluid viscosity. At the opposite extreme, Smoluchowski first derived U = QE∞ /4πκa2 µ for κa 1. Since Q = 4πεκa2 ζ for thin charge clouds, 10 where ε is the permittivity, we may rewrite the velocity as U = εζ E∞ /µ, which is perhaps the more familiar form. Numerical calculations by Wiersema et al.

11

and O’Brien and White 12 lifted the

restriction of small zeta potential. These studies, and indeed the majority of theoretical analyses of electrophoresis, assume that the applied field is sufficiently ‘weak’ that it only slightly distorts the equilibrium charge cloud. Here, by weak it is meant that the applied voltage drop across the particle, aE∞ (where E∞ is the strength of the applied field), is much less than the thermal voltage. 13 Consequently, the particle velocity is linear in the applied field; the electrophoretic mobility (i.e., the ratio of the magnitudes of the velocity and field) is independent of the field strength. Several studies have considered electrophoresis beyond this weak-field limit, for which the particle velocity is a nonlinear function of the field strength. The first (weakly) nonlinear contribution to the velocity in a binary electrolyte with equal, in magnitude, cation and anion charge numbers is proportional to the cube of the field strength. 14 An analytical expression for this contribution was calculated by Shilov et al.

15

at κa 1, for an electrolyte

with identical ionic diffusivities. Schnitzer and Yariv 13 corrected deficiencies in the analysis of Shilov et al.

15

and also derived a general expression for an electrolyte with unequal ionic

diffusivities. Schnitzer and Yariv 16,17 have also considered other facets of strong-field electrophoresis in the thin-charge-cloud limit. Here, we consider the opposite limit of a thick charge cloud at small surface potentials, and we will derive an analytical expression for the leading or4 ACS Paragon Plus Environment

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der, O(κa), electrokinetic contribution to the particle velocity. This contribution is a nonlinear function of the Péclet number, Pe = U/κD, in which the velocity scale U = QE∞ /6π µa. Therefore, the particle velocity is a nonlinear function of the field strength. We will predict that the electrophoretic mobility monotonically increases with increasing E∞ , which is in qualitative agreement with experiments on electrophoresis in surfactant doped nonpolar fluids. 18–22 Furthermore, we will demonstrate a quantitative agreement to the experiments of Stotz. 18 Thus, the theory developed herein may be useful to interpret electrophoresis measurements in nonpolar media, which are employed to infer particle charge. Finally, we will discuss our results in relation to the Wien effect, which refers to the field-dependent conductance of an electrolyte. In section 2 we present the equations governing the electrolyte dynamics and particle motion. These equations are simplified in the thick-charge-cloud limit in section 3. The sedimentation and electrophoresis problems are analyzed in sections 4 and 5, respectively. Concluding remarks are offered in section 6.

2

Governing equations

A spherical particle with uniform surface charge density q is immersed in a dilute binary symmetric electrolyte. The surface charge is screened by a diffuse charge cloud, whose linear dimension is characterized by the Debye length s κ −1 =

εkB T , 2(ze)2 nb

(1)

where z is the magnitude of the charge number of the ions, and nb is the bulk ion concentration at large distances from the particle. In what follows, we consider the particle motion under a gravitational field or electric potential gradient; we assume that the viscosity and permittivity are unaffected by these stimuli. A reference frame moving with the velocity of the particle U 5 ACS Paragon Plus Environment

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is adopted. At steady state the ion concentrations n± (where + denotes cation and − denotes anion) satisfy the conservation law ∇·j± = 0,

(2)

in which j± is the flux density. In writing (2) we have neglected the generation of ions via reactions in the electrolyte. In dilute solutions the flux density

j± = ∓

D± ze n± ∇φ − D± ∇n± + n± u, kB T

(3)

where D± are the ion diffusion coefficients, and henceforth we assume D+ = D− ≡ D; φ is the electric potential; and u is the fluid velocity vector. The first term on the right-hand side of (3) describes the electro-migration of ions under a potential difference; the second describes diffusion down a concentration gradient; and the third describes advection with the local flow velocity. The potential satisfies the Poisson equation

−ε∇2 φ = ρ,

(4)

where ρ = ze(n+ − n− ) is the ionic space-charge density. Inertial forces can be typically neglected for colloidal scale particles; for example, a particle with a = 1µm moving at a speed of 1µm/s in water has a Reynolds number of O(10−6 ). Hence, the fluid mechanics is described by the Stokes equations µ∇2 u − ∇p = ρ∇φ and ∇·u = 0,

(5)

where the first equation reflects a balance of viscous and electric forces on a fluid element; and the second equation stipulates that the fluid is incompressible. At large distances φ → 0, n± → nb , and u → −U as r → ∞,

(6)

where r is the position vector anchored at the centroid of the particle, and r = |r|. The particle 6 ACS Paragon Plus Environment

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velocity U in the sedimentation problem can be prescribed a priori; one calculates the external force to sustain the motion. However, the particle velocity in electrophoresis is unknown a priori: it is determined from the constraint that the particle is free from an external force. At the particle surface, r = a, we impose

n · j± = 0 and − ε n · ∇φ = q,

(7)

where n is the unit normal vector pointing into the fluid. The first condition in (7) states that there are no ionic fluxes across the particle surface, and the second condition specifies a fixed, uniform surface charge density, which is unaltered by the particle motion. The total charge on the particle Q = 4πa2 q. We seek to determine the O(κa) contribution to the particle velocity. Previous work at small Pe has shown that this contribution is independent of the ratio of the permittivity of the particle to the fluid. 9 Hence, one can neglect the field in the particle in the second condition of (7). A physical explanation is that the charge cloud is so much larger than the particle that the precise electrostatic boundary condition at the particle surface does not matter at O(κa). In the absence of an external field or flow (u = 0), the equilibrium ion concentrations follow a Boltzmann distribution n± (r) = nb e∓zeφ (r)/kB T ,

(8)

and, consequently, the potential φ satisfies the (nonlinear) Poisson-Boltzmann equation. 10 Henceforth, it is assumed that the surface potential, ζ , is small compared to the thermal voltage scale kB T /e. Therefore, the potential in the fluid satisfies the (linear) Debye-Huckel equation ∇2 φ = κ 2 φ .

