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Proc. R. Soc. A (2009) 465, 709–723
doi:10.1098/rspa.2008.0322
Published online 18 November 2008
Boundary-induced electrophoresis of
uncharged conducting particles: remote
wall approximations
B Y E HUD Y ARIV *
Faculty of Mathematics, Technion – Israel Institute of Technology,
Technion City, Haifa 32000, Israel
An initially uncharged ideally polarizable particle is freely suspended in an electrolyte
solution in the vicinity of an uncharged dielectric wall. A uniform electric field is
externally applied parallel to the wall, inflicting particle drift perpendicular to it.
Assuming a thin Debye thickness, the electrokinetic flow is analysed for large particle–
wall separations using reflection methods, thereby yielding an asymptotic approximation
for the particle velocity. The leading-order correction term in that approximations stems
from wall polarization.
Keywords: electrokinetics; asymptotic methods; Stokes flow
1. Introduction
In the classical view of electrokinetics (Saville 1977), surface charge (and whence
zeta potential) is considered a fixed physicochemical property of the solid–
electrolyte interface, unaffected by the applied field. Implicit in this view is the
assumption of a non-polarizable surface (‘a perfect dielectric’). The resulting
mathematical model in the thin-Debye-layer limit is described by Keh &
Anderson (1985).
This model is inapplicable to conducting solid surfaces, which are effectively
infinitely polarizable. The analysis of electrokinetic flows about conducting
particles began with Levich (1962) and was followed by other researchers in the
former USSR (Simonov & Dukhin 1973; Shilov & Simonova 1981; Dukhin &
Murtsovkin 1986; Gamayunov et al. 1986). On conducting walls the surface
charge is mobile, and it arranges itself to ensure zero interior electric field. This
polarization mechanism affects the zeta potential, which can no longer be
considered a fixed quantity. A similar polarization mechanism also appears at
electrokinetic flows about dielectric surfaces (Murtsovkin 1996; Nadal et al.
2002) which possess a finite polarizability (represented by their dielectric
constant). Indeed, surface polarization was shown to be responsible to observed
vortices around sharp corners in micro-channels (Thamida & Chang 2002;
Yossifon et al. 2006). A thin-Debye-layer macroscale formulation for flows about
polarizable surfaces was developed by Yossifon et al. (2007).
*[email protected]
Received 3 August 2008
Accepted 20 October 2008
709
This journal is q 2008 The Royal Society
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710
E. Yariv
Squires & Bazant (2004) coined the term ‘induced-charge’ flows to describe
the entire host of electrokinetic processes in which surface polarization affects the
zeta potential. The archetypical configuration of induced-charge electro-osmosis
consists of an initially uncharged ideally polarizable spherical particle that is
suspended in an unbounded electrolyte and is exposed to an otherwise uniform
faradaic current. It is common to assume that the particle boundary is
chemically inert; thus, the dipolar charge distribution that is induced on it
corresponds to zero net charge. The steady-state flow in this configuration was
studied by Gamayunov et al. (1986). In view of the resulting flow symmetry, such
a particle does not experience any hydrodynamic force, and would therefore
remain stationary.
When the preceding symmetry is violated, it is possible to affect particle
motion despite the zero net charge (Bazant & Squires 2004). The theoretical
possibility of animating induced-charge electrophoresis has led to a series of
theoretical investigations of non-spherically symmetric configurations. Using
general symmetry arguments, Yariv (2005) discussed flows about arbitrary
particle shapes. Squires & Bazant (2006) employed regular perturbations
to analyse near spheres and near cylinders. These authors also considered
other modes of asymmetry for spherical particles; the electrophoretic motion in
one of these, Janus-type particles, was experimentally observed by Gangwal et al.
(2008). Spheroids were studied by Saintillan et al. (2006a) in the slender limit
and by Yossifon et al. (2007) in general. Spheroids exhibit many features that are
absent in spherical geometries; having both fore–aft and axial symmetries,
however, they do not experience electrophoresis when exposed to a uniformly
applied field (Yariv 2005). This limitation motivated the recent investigation of
arbitrarily shaped slender particles (Yariv 2008b); when lacking fore–aft
symmetry, such particles do experience electrophoretic motion. Asymmetry
can also be animated by the presence of neighbouring particles. Interactions
between spherical particles were investigated using both analytic approximations
(Dukhin & Murtsovkin 1986; Gamayunov et al. 1986) and numerical methods
(Saintillan 2008). Saintillan et al. (2006a) used slender-body approximations to
calculate the interactions between elongated spheroids; these were employed in
the subsequent analyses of rod-like particle suspensions (Saintillan et al. 2006b;
Rose et al. 2007).
