3341
Electrophoresis 2011, 32, 3341–3347
Chien C. Chang1,2
Chih-Yu Kuo1
Chang-Yi Wang1,3
1
Division of Mechanics, Research
Center for Applied Sciences,
Academia Sinica, Taipei,
Taiwan, ROC
2
Institute of Applied Mechanics
and Taida Institute of
Mathematical Sciences, National
Taiwan University, Taipei,
Taiwan, ROC
3
Department of Mathematics and
Department of Mechanical
Engineering, Michigan State
University, East Lansing, MI,
USA
Received March 21, 2011
Revised August 7, 2011
Accepted August 16, 2011
Research Article
Unsteady electroosmosis in a microchannel
with Poisson–Boltzmann charge
distribution
The present study is concerned with unsteady electroosmotic flow (EOF) in a microchannel with the electric charge distribution described by the Poisson–Boltzmann (PB)
equation. The nonlinear PB equation is solved by a systematic perturbation with respect
to the parameter l which measures the strength of the wall zeta potential relative to the
thermal potential. In the small l limits (l51), we recover the linearized PB equation –
the Debye–Hückel approximation. The solutions obtained by using only three terms in
the perturbation series are shown to be accurate with errors o1% for l up to 2. The
accurate solution to the PB equation is then used to solve the electrokinetic fluid
transport equation for two types of unsteady flow: transient flow driven by a suddenly
applied voltage and oscillatory flow driven by a time-harmonic voltage. The solution for
the transient flow has important implications on EOF as an effective means for transporting electrolytes in microchannels with various electrokinetic widths. On the other
hand, the solution for the oscillatory flow is shown to have important physical implications on EOF in mixing electrolytes in terms of the amplitude and phase of the resulting
time-harmonic EOF rate, which depends on the applied frequency and the electrokinetic
width of the microchannel as well as on the parameter l.
Keywords:
Electroosmosis / Oscillatory EOF / Poisson–Boltzmann equation / Transient EOF
DOI 10.1002/elps.201100181
1 Introduction
The Poisson–Boltzmann (PB) equation is a highly nonlinear
differential equation that describes the charge distribution
of ions in an aqueous solution due to a charged solid
surface. The nonlinearity of the equation is adequately
measured by the parameter l which expresses the strength
of the wall zeta potential relative to the thermal potential.
The PB equation is important in numerous fields, especially
in the electrostatic energy of molecules and electrokinetics
[1, 2]. Most of the literatures are concerned with surfaces
with low charge potentials (l51) in which the PB equation
can be reduced to the simpler linear Debye–Hückel
approximation (DHA). For higher charge potentials, the
PB equation should be used for accuracy. Another
important parameter is the electrokinetic width denoted by
K, which measures the channel width relative to the
Correspondence: Dr. Chien C. Chang, Division of Mechanics,
Research Center for Applied Sciences, Academia Sinica, Taipei
115, Taiwan, ROC
E-mail: mechang@iam.ntu.edu.tw
Fax: 1886-2-23625238
Abbreviations: DHA, Debye–Hückel approximation; EDL,
electric double layer; PB, Poisson–Boltzmann
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
thickness of the electric double layer (EDL) – Debye length.
Steady EOF, a fully developed flow by neglecting the
entrance effect, exhibits very much different efficiencies in
transporting electrolytes at different Ks. It has been shown
from the linear DHA [3, 4] that for large K we have a nearly
plug flow of which the dimensionless flow rate Q is
proportional to the cross-sectional area of the channel,
while for small K the dimensionless flow rate Q is quadratic
as K2 in the leading-order behavior. In other words, a very
fine partition of a microchannel would give a much smaller
total flow rate since a finer channel makes K smaller, and
the flow rate even smaller (scaled as K2). The same issue will
be examined in the present study for higher charge potentials.
In order to understand the structure of the solutions
with the physical parameters involved, it is very useful and
helpful if we may obtain solutions to the PB equation in
analytical forms. In the literature, there are very few analytic
solutions to the PB equation though the equation has been
solved numerically for a variety of problems. There is a
closed-form solution for the single infinite plate and some
complicated, infinite series solutions are available for the
potential outside a circular cylinder [5, 6]. The authors of [7]
suggested a patching of two regions, each described by an
approximation of the PB equation. Subsequently, their
Colour online: See the article online to view Figs. 1–8 in colour.