Formally, the Debye-Huckel equation is valid for ζ ez/kB T 1. Ohshima et al.

(9)

4

computed

the sedimentation velocity of a spherical particle under a weak force field for arbitrary ζ ez/kB T 7 ACS Paragon Plus Environment

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and κa. They demonstrated that the sedimentation velocity calculated using the Debye-Huckel model of the double layer closely matches the velocity obtained using the Poisson-Boltzmann equation up to ζ ez/kB T ≈ 2, corresponding to a dimensional surface potential of approximately 50mV (for z = 1). The agreement extends to even larger ζ ez/kB T for thick charge clouds, which is the focus of the present study. Wiersema et al.

11

made an equivalent observation for

electrophoresis. Therefore, the practical utility of the Debye-Huckel approximation extends beyond its formal range of validity. The solution to (9) for a spherical particle is 10,23

φ=

a2 qe−κ(r−a) . ε(1 + κa)r

(10)

Hence, the surface potential ζ = Q/[4πεa(1 + κa)]. Thus, the Debye-Huckel approximation requires, formally, that Qez/[4πεa(1 + κa)kB T ] 1. Out of equilibrium, the small-potential approximation amounts to replacing n± by the bulk ion concentration, nb , within the electro-migration contribution to the flux density (3). 23 Hence, taking the difference of the conservation equations for the cations and anions leads to the following equation for the space-charge density

u·∇ρ = D∇2 ρ − Dκ 2 ρ.

(11)

Equation (11) encapsulates the effects of: (i) advection due to the flow in deforming the charge cloud (left-hand side); (ii) diffusion of ions in resisting this deformation (first term on right-hand side); and (iii) and attraction of the cloud to the particle via electrostatic screening, which also resists the deformation. The linearized boundary conditions on (11) are ze ∂ φ ∂ρ = −2zenb at r = a. ρ → 0 as r → ∞, and ∂r kB T ∂ r


(12)

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The radial component of the electric field, −∂ φ /∂ r, at the particle surface is not altered by the particle motion, since we have assumed a fixed surface charge distribution (7). Hence, using (10), the boundary condition on the charge density at the particle surface can be rewritten as ∂ (ρ − ρeq )/∂ r = 0, where ρeq =

−(κa)2 qe−κ(r−a) , (1 + κa)r

(13)

is the equilibrium space charge density. 24 Addition of the linearized conservation equations for cation and anions leads to an advectiondiffusion equation for the ionic strength, I = z2 (n+ + n− )/2, which reads u·∇I = D∇2 I,

(14)

subject to I → z2 nb as r → ∞, and ∂ I/∂ r = 0 at r = a. The solution is I = z2 nb throughout the electrolyte; hence, the ionic strength is not altered from its bulk value for small surface potentials, although the space charge density can vary. In anticipation of an analysis for thick charge clouds, κa 1, we render the governing equations dimensionless by normalizing distance with κ −1 ; charge density by Qκ 3 ; electric potential by Qκ/ε; velocity by U = |U|; and pressure by µUκ. Therefore the dimensionless versions of (4), (5), and (11), are −∇0 2 φ 0 = ρ 0 ,

(15)

∇0 2 u0 − ∇0 p0 = Ha ρ 0 ∇0 φ 0 ,

(16)

Pe u0 ·∇0 ρ 0 = ∇0 2 ρ 0 − ρ 0 ,

(17)

respectively, where the primes denote dimensionless variables. In (17)

Pe = U/Dκ


(18)

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is a Péclet number, which characterizes the importance of advection versus diffusion in establishing the space-charge distribution. In (16), (κa)2 Ha = 8π m Pe 2

zeζ kB T

2 (19)

is a Hartmann number, which characterizes the importance of the electric versus viscous forces in the fluid. The dimensionless quantity m = 2ε(kB T /ze)2 /µD is of order unity for aqueous electrolytes. Following Russel 23 and Lever, 24 we neglect the effect of electric forces on the flow, which requires that (κa)2 (zeζ /kB T )2 Pe/8π 2 . This is a reasonable action, since zeζ /kB T has already been assumed to be small (formally, at least), and we will shortly invoke κa 1. The boundary conditions on (15)-(17) are φ 0 → 0, ρ 0 → 0, 0 ∂ ρ 0 ∂ ρeq = ∂ r0 ∂ r0

and

u0 → −U0 as r0 → ∞,

and

−

∂φ0 1 = at r0 = κa, 0 ∂r 4π(κa)2

(20) (21)

0

0 = −e−(r −κa) /(4πr 0 ), and U0 is a unit vector along the direction of particle translation. where ρeq

3

Thick-charge-cloud limit

We now assume that κa 1. In this limit, the particle appears as a point charge of magnitude Q on the scale of κ −1 . The boundary conditions at the particle surface (21) are at r0 = κa 1. Hence, one may replace the finite-sized particle by a monopole singularity at r0 = 0, corresponding to a charge density Qδ (r), where δ (r) is the Dirac delta function. As noted by Lever, 24 this approximation yields the leading order effect of the charge cloud on the particle motion for small κa. To obtain higher order corrections one must analyze the problem with separate asymptotic expansions on the scales of κ −1 and a (outer and inner regions, respectively), which must then be matched in a domain of overlap, and the detailed particle geometry must be re10 ACS Paragon Plus Environment

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tained in the inner region. Since we have assumed Ha = 0, the particle generates a Stokeslet disturbance at distances r/a 1, such that the velocity u0 = −U0 + O(κa/r0 ). Therefore, we can neglect this disturbance in the thick-charge-cloud limit and take u0 = −U0 as the solution of (16) throughout the fluid. Hence, the dimensionless equations (15) and (17) are simplified to −∇0 2 φ 0 = ρ 0 + δ 0 (r0 ),

(22)

−Pe U0 ·∇0 ρ 0 = ∇0 2 ρ 0 − ρ 0 − δ 0 (r0 ).

