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Characterization of electroosmotic ﬂow in rectangular microchannels
Cheng Wang, Teck Neng Wong
*
, Chun Yang, Kim Tiow Ooi
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue 50, Singapore 639798, Singapore
Received 5 August 2005; received in revised form 18 July 2006Available online 22 February 2007
Abstract
In this paper, the electroosmotic displacing process between two solutions (namely the same electrolyte of diﬀerent concentrations) ina rectangular microchannel is studied theoretically and experimentally. Firstly, the electric potential and velocity ﬁeld in a rectangularmicrochannel are obtained by solving the governing equations. Fourier transform method is used to solve the electrolyte concentrationproﬁle equation. The electric current versus time curve through the microchannel is predicted based on the concentration proﬁleobtained. The current monitoring technique is then used to study the electroosmotic displacing process. The results from the measuredcurrent–time relations agree well with those from the prediction, suggesting a reliable theoretical model developed in this study.
2007 Elsevier Ltd. All rights reserved.
Keywords:
Theoretical model; Electroosmotic; Rectangular microchannel; Solution displacement
1. Introduction
The study in microﬂuidics is rapidly becoming a veryimportant area of research due to numerous potentialapplications in separation and analysis. Performing biolog-ical analysis on a microﬂuidics based bioMEMS or lab-on-a-chip usually involves sample preparation, treatment,injection, delivery, separation and detection. Most sub-stances acquire surface electric charges when in contactwith an aqueous (polar) medium. The rearrangement of the charges on the solid surface and the balancing chargesin the liquid results in the formation of the electrical doublelayer (EDL) [1]. If an electric ﬁeld is applied tangentiallyalong a charged surface, the electric ﬁeld will exert a bodyforce on the ions in the diﬀuse layer, resulting in electroos-motic ﬂow (EOF). EOF can provide a very ﬂat velocityproﬁle, thus avoiding smearing [2]. Besides, other advanta-ges of electroosmotic pumping such as valve-less switching,accurate control of transportation and manipulation of liquid sample by an electrical ﬁeld and with no solidmoving parts make electroosmosis a preferred method fortransporting liquids in microﬂuidics.Burgreen and Nakache [3] studied the eﬀect of the sur-face potential on liquid transport through ultraﬁne capil-lary slits assuming Debye–Hu¨ckel linear approximationfor the electrical potential distribution under an imposedelectrical ﬁeld. Rice and White [4] discussed the same prob-lem in narrow cylindrical capillaries. Levine et al. [5]extended the Rice and Whitehead’s model to cases involv-ing high zeta potentials. Yang and Li [6] studied the elec-trokinetic eﬀects of pressure-driven ﬂow in rectangularmicrochannels using a ﬁnite diﬀerence scheme. In Yang’sstudy, the motion of the liquid with electrokinetic eﬀectswas analytically solved by employing the Green functionformulation. Mala et al. [7] reported a study of microchan-nel ﬂow and heat transfer in two parallel plates. Tsao [8]studied the electroosmosis through an annulus withDebye–Hu¨ckel linear approximation and Kang et al. [9]studied the electroosmosis in annuli with high zeta poten-tials by introducing a correction to account for the ﬁnitethickness of the EDL and geometrically related factors.More recently, the alternating current (AC) electroosmosisin rectangular microchannels has been studied by Yanget al. [10], Erickson and Li [11], and Marcos et al. [12].
0017-9310/$ - see front matter
2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2006.11.035
*
Corresponding author. Tel.: +65 67905877; fax: +65 67911895.