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Electrophoresis 2011, 32, 3341–3347
C. C. Chang et al.
method has been used by others [8–10]. Semi-empirical [11]
and numerical solutions [12–14] have also been done. In the
present study, a systematic perturbation in terms of l will be
used to obtain accurate solutions for the PB equation which
are uniformly valid for all K. Then the EOF due to an
unsteady applied axial electric field is determined for a
parallel-plate microchannel. The unsteady electric fields
considered include a suddenly applied voltage which causes
a transient flow, and a time-harmonic voltage which causes
an oscillatory flow.
The transient flow is important as the time for a starting
electroosmosis to reach the steady state or to become fully
developed is proportional to the square of the channel width
L with other physical parameters fixed (e.g. [15]). In other
words, a channel of width L 5 1 mm compared with a
channel of width L 5 10 nm takes 10 000 longer elapse time
for a steady state or fully developed EOF. It is therefore
appealing that one should make use of many smaller
channels instead of a large one with the same total area.
However, this involves the issue signified by the small
electrokinetic width that prevents us from using a much
finer partition of channel.
On the other hand, time-harmonic voltages are applied to
mix liquids. Actually, it has been shown [16] that that any
EOF-driven system can incorporate mixing, just by making
the voltage vary with time. Mixing is a crucial process in
many microfluidic applications which require that reagents
be input and samples diluted. EOF is a good choice for flow
control in microfluidic and nanofluidic systems since the
flow is manipulated by an electric field, instead of being by
mechanical moving parts (e.g. [17]). Rapid phase changes
controlled by the time-varying voltages would imply rapid
mixing of the electrolytes. It is therefore as importance to
understand the oscillatory EOF in terms of its amplitude and
phase, and their dependence on the relevant physical parameters. In an earlier study [18], the present authors studied
the oscillatory flow in a sector microchannel by the applied
time harmonic voltage based on the linear DHA. In this
respect, the present study extends our previous method of
analysis to the regime of fully nonlinear PB charge distribution for a wider range of applications. To the best knowledge
of the authors, this is the first analytical study of transient/
unsteady nonlinear PB equation.
2 The PB equation
The PB equation for a symmetric electrolyte is (e.g. [1])
zef0
ð1Þ
re ¼ eH02 f0 ¼ 2zen0 sinh
kb T
where re is the charge density, f0 is the electric potential, e
is the electric permittivity of the fluid, z is the valence, e is
the proton charge, n0 is the bulk electrolyte concentration, kb
is the Boltzmann constant, and T is the absolute temperature.
Assume that the wall zeta potential z is constant on the
boundary. Let f 5 f0 /z and normalize all the lengths,
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
including those in the Laplacian, by a length scale L which
characterizes the dimension of the domain. Equation (1) can
be written in the non-dimensional form
H2 f ¼
K2
sinhðlfÞ
l
ð2Þ
Notice that there are two independent non-dimensional
parameters. The parameter
zeB
l¼
ð3Þ
kb T
measures the strength of the zeta potential relative to the
thermal potential, representing the degree of nonlinearity of
the PB equation. The range of l-values is of practical
concern.