(23)

The solution to (23) is obtained by introducing the Fourier transform pair 0

Z

ρ 0 (r0 ) =

Z

ˆ0

ρ (k ) =

0 0

ρ 0 (r0 )e−2πik ·r dr0 ,

(24)

0 0

ρˆ 0 (k0 )e2πik ·r dk0 ,

(25)

where k0 is a dimensionless wave vector scaled on κ 3 , and i =

√ −1. Upon taking the Fourier

transform of (23), it is readily shown that the charge density in Fourier space is ρˆ 0 (k0 ) = −

1 , (2πk02 ) + 1 − 2πiPe k0 · U0

(26)

where k0 = |k0 |. In Appendix A we invert (26) to yield the charge density in real space, 1 − Pe r0 ·U0 − r0 2 e 2 ρ (r ) = − 4πr0 0

0

√

2

4+Pe

.

(27)

It is instructive to examine the limits of (27) at small and large Pe: the former is 0 e−r Pe 0 0 2 ρ (r ) ∼ − 1 − r · U + O(Pe ) , 4πr0 2 0

0

(28)

which represents a weak, O(Pe), dipolar deformation of the equilibrium cloud, such that the space charge density is advected downstream of the particle (Figure 1). At the opposite limit of


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-10 10

-5

0

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5

10 (a)

5 0 -5 -U -10 (b) 5 0 -5 -U -10 -10

-5

0

5

10

Figure 1: (a) Plot of the charge density ρ 0 at Pe = 0.2 from (27). Contours are shown at ρ 0 = −0.1, −10−2 , −10−3 , −10−4 , −10−5 and −10−6 , moving from the origin outward. (b) Plot 0 , which takes on a dipolar of the deformation of the cloud from its equilibrium state, ρ 0 − ρeq 0 = ±10−3 , ±10−4 , ±10−5 and ±10−6 , structure at small Pe. Contours here are for ρ 0 − ρeq moving from the origin outward, where positive(negative) values are to the right(left) of the origin. The arrow points in the direction of the flow at infinity, −U, in the co-moving frame. Pe 1, ρ 0 (r0 ) ∼ −

1 − Pe (r0 +r0 ·U0 ) e 2 + O(Pe−1 ). 4πr0

(29)

The leading order term in (29) is identical to the temperature distribution around a point source of heat in a uniform stream. 8,25 Physically, advection overwhelms electrostatic screening and Brownian diffusion at large Pe; consequently, the cloud is strongly deformed from its equilibrium state (Figure 2). The charge density is exponentially small almost everywhere [provided r0 > O(Pe−1 )], except for a thin parabolic wake downstream, in which the decay is algebraic (∼ 1/r0 ). This structure reinforces the claim that the charge cloud can be “stripped” from a particle during electrophoresis at large field strengths. 18,26 Here, the word “stripped” refers to 12 ACS Paragon Plus Environment

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-U

10

0

-10 -70

-60

-50

-40

-30

-20

-10

0

Figure 2: Plot of the charge density ρ 0 at Pe = 10 from (27). Contours are shown at ρ 0 = −0.1, −10−2 , −10−3 , −10−4 , −10−5 and −10−6 , moving outward from the origin. The arrow points to the direction of the flow at infinity, −U, in the co-moving frame. the fact that the charge density is vanishingly small in the majority of the volume around the particle (excepting the wake), as if it were moving through a dielectric medium. Here, we have additionally shown that (29) is common to electrophoresis and sedimentation for Pe 1. Having determined the structure of the charge cloud, we proceed to examine the sedimentation and electrophoresis problems separately.

4

Sedimentation

The goal of the sedimentation problem is to calculate the external force, F, required to keep the particle fixed in the uniform stream −U. (Of course, the same force is required for the particle to translate at a velocity U relative to a fluid that is stationary at large distances.) If the particle were uncharged the force would be given by Stokes’ law, F = 6π µaU. The deformed charge cloud gives rise to two additional contributions: (i) an electrical force, originating from Maxwell stress; and (ii) another hydrodynamic force arising from electro-osmotic flow. The force balance on the particle is F +FH +FE = 0, where the hydrodynamic FH and electrical FE forces are H

F =

Z

H

E

σ · n dS and F =

Z

σ E · n dS,


(30)

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in which the integral is over the particle surface S. The hydrodynamic σ H and electrical σ E stress tensors are 1 T E σ = −pI + µ ∇u + (∇u) and σ = ε ∇φ ∇φ − (∇φ · ∇φ )I , 2 H

(31)

where I is the identity tensor, and the superscript T denotes transposition. Using the divergence theorem and Poisson’s equation (4) yields

H

E

Z

F +F =

H

σ · n dS +

Z

ρ∇φ dr,

(32)

where the second integral is over the fluid volume, and dr is a volume element. The hydrodynamic force FH contains contributions from the imposed flow and electro-osmosis. Following Sherwood 27 we use the Lorentz reciprocal theorem to collect both electrical effects — Maxwell stress and electro-osmotic flow — into a single term. Thus, consider two Stokes flows, u1 and u2 , with stress fields σ1 and σ2 , in the presence of body forces f1 and f2 , respectively. The reciprocal theorem states Z

u1 · σ2 · n dS +

Z

u1 · f2 dr =

Z

u2 · σ1 · n dS +

Z

u2 · f1 dr.

(33)

We choose flow “1” to be that of an uncharged sphere translating at speed U with f1 = 0; hence, u1 = P · U, where P is a second-order tensor independent of U. We choose flow 2 to be that produced by a sphere at rest in a distribution of electric body forces f2 = ρ∇φ . Hence from (33) the force on the sphere in flow “2” equals

F2 = −

Z

P · ρ∇φ dr.

(34)

Now, F2 is precisely due to the body forces that drives the electro-osmotic flow. Thus, we can


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write the total electric force (due to Maxwell stress and electro-osmosis) on the particle as

FEtot = −

Z

R · ρ∇φ dr,

(35)

where R = P − I is the tensor describing flow past an uncharged sphere, 10 3a 3a3 rr 3a a3 R = −1 + + 3 I + − . 4r 4r 4r 4r3 r2

(36)

The total external force F is therefore Z

F = 6π µaU +

R · ρ∇φ dr.