Another category of asymmetric geometries comprises bounded configurations. This category is of special importance since all practical devices are
bounded in one or more dimensions; a dielectric wall, for example, can represent
the boundary of a microfluidic channel. The simplest scenario entails a dielectric
plane wall, the applied current directed parallel to it. Even in the absence of
electrokinetic flow, and despite its zero net charge, the particle experiences a net
electric force that tends to repel it from the wall (Yariv 2006). It is plausible
that the induced-charge electro-osmosis will result in an additional force along
that direction; it was already speculated by Gangwal et al. (2008) that such a
force may explain a recent theory–experiment discrepancy in the motion
of Janus-type particles. The goal of this paper is to investigate this inducedcharge phenomenon.
Wall effects were analysed by Zhao & Bau (2007) for a cylindrical particle. In
the thin-Debye-layer limit, the electrostatic and flow problems were solved using
eigenfunction expansions in bipolar coordinates. In principle, this procedure can
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Boundary-induced electrophoresis
711
be adapted to a spherical particle via an appropriate use of bi-spherical
coordinates (see, e.g. Keh & Chen 1989). Since these eigenfunction expansions do
not provide direct mathematical insight, we here adopt a different approach,
following Keh & Anderson (1985). Rather than considering arbitrary particle–
wall separations, we focus from the start upon the remote wall scenario. This
allows to obtain closed-form analytic approximations, which, in turn, can be
used in modelling of more complicated bounded systems. Our approach is
motivated by the existing approximations for remote particle–particle
interactions (Dukhin & Murtsovkin 1986; Gamayunov et al. 1986) which were
recently improved by Saintillan (2008).
Towards this end, we consider the simplest particle–wall configuration,
consisting of an initially uncharged ideally polarizable (i.e. perfectly conducting)
spherical particle (radius a) that is suspended in a symmetric (valency Z, ionic
strength Z 2nN) electrolyte solution (viscosity m, electrical permittivity e) in the
vicinity of an uncharged non-polarizable plane wall. At time zero, a uniform
faradaic current is externally driven through the solution in a direction parallel
to the wall.
Our interest lies in the motion of the particle following the transient period
(Squires & Bazant 2004; Chu & Bazant 2006; Yossifon et al. in press) during
which the induced Debye layer about it is formed. At temperature T, this layer
is characterized by the Debye–Hückel parameter k, defined by (k being
Boltzmann’s constant and e the elementary charge)
2e2 Z 2 nN
:
ð1:1Þ
ekT
Throughout our investigation, we will assume that the Debye thickness 1/k is
small compared with particle size
ka[ 1:
ð1:2Þ
k2 Z
Following Keh & Anderson (1985), we introduce an iterative scheme that
naturally handles the particle–wall geometry. When focusing upon the remote
wall limit, it is necessary to calculate only several terms in that scheme. When
limiting our attention to the leading-order term and to its leading-order
correction, we find that the force experienced by the particle is not affected by
Maxwell stresses. When focus lies at these asymptotic orders, it is possible to
employ the Robin condition of Yossifon et al. (2007) so as to generalize the
analysis to polarizable walls. It is found that a finite wall polarizability affects the
leading-order correction to the force.
The paper is arranged as follows: In §2, we formulate the dimensionless
electrokinetic problem. The iterative scheme is delineated in §3. The remote wall
approximation is obtained in §4. In §5, we derive a generalization for polarizable
walls. Conclusions appear in §6.
2. Problem formulation
The system that we consider is described in figure 1. It comprises an electrolyte
solution that is bounded by a non-polarizable planar wall and a spherical
conducting particle of radius a. The particle centre O is instantaneously
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712
E. Yariv
az
ar
E∞
a
ax
n
Figure 1. Schematic of the particle–wall configuration and coordinate systems.
positioned at distance a/l (l!1) from the wall. The system is exposed to a
uniform and constant external electric field ENZENÊ (Ê being a unit vector in
the field direction), which is applied parallel to the wall.
2
We employ a dimensionless notation, using a, EN, aEN and eE N
as the
respective units of length, electric field, electric potential and stress; velocities
2
are accordingly normalized by eE N
a=m. It is convenient to employ a Cartesian
coordinate system centred about O, with the z -axis lying perpendicular to wall
(which is then given by zZK1/l) and the x -axis lying in the applied field
direction (ÊZêx). In addition, we also employ spherical polar coordinates, the
radial coordinate r measured from O and the polar angle q measured from the
x -axis. We analyse the electrokinetic flow using the thin-Debye-layer limit (1.2),
where it is understood that the description in the preceding coordinates is a
‘coarse-grained’ one. Accordingly, the no-flux boundary condition and the
Smoluchowski slip condition apply at both the sphere boundary rZ1 and the
wall zZK1/l.