Take,
for
example,
T 5 288K
(151C),
kbT 5 0.0248 eV (kb 5 8.62 105 eV K1). The z-value of
25 mV (either positive or negative), can be taken as a useful
reference that separates low-charged surfaces from highly
charged surfaces. In CE applications, the wall zeta potential
z may vary in a wide range of values (say, from 0 to
775 mV, i.e. l 5 0–3z), which requires solution of the
nonlinear PB equation for the full range of applications. The
other parameter K, defined by
K2 ¼
L2 2z2 e2 n0 L2
¼
ekb T
l2D
ð4Þ
is called the non-dimensional electrokinetic width, as
mentioned above, indicating a relatively thin EDL if K is
large and a thick one if K is small. For example, the CE
applications usually have K at the orders of about 103–105
with the Debye lengths (lD) in the range of about 1–10 nm
in a microfluidic channel (with characteristic widths:
10–100 mm). However, a smaller K (with overlapped EDLs)
can be produced by using a smaller microchannel (i.e. L on
the nano-scales) or decreasing the bulk electrolyte concentration (low n0). The boundary condition for Eq. (2) is
f¼1
ð5Þ
on the boundary. Except for the single infinite plate, Eq. (2)
subject to Eq. (5) has no closed-form solution. For low
surface potentials (l-0), Eq. (2) is reduced to the linear
DHA
H2 f ¼ K 2 f
ð6Þ
upon which most published reports are based. The fully
nonlinear PB equation (2) is more difficult to solve. Here,
we briefly describe the method in [5] for large l or high
surface potentials. The method is to approximate the
hyperbolic sine in Eq. (2) by
(
ec =2 c41
sinh c ð7Þ
c
co1
where c 5 lf. Apply the approximation to the PB equation,
(
elf =2l lf41
H2 f sinhðlfÞ
D¼ 2 ¼
ð8Þ
K
l
f
lfo1
Figure 1 shows the exact value for D versus the
approximation for l 5 2 Notice the discontinuity and the
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Microfluidics and Miniaturization
Electrophoresis 2011, 32, 3341–3347
3343
Table 1. The values of D at f 5 1 for the series solution
lnN
1
2
5
1
2
3
4
1
1
1
1.167
1.667
5.167
1.175
1.800
10.38
1.813
13.48
5
14.55
6
Exact
14.80
1.175
1.813
14.84
N is the number of terms retained.
respect to l2 without loss of generality:
f ¼ f0 1l2 f1 1l4 f2 1 Figure 1. The function D versus f for l 5 2. The kinked curve
(broken line) is the approximation [7]. The two smooth curves,
almost indistinguishable, are from the exact form Eq. (8) and the
three-term series Eq. (9), respectively.
Successive orders give
f0 ¼
f1 ¼
large error involved near lf 5 1 (here f 5 0.5). Other values
of l41 show a similar behavior.
Here, we propose to use a systematic perturbation in l
to approximate D for use in solving the PB equation. An
asymptotic series of the hyperbolic sine gives
D ¼ f1
l2 f3 l4 f5
1
1
3!
5!
ð9Þ
A ratio test shows the series expansion is absolutely
convergent, and approaches the exact D value from below.
The three-term expansion equation (9) is also plotted in
Fig. 1 for l 5 2 It is almost indistinguishable from the exact
D values. The largest error is at f 5 1. For larger l more
terms in the series can be retained until the value of D at
f 5 1 is within a specified error bound. Table 1 shows the
convergence for various l values from which we see good
accuracy with the error o1% for lr2 by retaining only
three terms in the series.
It is also worthwhile to mention the work in [11] which
also provides an analytical approach to solving the PB
equation. It is empirical-curve fitting the sinh by a polynomial, followed by curve fitting the Debye–Hückel solution
by adjusting a parameter. In comparison, our perturbation
method is systematic and does not need any curve fitting,
and can be accurate to any order.
3 Starting electroosmotic flow in a
microchannel
Consider a long microchannel bounded by two parallel
plates at y0 5 7L (i.e. y 5 71). The fluid dissociates with the
charge density given by the PB equation. From Eqs. (8) and
(9), we have the electric potential governed by
d2 f
l2 f3 l4 f5
2
¼
K
f1
1
1
; fð1Þ ¼ 1
ð10Þ
3!