(37)

Equation (37) is valid for arbitrary κa. However, our focus is on small κa, for which (36) gives R = −I + O(κa/r0 ); hence, (37) can be approximated as F 8πεζ 2 κa 0 = 1− U· 6π µaU 3µD Pe

Z

ρ 0 ∇0 φ 0 dr0 ,

(38)

where F = |F|, and the integral has been non-dimensionalized via the normalizations introduced in section 2. Recall that the particle is replaced by a point charge for κa 1; hence, it is convenient to express the volume integral in Fourier space, yielding 4πiεζ 2 κa 0 F = 1+ U· 6π µaU 3µD Pe

Z

k0 ρˆ 0 (k0 )φˆ 0 (−k0 ) dk0 ,

(39)

where dk0 is the volume element in Fourier space and, from (22), φˆ 0 = (1 + ρˆ 0 )/(2πk0 )2 is the Fourier transform of the potential, with ρˆ 0 given by (26). The integral in (39) is evaluated in Appendix B to yield F 2εζ 2 = 1+ κa G(Pe), 6π µaU 3µD


(40)

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10-1 G(Pe) 10-2

10-2

10-1

100 Pe

101

102

Figure 3: Plot of G(Pe) versus Péclet number, Pe. The full expression for G(Pe) (41) is plotted as the solid line. The small and large Pe asymptotes of G(Pe) (as stated in the text) are shown as the dash and dash-dot lines, respectively. where " p # p 2 4 + Pe2 4 + Pe 1 Pe − 3 tanh−1 . G(Pe) = 2 2Pe Pe 2 + Pe2

(41)

For small Pe we have G(Pe) ∼ 16 + O(Pe2 ), and therefore εζ 2 F ∼ 1+ 1 + O(Pe2 ) κa, 6π µaU 9µD

(42)

which agrees with Booth, 2 who calculated the force for arbitrary κa and Pe 1. For strong advection, Pe 1, we have G(Pe) ∼ 1/(2Pe) + O(Pe−3 ln Pe), and therefore κa F εζ 2 ∼ 1+ 1 + O(Pe−2 ln Pe) , 6π µaU 3µD Pe

(43)

which indicates that electrokinetic effects on the particle motion vanish as Pe → ∞ (for fixed κa), i.e. the force asymptotes to Stokes’ law. The function G(Pe) is plotted in Figure 3, which demonstrates that the force monotonically transitions between the limiting values of (42) and (43). It is not the case, however, that electrokinetic effects vanish entirely in the sedimentation problem at large Pe, which will become evident from an examination of the electric potential.


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Langmuir

From (22), the potential is expressed as the convolution φ 0 (r0 ) =

1 1 + 4πr0 4π

Z

ρ 0 (z0 ) 0 dz , |r0 − z0 |

(44)

where the first term on the right-hand side of (44) is the potential due to the particle and the second term is the potential due to the cloud. The potential at large distances from the particle, |r0 | 1, is obtained from a multipole expansion as 1 1 φ (r ) ∼ + 0 4πr 4πr0 0

0

Z

r0 ρ (z )dz + · 4πr0 3 0

0

0

Z

z0 ρ 0 (z0 )dz0 + O(r0 −3 ),

(45)

where the first integral in (45) represents the net (dimensionless) charge in cloud, which equals −1 and thus cancels the charge on the particle. Consequently, the potential cannot have a monopole contribution at large distances. The second integral in (45) is the dipole moment of the charge in the cloud. That integral is readily evaluated by utilizing (27) and performing the integration in spherical coordinates, which yields φ 0 (r0 ) ∼

Pe r0 · U0 + O(r0 −3 ), 4π r0 3

(46)

which corresponds to a dipolar disturbance at large distances. The strength of the dipole grows linearly with Pe, due to the strengthening advection of the negative diffuse charge downstream of the particle. In a dilute (non-interacting) suspension of identical particles, the local fields from each particle combine to generate a macroscopic sedimentation field, Esed , which is given by 28 Esed = −N

Z

NQ ∇φ dr = − κε

Z

∇0 φ 0 dr0 ,

(47)

where N is the particle number density, and the integral is over a single particle. This integral is readily evaluated by noting that Z

∇0 φ 0 dr0 = 2πi lim k0 φˆ 0 (k0 ), 0 k →0


(48)

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which yields c Esed = − a

1 κa

2

Q κε

Pe U0 ,

(49)

where c = 4πNa3 /3 is the particle volume fraction. Note that the assumption of a non-interacting suspension requires that the volume fraction based on the Debye length (not the particle size, since κa 1) must be small; that is, 4πn/3κ 3 1. For a negatively(positively) charged particle, the sedimentation field is along(opposite) the direction of particle motion. It is instructive to estimate the magnitude of the sedimentation field that could be realized in an experiment. To this end, we select parameters that are representative of surfactant-doped nonpolar fluids. 18 We assume Qκ/ε = 25mV; c = 0.05; κa = 0.1; and a = 100nm. Hence, from (49) Esed = (1.25 × 106 )Pe

V . cm

(50)

Let us first consider a suspension settling under gravity. From Stokes’ drag, the settling velocity is U = 2a2 ∆σ g/9µ, where ∆σ is the difference in density between the particle and fluid, and g = 9.81m/s2 is the gravitational acceleration. For polystyrene particles (again with a = 100nm) in water ∆σ = 0.05 × 103 kg/m3 at T = 298K, which yields U = 1.1nm/s. The ion diffusion coefficient is obtained via the Stokes-Einstein relation as D = kB T /6π µaion ; taking the ion radius aion = 1nm yields D = 2.2 × 10−10 m2 /s. Therefore, Pe = 5.0 × 10−6 . Finally, (50) yields Esed = 6.3 V/cm, which could be measurable. The smallness of Pe dictates that the deformation of the charge cloud is weak; the settling velocity would be accurately predicted by the small-Pe formula (42). However, much larger values of Pe could be realized in an centrifugal field, which can generate accelerations up to O(105 g). Here, one may reach Pe ∼ O(1); thus, the effect of deformation of the charge cloud on the particle velocity could be significant. Moreover, at such values of Pe equation (50) suggests a huge, O(106 V/cm), sedimentation field, which is a dramatic, macroscopic consequence of the strong deformation of the charge cloud around a single particle.