The electric potential is governed by (i) Laplace’s equation in the fluid domain
V2 4 Z 0;
ð2:1Þ
(ii) the no-flux condition on both the particle boundary
v4
Z0
vr
at r Z 1
ð2:2Þ
at z Z K1=l
ð2:3Þ
as r /N:
ð2:4Þ
and the wall
v4
Z0
vz
and (iii) the far-field condition
V4/KE^
The above Neumann-type boundary-value problem uniquely defines 4 up to a
physically meaningless integration constant. It is readily verified that the electric
potential can be made an odd function of x by a proper choice of that constant.
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Boundary-induced electrophoresis
713
The induced zeta potential on the particle is
ð2:5Þ
z Z FK4jrZ1 ;
in which F is the uniform particle potential.1 The value of F is determined from
an integral constraint representing the zero net charge of the particle (Yariv
2005). In view of the oddness of 4 and the odd dependence of the Debye-layer
capacitance upon z (Yariv 2008a), that constraint is trivially satisfied by
choosing FZ0. We calculate the electrokinetic flow assuming a stationary
particle. Once the loads on such a particle are calculated, the velocities of a
comparable freely suspended particle are readily obtained using the known
mobility relations of the sphere–wall configuration (Happel & Brenner 1965).
The velocity field v and the pressure field p are calculated in an inertial reference
system attached to the wall. Thus, the hydrodynamics are described by (i) the
Stokes equations,
V$v Z 0;
Vp Z V2 v;
ð2:6Þ
(ii) Smoluchowski’s slip condition on the particle, which, upon using (2.5),
appears as
v ZK4V4
at r Z 1;
ð2:7Þ
(iii) the no-slip condition on the wall (representing the presumed zero zeta
potential there)
v Z 0 at z ZK1=l
ð2:8Þ
and (iv) the condition of velocity decay at large distances from the particle.
Once the electric and velocity fields are evaluated, it is possible to calculate
2 2
2 3
the force and torque (respectively normalized with eE N
a and eE N
a ) exerted on
the stationary particle. These loads consist of (i) hydrodynamic contributions
FZ
#
rZ1
e^r $s dA;
GZ
#
rZ1
e^r !ðe^r $sÞdA;
ð2:9Þ
which result from the tractions caused by the Newtonian stresses (I being the
idem factor and † denoting transposition)
s ZKpI C Vv C ðVvÞ†
ð2:10Þ
and from (ii) electric contributions
F~ Z
#
e^r $~
s dA;
rZ1
~Z
G
#
e^r !ð^
er $~
sÞdA;
ð2:11Þ
rZ1
which result from the tractions caused by the Maxwell stresses
1
ð2:12Þ
s~ Z V4V4K ðV4$V4ÞI:
2
~
Our interest lies in the total force and torque, F CF~ and GCG.
Even without solving the governing equations, it is possible to use symmetry
arguments so as to predict the force and torque directions. Since the electrical
problem is linear and homogeneous in the constant vector Ê, the electric
1
We here assume for simplicity that all the potential drop in the double layer occurs in its diffuse
part, thereby neglecting the Stern layer voltage.
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714
E. Yariv
potential must be linear in it. In view of the quadratic slip structure (2.7) and the
linearity of the flow problem, it becomes clear that all the flow variables are
quadratic in Ê and then so must also be the hydrodynamic loads (2.9). These
loads can therefore be represented in the invariant tensorial notation
^
F Z F : E^E;
^
G Z G : E^E;
ð2:13Þ
in which F is a third-order tensor and G a third-order pseudo-tensor. These
dimensionless coefficients can only depend upon the instantaneous configuration
of the particle–wall system. This configuration introduces only a single constant
^ a unit normal to the wall, which points into the fluid (nZ
^ e^z ). Thus,
vector: n,
^ while
the only candidates for F are n^ n^ n^ as well as the three permutations of In,
^ and
the only candidates for G are the alternating pseudo-tensor e as well as e$n^ n
^
n^ n$e.
In general, all of these candidates are multiplied by functions of l, the
single scalar parameter in the problem. In view of the contraction with ÊÊ, it
becomes evident that the sphere does not experience a hydrodynamic torque and
that the hydrodynamic force is of the form
^
F Z FðlÞn:
ð2:14Þ
2
~ vanishes and that F~ is of the form FðlÞ
~ n.