5!
dy2
Note that Eq. (10) is invariant when l is replaced by l,
i.e. is even in l. This suggests that we can expand f with
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ð11Þ
f2 ¼
coshðKyÞ
coshðKÞ
ð12Þ
1
192 cosh3 ðKÞ
8
9
coshð3KyÞ112Ky sinhðKyÞ
>
>
<
=
coshðKyÞ
>
>
: ½coshð3KÞ112K sinhðKÞ
;
coshðKÞ
ð13Þ
1
122880 cosh5 ðKÞ
9
8
240K 2 y2 coshðKyÞ16 coshð5KyÞ
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
coshð3KÞ
K
sinhðKÞ
>
>
>
>
>
>
10
31
112
>
>
>
>
coshðKÞ
coshðKÞ
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
coshð3KyÞ1240Ky
coshð2KyÞ
sinhðKyÞ
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
<
120K
½
2
coshðKÞ
coshð3KÞ
12K
sinhðKÞ
y
sinhðKyÞ
1
coshðKÞ
>
>
>
>
>
>
>
>
>
>
3
2
>
>
>
>
2
2
>
>
5
840K
115ð1140K
Þ
coshð2KÞ112
coshð4KÞ
>
>
>
>
>
>
7
6
>
> 14
>
>
5
>
>
>
>
>
>
>
>
>
>
12
coshð6KÞ
240K
sinhð2KÞ160K
sinhð4KÞ
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
coshðKyÞ
>
>
>
>
: ;
2
cosh ðKÞ
ð14Þ
Note that f0 of Eq. (12) is the solution of the DHA. As in
Eq. (9), the higher order corrections have smaller and
smaller coefficients, assuring the convergence of the series
even for moderately larger l values. Figure 2 shows a
comparison between the accurate numerical solution and
the two-term and three-term series solutions for l 5 2 as
well as the solution of the DHA. The accurate numerical
solution is obtained by solving Eq. (8) using the Runge–
Kutta shooting method. It is seen that the three-term series
solution compares well with the accurate numerical solution
with the maximum error o1%, occurring at the channel
center y 5 0 where the two-term series solution has the
maximum error about 2.5%. In contrast, the Debye–Hückel
solution deviates from the accurate numerical solution by
12% at y 5 0. It is also noted that the perturbation solution
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Electrophoresis 2011, 32, 3341–3347
C. C. Chang et al.
Figure 2. The electric potential f versus distance y for the PB
Equation and with l 5 2 and K 5 1. From top we have the
Debye–Hückel solution equation (12), the three-term series
solution Eq. (11) with Eqs. (12)–(14), the accurate numerical
solution as well as the two-term series solution.
Figure 3. The velocity profile u as a function of distance y (K 5 1,
l 5 2) for the starting EOF. From top we have the solutions:
steady state (t 5 N), and at the different times t 5 1, 0.5, 0.2, 0.1.
The solution obtained by using separation of variables is
1
X
2
u~ ¼
An cosðan yÞean t
ð21Þ
n¼1
equation (11) with (12)–(14) is valid for all K. The value of K
determines the structure or pattern of the solution, not the
validity/convergence of the perturbation. As a matter of fact,
the corrections l2j11l4j2 are very small for very large K by
noting that the denominators of j1 and j2 have the fast
growing functions cosh3(K) and cosh5(K) with increasing K.
If an axial unsteady electric field E(t0 ) is applied, the
electrolyte starts to move due to electroosmosis. The
governing equation is (e.g. [19])
r
@u0
¼ mH02 u0 1re Eðt0 Þ
@t0
ð15Þ
where u0 denotes the axial velocity, and r, m are the mass
density and dynamic viscosity of the electrolyte, respectively.
Consider the starting flow due to a suddenly applied
constant axial voltage with the electric field given by
(
0
t0 o0
0
Eðt Þ ¼
ð16Þ
E0 t0 0
The solutions we seek are u0 5 u0 (y0 ) for parallel flows,
the continuity equation is automatically satisfied. Normalize
the velocity by E0ejb/m, the time by rL2/m and drop primes.
