J. Fluid Mech. (2004), vol. 509, pp. 217–252.
c
2004 Cambridge University Press
DOI: 10.1017/S0022112004009309 Printed in the United Kingdom
217
Induced-charge electro-osmosis
By TODD M. SQUIRES
1
AND MARTIN Z. BAZANT
2
1
Departments of Applied and Computational Mathematics and Physics,
California Institute of Technology, Pasadena, CA 91125, USA
2
Department of Mathematics and Institute for Soldier Nanotechnologies,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Received 5 May 2003 and in revised form 13 February 2004)
We describe the general phenomenon of ‘induced-charge electro-osmosis’ (ICEO) –
the nonlinear electro-osmotic slip that occurs when an applied field acts on the ionic
charge it induces around a polarizable surface. Motivated by a simple physical picture,
we calculate ICEO flows around conducting cylinders in steady (DC), oscillatory (AC),
and suddenly applied electric fields. This picture, and these systems, represent perhaps
the clearest example of nonlinear electrokinetic phenomena. We complement and
verify this physically motivated approach using a matched asymptotic expansion to the
electrokinetic equations in the thin-double-layer and low-potential limits. ICEO slip
velocities vary as u
s
∝E
2
0
L, where E
0
is the field strength and L is a geometric length
scale, and are set up on a time scale τ
c
= λ
D
L/D, where λ
D
is the screening length
and D is the ionic diffusion constant. We propose and analyse ICEO microfluidic
pumps and mixers that operate without moving parts under low applied potentials.
Similar flows around metallic colloids with fixed total charge have been described in
the Russian literature (largely unnoticed in the West). ICEO flows around conductors
with fixed potential, on the other hand, have no colloidal analogue and offer further
possibilities for microfluidic applications.
1. Introduction
Recent developments in micro-fabrication and the technological promise of micro-
fluidic ‘labs on a chip’ have brought a renewed interest to the study of low-Reynolds-
number flows (Stone & Kim 2001; Whitesides & Stroock 2001; Reyes et al. (2002)).
The familiar techniques used in larger-scale applications for fluid manipulation, which
often exploit fluid instabilities due to inertial nonlinearities, do not work on the micron
scale due to the pre-eminence of viscous damping. The microscale mixing of miscible
fluids must thus occur without the benefit of turbulence, by molecular diffusion alone.
For extremely small devices, molecular diffusion is relatively rapid; however, in typical
microfluidic devices with 10–100 µm features, the mixing time can be prohibitively
long (of order 100 s for molecules with diffusivity 10
−10
m
2
s
−1
). Another limitation
arises because the pressure-driven flow rate through small channels decreases with
the third or fourth power of channel size. Innovative ideas are thus being considered
for pumping, mixing, manipulating and separating on the micron length scale (e.g.
Beebe, Mensing & Walker 2002; Whitesides et al. 2001). Naturally, many focus on
the use of surface phenomena, owing to the large surface to volume ratios of typical
microfluidic devices.
Electrokinetic phenomena provide one of the most popular non-mechanical tech-
niques in microfluidics. The basic idea behind electrokinetic phenomena is as follows:
218 T. M. Squires and M. Z. Bazant
locally non-neutral fluid occurs adjacent to charged solid surfaces, where a diffuse
cloud of oppositely charged counter-ions ‘screens’ the surface charge. An externally
applied electric field exerts a force on this charged diffuse layer, which gives rise to
a fluid flow relative to the particle or surface. Electrokinetic flow around stationary
surfaces is known as electro-osmotic flow, and the electrokinetic motion of freely
suspended particles is known as electrophoresis. Electro-osmosis and electrophoresis
find wide application in analytical chemistry (Bruin 2000), genomics (Landers 2003)
and proteomics (Figeys & Pinto 2001; Dolnik & Hutterer 2001).