^
Identical arguments imply that G
Lastly, the preceding tensorial arguments can be repeated for a freely suspended
particle, showing that it must acquire a rectilinear velocity of the form
^
U Z U ðlÞn
ð2:15Þ
and no angular velocity.
3. Iterative reflections
Following Keh & Anderson (1985), we employ an iterative reflection scheme. The
electric potential is provided by the following series:
h
i h
i h
i
ð0Þ
ð0Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
4 Z 4ð0Þ C 4ð1Þ C 4ð2Þ C/Z 4W C 4P C 4W C 4P C 4W C 4P C/;
ð3:1Þ
where we define
ðnÞ
ðnÞ
4ðnÞ Z 4W C 4P :
ð3:2Þ
ð0Þ
The potential 4(0) is the solution in the absence of a wall. Specifically, 4W ZKx
(which automatically satisfies the no-flux condition (2.3) on the wall) corresponds
ð0Þ
to the applied field and 4P is the dipole
ð0Þ
4P ZK
cos q
x
ZK
;
2
2r
2ðx 2 C y 2 C z 2 Þ3=2
ð3:3Þ
required to satisfy the boundary condition (2.2) on the particle.
ðnÞ
ðnÞ
The harmonic corrections 4W and 4P for nO0 represent successive
reflections that alternately satisfy the no-flux conditions on the two surfaces:
ðnÞ
The ‘wall correction’ 4W decays at large distances from the wall and restores the
2
This is also evident from (2.2) and (2.12), which together imply that the Maxwell tractions e^r $~
s
are radial.
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Boundary-induced electrophoresis
715
ðnK1Þ
no-flux condition (2.3) violated by 4P
ðnÞ
ðnK1Þ
v4W
v4
ZK P
vz
vz
for z Z K1=l;
ð3:4Þ
ðnÞ
similarly, the ‘particle correction’ 4P decays at large distances from the particle
ðnÞ
and restores the no-flux condition (2.2) violated by 4W
ðnÞ
ðnÞ
v4
v4P
ZK W for r Z 1:
vr
vr
We also introduce the iterative expansion for the velocity field:
h
i h
i
ð0Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
v Z vP C v W C vP C vW C v P C/;
ð3:5Þ
ð3:6Þ
with a similar series for the pressure p. Each corresponding pair in these
expansions separately satisfies the Stokes equations (2.6); then, due to the
linearity of the hydrodynamic stress in v and p (see (2.10)), a similar series is
automatically induced for s.
ð0Þ
The first term in (3.6), vP , represents the flow in the absence of a wall.
It decays at large distances from the particle, and is driven by the slip condition
(cf. (2.7))
ð0Þ
v P ZK4ð0Þ V4ð0Þ at r Z 1:
ð3:7Þ
This field was calculated by Gamayunov et al. (1986) who obtained the
quadrupolar profile
9 1
1
9
ð0Þ
vP Z e^r
K 2 ð3 cos2 q K1Þ Ce^q 4 sin q cos q:
ð3:8Þ
4
8 r
r
4r
ðnÞ
The field vW (nR1) decays at large distances from the wall and satisfies the
boundary condition
ðnÞ
ðnK1Þ
vW ZKv P
for z ZK1=l;
ð3:9Þ
ðnK1Þ
) of v on the wall.
which restores the null value (violated by v P
ðnÞ
The field vP (nR1) is split into two sub-fields (with a similar decomposition
ðnÞ
ðnÞ
being applied to both pP and sP )
ðnÞ
ðnÞ
ðnÞ
vP Z v P;W C vP;4 :
ð3:10Þ
Both sub-fields satisfy the Stokes equations and decay at large distances from the
ðnÞ
ðnÞ
particle; The sub-field v P;W , triggered by the distribution of vW on the particle,
satisfies the boundary condition
ðnÞ
ðnÞ
vP;W ZKvW
for r Z 1;
ð3:11Þ
ðnÞ
v P;4 ,
the sub-field
triggered by the additional electrokinetic slip animated by 4(n),
satisfies the boundary condition (cf. (3.7))
nK1 h
i
X
ðnÞ
vP;4 ZK4ðnÞ V4ðnÞ K
4ðiÞ V4ðnÞ C 4ðnÞ V4ðiÞ :
ð3:12Þ
i Z0
ð0Þ
The decomposition (3.10) also applies for nZ0 when it is understood that v P;W is
ð0Þ
ð0Þ
null (i.e. v P Z vP;4 ).