Equation (15) becomes
@u @2 u d2 f
¼ 2 1 2 HðtÞ
@t
@y
dy
ð17Þ
where H(t) is the unit step function. Let
~ tÞ
uðy; tÞ ¼ uðyÞ
uðy;
ð18Þ
~ tÞ is the
where uðyÞ
is the steady-state solution and uðy;
transient which decays for large times. From Eq. (17) the
steady-state solution is
uðyÞ
¼ 1 fðyÞ
ð19Þ
The transient velocity satisfies
@u~ @2 u~
¼ 2;
@t
@y
~ 0Þ ¼ u;
uðy;
~
uð1;
tÞ ¼ 0
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ð20Þ
where an 5 (n0.5)p and the initial condition gives
1
X
An cosðan yÞ ¼ uðyÞ
ð22Þ
1
Inversion of Eq. (22) yields
Z 1
cosðan yÞ½1 fðyÞdy
An ¼ An ðKÞ ¼ 2
ð23Þ
0
The integral can be analytically integrated since the
series form of f consists of simple exponentials. A typical
sequence for K 5 1 and l 5 2 is An 5 0.4401, 0.0311,
0.0080, 0.0031, 0.0015, 0.0008, 0.0005, y. The corresponding velocity profiles are shown in Fig. 3 where the
flow velocity increases monotonically to its steady values as
time evolves. Notice the location of velocity maximum is
near the walls for tr0.1.
The EOF rate, normalized by E0ejbL2/m, is
Z 1
uðy; tÞdy
ð24Þ
Q ¼2
0
Figure 4 shows that the flow rate Q approaches its
steady state value within tr2. The flow rate increases with
increasing electrokinetic width K because a larger K indicates a thinner EDL and thus a wider region of bulk flow.
The flow rate Q also increases with increasing nonlinearity l
because a larger l with other physical conditions unchanged
would correspond to a larger zeta potential. In other words,
the EOF rate of the PB equation is always larger than the
flow rate of the DHA. Note also that the effect of nonlinearity on the relative increase of the flow rate is most
pronounced for KE1 or less. This is because in the present
study, the value of l is moderate (0olo2), and the
contribution of nonlinearity comes largely from l2j1
(cf. Eqs. 11 and 13). If K is large, the effect is limited to the
very thin EDLs; and only when K is of the order unity or
smaller (with EDLs overlapped), the effect of nonlinearity
extends over the entire channel.
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Electrophoresis 2011, 32, 3341–3347
3345
Figure 4. The flow rate Q versus time t for the starting EOF. The
values of (K, l) from bottom are (0.5, 0), (0.5, 1), (0.5, 1.5), (0.5, 2);
(1, 0), (1, 1), (1, 2); (2, 0), (2, 2), respectively.
4 Oscillatory EOF in a microchannel
We consider an oscillatory flow caused by a time-harmonic
applied voltage with its electric field given by
Eðt0 Þ ¼ E0 cosðOt0 Þ
ð25Þ
where E0 is the amplitude and O is the frequency. Equation
(15), normalized as before, becomes
@u @2 u d2 f
¼ 2 1 2 cosðstÞ
@t
@y
dy
ð26Þ
where s 5 rL2O/m is the non-dimensional frequency. It is
more convenient to solve (26) by casting it in the complex
form
2
@w @2 w d f ist
¼ 21 2e
@t
@y
dy
ð27Þ
Since f is nonlinear, we seek the solution by setting
u ¼ ReðwÞ with
"
w ¼ 1 fðyÞ 1
X
#
Bn cosðan yÞ eist
ð28Þ
n¼1
pffiffiffiffiffiffiffi
where i ¼ 1, and the real part of the product is implied.
Equation (27) then reduces to
1
X
ða2n 1isÞBn cosðan yÞ ¼ isð1 fÞ
ð29Þ
1
Substituting Eqs. (19) and (22) for 1f, we obtain
is
An ðKÞ
ð30Þ
Bn ¼ 2
an 1is
where the coefficients An 5 An(K) are found previously (cf.
23). The EO velocity is thus
u ¼ ½1 fðyÞ cosðstÞ
1
X
n¼1
cosðan yÞ
An s
½s cosðstÞ a2n sinðstÞ
ða4n 1s2 Þ
ð31Þ
where we note that u has the functional dependence on s, t, y
as well as on K where the effects of K come in through f(y)
and An 5 An(K)
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 5. (A) The velocity profiles with K 5 0.1, l 5 2 and s 5 1 for
the oscillatory EOF. From top we have the profiles at the four
different phases st 5 0, 0.5p, 1.5p, p of a time period. The velocity
profiles with K 5 0.1, l 5 2 and s 5 50 for the oscillatory EOF.