E-mail address:
mtnwong@ntu.edu.sg (T.N. Wong).www.elsevier.com/locate/ijhmtInternational Journal of Heat and Mass Transfer 50 (2007) 3115–3121
In applications, electroosmotic pumps have been fabricatedby packing micron sized particles into fused silica capillar-ies [13] and by using standard micromachining techniques[14].In studying the electrokinetic phenomena, one of themost important characterization parameters is zeta poten-tial. In many practical situations, it is diﬃcult to obtain areliable estimate of zeta potential, because zeta potentialcannot be measured directly. Diﬀerent ways to measurethe value of zeta potential have been proposed, theseinclude a conventional microelectrophoresis method [15]and the micro particle image velocimetry (micro-PIV) tech-nique [16]. In the latter, Yan et al. [16] formulated an
expression to determine the electrophoretic velocity of thetracer particles and the EOF ﬁeld of the microchannelthrough the measurement of the steady velocity distribu-tions of the tracer particles in both open- and closed-endrectangular microchannels under the same water chemicalconditions. Another simple experimental method, termedthe current monitoring method, was ﬁrst suggested byHuang et al. [17] in which the electroosmotic ﬂow ratewas monitored using the electric current change, whenone solution is electrokinetically displaced by the samesolution with a slight diﬀerence in ionic concentration.Arulanandam and Li [18] employed this method to evalu-ate the zeta potential and the surface conductance. In theseexperiments, the average velocity of electroosmosis ﬂowwas determined by measuring the time required to com-pletely displace a solution by another similar solution inthe capillary tube. The zeta potential value was determinedby ﬁtting the measured average velocity to the theoreticalmodel developed in the study [18]. Ren et al. [19] improved
the current monitoring method by using the slope of themeasured correlation of the current versus time to achievea better accuracy. The electroosmotic displacing process ina circular capillary was later studied numerically by Renet al. [20] using a ﬁnite control volume method to solve amathematical model accounting for three zones of solution,namely solution 1 zone, mixing zone and solution 2 zone.Analytically, Ren et al. [21] developed mathematicalformulations for the displacing process in the cylindricalcapillaries based on two models: the sharp interface andthe mixing zone models.The above mentioned cases studied the electroosmoticﬂow displacing processes in cylindrical capillaries. How-ever, in practice, the microchannel networks in lab-on-a-chip platforms are usually fabricated by microelectronicfabrication techniques [22,23], and these channels are rect-angular rather than cylindrical in cross section. This paperpresents an in-depth analysis of electroosmotic displacingprocess of one solution with another in rectangular micro-channels. A mathematical model is developed to describethe electroosmotic ﬂow in a rectangular microchannel.The model includes solving the Poisson–Boltzmann’s equa-tion for EDL potential distribution and the modiﬁedNavier–Stokes equation for the electroosmotic ﬂow ﬁeldby the separation of variables method, and the mass trans-port equation using Fourier transform method. Based onthe mass transport equation, the average electroosmoticvelocity is determined by measuring the time needed totransport the electrolyte across the microchannel. The zetapotential is also obtained. The theoretical predictions forthe current–time relationship are compared with measuredresults. Good agreement is obtained.
2. Mathematical model
2.1. Electrical double layer potential distribution
A straight rectangular microchannel of width 2
W
,height 2
H
and length
L
is shown in Fig. 1. In this theoret-ical model, the channel wall is assumed to be uniformlycharged so that the electrical potential in the EDL variesin the
x
and
y
directions only. Due to the symmetry of the potential and velocity ﬁelds, the solution domain canbe reduced to a quarter cross section of the channel. It isfurther assumed that the electric charge density is uninﬂu-enced by the external electric ﬁeld due to thin EDLs; there-fore the charge convection can be ignored and thus theelectric ﬁeld equation and the ﬂuid ﬂow equation aredecoupled [24]. Introducing the dimensionless parameters:
X
¼
x D
h
;
Y
¼
y D
h
;
W
¼
z
0
ekT
w
and
D
h
¼
4
HW H
þ
W
, and assuming asmall zeta potential, the electric potential due to chargedwall can be described by the linearized Poisson–Boltz-mann’s equation which can be written in terms of dimen-sionless variables [12] as
r
2
W
¼
K
2
W
;
ð
1
Þ
where
K
¼
j
D
h
is the ratio of the length scale
D
h
tothe characteristic double layer thickness 1/
j
. Here
j
isthe Debye–Hu¨ckel parameter, given by
j
¼
e
r
e
0
k
b
T
2
z
20
e
2
n
0
1
=
2
;
where
z
0
is the valence,
e
is the fundamental electric charge,
w
is the electric potential,
e
r
is the relative permittivity, and
e
0
is the permittivity in vacuum.