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Before continuing, we note that Rasa and Philipse 29 proposed the existence of a macroscopic electric filed in experiments on centrifugation of colloidal silica particles in ethanol at varying concentration of salt (lithium nitrate). The evidence for this electric field is a measured ‘non-barometric’ variation of the particle number density with position in the centrifuge, which is supported by theoretical arguments. (By ‘non-barometric’ it is meant that the density deviates from the conventional Boltzmann distribution.) Rasa and Philipse explicitly state that the electric field they consider is not the same as the sedimentation field that we calculate above. Rather, their field is an equilibrium quantity that arises to ensure bulk electroneutrality of the suspension comprised of silica particles and their molecular counterions and coions. Specifically, the colloids are much heavier than the ions and thus an electric field arises to compensate the attempted charge separation due to gravity. In contrast, the sedimentation field considered herein is a non-equilibrium quantity, arising from convective distortion of the Debye charge cloud around a settling particle.

5

Electrophoresis

We now address the electrophoresis of a particle instigated by a uniform and steady electric field E∞ . Our aim is to calculate the particle velocity U, which, unlike in sedimentation, cannot be specified a priori. Hence, the Péclet number Pe = U/Dκ also cannot be specified in advance, because it is dependent on the magnitude of the particle velocity, U. However, progress can be made in the thick-charge-cloud limit. If κa ≡ 0 the effects of the charge cloud are absent; consequently, the particle velocity equals QE∞ /6π µa (Huckel’s result), which expresses a balance of Stokes drag and the electric force on a charged particle in a dielectric medium. 10 If κa is small we assume that the particle speed, U, appearing in the Péclet number is given by Huckel’s formula, such that Pe = QE∞ /(6πDκ µa), where E∞ is the magnitude of the applied field. This is a legitimate approximation, since corrections to the particle speed due to the presence of the


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charge cloud will be asymptotically small in κa and hence have a subdominant effect on the deformation of the cloud. A formula for the particle velocity is found by setting the external force, F, to zero in (37), thereby yielding 1 U=− 6π µa

Z

R · ρ∇φ dr.

(51)

Equation (51) can be simplified in the thick-charge-cloud limit. First, we assume that the particle and cloud do not affect the macroscopic electric field, such that ∇φ ≈ −E∞ . Second, we approximate the hydrodynamic tensor for κa 1 as 3κa r0 r0 R = −I + 0 I + 0 2 + O(κa)3 . 4r r

(52)

Substituting these two approximations into (51) yields QE∞ Q 3κa U= + E∞ · 6π µa 6π µa 4

r0 r0 I + 02 r

Z

ρ0 0 dr . r0

(53)

The first term on the right-hand side of (53) is Huckel’s result; the second term is the contribution due to the cloud. The integrand of this term is the charge density weighted by the (dimensionless) Oseen tensor for Stokes flow. This term represents the net effect of electroosmotic flow in altering the drag on the particle: a non-zero contribution can only arise if the charge distribution is an even function of position, i.e. ρ 0 (r0 ) = ρ 0 (−r0 ); any components of ρ 0 that are odd in r0 (e.g. a dipole disturbance) yield canceling contributions to the drag from the downstream and upstream faces of the particle. Note that relaxing the assumption of neglecting the effect of the particle on the electric field would lead to a contribution to U that is nonlinear in the particle charge, or, equivalently, the surface potential ζ . It is also worthwhile to note that the leading contribution of the charge cloud for small ζ to the particle velocity is linear in ζ , and thus the velocity is reversed by a sign change in ζ . In contrast, the leading contribution to the force in sedimentation is proportional to ζ 2 (see (38)) and is thus invariant to a sign change 20 ACS Paragon Plus Environment

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100

L(Pe) 10-1

10-2

B(Pe)/2

10-1

100 Pe

101

102

Figure 4: Plot of L(Pe) versus Péclet number, Pe. The full expression for L(Pe) (55) is plotted as the solid line. The small and large Pe asymptotes of L(Pe) (as stated in the text) are shown as the large dash and dash-dot lines, respectively. The function B(Pe)/2 from (59) is plotted as the small dash line. in ζ . The electrophoretic velocity is computed by substituting the charge density (27) into (53) and performing the integration in spherical coordinates. The final result is QE∞ 3 , U = 1 − κa L(Pe) 2 6π µa

(54)

"p # p 2 4 + Pe2 4 + Pe 1 4 + Pe2 1 −1 1+ coth − . L(Pe) = Pe Pe Pe Pe Pe2

(55)

where

At small Pe, we have L(Pe) ∼ 23 + O(Pe2 ); therefore, QE∞ U ∼ 1 − κa 1 + O(Pe2 ) , 6π µa

(56)

which agrees with Henry’s analysis of weak-field electrophoresis 9 at small κa. For Pe 1 we have L(Pe) ∼ Pe−1 (2 ln Pe − 1) + O(Pe−3 ln Pe). Thus, in this regime, QE∞ 3 −1 −3 U ∼ 1 − κa Pe (2 ln Pe − 1) + O(Pe ln Pe) . 2 6π µa


(57)