From top we have the profiles at four different phases st 5 0.5p,
0, p, 1.5p of a time period.
Figure 5A and B shows the velocity profiles at different
time instants for the narrower channel with small K 5 0.1.
Due to symmetry about the centerline, only half the channel
is shown. For the small s 5 1 the velocity is almost parabolic
and monotonic from the centerline to the wall at each time
instant. For the large s 5 50 the velocity maximum shifts
towards the walls, and at some instants there is almost no
flow in a middle region of the channel.
Figure 6A and B shows the velocity profiles at different
time instants for the wider channel with large K 5 10. The
interior velocity tends to flatten in the middle channel. But
the profiles are quite different from being plug-like as in the
Smoluchowsky limit (K-N). For the small s 5 1, we see
that at some instants the velocity has its maximum near to
the wall while at some instants the profile is monotonic
from the channel center to the wall. For the large s 5 50, the
interior is essentially at zero velocity and the oscillatory flow
occurs in the EDL. The larger the amplitude of oscillation is,
the closer the velocity maximum is shifting toward the wall.
The instantaneous EOF rate is obtained by integrating
(31) over the cross-section
Z 1
Q¼2
ð1 fÞdy cosðstÞ
0
2
1
X
sinðan ÞAn s
1
an ða4n 1s2 Þ
½s cosðstÞ a2n sinðstÞ ¼ q cosðst bÞ
ð32Þ
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C. C. Chang et al.
Electrophoresis 2011, 32, 3341–3347
Figure 7. The amplitude q of the oscillatory EOF versus the
externally applied frequency s with K 5 1, 10 and l 5 0, 2. At an
even larger K (say, 100 or 1000), the amplitudes q for l 5 0,2 are
very close to each other in values and indistinguishable in the
plots.
Figure 6. (A) The velocity profiles with K 5 10, l 5 2 and s 5 1 for
the oscillatory EOF at different time instants. From top we have
the profiles at four phases st 5 0, 0.5p, 1.5p, p of a time period. At
an even larger K (say, 100 or 1000), the behaviors are similar
except near the channel wall (y 5 1) where the velocity profile
changes more abruptly. (B) The velocity profiles with K 5 10,
l 5 2 and s 5 50 for the oscillatory EOF. From top near y 5 1 we
have the profiles at four different phases st 5 0, 0.5p, 1.5p, p of a
time period. At an even larger K (say, 100 or 1000), the oscillatory
behaviors of the velocity are similar but occur closer to the
channel wall (y 5 1).
where q is the flow amplitude, and b is the phase lag with
respect to the applied electric field. Both q and b depend on
the non-dimensional electrokinetic width K. It is noted that
q is the amount of fluid that can be manipulated by the
applied time-harmonic voltage, and efficient control of b
implies the ability of mixing the liquids by the voltage.
In order to examine the effects of the applied voltage, we
plot q and b versus the applied frequency s for three cases
K 5 1 and 10, and compare the differences between the
nonlinear l 5 2 and linear l 5 0. The results are shown in
Figs. 7 and 8, respectively. In all the cases, the amount q that
can be manipulated is shown to be decreased by increasing
the frequency s for both PB and DHA charge distributions.
This is because of the nature of the EDL, and thus the entire
flow over the channel. Notice that the steady-state flow rate
is always larger than the transient flow rate. If s is large, the
flow has little time to reach the stage of full development
(steady-state), hence the amplitude q of the flow rate
becomes smaller with increasing applied frequency.
