Summary This paper presents a theoretical investigation into chaotic mixing in low Reynolds number electro-osmotic flows. In this mixing system, the primary flow is the pluglike electro-osmotic flow contributed by the permanent surface charge at the wall, and the secondary electro-osmotic flows (or electro-osmotic recirculating rolls) contributed by the field-effect-induced surface charge act as the perturbed flow. By time-periodic switching two different secondary electro-osmotic flows, it makes streamlines to cross at successive intervals and results in chaotic mixing. Dynamic system techniques such as Poincaré map and finite-time Lyapunov exponent analyses are employed to describe the behaviors of particle motion in this mixing system. Finally, the optimum operating condition (e.g. amplitude and time-switching period) for the mixing is identified

Chih-Chang Chang

Ruey-Jen Yang

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Copyright © 2009 ICCES ICCES, vol.10, no.2, pp.77-86
Chaotic Mixing in a Microfluidic Device
Chih-Chang Chang, Ruey-Jen Yang1
Summary
This paper presents a theoretical investigation into chaotic mixing in low Reynolds
number electro-osmotic flows. In this mixing system, the primary flow is the plug-
like electro-osmotic flow contributed by the permanent surface charge at the wall,
and the secondary electro-osmotic flows (or electro-osmotic recirculating rolls)
contributed by the field-effect-induced surface charge act as the perturbed flow. By
time-periodic switching two different secondary electro-osmotic flows, it makes
streamlines to cross at successive intervals and results in chaotic mixing. Dynamic
system techniques such as Poincaré map and finite-time Lyapunov exponent analy-
ses are employed to describe the behaviors of particle motion in this mixing system.
Finally, the optimum operating condition (e.g. amplitude and time-switching pe-
riod) for the mixing is identified.
Introduction
Recently, micro-scale devices have been proposed as a means of construct-
ing micro-total analysis systems (μ-TAS) or Lab-on-a-Chip (LOC) devices. These
devices have received much attention in a variety of chemical and biological ap-
plications [1]. Mixing of different reactants is an important process for most bio-
chemical applications such as polymer chain reaction (PCR) and DNA hybridiza-
tion. In convectional laboratories, mixing is usually performed by either manually
shaking the test tube or mixing using a stirrer. Unfortunately, these convectional
mixing methods are difficult to implement in miniaturized systems. Therefore, de-
veloping novel methods for rapid and efficient mixing is crucial for the development
of μ-TAS.
In large-scale devices, turbulent enhancement has been an effective method to
achieve efficient mixing. However, it is much more difficult to induce turbulence
in micro-scale devices due to the viscous forces dominate the flow at low Reynolds
number. Laminar mixing through molecular diffusion usually is too slow for most
practical applications. Therefore, the development of micro-mixing approaches
for microfluidic applications has attracted the attention of many research groups
worldwide, and a quite large number of micro-mixers based on different strategies
have been proposed [2]. In particular, mixing by chaotic advection is considered
to be one of the most promising approaches for rapid and efficient mixing in lami-
nar flows [3]. In chaotic advection, fluid elements always are stretched and folded
repeatedly. It provides an effective increase in the interfacial contact area and con-
1Department of Engineering Science, National Cheng Kung University E-mail:
rjyang@mail.ncku.edu.tw
Chaotic Mixing in a Microfluidic Device 78
centration gradient due to the reduction of striation thickness (i.e. diffusion length).
In this way, the mixing time and length can be considerably reduced.
In this study, we propose a novel technique to create chaotic mixing in low
Reynolds number electro-osmotic flows. Electro-osmotic flow refers to the move-
ment of bulk liquid relative to a stationary charged surface (e.g. microchannel) un-
der an externally applied electric field. Electro-osmotic flow has became a popular
means to drive flow in microfluidic devices in recent years [4], which do not involve
mechanical moving parts and are readily integrated within microfluidic devices us-
ing standard micro-fabrication methods. In the mixing system (see Figure 1), the
plug-like electro-osmotic flow is a throughput flow contributed by the permanent
zeta potential of the microchannel, and time-wise periodic alternating secondary
electro-osmotic (SEO) flows (or electro-osmotic recirculating rolls) contributed by
the field-effect-induced zeta potential act as the perturbed flow. Chaotic mixing
occurs when appropriate operating conditions (e.g. amplitude of perturbation and
frequency) are applied. Several techniques such as Poincaré map, finite-time Lya-
punov exponent, and back-trace imaging analyses are used to examine the existence
of chaos and evaluate the mixing performance of this mixing system.
el
w2
xE
≈
≈
x
y
Figure 1: Schematic illustration of electro-osmotic flow mixing in a microchannel
with spatiotemporal zeta potential modulation using the field-effect control.
Mathematical model
Governing equation
We consider mixing in a microchannel with high aspect ratio (i.e., channel
height is much larger than channel width) in our work. The two-dimensional flow
is assumed. Furthermore, the Reynolds number usually is much smaller than unity
in electro-osmotic flows, and the flow can be modeled with Stokes equation. Due
to the time-switching period of two different SEO flows is much larger than three
time scales: (1) the charge relaxation time of electrical double layer (EDL) ca-
pacitor, (2) time required for fully charging the capacitor of the wall (field-effect),
and (3) viscous diffusion time of fluid in this study, the time-dependent flow can
be regarded as quasi-steady Stokes flow [5]. With the thin double layer assump-
tion, the Helmholtz-Smoluchowski slip boundary condition is considered as the
electro-osmotic effect. Consequently, the flow can be described by the following
Chaotic Mixing in a Microfluidic Device 79
bi-harmonic equation:
∇4Ψ = 0 (1)
where Ψ is the dimensionless stream function. The dimensionless velocity field
can be represented by {U,V} = {∂Ψ/∂Y ,−∂Ψ/∂X}. Equation (1) can be easily
solved by separation of variables and series expansion. The secondary electro-
osmotic flow fields corresponding to three different spatial field-effect-induced zeta
potential distributions are shown in Figure 2, respectively.
xE
 
