Numerical Simulation of Electroosmotic Flow
Neelesh A. Patankar
�
and Howard H. Hu*
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania,
Philadelphia, Pennsylvania 19104-6315
We have developed a numerical scheme to simulate
electroosmotic flows in complicated geometries. We
studied the electroosmotic injection characteristics of a
cross-channel device for capillary electrophoresis. We
found that the desired rectangular shape of the sample
plug at the intersection of the cross-channel can be
obtained when the injection is carried out at high electric
field intensities. The shape of the sample plug can also
be controlled by applying an electric potential or a pres-
sure at the side reservoirs. Flow induced from the side
channels into the injection channel squeezes the stream-
lines at the intersection, thus giving a less distorted
sample plug. Results of our simulations agree qualita-
tively with experimental observations.
Extensive experimental studies of capillary electrophoresis
using micromachined channels have been performed recently.
Harrison et al.
1
integrated a capillary electrophoresis and sample
injection system on a planar glass chip. They observed that the
solvent flow could be directed along a specified capillary by the
application of appropriate voltages, so that valveless switching of
fluid flow between the capillaries could be achieved. Seiler et
al.
2
presented improvements in the instrumentation and experi-
mental method for the device described by Harrison et al.
1
Harrison et al.
3
performed electroosmotic pumping and electro-
phoretic separation of samples using a device consisting of two
intersecting channels micromachined in silicon. Fan and Harri-
son
4
used a similar cross-channel device integrated on a glass chip
and evaluated its performance. Seiler et al.
5
used a device in which
the electroosmotic flow was driven within a network of intersecting
capillaries integrated on a glass chip. Kirchoff's rules for resistive
networks were found to predict the fluid flow within the capillaries.
Effenhauser et al.
6
performed high-speed separation of antisense
oligonucleotides on a micromachined capillary electrophoresis
device integrated on a glass plate. Burggraf et al.
7
presented a
novel approach for the ion separation in solutions using the
concept of synchronized cyclic capillary electrophoresis. Repeated
column switching during the capillary electrophoresis eliminated
the unwanted sample components and separated the species
having very similar mobilities. Jacobson et al.
8
did capillary
electrophoresis in a cross-channel device with an integrated
postcolumn reactor. The device was fabricated on a glass plate.
A postcolumn reactor was added to conduct postseparation
derivatization using a fluorescent �tag� for amino acids. Culbert-
son and Jorgenson
9
suggested a new approach for increasing the
efficiency and resolution of the capillary electrophoresis. The
approach utilized a pressure-induced counterflow to actively retard,
halt, or reverse the electrokinetic migration of an analyte, thus
keeping the analytes of interest in the separation field much longer
than under the normal separation conditions.
In most of these experiments, the injection of the sample into
the separation channel was achieved using the electroosmotic flow
driven by an applied potential along the injection channel. The
injection channel and the separation channel were perpendicular
to each other, although the overall design of the device might
not be as simple as a cross-channel. The shape of the inserted
sample, which is an important parameter that influences the
resolution of the separated zones during the electrophoresis,
depends primarily on the electroosmotic flow pattern at the
intersection of the channels. The electroosmotic flow at the
intersection itself is influenced by various parameters. Until now
only experimental investigations have been performed to deter-
mine the parameters that give the most desirable shape of the
injected sample. However, it would be more suitable to do such
investigations through numerical simulations since one can better
control various parameters involved.
Rice and Whitehead
10
formulated the equations for the elec-
troosmotic flow in a capillary. Jorgenson and Lukacs
11
developed
a one-dimensional model for the capillary electrophoresis. An-
dreev and Lisin
12
presented a mathematical model for one-
dimensional capillary electrophoresis and studied the influence
of the electroosmotic flow profile on the efficiency of separation.
A three-dimensional model is required to study the electroosmotic
flow in more complicated geometries.
In this work, we tested a three-dimensional model for elec-
troosmotic flows. The primary aim is to develop a robust
�
Present address: Department of Aerospace Engineering and Mechanics,
University of Minnesota, Minneapolis, MN 55455.
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(3) Harrison, D. J.; Glavina, P. G.; Manz, A. Sens. Actuators, B 1993, 10, 107-
116.
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(5) Seiler, K.; Fan, Z. H.; Fluri, K.; Harrison, D. J. Anal. Chem. 1994, 66, 3485-
3491.
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66, 2949-2953.
(7) Burggraf, N.; Manz, A.; Verpoorte, E.; Effenhauser, C. S.; Widmer, H. M.
Sens. Actuators, B 1994, 20, 103-110.
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J. M. Anal. Chem. 1994, 66, 3472-3476.
(9) Culbertson, C. T.; Jorgenson, J. W. Anal. Chem. 1994, 66, 955-962.