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For fixed κa it is seen that the contribution of the charge cloud to the electrophoretic velocity vanishes as Pe → ∞. Consequently, the particle velocity asymptotes to Huckel’s result. The function L(Pe) is plotted in Figure 4, which demonstrates that the velocity monotonically transitions between the limiting values of (56) and (57). The electrophoretic mobility Me = U/E∞ , where U is the magnitude of the particle velocity from (54). Our results imply that Me increases in magnitude with increasing field strength, since L(Pe) monotonically decreases with increasing Pe and the O(κa) term in (54) is negative. This prediction is qualitatively consistent with experimental observations that the mobility of colloidal particles in nonpolar liquids increases with field strength. 18–22 Such low conductivity fluids can withstand large electric fields without Joule heating, which can lead to significant variation in the physical properties (e.g., permittivity) of aqueous electrolytes, thereby complicating the analysis of electrophoresis measurements. A quantitative comparison of the present theory can be made against the experiments of Stotz, 18 who measured the mobility of polystyrene particles in “Shellsol T,” which is an aliphatic hydrocarbon from Shell Chemical. The polystyrene particles acquired a positive charge through addition of oil-soluble ionic additives. Stotz observed the sedimentation of particles parallel to a pair of planar electrodes that supplied a pulsed electric field, resulting in electrophoretic drift normal to the electrodes, from which the mobility was inferred. Stotz examined several particle sizes and solution conductivities. In particular, he claimed that a transition in mobility from low- to high-field strength regimes, where the latter corresponds to stripping of the charge cloud from the particle, is observed for measurements on particles with average radii a = 3µm in a solution with conductivity of 3.2−10 Ω−1 m−1 . A Debye length of κ −1 = 3µm was inferred from this value of conductivity and the assumption that the ionic additives were in a pre-micellar state with a radius of aion = 1nm. The field strength was varied from E∞min = 0.25 × 105 V/m to E∞max = 106 V/m. Stotz estimated the particle charge by applying Huckel’s result, Me = Q/6π µa, to the mobility measured at E∞max , yielding Q = 0.48−15 C. If it is assumed that the ion diffusion coefficient follows the Stokes-Einstein relation, D = kB T /6π µaion , then the Péclet number can be written 22 ACS Paragon Plus Environment

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Electrophoretic mobility (m2/V s)

Page 23 of 33

10-8

10-9

105 Electric field (V/m)

106

Figure 5: Comparison of present theory (54) (solid line) to experiments of Stotz 18 (circles). Experimental data is taken from Table III of Stotz’s paper. The prediction of Onsager and Kim, 32 namely equation (58), is shown as the dash line. as Pe = aion QE∞ /kB T κ a. Therefore, the minimum and maximum field strength correspond to Péclet numbers of Pemin ≈ 3 and Pemax ≈ 117. The effect of advection of the cloud in increasing the mobility should be evident over this range of Pe (Figure 4). In Figure 5 we compare the mobility from (54) with Stotz’s measurements; good agreement is seen. However, this agreement should perhaps be viewed with caution, for a couple of reasons. First, we have assumed that the particle charge is constant, which neglects the field-induced charging that has been suggested to occur in certain nonpolar fluids. 20,30 Second, κa = 1 in the experiments, whereas (54) was derived under the assumption κa 1. Nonetheless, the agreement displayed in Figure 5 gives encouragement that the present theory may be useful in interpreting measurements of electrophoresis in nonpolar fluids. Finally, we discuss the relation of the present analysis to the Wien effect, which refers to the dependence of the conductance of an electrolyte upon electric field strength. A field-dependent conductance of a strong (i.e., fully ionized) electrolyte is due to nonlinear variation of the velocity of an ion with field strength, which itself arises due to the deformation of the charge cloud surrounding that ‘central’ ion. It is reasonable to suppose a connection with the present


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work, in which the electrophoretic velocity varies nonlinearly with field strength, again due to deformation of the cloud surrounding a particle, which is treated as a point charge in the limit κa 1. Onsager and Kim 32 state that the presence of a charge cloud alters the velocity of a central ion via two mechanisms: (i) a modification of the field around the ion from the imposed value of E∞ , which alters the electric force on the ion; and (ii) the action of the imposed field on the cloud, which generates a (electro-osmotic) fluid flow that advects the central ion. The first effect is quadratic in the ion charge and thus not relevant to our discussion. Interestingly, the second effect is called the “electrophoretic velocity” by Onsager and Kim. 32 In the present notation, their result for the velocity of the central ion is 3 B(Pe) QE∞ , UOK = 1 − κa 2 2 6π µa

(58)

where √ p √ 1 h 2 8 −1 2− 2 + B(Pe) = 2Pe sinh Pe − Pe 1 + Pe 3 2Pe3( !)# √ Pe . − (1 + 2Pe2 ) tan−1 2Pe − tan−1 p 1 + Pe2

(59)

The O(κa) term in (58) is the additional “electrophoretic velocity” due to electro-osmotic flow. Note that the expression for B(Pe) given by Onsager and Kim (equation 6.8 in their paper) is incorrect; the correct expression (59) follows from integration of their equation 6.6. For Pe 1 we have B(Pe) ∼ 4/3 + O(Pe2 ); hence (58) matches (56) in the small-Pe limit. However, √ B(Pe) ∼ 2/3 + O(Pe−1 ln Pe) for Pe 1, which yields κa QE∞ UOK ∼ 1 − √ . 2 6π µa

(60)

The function B(Pe) decreases monotonically between these limits; hence, the velocity of the central ion increases with field strength. However, (60) is different to (57); more generally, the functions L(Pe) and B(Pe)/2 are unequal, apart from at Pe = 0 (Figure 4). In particular, 24 ACS Paragon Plus Environment

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(60) implies that the cloud makes a non-zero contribution as Pe → ∞; thus, the ion velocity does not limit to Huckel’s result. In contrast, (57) indicates that Huckel’s result is recovered at large Pe. Onsager and Kim calculate the O(κa) contribution to the ion velocity by first taking the Fourier Transform of the Stokes equations, with the electric body force approximated as −ρE∞ , i.e. the imposed field acting on the deformed charge cloud, the latter in Fourier space is given by an expression equivalent to (26). (Onsager and Kim do not present an expression for the cloud in real space akin to (27).) The Fourier space velocity field is then inverted to evaluate the fluid velocity at the position of the central ion, which is added to the velocity of the ion in isolation (i.e. Huckel’s result) to yield (58). Their approach is evidently different from our evaluation of the particle velocity via the force balance (51). Therefore, it is unsurprising that the two calculations give differing results. The present analysis yields a superior match to Stotz’s experiments (Figure 5), which lends confidence to our calculations.