 
xE
 
 
xE
 
 
 
Figure 2: Secondary electro-osmotic flow patterns resulting from three different
spatial field-effect-induced zeta potential distributions.
Kinematic model for the mixing system
The equation for the motion of advected passive particle in Largangian repre-
sentation is given below:
dX
dτ
= UA (X ,Y)gA (τ)+UB or C (X ,Y )gB or C (τ) (2)
where
gA (τ) =
{
1, (k−1)T < τ < (2k−1)T/2
0, (2k−1)T/2 < τ < kT
gB or C (τ) =
{
0, (k−1)T < τ < (2k−1)T/2
1, (2k−1)T/2 < τ < kT
Chaotic Mixing in a Microfluidic Device 80
X is the location of the advected passive particle, the dimensionless time τ =
tuo/w, the dimensionless time period T = T ∗uo/w, and the number of time peri-
ods k = 1,2,3..... Note that UA, UB, and UC represent the total velocity fields of
plug-like electro-osmotic flow and secondary electro-osmotic flow for three differ-
ent spatial zeta potential modulations, respectively. Two different temporal zeta
potential modulations are considered to explore the chaotic advection in electro-
osmotic flows. One is the time-wise alternation of flow patterns A and B (namely
Design I mixing system), and another is the alternation of flow patterns A and C
(namely Design II mixing system).
Results
Poincaré map analysis
The Poincaré maps for Design I and Design II are shown in Figures 3 and 4,
respectively. The lines represent the particles move in periodic manners. These
lines are termed as Kolmogorov-Arnold-Mose (KAM) curves or boundaries [3].
The white region is isolated due to the presence of KAM curve. It could be an
ordered area, or it could contain both ordered and chaotic areas. Any trajectory of a
point initiated in one attractor (periodic area or chaotic area) cannot cross over the
KAM curves to another attractor. In other words, the KAM curves act as bound-
aries that prevent the mixing of material elements between the two separated areas
without considering the molecular diffusion effect. In Design I (Figure 3), it can
be seen that KAM boundaries are significant at any time period (T ), which implies
poor mixing when molecular diffusion effect is insignificant. From Poincaré maps
analysis, the better time periods for mixing in Design I are T = 3.5and T = 4.0due
to the smaller white region (unmixed region). From Figure 4, it can be seen clearly
that KAM boundaries or unmixed regions are not significant compared to Figure 3
expect the cases of T = 1.0 and T = 2.0. When T = 3.5, T = 4.0, T = 4.5, T = 5.0,
and T = 8.0, the results indicate that the particles move in more chaotic behaviors.
The system seems nearly fully chaotic under these operating conditions. This also
implies the better choice of time period for mixing among them, but an optimal
time period for the mixing has undecided yet.
Finite-time Laypunov exponent analysis
Lyapunov exponent refers to the average exponential rates of divergence or
convergence of nearby orbits in the phase space. In other words, Lyapunov expo-
nent can describe the stretching rates of material elements in a quantitative manner.
The Lyapunov exponent is expressed as Equation (3), which is given by
λ (τ) = lim
|dX0|→0
1
τ
ln
[ |dX(τ)|
|dX0|
]
(3)
where |dX0| and |dX(τ)| are the distance between two particles at initial condi-
Chaotic Mixing in a Microfluidic Device 81
(a) 
 