(10) Rice, C. I.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017-4024.
(11) Jorgenson, J. M.; Lukacs, K. D. Anal. Chem. 1981, 53, 1298-1302.
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Anal. Chem. 1998, 70, 1870-1881
1870 Analytical Chemistry, Vol. 70, No. 9, May 1, 1998 S0003-2700(97)00846-9 CCC: $15.00 1998 American Chemical Society
Published on Web 03/24/1998

numerical scheme that can be applied to various flow conditions.
As an example we simulate the electroosmotic flow at the
intersection of a cross-channel during the injection process. These
simulations help to determine the parameters that are important
in controlling the shape of the injected sample. Fan and Harrison
4
studied such a device experimentally. Our findings will be
qualitatively compared with their conclusions. Such a comparison
will validate the predictions made by our simulations, thus
establishing a reliable scheme to simulate electroosmotic flows
in more complicated geometries.
DESCRIPTION OF THE GEOMETRY
The geometry of the device used by Fan and Harrison
4
is
similar to that shown in Figure 1. There is an intersection of cross-
channels with four reservoirs numbered 1-4. The cross-channels
are closed by walls from above and below. The injection of the
sample solution (containing various species) is effected by
applying an electric potential between reservoirs 1 and 2. This
potential drives the sample solution from reservoir 1 across the
intersection, creating a sample plug in the path between reservoirs
3 and 4. Sample separation does not occur during the injection,
since the sample solution runs continuously in the injection
channel. The separation of the species in the sample requires
the sample to be arranged initially in a zone so that the species
can move away from each other due to the difference in their
electrophoretic mobilities (as the zone spreads).
13
During the
separation process, the electric potential between reservoirs 3 and
4 drives the sample plug to move toward reservoir 4 and causes
the separation of various species.
In our numerical simulation, we intend to study the flow pattern
in the injection phase, i.e., when the sample is injected by applying
a potential between reservoirs 1 and 2. Clearly the shape of the
sample at the intersection depends largely on the flow pattern at
the intersection and on the diffusion of the sample into the buffer
solution. For reasons to be described later, we will be neglecting
the diffusion of the sample. Besides, we will only study the steady-
state flow pattern at the intersection.
PROBLEM FORMULATION
Unless the sample is allowed to stay at the intersection for a
long time, the diffusion of the sample solution from the injection
channel into the separation channel is not important, since the
diffusion time is much longer than the relevant convective time
scale of the flow. This will become clear from the subsequent
discussion. The equation for the diffusion of the sample is
where C is the sample concentration, V is the flow velocity, and
D is the diffusion coefficient. Let U be the convective velocity
scale and h be some length scale, e.g. the channel width. If we
nondimensionalize the velocity by U, length by h, and time by
h/U, (1) becomes (we retain the same symbols for the nondi-
mensional variables)
where the nondimensional parameters Sc ) ν/D is the Schmidt
number and Re ) Uh/ν is the Reynolds number. ν is the
kinematic viscosity of the sample solution. For the problem at
hand, D is of the order of 10
-11
m
2
/s, and ν is of the order of 10
-6
m
2
/s; thus Sc is of the order of 10
5
and Re is of order one.
Therefore, the diffusion is negligible as compared to the convec-
tion, since Sc‚Re is large. This means that the sample species
convect with the flow. Consequently, the solution of the flow field
should give a fairly good idea regarding the concentration
distribution of the sample.
An efficient and accurate separation requires a rectangular
sample plug at the intersection of the channels. Straight stream-
lines (or particle path lines) at the intersection are desirable.
However, the streamlines at the intersection will generally be
�bent� (Figure 2). After the injection starts, the initially straight
streamlines will bend more and more at the intersection as the
flow progresses toward the steady state. The flow pattern at the
steady state will exaggerate the distortion of the sample shape at
the intersection and will give a �conservative� prediction regarding
the parameters most suitable for the desired rectangular sample
shape. Unsteady simulations are computationally costly since the
flow field is to be solved at each time step instead of just once as
in the steady case. Given the above argument, we choose to
perform only steady simulations. It should be noted that unsteady
simulations can provide information regarding the duration of
injection (discussed later). This information is not accessible
through the steady simulation.
During the injection, some part of the cross-channel will be
occupied by the buffer solution while the rest will be occupied by(13) Saville, D. A.; Palusinski, O. A. AIChE J. 1986, 32, 207-214.
Figure 1. Cross-channel device.
Figure 2. Bent streamlines at the intersection.
∂C/∂t + (V‚∇)C ) D∇
2
C (1)
∂C
∂t
+ (V‚∇)C )
1
ScRe
D∇
2
C (2)
Analytical Chemistry, Vol. 70, No. 9, May 1, 1998 1871