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716
E. Yariv
The iterative decomposition (3.6) directly induces a comparable decomposition for the hydrodynamic force,
h
i h
i
ð0Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
F Z F P;4 C F P;W C F P;4 C F P;W C F P;4 C/;
ð3:13Þ
in which we naturally define
ðnÞ
F P;4 Z
#
ðnÞ
ðnÞ
e^r $sP;4 dA;
F P;W Z
rZ1
#
ðnÞ
ð3:14Þ
e^r $sP;W dA:
rZ1
ðnÞ
Since the wall reflections are regular for zOK1/l, V$sW vanishes inside the
particle; thus, these reflections do not contribute to the hydrodynamic force.
ðnÞ
In view of the boundary condition (3.11), the contribution F P;W is simply
provided by Faxén’s laws (Happel & Brenner 1965) applied upon the wall
ðnÞ
ðnÞ
reflection v W that ‘triggered’ the field v P;W
h
i
ðnÞ
ðnÞ
2 ðnÞ
F P;W Z 6pv W C pV v W
:
ð3:15Þ
rZ0
ðnÞ
F P;W
ðnÞ
In what follows, we will refer to
as the force ‘provoked’ by v W .
In view of the quadratic dependence of the Maxwell stresses (2.12) upon the
electric field, we define
nK1
X
1
s~ðnÞ Z V4ðnÞ V4ðnÞ K V4ðnÞ $V4ðnÞ I C
½V4ðiÞ V4ðnÞ C V4ðnÞ V4ðiÞ
2
i Z1
KV4ðnÞ $V4ðiÞ I:
ð3:16Þ
The contribution of s~ðnÞ to the electric force is
~ ðnÞ Z
F
#
e^r $~
sðnÞ dA:
ð3:17Þ
rZ1
4. Remote wall approximation
We focus upon the remote wall limit, l/1. While consecutive terms in the
iterative representations are not asymptotically ordered; they eventually
generate separate asymptotic expansions in two asymptotic regions. The first,
characterized by the ‘particle scale,’ lies at the O(1) neighbourhood of the
particle; the second, characterized by the ‘gap scale,’ lies at O(1/l) distances
from the particle. Following Ho & Leal (1974), the gap region is treated using
the stretched coordinates
X Z lx;
Y Z lz;
Z Z lz;
R Z lr:
ð4:1Þ
In these coordinates the wall is described by the plane ZZK1, while the particle
boundary is the sphere RZl.
In the particle scale, the leading-order approximation is provided by the
solution of Gamayunov et al. (1986) for a particle in an unbounded fluid domain.
This solution is highly symmetric and does not result in either a hydrodynamic or
~ ð0Þ Z 0.3
an electric force F ð0Þ Z F
3
This is also evident from tensorial arguments: in the absence of a wall the system is istropic,
whence no candidates exist for F in (2.13).
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Boundary-induced electrophoresis
717
ð0Þ
Owing to the r – 2 type decay of the vP (see (3.8)), it transforms from O(1)
in the particle scale to O(l2) in the gap scale. In view of (3.9), it becomes evident
ð1Þ
that vW is also O(l2), and then, following Faxén’s law (3.15), so must
be the hydrodynamic force provoked by it. We will therefore focus upon
obtaining a leading-order O(l2) approximation for F together with an O(l3)
correction term.
ð1Þ
Evaluating v W requires first expressing
ð0Þ
ð0Þ
ð0Þ
ð0Þ
vP Z e^x u P Ce^y vP Ce^z wP ;
in terms of the gap-scale variables
9
9 2 X
3X 3
>
ð0Þ
4
>
uP Z l
K 5 C Oðl Þ; >
>
3
>
8
R
R
>
>
>
=
2
9 2 Y
3YX
ð0Þ
4
vP Z l
K
C Oðl Þ;
>
8
R3
R5
>
>
>
>
2
>
9 2 Z
3ZX
ð0Þ
4 >
>
wP Z l
K
Þ:
C
Oðl
;
3
5
8
R
R
ð4:2Þ
The O(l4) error in the above expressions stems from terms that decay at an r K4
ð0Þ
rate in the particle-scale description (3.8) of v P .
ð1Þ
Since v W satisfies the Stokes equations, its Cartesian components can be
expressed as Fourier transforms (Happel & Brenner 1965). Following Ho & Leal
(1974) we express the Cartesian components of
ð1Þ
ð1Þ
ð1Þ
ð1Þ
vW Z e^x u W Ce^y vW Ce^z wW ;
in the form
9
x2
KkZ
>
Z F g1 C 2 ðg2 C kZg3 Þ e
; >
>
>
>
k
>
>
=
xh
ð1Þ
KkZ
;
½g2 C kZg3 e
vW Z F
>
k2
>
>
>
>
>
ix
ð1Þ
KkZ
;
½g1 C g2 C g3 ð1 C kZÞe
:>
wW Z F
k
ð1Þ
uW
ð4:3Þ
Here, F denotes a two-dimensional Fourier transform, defined generically by
ð ð
1 N N
f ðx; hÞeiðxXChY Þ dx dh;
ð4:4Þ
F ff g Z
2p KN KN
kZ(x2Ch2)1/2; and g1, g2 and g3 are arbitrary functions of x and h. These
functions are determined by imposing (3.9) for nZ1.