(b) 
 
(c) 
(d) 
 
(e) 
 
(f) 
(g) 
 
(h) 
 
(i) 
Figure 3: Poincaré maps for the different time periods when Ap = 2.0andLe = 2.0in
Design I (a)T = 1.0, (b)T = 2.5, (c)T = 3.0, (d)T = 3.5, (eT = 4.0, (f)T = 4.5,
(g)T = 5.0, (h)T = 7.0, and (i)T = 8.0. Note Apis the amplitude of perturbed flow
defined as the Smoluchowski velocity ratio of secondary EO to plug-like EO.
(a) 
 
(b) 
 
(c) 
(d) 
 
(e) 
 
(f) 
(g) 
 
(h) 
 
(i) 
Figure 4: Poincaré maps for the different time periods when Ap = 2.0andLe = 2.0in
Design II (a)T = 1.0, (b)T = 2.0, (c)T = 3.0, (d)T = 3.5, (eT = 4.0, (f)T = 4.5,
(g)T = 5.0, (h)T = 7.0, and (i)T = 8.0.
tion and time τ , respectively. This study adopts Sprott’s method to calculate the
value of the Lyapunov exponent (i.e. largest Lyapunov exponent). When τ → ∞,
Lyapunov exponent converges to a value termed as infinite-time Lyapunov expo-
nent. The calculation of infinite-time LE usually is a time-consuming work. In a
fully chaotic system, infinite-time LE is not dependent on the initial positions of
particles. Due to the mixing system may not be fully chaotic (see Figures 3 and 4),
the choice of initial positions to calculate infinite-time LEs of different attractors is
Chaotic Mixing in a Microfluidic Device 82
difficult. In addition, infinite-time LE is not a practical mixing index because the
complete mixing within a short mixing time is usually required in practical applica-
tions. Finite-time LE may be a more practical mixing index than infinite-time LE.
Unlike infinite-time LE, Finite-time LE is dependent on the initial position of par-
ticle and time. The domain of one period unit of mixing channel was divided into
uniform grids (ΔX = 0.025 and ΔY = 0.025), which are used as initial points to cal-
culate FTLEs. The mean FTLE of τ = 100 as a function of time period at different
amplitudes for Designs I and II are shown in Figure 5. The results indicate that the
mean FTLE increases significantly as the amplitude increases in Design II. It also
reveals the mixing performance in Design II is better than Design I due to larger
FTLE implies larger stretching rate of material element. The better operating time
period in Design II seems is between 3.0 and 5.0 for any amplitude. T = 4.0seems
to be the optimal time period at the amplitude Ap = 2.0in Design II.
(a) 
T
M
ea
n
FT
LE
0 2 4 6 80
0.05
0.1
0.15
0.2
0.25
0.3
0.5
1.0
1.25
1.5
1.75
2.0
Ap
 