The q value for the PB charge distribution (l40) is
always larger than that for the DHA charge distribution
(l 5 0), but the difference becomes smaller with increasing
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 8. The phase lag b of the oscillatory EOF versus the
externally applied frequency s with K 5 1, 10 and l 5 0,2. At an
even larger K (say, 100 or 1000), the phase lags b for l 5 0, 2 are
very close to each other in values and indistinguishable in the
plots.
the applied frequency s. This also depends on the value of K:
the relative increase in q by the nonlinearity is most
significant for small K. For example, with the zero applied
frequency s 5 0, the amplitude q of l 5 2 is 1.53 times that
of l 5 0 at K 5 0.5, 1.21 times at K 5 1, and nearly 1 at
K 5 10. The phase lag b always rises from 0 for s 5 0 to p/2
for large s, but is little sensitive to l no matter if we consider
PB or DHA charge distribution. The value of b 5 p/2, totally
out of phase, occurs at large values of s, indicating
substantial damping of the flow with higher applied
frequencies. The results for the phase of the EOF rate show
that b changes drastically in the range of so10 for K 5 1;
and in the range of so5 for K 5 10. This has the implication
that it is easier and sufficient for mixing electrolytes by
varying b in a smaller range of applied frequencies for a
microchannel with larger electrokinetic widths K.
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Electrophoresis 2011, 32, 3341–3347
5 Concluding remarks
The present work provides an analytical study of transient/
unsteady EOF with using the nonlinear PB equation. The
nonlinear PB equation for the electrostatic charge distribution was solved by a systematic perturbation based on a
series expansion of the hyperbolic sine function. The
solution for the PB equation was then substituted in the
electrokinetic transport equation to solve for the unsteady
EOF rates of the electrolyte by the method of eigenfucntion
expansion.
The nonlinearity of the PB equation is measured by the
parameter l. In the small l limits (l51), we recover the
linear DHA. The two-term perturbation solution was illustrated to have relative errors within 2.5% compared with the
accurate numerical solution of the PB equation for l 5 2,
while the three-term solution has relative errors o1%. The
two types of unsteady EOF considered are the flow driven by a
suddenly applied voltage and the flow driven by a timeharmonic voltage, respectively; the former results in a transient EOF, while the latter results in an oscillatory flow in
time. Some interesting physical behaviors were observed and
the salient features can be summarized as follows.
For the transient flow, the time scales to reach steady
states are the same for the solutions of the nonlinear PB
equation and the linear DHA because the time scales are
determined by the transient velocity Eq. (20) not by the PB
equation (2). Besides l, another important physical parameter is the dimensionless electrokinetic width K. It was
shown that the higher zeta potentials (PB Eq) always
increases the flow rate compared with the lower zeta
potentials (DHA), but the increase is most pronounced for K
near 1 or less. From the velocity profiles at different times, it
is also clearly seen that the flow is initially developed from
the EDL driven by the applied voltage. The maximum of the
velocity occurs initially in the EDL, and then moves gradually to the center of the microchannel.
For the oscillatory flow, the effects of the applied voltage
are many-fold. The dimensionless EOF rate Q can be written
in the time-harmonic form Q 5 qcos(stb), where s is the
normalized frequency of the applied electric field and b is
the phase lag. The amount q that can be manipulated is
decreased by increasing the frequency s for both PB and
DHA charge distributions. The q value for the PB charge
distribution (l40) is always larger than that for the DHA
charge distribution (l 5 0), but the difference becomes
smaller with increasing the applied frequency s. The flow
amplitude also depends on the value of K: the increase in q
by the nonlinearity is most significant for small K, while the
increase is relatively small for large K in the present range of
0olo2. On the other hand, the ability of mixing liquids by
electroosmosis depends on the efficient control of the phase
lag b. The phase lag b always rises from 0 at s 5 0 to p/2 at
s 5 N, but is much less sensitive to l (no matter whether it
is PB or DHA). The results show that b changes drastically
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Microfluidics and Miniaturization
3347
in the range of so10 for K 5 1; and in the range of so5 for
K 5 10. This has the implication that it is easier and sufficient for mixing electrolytes by varying b in a smaller range
of applied frequencies for channels with larger electrokinetic
widths K. If a large s is applied to the case with a large K, the
maximum of the velocity occurs within the EDL with very
little flow in the electrically neutral middle of the microchannel.
The work was supported in part by the National Science
Council of the Republic of China (Taiwan) under the Contract
nos. NSC99-2628-M-002-003 and NSC100-2221-E-002-152MY3.
The authors have declared no conflict of interest.
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