It is therefore necessary to express the various terms in (4.2) as Fourier
transforms (evaluated at ZZK1). Manipulating the identity (Happel &
Brenner 1965)
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718
E. Yariv
(
)
1
e KkjZj
;
ZF
R
k
ð4:5Þ
yields the following relations at ZZK1 (see Yariv & Miloh in press)
9
(
)
(
)
Kk
Kk
>
1
X
xe
Y
he
>
>
;
;
Z F fe Kk g;
ZKiF
ZKiF
>
>
3
3
3
>
k
k
R
R
R
>
>
(
!
) >
>
>
2
2
2
2
=
X
1
YX
i
1 x
x
2 Kk
Kk
e
;
Z
Þe
g;
ZK
K
F
fð1Kx
F
h
K
3
3
k k3 k2
R5
R5
>
>
>
>
(
!
)
>
>
3
3
2
3 2
5
>
>
X
i
2x
3xh
xh
x
Kk
>
>
F
e
:
ZK
C
K
K
>
5
3
3
4
4
;
3
R
k
k
k
k
ð4:6Þ
For future reference, we also find that
X
i
ZK F fxeKk g at Z Z K1:
5
3
R
ð4:7Þ
Applying the boundary condition (3.9) for nZ1 yields g1, g2 and g3;
straightforward integration over the (x, h)-plane yields
27
ð1Þ ð4:8Þ
vW r Z0 Z l2e^z C Oðl4 Þ:
64
ð1Þ
In view of (4.1), the Laplacian of v W is O(l4); thus, Faxén’s law (3.15) gives
ð1Þ
F P;W Z
81p 2
l e^z C Oðl4 Þ:
32
ð4:9Þ
ð1Þ
Consider now the contribution of v P;4 , whose evaluation requires the
ð1Þ
ð1Þ
ð1Þ
calculation of 4W and 4P at the particle region. The first wall reflection 4W
represents a mirror dipole to (3.3), positioned at zZK2/l (Keh & Anderson
1985). In the gap-scale variables
X
ð1Þ
4W ZKl2
2½X 2 C Y 2 C ðZ C 2Þ2 3=2
:
ð4:10Þ
Expanding (4.10) into a Taylor series about O (Keh & Anderson 1985) yields
at the particle region
ð1Þ
4W wKl3
x
C Oðl4 Þ:
16
ð4:11Þ
To leading order, this expression represents a uniform electric field in the
x -direction of magnitude l3/16. The following terms in the Taylor expansion
are of progressively smaller asymptotic magnitude. At O(l3), the evaluation of
ð1Þ
ð0Þ
4P is similar to that of 4P ; thus, the leading-order term in the particle-scale
ð1Þ
ð0Þ
expansion of 4P is a dipole in the x -direction, identical to 4P (see (3.3)) with
a l3/16 multiplicative factor.
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Boundary-induced electrophoresis
We therefore conclude that
4
ð0Þ
C4
ð1Þ
l3
Z4
1C
C Oðl4 Þ:
16
ð0Þ
719
ð4:12Þ
ð0Þ
Recall that the field vP;4 , triggered by the quadratic interaction (3.7) in
the leading-order potential 4(0), does not result in a force. It is therefore evident
ð1Þ
from (4.12) that the slip-driven field vP;4 , triggered by quadratic interactions
(0)
(1)
(3.12) in 4 and 4 , produces a force that is O(l4) at most. This must also be
ð1Þ
the order of magnitude of the electrical force F~ , which also results from quadratic
(0)
(1)
interactions in 4 and 4 (see (3.16) for nZ1).4
ð2Þ
ð1Þ
ð1Þ
Consider now the field vW , induced by the two components of vP : v P;W and
ð1Þ
ð1Þ
vP;4 . In the particle region, vP;4 is O(l3); in view of its r K2 decay, it is O(l5) at
ð2Þ
the gap region; this is then the order of magnitude of the reaction to it in v W .
ð1Þ
Accordingly, we need only consider the effect of v P;W .