The standard picture for electrokinetic phenomena involves surfaces with fixed, con-
stant charge (or, equivalently, zeta-potential ζ , defined as the potential drop across
the screening cloud). Recently, variants on this picture have been explored. Anderson
(1985) demonstrated that interesting and counter-intuitive effects occur with spatially
inhomogeneous zeta-potentials, and showed that the electrophoretic mobility of a
colloid was sensitive to the distribution of surface charge, rather than simply the total
net charge. Anderson & Idol (1985) explored electro-osmotic flow in inhomogeneously
charged pores, and found eddies and recirculation regions. Ajdari (1995, 2002) and
Stroock et al. (2000) showed that a net electro-osmotic flow could be driven either
parallel or perpendicular to an applied field by modulating the surface and charge
density of a microchannel, and Gitlin et al. (2003) have implemented these ideas to
make a ‘transverse electrokinetic pump’. Such transverse pumps have the advantage
that a strong field can be established with a low voltage applied across a narrow
channel. Long & Ajdari (1998) examined electrophoresis of patterned colloids, and
found example colloids whose electrophoretic motion is always transverse to the
applied electric field. Finally, Long, Stone & Ajdari (1999), Ghosal (2003), and others
have studied electro-osmosis along inhomogeneously charged channel walls (due
to, e.g., adsorption of analyte molecules), which provides an additional source of
dispersion that can limit resolution in capillary electrophoresis.
Other variants involve surface charge densities that are not fixed, but rather
are induced (either actively or passively). For example, the effective zeta-potential of
channel walls can be manipulated using an auxillary electrode to improve separa-
tion efficiency in capillary electrophoresis (Lee, Blanchard & Wu 1990; Hayes &
Ewing 1992) and, by analogy with the electronic field-effect transistor, to set up ‘field-
effect electro-osmosis’ (Ghowsi & Gale 1991; Gajar & Geis 1992; Schasfoort et al.
1999).
Time-varying, inhomogeneous charge double layers induced around electrodes give
rise to interesting effects as well. Trau, Saville & Aksay (1997) and Yeh, Seul &
Shraiman (1997) demonstrated that colloidal spheres can spontaneously self-assemble
into crystalline aggregates near electrodes under AC applied fields. They proposed
somewhat similar electrohydrodynamic mechanisms for this aggregation, in which an
inhomogeneous screening cloud is formed by (and interacts with) the inhomogeneous
applied electric field (perturbed by the sphere), resulting in a rectified electro-osmotic
flow directed radially in toward the sphere. More recently, Nadal et al. (2002a)perfor-
med detailed measurements in order to test both the attractive (electrohydrodynamic)
and repulsive (electrostatic) interactions between the spheres, and Ristenpart, Aksay &
Saville (2003) explored the rich variety of patterns that form when bidisperse colloidal
suspensions self-assemble near electrodes.
A related phenomenon allows steady electro-osmotic flows to be driven using AC
electric fields. Ramos et al. (1998, 1999) and Gonzalez et al. (2000) theoretically
and experimentally explored ‘AC electro-osmosis’, in which a pair of adjacent, flat
electrodes located on a glass slide and subjected to AC driving, gives rise to a steady
Induced-charge electro-osmosis 219
electro-osmotic flow consisting of two counter-rotating rolls. Around the same time,
Ajdari (2000) theoretically predicted that an asymmetric array of electrodes with
applied AC fields generally pumps fluid in the direction of broken symmetry (‘AC
pumping’). Brown, Smith & Rennie (2001), Studer et al. (2002), and Mpholo, Smith &
Brown (2003) have since developed AC electrokinetic micro-pumps based on this
effect, and Ramos et al. (2003) have extended their analysis.
Both AC colloidal self-assembly and AC electro-osmosis occur around electrodes
whose potential is externally controlled. Both effects are strongest when the voltage
oscillates at a special AC frequency (the inverse of the charging time discussed below),
and both effects disappear in the DC limit. Furthermore, both vary with the square
of the applied voltage V
0
. This nonlinear dependence can be understood qualitatively
as follows: the induced charge cloud/zeta-potential varies linearly with V
0
, and the
resulting flow is driven by the external field, which also varies with V
0
. On the other
hand, DC colloidal aggregation, as explored by Solomentsev, Bohmer & Anderson
(1997), requires large enough voltages to pass a Faradaic current, and is driven by a
different, linear mechanism.
Very recently in microfluidics, a few cases of nonlinear electro-osmotic flows around
isolated and inert (but polarizable) objects have been reported, with both AC and
DC forcing. In a situation reminiscent of AC electro-osmosis, Nadal et al. (2002b)
studied the micro-flow produced around a dielectric stripe on a planar blocking
electrode. In rather different situations, Thamida & Chang (2002) observed a DC
nonlinear electrokinetic jet directed away from a protruding corner in a dielectric
microchannel, far away from any electrode, and Takhistov, Duginova & Chang (2003)
observed electrokinetically driven vortices near channel junctions. These studies (and
the present work) suggest that a rich variety of nonlinear electrokinetic phenomena
at polarizable surfaces remains to be exploited in microfluidic devices.