Fig. 1. Schematic of the rectangular microchannel and the coordinatesystem used for modeling.3116
C. Wang et al./International Journal of Heat and Mass Transfer 50 (2007) 3115–3121
The boundary conditions along the symmetrical lines is
o
W
o
X
¼
0 at
X
= 0 and
o
W
o
Y
¼
0 at
Y
= 0. At the channel wall,
W
¼
f
at
X
¼
W D
h
and
Y
¼
H D
h
. Using the separation of vari-able method, the solution to the linearized Poisson–Boltz-mann equation gives
W
ð
X
;
Y
Þ¼
4
f
X
1
n
¼
1
ð
1
Þ
n
þ
1
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
n
1
Þ
2
p
2
D
2h
4
K
2
W
2
q
KY
ð
2
n
1
Þ
p
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
n
1
Þ
2
p
2
D
2h
4
K
2
W
2
q
KH D
h
cos
ð
2
n
1
Þ
D
h
p
2
W X
þ
4
f
X
1
m
¼
1
ð
1
Þ
m
þ
1
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
m
1
Þ
2
p
2
D
2h
4
K
2
H
2
q
KX
ð
2
m
1
Þ
p
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
m
1
Þ
2
p
2
D
2h
4
K
2
H
2
q
KW D
h
cos
ð
2
m
1
Þ
D
h
p
2
H Y
:
ð
2
Þ
The ionic net charge density in the EDL can be expressedby [25]
q
e
¼
2
W
:
ð
3
Þ
2.2. Electroosmotic ﬂow ﬁeld
The motion of an incompressible ﬂuid is governed bythe Navier–Stokes equation, which is expressed as
q
o
V
o
t
þ
q
ð
V
rÞ
V
¼r
p
þ
F
þ
l
r
2
V
;
ð
4
Þ
where
V
is the velocity vector,
p
is the pressure,
F
is theexternal force,
q
and
l
are the density and dynamics viscos-ity of the ﬂuid [26].Using the following assumptions(1) The ﬂuid is Newtonian;(2) the properties of the ﬂuid are independent of localelectric ﬁeld, thus only diluted solutions are consid-ered in this study;(3) the ﬂuid’s properties are temperature independent.Joule heating eﬀect could increase the temperature,but it can be negligible for diluted solution or underlow electric ﬁeld strength;(4) the ﬂow ﬁeld is steady, fully developed and obeyingno-slip conditions at the channel wall, and(5) there is no pressure gradient along the microchannel,and the two reservoirs are large enough to maintainthe same pressure level.Eq. (4) is thus reduced to
l
r
2
u
¼
q
e
E
;
ð
5
Þ
where
E
is the applied electric ﬁeld strength,
q
e
is the ionicnet charge density and
u
is the velocity [24].The boundary conditions are described by
u
j
x
¼
W
¼
0
;
u
j
y
¼
H
¼
0
;
o
u
o
x
x
¼
0
¼
0
;
o
u
o
y
y
¼
0
¼
0
:
ð
6
Þ
Introducing dimensionless parameters,
U
¼
uU
0
and usingthe separation of variables method, the analytical velocityﬁeld inside the quarter domain is obtained as [25,27]
U
ð
X
;
Y
Þ¼
4
E
e
r
e
0
kT
l
zeU
0
f
X
1
m
¼
1
ð
1
Þ
m
þ
1
cos
ð
2
m
1
Þ
D
h
p
2
H
Y
h i
ð
2
m
1
Þ
p
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
m
1
Þ
2
p
2
D
2h
4
K
2
H
2
q
KX
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
m
1
Þ
2
p
2
D
2h
4
K
2
H
2
q
KW D
h
cosh
ð
2
m
1
Þ
D
h
p
2
H
X
h i
cosh
ð
2
m
1
Þ
D
h
p
2
H
h i8>><>>:9>>=>>;
4
E
e
r
e
0
kT
l
zeU
0
f
X
1
n
¼
1
ð
1
Þ
n
þ
1
cos
ð
2
n
1
Þ
D
h
p
2
W
Y
h i
ð
2
n
1
Þ
p
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
n
1
Þ
2
p
2
D
2h
4