ð1Þ
Expanding v W to a Taylor series about O yields
ð1Þ
ð1Þ ð4:13Þ
v W wvW RZ0 C Oðl3 Þ;
in which the leading-order term is O(l2). We are interested in the reaction of
ð1Þ
vP;W to that term—namely the disturbance caused by a sphere that is positioned
within an O(l2) uniform stream in the z -direction. This reaction consists of two
parts (Happel & Brenner 1965), both O(l2) in the particle region: the first is a
Stokeslet that decays at an r K1 rate; the second is a dipole that decays like r K3.
At the gap region, these terms are O(l3) and O(l5). The leading-order O(l3)
ð2Þ
ð1Þ
ð1Þ
reaction in v W to vP is therefore triggered by the O(l2) Stokeslet of vP;W .
ð2Þ
ð1Þ
3
Thus, to obtain F P;W to O(l ), we only need to consider the Stokeslet of vP;W
ð2Þ
that is triggered by the leading-order term in (4.13), and then the reaction in vW to
that Stokeslet. As a matter of fact, no calculations are required: to leading order,
ð2Þ
the ratio of the force provoked by the reaction in v W to the Stokeslet and that
provoked by the leading-order uniform-stream term in (4.13) is 9l/8: this is the
well-known (Happel & Brenner 1965) leading-order wall-effect appearing in
the classical drag problem of a sphere translating away from a wall (cf. (6.1)).5
Since the contribution of all the other reflections is o(l3), we conclude that
81p 2
9
2
Fw
F~ wOðl4 Þ:
ð4:14Þ
l 1 C l C Oðl Þ ;
32
8
5. Generalization to a polarizable wall
Implicit in the no-slip condition (2.8) on the dielectric wall is the assumption of
zero zeta potential. This assumption is tantamount to that of an ideally nonpolarizable wall. In reality, the dielectric wall material possesses a finite
~ ð1Þ to this order was carried out by Yariv (2006).
The calculation of F
The two problems are not completely analogous owing to the differences in particle motion. The
mere effect of this motion on the velocity field, however, is the additional of a dipole term; this term
decays as r K3 and does not affect the O(l) leading wall effect.
4
5
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720
E. Yariv
polarizability and a zeta potential can be induced at the wall–fluid interface
(Squires & Bazant 2004). Here, we analyse the effect of the wall polarization
upon the hydrodynamic force exerted on a stationary particle.
Consistently with the thin-Debye-layer limit, the zeta potential on the wall
is simply
zW Z 4K4
at z Z K1=l;
ð5:1Þ
is the electric potential within the wall. Since the interior of the
wherein 4
is harmonic. This potential needs to match the
dielectric wall is charge free, 4
electric potential inside the induced Debye layer surrounding the wall. Assuming
small zeta potentials, it was shown by Yossifon et al. (2007) that the requisite
matching is equivalent to the macroscale Robin-type condition
ð5:2Þ
^ 4
C an$V
Z 4:
4
Here, aZ ð
e=eÞð1=kaÞ, where e is the dielectric permittivity of the wall and 1/k is
the Debye thickness, see (1.1).
The common model of an ideally non-polarizable wall ðe=e/ 0Þ corresponds to
4 and the zeta potential vanishes. Within the
vanishingly small a, whereby 4Z
thin-Debye layer regime (1.2) it seems plausible to assume small a even for
polarizable walls, see Yossifon et al. (2006).6 When considering, however, the
entire range of dielectric constants that appear in specific applications, we find
that moderate a-values can appear as well. (For certain ceramic materials e=e is
quite large, see Rodriguez & Markx (2006).) In what follows, we follow Yossifon
et al. (2007) and present a general analysis for arbitrary a-values.
using the gap-scale variables, whereby condition
It is natural to evaluate 4
(5.2) appears as
v4
C al
4
Z 4 at Z ZK1:
ð5:3Þ
vZ
Then, in view of condition (5.3), the zeta potential is
zW ZKal
v4
:
j
vZ ZZK1
ð5:4Þ
The no-slip condition (2.8) is therefore modified to
v ZKal
v4
V4
vZ
at Z ZK1:
ð5:5Þ
is an odd function of X. We postulate the iterative solution
Just like 4, 4
Z4
ð0Þ C 4
ð1Þ C 4
ð2Þ C/;
4
ð5:6Þ
ð0Þ
ð0Þ Z 4W . The harmonic corrections 4
ðnÞ (nR1) are driven by
where 4
corresponding reflections in 4 through the Robin condition (5.3), that is
ðnÞ C al
4
ðnÞ
v4
ðnK1Þ
ðnÞ
C 4W
Z 4P
vZ
at Z Z K1;
ð5:7Þ
6
Physically, the smallness of a represents the intensive electric displacement within the thin Debye
layer, as compared with the moderate displacement in the wall. The dominance of the former
in Gauss’s electrostatic boundary condition decouples the bulk electrostatics from the
wall polarization.