6

Conclusion

The sedimentation and electrophoresis of a charged spherical colloid in a binary electrolyte have been quantified, under the following assumptions: (i) the zeta potential is small relative to the thermal voltage; (ii) the Debye length is much larger than the particle radius; (iii) the electrolyte is completely ionized; and (iv) the cations and anions have equal diffusion coefficients. These assumptions were invoked to allow an analytical investigation of the transition from weak to strong deformation of the charge cloud with increasing Péclet number, Pe. The strongly-deformed cloud at large Pe results in a vanishing electrokinetic effect on the particle motion. In sedimentation this strong deformation leads to a concomitant increase in the macroscopic sedimentation field in a dilute suspension. As mentioned above, our analysis of electrophoresis should be of value in analyzing mobility measurements in nonpolar fluids. The present analysis could be extended in a few directions. First, a natural extension is to relax the


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assumption of small κa (and small Ha) by solving numerically the coupled nonlinear system (15)-(17). This would determine the range of κa over which our asymptotic analysis for thick charge clouds is accurate. A second extension would be to consider asymmetric electrolytes with unequal diffusion coefficients or unequal anion and cation charge numbers. Finally, a third extension is to account for bulk ion dissociation-association reactions, which can impact electrokinetic phenomena in weak electrolytes at large field strengths. 33

Acknowledgement The author acknowledges National Science Foundation CAREER support under CBET-1350647.

Appendix A: Derivation of (27) To invert (26) we adopt a coordinate system wherein the wave vector k0 = k30 e3 + kt0 , in which e3 is a Cartesian unit vector aligned with U0 , and kt0 = k10 e1 + k20 e2 is a vector in the plane transverse to U0 , where e1 and e2 are Cartesian unit vectors along the 1-axis and 2-axis, respectively. Since the charge density is rotationally symmetric about U0 , we take the position vector in real space as r0 = r0 (sin θ e1 + cos θ e3 ),

(61)

where θ is the polar angle between r0 and e3 axis. Therefore, inserting (26) into (25) and setting q0 = 2πk0 yields 1 ρ (r ) = − (2π)3 0

0

Z

iq01 r0 sin θ

e

dqt0

0 0

eiq3 r cos θ dq03 , 02 02 0 −∞ q3 + qt + 1 − iPeq3

Z ∞


(62)

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where qt0 =

q q012 + q022 . The integral over q03 , to be denoted as K, is performed in the complex

q03 plane, over a contour that spans the real axis and is closed by a semicircular arc in the upper half plane. Jordan’s lemma guarantees that the contribution to the integral from the arc vanishes as the radius of the arc goes to infinity (for cos θ > 0). The integrand has a single simple pole in the upper half plane, located on the imaginary axis at q03

q i 2 0 2 = Pe + Pe + 4(1 + qt ) . 2

(63)

Therefore, K equals 2πi multiplied by residue at this pole, which yields 0

− r2 cos θ

K=

2πe

q

Pe+ Pe2 +4(1+qt0 2 )

p Pe2 + 4(1 + qt0 2 )

.

(64)

The integrations in (62) over q01 and q02 are carried out in plane polar coordinates, q01 = qt0 cos ϕ and q02 = qt0 sin ϕ. The integral over ϕ equals J0 (rqt0 sin θ ), where J0 is the zeroth order Bessel function of the first kind. Therefore, (62) simplifies to

ρ 0 (r0 ) = −

q

2 r0 02 Z e−Pe cos θ ∞ e− 2 cos θ Pe +4(1+qt ) r0 2

4π

0

p J0 (rqt0 sin θ )qt0 dqt0 2 0 2 Pe + 4(1 + qt )

(65)

The integral in (65) can be evaluated in closed form, 31 after which using cos θ = r0 · U0 yields (27).


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Appendix B: Derivation of (40) and (41) Let I = U0 · k0 ρˆ 0 (k0 )φˆ 0 (−k0 ) dk0 denote the integral appearing in (39). Substituting the Fourier R

space charge density (26) and using φˆ 0 = (1 + ρˆ 0 )/(2πk0 )2 yields

I=−

Z

U0 · k0 (2πk0 )2

(2πk0 )2 + 2πiPeU0 · k0 (2πk0 )2 + 1 + 2πiPeU0 · k0

1 0 2 (2πk ) + 1 − 2πiPeU0 · k0

dk0 .

(66)

We again adopt the Fourier space coordinate system k0 = k30 +kt0 . Hence, after setting q0 = 2πk0 , (66) becomes 1 I=− (2π)4

Z

0 q0 2 + q0 2 + iPeq0 q t 3 3 3 . dqt0 dq03 0 2 q3 + qt0 2 q032 + qt0 2 + 1 + iPeq03 q032 + qt0 2 + 1 − iPeq03 −∞ Z ∞

(67)

The integral over q03 , to be denoted as J, is performed in the complex q03 plane, over a contour that spans the real axis and is closed by a semicircular arc in the upper half plane. The contribution to the integral from the arc vanishes as the radius of the arc goes to infinity, since the integrand scales as |q03 |−3 at large distances. The integrand has three simple poles in the upper half plane, located on the imaginary axis at q03a = iqt0 , q i 0 2 0 2 −Pe + Pe + 4(1 + qt ) , q3b = 2 q i 0 2 q3c = Pe + Pe + 4(1 + qt0 2 ) . 2

(68) (69) (70)

Therefore, J equals 2πi multiplied by the sum of the residues at each pole. After evaluating these residues one finds qt0 Pe(1 + 2qt0 2 ) J = πiPe − . (Peqt0 − 1)(Peqt0 + 1) (4 + Pe2 + 4qt0 2 )(Pe2 qt0 2 − 1)


(71)

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The integrations in (67) over q01 and q02 are carried out in plane polar coordinates. The integration over ϕ simply yields 2π (due to the axisymmetry of the charge density and potential about U0 ), leaving ∞ 1 Jqt0 dqt0 , (2π)3 0 " ! p # p 2 2 1 i Pe 4 + Pe 4 + Pe tanh−1 − . = (2π)2 Pe2 2Pe 2 + Pe2

I = −

Z

(72) (73)

Inserting (73) into (39) yields (40), with G(Pe) defined in (41).