K
2
W
2
q
KY
cosh
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
þ
ð
2
n
1
Þ
2
p
2
D
2h
4
K
2
W
2
q
KH D
h
cosh
ð
2
n
1
Þ
D
h
p
2
W
Y
h i
cosh
ð
2
n
1
Þ
D
h
p
2
W
h i8>><>>:9>>=>>;
:
ð
7
Þ
The average electroosmotic velocity can be found as
U
¼
1
A
Z
U
ð
X
;
Y
Þ
d
A
;
ð
8
Þ
where the integration is over the quarter domain of themicrochannel.
2.3. Solution displacement in the microchannel
The solution displacing model studied is shown in Fig. 2.There are two reservoirs containing the same electrolytesolution but with two diﬀerent concentrations,
c
1
and
c
2
,where
c
1
and
c
2
are not signiﬁcantly diﬀerent, (e.g.,
c
1
= 70%
c
2
). A rectangular microchannel connects thetwo reservoirs. Initially, the connecting channel is ﬁlledwith solution of concentration
c
1
. Immediately after anelectric ﬁeld is applied along the channel, electroosmoticﬂow is generated. Gradually, the solution of higher concen-tration from the reservoir 2 displaces the solution of lowerconcentration towards the reservoir 1.
Fig. 2. Schematic of the solution displacement inside the microchannel.
C. Wang et al./International Journal of Heat and Mass Transfer 50 (2007) 3115–3121
3117
The solution displacement along
z
direction is governedby the mass transport equation [28]
o
c
o
t
þ
u
o
c
o
z
¼
D
o
2
c
o
z
2
;
ð
9
Þ
where
c
is the bulk concentration,
t
is the time,
u
is theaverage velocity and
D
is the diﬀusion coeﬃcient. By intro-ducing non-dimensional parameters:
Pe
¼
uD
h
D
,
t
¼
tD D
2h
,
Z
¼
z D
h
and
C
¼
c
c
1
c
2
c
1
, Eq. (9) in the dimensionless formcan be written as
o
C
o
t
þ
Pe
o
C
o
Z
¼
o
2
C
o
Z
2
;
ð
10
Þ
where
Pe
is Peclect number. Eq. (10) is subjected to the ini-tial and boundary conditions shown below:
C
j
t
¼
0
¼
0
;
C
j
z
¼
0
¼
1
;
where
L
¼
l D
h
is the dimensionless channel length.Introducing new variables
g
¼
Z
Pe
t
and
s
¼
t
, themass transport equation can be simpliﬁed to that shownin Eq. (11)
o
C
o
s
¼
o
C
2
o
g
2
:
ð
11
Þ
Using the Fourier transform theorem, the Fourier trans-form [29] of a function
f
(
x
) is
F
ð
a
Þ¼
1
ﬃﬃﬃﬃﬃﬃ
2
p
p
Z
11
f
ð
t
Þ
e
i
a
t
d
t
;
and the inverse Fourier transform of
F
(
a
) is
f
ð
x
Þ¼
1
ﬃﬃﬃﬃﬃﬃ
2
p
p
Z
11
F
ð
a
Þ
e
i
xt
d
a
:
Then, it can be shown that the solution to Eq. (11) can beexpressed as
C
ð
g
;
s
Þ¼
12
1
þ
erf
g
2
ﬃﬃ
s
p
h i
for
g
<
0
;
12
1
erf
g
2
ﬃﬃ
s
p
h i
for
g
>
0
:
8><>:
ð
12
Þ
In the
z
and
t
coordinates, Eq. (12) can be rewritten as
c
ð
z
;
t
Þ¼
12
ð
c
2
c
1
Þ
1
þ
erf
ut
z
2
ﬃﬃﬃﬃ
Dt
p
h i
þ
c
1
for
ut
>
z
;
12
ð
c
2
c
1
Þ
1
erf
z
ut
2
ﬃﬃﬃﬃ
Dt
p
h i
þ
c
1
for
ut
<
z
:
8><>:
ð
13
Þ
From the current-monitoring method, the total displace-ment time
t
max
can be obtained from measurementvia the recorded current–time relationship, and theoreti-cally it also satisﬁes
c
ð
l
;
t
max
Þ¼
c
2
. Together with Eq.(13), the average velocity can be determined. Finally, thecorresponding zeta potential can be found from Eqs. (7)and (8).
