J. Fluid Mech. (2002), vol. 459, pp. 103{128.
c
© 2002 Cambridge University Press
DOI: 10.1017/S0022112002007899 Printed in the United Kingdom
103
Lubrication theory for electro-osmotic flow in a
microfluidic channel of slowly varying
cross-section and wall charge
By S A N D I P G H O S A L
Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road,
Evanston, IL 60208, USA
(Received 24 August and in revised form 2 November 2001)
Electro-osmotic flow is a convenient mechanism for transporting fluid in microfluidic
devices. The flow is generated through the application of an external electric eld
that acts on the free charges that exist in a thin Debye layer at the channel walls.
The charge on the wall is due to the particular chemistry of the solid{fluid interface
and can vary along the channel either by design or because of various unavoidable
inhomogeneities of the wall material or because of contamination of the wall by
chemicals contained in the fluid stream. The channel cross-section could also vary in
shape and area. The eect of such variability on the flow through microfluidic channels
is of interest in the design of devices that use electro-osmotic flow. The problem of
electro-osmotic flow in a straight microfluidic channel of arbitrary cross-sectional
geometry and distribution of wall charge is solved in the lubrication approximation,
which is justied when the characteristic length scales for axial variation of the wall
charge and cross-section are both large compared to a characteristic width of the
channel. It is thereby shown that the volume flux of fluid through such a microchannel
is a linear function of the applied pressure drop and electric potential drop across it,
the coecients of which may be calculated explicitly in terms of the geometry and
charge distribution on the wall. These coecients characterize the ‘fluidic resistance’
of each segment of a microfluidic network in analogy to the electrical ‘resistance’ in
a microelectronic circuit. A consequence of the axial variation in channel properties
is the appearance of an induced pressure gradient and an associated secondary flow
that leads to increased Taylor dispersion limiting the resolution of electrophoretic
separations. The lubrication theory presented here oers a simple way of calculating
the distortion of the flow prole in general geometries and could be useful in studies
of dispersion induced by inhomogeneities in microfluidic channels.
1. Introduction
Many solid substrates, (such as glass, silicon, polymeric materials, minerals) acquire
a surface charge when in contact with electrolytes. The charged surface attracts free
ions of the opposite sign creating a thin ( 1{10 nm) Debye layer of mobile charges
next to it. In the presence of an external electric eld, the fluid in this Debye
layer acquires a momentum which is then transmitted to adjacent layers of fluid
through the eect of viscosity. The resulting fluid motion is known as electro-osmotic
flow (Probstein 1994). Since the force per unit length of channel is proportional
to the circumference of the channel while the mass of fluid that must be moved is
proportional to the cross-sectional area, the eect is signicant in very narrow channels

104 S. Ghosal
such as those etched on substrates ( 10{100 m diameter or less) in microfluidic
devices, or, in the interparticle spaces in porous media. Electro-osmotic flow was rst
reported by Reuss in 1809 in experiments that demonstrated that water could be
made to percolate through porous clay diaphragms through the application of an
electric eld (Reuss 1809).
In recent years, electro-osmotic flow (EOF) has found wide application in micro-
fluidic devices as an ecient method for transporting fluid (Jakeway, de Mello &
Russell 2000; Whitesides & Stroock 2000). The convenience of being able to move
fluid by applying voltages along the channels, has the advantage that electric and
fluidic circuits can be integrated on the same microchip to build complex miniaturized
devices without moving parts. In addition, EOFs in uniform channels have a constant
velocity over the channel cross-section (except within the thin Debye layer at the wall).
This is in contrast to the situation in pressure-driven flows, such as the Poiseuille
flow in circular pipes, where the velocity distribution has a parabolic prole. As a
result, in EOF Taylor dispersion (Probstein 1994) of solutes is very small. This is
a great advantage in many bioanalytical applications of microfluidics. For example,
the resolution in electrophoretic separation of biomolecules of only slightly diering
mobilities is limited by Taylor dispersion (Culbertson, Jacobson & Ramsey 1998).
The uniformity of the flow prole and the resulting low Taylor dispersion are
characteristic of microfluidic channels with a uniform wall charge. In the presence
of inhomogeneities in the wall charge, induced pressure gradients are created that
distort the uniformity of the flow prole (Herr et al. 2000), reducing the eciency of
microfluidic devices that use EOF for electrophoretic separations. A common cause
of non-uniformity of the wall charge is the adsorption of certain organic molecules
onto the wall during analysis (Towns & Regnier 1991, 1992). Further, the wall charge
depends strongly on the pH of the buer and is known to exhibit hysterisis eects
when the pH is changed (Lambert & Middleton 1990). Various synthetic materials
such as acrylic and Poly(dimethylsiloxane) (PDMS) are being investigated as possible
replacements for glass or silicon substrates on account of their lower cost among
other advantages (Anderson et al. 2000). A diculty with the use of such materials
is that the wall charge is not as uniform as in the glass and silicon-based devices.
In order to overcome such diculties and allow more precise control over the wall
charge, various synthetic coatings are being investigated (Liu et al. 2000). With such
techniques, channels with a specied variation of the wall charge could be engineered
to build novel fluidic properties into micro devices (Barkar et al. 2000). Control of
wall charge using externally applied voltages has also been studied (Lee, Blanchard
& Wu 1990; Hayes & Ewing 1992).
In light of these continuing developments in microfluidic technology, the problem
of EOF in microfluidic channels of variable wall charge is of great interest. Anderson
& Idol (1985) considered the problem of EOF through a uniform, innite, straight
cylindrical capillary with a wall charge that varies solely in the axial direction. A
uniform external electric eld and zero imposed pressure gradient was assumed. An
exact solution to the Stokes flow problem was derived by means of separation of
variables and series expansion. More recently, Herr et al. (2000) studied the problem
of flow through a cylindrical capillary tube with a wall charge that undergoes
a stepwise change in the axial direction. This problem is a special case of that
considered by Anderson & Idol. The requirement of mass continuity forces the
appearance of a pressure gradient and associated Poiseuille flow. Two capillaries
with dierent surface coatings were joined together to produce a capillary with a
stepwise variation in wall charge. The flow prole was measured experimentally using