References (1) Prieve, D. C.; Hoggard, J. D.; Fu, R.; Sides, P. J.; Bethea, R. Two independent measurements of Debye lengths in doped nonpolar liquids. Langmuir 2008, 24, 1120-1132. (2) Booth, F. Sedimentation potential and velocity of solid spherical particles. J. Chem. Phys. 1954, 22, 1956-1968. (3) Saville, D. A. Electrokinetic effects with small particles. Ann. Rev. Fluid Mech. 1977, 9, 321-337. (4) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. Sedimentation velocity and potential in a dilute suspension of charged spherical colloidal particles. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299-1317. (5) Dukhin, S. S. Non-equilibrium electric surface phenomena. Adv. Colloid Interface Sci. 1993, 44, 1-134. (6) Keller, F.; Feist, M.; Nirschl, H.; Dörfler, W. Investigation of the nonlinear effects during the sedimentation process of a charged colloidal particle by direct numerical simulation. J. Colloid Interface Sci. 2010, 344, 228-236. 29 ACS Paragon Plus Environment

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(7) Bhattacharya, S.; Gopmandal, P. P. Migration of a charged sphere at an arbitrary velocity in an axial electric field. Colloids Surf., A 2011, 390, 86-94. (8) Acrivos, A.; Taylor, T. D. Heat and mass transfer from single spheres in Stokes Flow. Phys. Fluids. 1962, 5, 387-394 (1962). (9) Henry, D. C. The cataphoresis of suspended particles. Proc. R. Soc. London Ser. A 1931, 133, 106-129. (10) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions 1989 Cambridge University Press. (11) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. Calculation of the electrophoretic mobility of a spherical colloid particle. J. Colloid Interface Sci. 1966, 22, 78-99 (1966). (12) O’Brien, R. W.; White, L. R. Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 2 1978, 74, 1607-1626. (13) Schnitzer, O.; Zeyde, R.; Yanveh, I.; Yariv, E. Weakly nonlinear electrophoresis of a highly charged colloidal particle. Phys. Fluids 2013, 25, 052004. (14) Dukhin, A. S.; Dukhin, S. S. Aperiodic capillary electrophoresis method using an alternating current electric field for separation of macromolecules. Electrophoresis. 2005, 26, 2149-2153. (15) Shilov, V.; Barany, C.; Grosse, C.; Shramko, O. Field-induced disturbance of the double layer electro-neutrality and non-linear electrophoresis. Adv. Colloid Interface Sci. 2003, 104, 159-173. (16) Schnitzer, O.; Yariv, E. Strong-field electrophoresis. J. Fluid Mech. 2012, 701, 333-351. (17) Schnitzer, O.; Yariv, E. Nonlinear electrophoresis at arbitrary field strengths: smallDukhin-number analysis. Phys. Fluids 2014, 26, 122002. 30 ACS Paragon Plus Environment

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(18) Stotz, S. Dependence of the electrophoretic mobility of particles suspended in lowconductivity liquids. J. Colloid Interface Sci. 1978, 65, 118-130. (19) Thomas, J. C.; Hanton, K. L.; Crosby, B. J. Measurement of the field dependent electrophoretic mobility of surface modified silica/AOT suspensions. Langmuir 2008, 1069810701. (20) Gacek, M. M.; Berg, J. C. Investigation of surfactant mediated acid-base charging of mineral oxide particles dispersed in apolar systems. Langmuir 2012, 28, 17841-17845 (2012). (21) Guo, Q.; Lee, J.; Singh, V.; Behrens, S. H. Surfactant mediated charging of polymer particles in a nonpolar liquid. J. Colloid Interface Sci. 2013, 392, 83-89 (2013). (22) Strubbe, F.; Beunis, F.; Brans, T.; Karvar, M.; Woestenborghs, W.; Neyts, K. Electrophoretic retardation of colloidal particles in nonpolar liquids. Phys. Rev. X 2013, 3, 021001. (23) Russel, W. B. Bulk stresses due to deformation of the electrical double layer around a charged sphere. J. Fluid Mech. 1978, 85, 673-683. (24) Lever, D. A. Large distortion of the electric double layer around a charged particle by a shear flow. J. Fluid Mech. 1979, 92, 421-433. (25) Wilson, H. A. On the convection of heat. Cambridge Phil. Soc. 1904, 12, 406-423. (26) Vincett, P. S. Demonstration of true electrophoresis in an insulating liquids at fields up to 3 × 107 Vm−1 . J. Colloid Interface Sci. 1979, 69, 354-357. (27) Sherwood, J. D. Primary electroviscous effect in a suspension of spheres. J. Fluid Mech. 1980, 101, 609-629. (28) Saville, D. A. The sedimentation potential in a dilute suspension. Adv. Colloid Interface Sci. 1982, 16, 267-279. 31 ACS Paragon Plus Environment

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(29) Rasa, M.; Philipse, A. P. Evidence for a macroscopic electric field in the sedimentation profiles of charge colloids. Nature 2004, 429, 857-860. (30) Espinosa, C. E.; Guo, Q.; Singh, V.; Behrens, S. H. Particle charging and charge screening in non polar dispersions with nonionic surfactants. Langmuir. 2010, 26, 16941-16948. (31) Gradshteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series, and Products 2000, Academic Press. (32) Onsager, L.; Kim, S. H. Wien effect in simple strong electrolytes. J. Chem. Phys. 1957, 61, 198-215. (33) Baygents, J. C.; Saville, D. A. Electrophoresis of small particles and fluid globules in weak electrolytes. J. Colloid Interface Sci. 146, 9-37 (1991).


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Strong Deformation of the Thick Electric Double Layer around a

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