2.4. Current prediction
Once the concentration proﬁle of the solution in thechannel is known, the resistance of the solution and thecurrent
I
(
t
) through the channel can be found. The resis-tance of the solution is then given by Ohm’s Law
R
¼
l
k
A
total
, in which
R
is the resistance,
l
is the length,
A
total
is the total cross section area of the conductor (equals thechannel cross section area), and
k
is the conductivity of the electrolyte solution. Noted that
k
depends on the con-centration of the electrolyte solution during the solutiondisplacement, the total resistance of the electrolyte in thechannel can be written as
R
total
ð
t
Þ¼
Z
l
0
d
x
k
i
ð
c
Þ
A
total
:
ð
14
Þ
The current is given by
I
ð
t
Þ¼
El R
total
ð
t
Þ
:
ð
15
Þ
3. Experimental setup and discussion
In the measurement setup, the current monitoringmethod is used to study the characteristics of the electroos-motic ﬂow in a rectangular microchannel. The experimen-tal setup consists of a high voltage power supply (CZE1000R, Spellman, USA), a personal computer (PC), a dataacquisition system (BNC 2110 unit, National InstrumentsCorporation), ﬂow reservoirs (manufactured using Teﬂonmaterial), and polyimide-fused silica capillaries (PolymicroTechnologies Incorporated, USA).The microcapillaries with square cross section of 100
l
m
100
l
m and 75
l
m
75
l
m were cut to 5 cm inlength and used to connect reservoir 2 (higher concentra-tion
c
2
) and reservoir 1 (lower concentration
c
1
). Initially,the microcapillary was ﬁlled with NaCl (Sigma–Aldrich)electrolyte solution of a lower concentration
c
1
. Platinumelectrodes were inserted in both reservoirs with the ground-ing to reservoir 1 and high voltage power supply unit toreservoir 2. Measurements were conducted using electro-lyte solution NaCl of diﬀerent concentrations, 10
2
M,10
3
M, in microchannel of diﬀerent dimensions, 75
l
mand 100
l
m, under diﬀerent applied electric ﬁelds of 200 V/cm, 400 V/cm, 600 V/cm and 700 V/cm.The current–time curves from both measured and pre-dicted are shown in Fig. 3. The electric current increaseswith time after an electric ﬁeld is applied. An EOF isinduced, whereby the higher concentration solution dis-places the lower concentration one causing the variationin electrical current along the ﬂow direction. The currentreaches a constant and maximum value when the higherconcentration solution completely displaces the lower con-centration solution. The stable current level indicates thecompletion of the displacement process. The good agree-ment between the prediction and the measured values sug-gests the validation of the theoretical model.
3118
C. Wang et al./International Journal of Heat and Mass Transfer 50 (2007) 3115–3121

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