In colloidal science, nonlinear electro-osmotic flows around polarizable (metallic
or dielectric) particles were studied almost two decades ago in a series of Ukrainian
papers, reviewed by Murtsovkin (1996), that has gone all but unnoticed in the West.
Such flows occur when the applied field acts on the component of the double-layer
charge that has been polarized by the field itself. This idea can be traced back at
least to Levich (1962), who discussed the dipolar charge double layer (using the
Helmholtz model) that is induced around a metallic colloidal particle in an external
electric field and touched upon the quadrupolar flow that would result. Simonov &
Dukhin (1973) calculated the structure of the (polarized) dipolar charge cloud in
order to obtain the electrophoretic mobility, without concentrating on the resulting
flow. Gamayunov, Murtsovkin & Dukhin (1986) and Dukhin & Murtsovkin (1986)
first explicitly calculated the nonlinear electro-osmotic flow arising from double-layer
polarization around a spherical conducting particle, and Dukhin (1986) extended this
calculation to include a dielectric surface coating (as a model of a dead biological cell).
Experimentally, Gamayunov, Mantrov & Murtsovkin (1992) observed a nonlinear
flow around spherical metallic colloids, albeit in a direction opposite to predicitions
for all but the smallest particles. They argued that a Faradaic current (breakdown of
ideal polarizibility) was responsible for the observed flow reversal.
This work followed naturally from many earlier studies on ‘non-equilibrium electric
surface phenomena’ reviewed by Dukhin (1993), especially those focusing on the
‘induced dipole moment’ (IDM) of a colloidal particle reviewed by Dukhin & Shilov
(1980). Following Overbeek (1943), who was perhaps the first to consider non-
uniform polarization of the double layer in the context of electrophoresis, Dukhin
(1965), Dukhin & Semenikhin (1970), and Dukhin & Shilov (1974) predicted the
220 T. M. Squires and M. Z. Bazant
electrophoretic mobility of a highly charged non-polarizable sphere in the thin-double-
layer limit, in good agreement with the later numerical solutions of O’Brien & White
(1978) (see, e.g., Lyklema 1991). In that case, diffuse charge is redistributed by surface
conduction, which produces an IDM aligned with the field, and some variations in
neutral bulk concentration, and secondary electro-osmotic and diffusio-osmotic flows
develop as a result. Shilov & Dukhin (1970) extended this work to a non-polarizable
sphere in an AC electric field. Simonov & Dukhin (1973) and Simonov & Shilov
(1973) performed similar calculations for an ideally polarizable, conducting sphere,
which typically exhibits an IDM opposite to the applied field. Simonov & Shilov
(1977) revisited this problem in the context of dielectric dispersion and proposed
a much simpler RC-circuit model to explain the sign and frequency dependence of
the IDM. O’Brien & White (1978) performed a numerical solution of the full ion,
electrostatic, and fluid equations with arbitrarily thick double layer, which naturally
incorporated the effects of double-layer polarization. O’Brien & Hunter (1981) and
O’Brien (1983), following Dukhin’s approach, arrived at a simpler, approximate
expression that incorporated double-layer polarization in the thin-double-layer limit,
and compared favourably with the numerical calculations of O’Brien & White (1978).
Nonetheless, it seems a detailed study of the associated electro-osmotic flows around
polarizable spheres did not appear until the paper of Gamayunov et al. (1986) cited
above.
In summary, we note that electrokinetic phenomena at polarizable surfaces share
a fundamental feature: all involve a nonlinear flow component in which double-layer
charge induced by the applied field is driven by that same field. To emphasize this
common mechanism, we suggest the term ‘induced-charge electro-osmosis’ (ICEO) for
their description. Specific realizations of ICEO include AC electro-osmosis at micro-
electrodes, AC pumping by asymmetric electrode arrays, DC electrokinetic jets around
dielectric structures, DC and AC flows around polarizable colloidal particles, and the
situations described below. Of course, other electrokinetic effects may also occur in
addition to ICEO in any given system, such as those related to bulk concentration
gradients produced by surface conduction or Faradaic reactions, but we ignore such
complications here to highlight the basic effect of ICEO.