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Boundary-induced electrophoresis
721
in addition, they are required to decay at large distances
ðnÞ / 0 as jXj/N or jY j/N or Z /KN:
4
ð5:8Þ
The iterative expansion (5.6) affects a comparable expansion for zW through (5.4).
Consider now the limit l/1. The gap-scale electric fields associated with
ð0Þ
ð1Þ
4P and 4W are both O(l3); it is therefore clear that zW , and then the slip on the
wall, begin at this asymptotic order. Consequently, it is sufficient to calculate
ð1Þ . Substitution of (3.3) and (4.10) into (5.7) for nZ1 gives
4
ð1Þ C al
4
ð1Þ
v4
X
ZKl2 2
vZ
ðX C Y 2 C 1Þ3=2
at Z Z K1:
ð5:9Þ
To leading order, this is simply a Dirichlet condition at ZZK1. Solving the
boundary-value problem to that order using Sine transforms yields:
X
ð1Þ ZKl2 3 C Oðl3 Þ:
4
ð5:10Þ
R
This is a dipole centred about O; aside from having twice the magnitude, it is
ð0Þ
identical to 4P (see (3.3)). Using (5.4), we then find
3X
ð1Þ
zW ZKl3 2
C Oðl4 Þ:
ð5:11Þ
ðX C Y 2 C 1Þ5=2
The small O(l3) zeta potential a posteriori justifies the use of condition (5.2).
In view of (5.5), the leading-order O(l3) wall slip results from interaction
between the O(l3) wall zeta potential and the O(1) leading-order electric field in
the bulk fluid, e^x :
3aX
ð5:12Þ
v ZK^
ex l3 5 C Oðl4 Þ at Z Z K1:
R
The velocity field generated by this slip condition is calculated using a Fourier
ð1Þ
representation, similar to that of vW (see (4.3)). The Fourier transform of the
slip condition (5.12) is obtained from (4.7). Evaluation at O yields
3a 3
ð1Þ ð5:13Þ
vW r Z0 Z
l e^z C Oðl4 Þ:
16
ð1Þ
The force provoked by v W is obtained using Faxén’s formula (3.15); it is
of magnitude
9pa 3
ð5:14Þ
l C Oðl4 Þ
8
and it is directed parallel to the z -axis.
Up to O(l3), the force induced by the dielectric wall is simply provided by
combining (4.14) and (5.14), the case of an ideally non-polarizable wall
corresponding to a/0.
6. Concluding remarks
We have calculated the wall-induced force acting on a stationary particle. When
the particle is freely suspended in the electrolyte, this force imparts it with the
velocity required to keep it force free (see (2.15)). Multiplying the mobility
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722
E. Yariv
(normalized with 1/am) of a spherical particle in a direction normal to solid wall
(Happel & Brenner 1965)
1
9
ð6:1Þ
1K l C Oðl3 Þ ;
6p
8
by the sum of (4.14) and (5.14) yields
27 2 3a 3
ð6:2Þ
l C l C Oðl4 Þ:
64
16
The leading-order term in this expression was independently found by Saintillan
(in preparation). It is different from that calculated for a pair of spherical
particles whose line of centres lies perpendicular to the applied field (Saintillan
2008); indeed, these two problems are not physically equivalent. Note that the
ð2Þ
O(l) wall effect in (6.1) cancels out the contribution provoked by v W . This is to
be expected, since that contribution represents a Stokeslet associated with a fixed
particle; this Stokeslet must disappear when a force-free particle is considered.
In principle, it is possible to improve the approximation (6.2). It should be noted
that once O(l4) terms are retained, this velocity becomes affected by the electric
force (Yariv 2006), and does not formally qualify as ‘electrophoretic’.
The present investigation of a sphere–wall system was motivated by the
bipolar calculation of Zhao & Bau (2007) for a two-dimensional cylinder–wall
system. It may appear that the present iterative scheme could be applied to the
comparable two-dimensional problem as well, thereby providing asymptotic
formulae that can supplement the numerical results of Zhao & Bau (2007).
Recall, however, that as l/0 the iterative reflection method represents a limit
process at which the particle–wall distance approaches infinity. In view of the
Stokes paradox, no such limit exists in the two-dimensional problem: specifically,
ðnÞ
the two-dimensional equivalents of v P;W do not exist. This, of course, is implicit
in the absence of a two-dimensional counterpart of Faxén’s law.
UZ
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