In the present work, we build upon this foundation of induced-charge electrokinetic
phenomena, specifically keeping in mind microfluidic applications. ICEO flows around
metallic colloids, which have attracted little attention compared to non-polarizable
objects of fixed zeta-potential, naturally lend themselves for use in microfluidic devices.
In that setting, the particle is replaced by a fixed polarizable object which pumps the
fluid in response to applied fields, and a host of new possibilities arise. The ability to
directly control the position, shape, and potential of one or more ‘inducing surfaces’ in
a microchannel allows a rich variety of effects that do not occur in colloidal systems.
Before we begin, we note the difference between ICEO and ‘electrokinetic pheno-
mena of the second kind’, reviewed by Dukhin (1991) and studied recently by Ben &
Chang (2002) in the context of microfluidic applications. Significantly, second-kind
electrokinetic effects do not arise from the double layer, being instead driven by
space charge in the bulk solution. They typically occur around ion-selective porous
granules subject to applied fields large enough to generate strong currents of certain
ions through the liquid/solid interface. This leads to large concentration variations
and space charge in the bulk electrolyte on one side, which interact with the applied
field to cause motion. Barany, Mishchuk & Prieve (1998) studied the analogous effect
for non-porous metallic colloids undergoing electrochemical reactions at very large
Faradaic currents (exceeding diffusion limitation). In contrast, ICEO occurs around
Induced-charge electro-osmosis 221
inert polarizable surfaces carrying no Faradaic current in contact with a homogeneous,
quasi-neutral electrolyte and relies on relatively small induced double-layer charge,
rather than bulk space charge.
The article is organized as follows: §2 provides a basic background on electrokinetic
effects, and §3 develops a basic physical picture of induced-charge electro-osmosis via
calculations of steady ICEO flow around a conducting cylinder. Section 4 examines
the time-dependent ICEO flow for background electric fields which are suddenly
applied (§4.1) or sinusoidal (§4.2). Section 5 describes some basic issues for ICEO
in microfluidic devices, such as coupling to the external circuit (§5.1) and the
phenomenon of fixed-potential ICEO (§5.2). Some specific designs for microfluidic
pumps, junction switches, and mixers are discussed and analysed in §5.3. Section 6
investigates the detrimental effect of a thin dielectric coating on a conducting surface
and calculates the ICEO flow around non-conducting dielectric cylinders. Section 7
gives a systematic derivation of ICEO in the limit of thin double layers and small
potentials, starting with the basic electrokinetic equations and employing matched
asymptotic expansions, concluding with a set of effective equations (with approxima-
tions and errors quantified) for the ICEO flow around an arbitrarily shaped particle
in an arbitrary space- and time-dependent electric field. The interesting consequences
of shape and field asymmetries, which generally lead to electro-osmotic pumping or
electrophoretic motion in AC fields, are left for a companion paper. The reader is
referred to Bazant & Squires (2004) for an overview of our results.
2. Classical (‘fixed-charge’) electro-osmosis
Electrokinetic techniques provide some of the most popular small-scale non-
mechanical strategies for manipulating particles and fluids. We present here a very
brief introduction. More detailed accounts are given by Lyklema (1991), Hunter (2000)
and Russel, Saville & Schowalter (1989).
2.1. Small zeta-potentials
A surface with charge density q in an aqueous solutions attracts a screening cloud
of oppositely charged counter-ions to form the electrochemical ‘double layer’, which
is effectively a surface capacitor. In the Debye–H
¨
uckel limit of small surface charge,
the excess diffuse ionic charge exponentially screens the electric field set up by the
surface charge (figure 1a), giving an electrostatic potential
φ =
q
ε
w
κ
e
−κz
≡ ζ e
−κz
. (2.1)
Here ε
w
≈80ε
0
is the dielectric permittivity of the solvent (typically water) and ε
0
is
the vacuum permittivity. The ‘zeta-potential’, defined by
ζ ≡
q
ε
w
κ
, (2.2)
reflects the electrostatic potential drop across the screening cloud, and the Debye
‘screening length’ κ
−1
is defined for a symmetric z:z electrolyte by
κ
−1
≡ λ
D
=
ε
w
k
B
T
2n
0
(ze)
2
, (2.3)
with bulk ion concentration n
0
, (monovalent) ion charge e, Boltzmann constant k
B
and temperature T . Because the ions in the diffuse part of the double layer are