(b) 
T
M
ea
n
FT
LE
0 2 4 6 80
0.05
0.1
0.15
0.2
0.25
0.3
0.5
1.0
1.25
1.5
1.75
2.0
Ap
Figure 5: Mean FTLE (λ̄ ) in terms of time period (T ) at different amplitudes of
perturbation source (Ap) for different mixing systems. (a) Design I and (b) Design
II.
The mean FTLE may not be an appropriate mixing index to evaluate the mixing
performance because the system may consist of low and high FTLE regions. The
uniformity of FTLE distribution should also be considered to evaluate the mixing
performance, i.e. the statistic of FTLE distribution is necessary. Figures 6 and 7
show the FTLE distribution within one period unit of mixing channel at different
amplitudes in Design I and Design II, respectively. The results show the white re-
gions shown in Figures 3 and 4 are approximately zero FTLE, i.e. zero stretching
rate. The statistical value: mean value, standard deviation (σ ), coefficient of vari-
ance (CV) for FTLE distribution are also shown in Figures 6 and 7. The value of
CV is used as a mixing index to evaluate the mixing performance. The smaller
CV implies the better mixing. The optimal time period for mixing is T = 3.5at
Ap = 2.0.
Chaotic Mixing in a Microfluidic Device 83
(a) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.019
0.031 161.7%
 
(b) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.077
0.031 72.6%
 
(c) 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.098
0.062 63.4%
(d) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.079
0.044 55.5%
 
(e) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.063
0.039 61.9%
 
(f) 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.050
0.041 81.9%
(g) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.059
0.051 87.4%
 
(h) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.052
0.039 76.3%
 
(i) 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.051
0.039 75.1%
Figure 6: Contour plots of FTLE distribution and probability density functions
(PDF) of FTLE for the different time periods when Ap = 2.0andLe = 2.0in Design
I (a)T = 1.0, (b)T = 2.5, (c)T = 3.0, (d)T = 3.5, (eT = 4.0, (f)T = 4.5, (g)T =
5.0, (h)T = 7.0, and (i)T = 8.0. Note that σand CV are standard deviation and
coefficient of variance (σ
/
λ̄ ×100%), respectively.
Visualization of mixing
We adopt back-trace imaging method to visualize mixing. At τ ≈ 40, the mix-
ing images for Designs I and II at different time-switching periods are shown in
Figures 8 and 9, respectively. It can be seen obviously that the mixing in Design II
is better than Design I. It is consistent with the result from Poincaré map and FTLE
Chaotic Mixing in a Microfluidic Device 84
(a) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.11
0.073 67.9%
 
(b) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.16
0.073 44.8%
 
(c) 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.19
0.068 35.9%
 
(d) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.2
0.051 25.3%
 
(e) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.21
0.055 26.1%
 
(f) 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.2
0.052 26.2%
 
(g) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.18
0.057 32.3%
 
(h) 
 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.14
0.055 39.1%
 
(i) 
FTLE
PD
F
0 0.1 0.2 0.310
-4
10-3
10-2
10-1
100
λ =
σ CV= =
0.14
0.049 35.4%
 
Figure 7: Contour plots of FTLE distribution and probability density functions
(PDF) of FTLE for the different time periods when Ap = 2.0andLe = 2.0in Design
II (a)T = 1.0, (b)T = 2.0, (c)T = 3.0, (d)T = 3.5, (eT = 4.0, (f)T = 4.5, (g)T = 5.0,
(h)T = 7.0, and (i)T = 8.0.
analyses. The poor mixing regions shown in Figure 9 are consistent with the low
FTLE value regions shown in Figure 7. The best time-switching period for mixing
is T = 3.5.
Conclusion
The Poincaré map and finite-time Lyapunov exponent analyses both have been
Chaotic Mixing in a Microfluidic Device 85
(a) 
 
(b) 
 
(c) 
 
(d) (e) (f) 
Figure 8: Mixing images for different time periods when Ap = 2.0andLe = 2.0in
Design I. (a)T = 3.0, (b)T = 3.5, (c)T = 4.0, (d)T = 4.5, (e)T = 5.0, (f)T = 8.0.
(a) 
 
(b) 
 
(c) 
(d) (e) (f) 
Figure 9: Mixing images for different time periods when Ap = 2.0andLe = 2.0in
Design II. (a)T = 3.0, (b)T = 3.5, (c)T = 4.0, (d)T = 4.5, (e)T = 5.0, (f)T = 8.0.
employed to investigate the chaotic mixing. The coefficient of variance for FTLE
distribution was used as a mixing index to evaluate the mixing performance. T =
3.5was identified to be the optimal time period for mixing at the amplitude 2.0 in
both Design I and Design II. The mixing performance of Design II is better than
that of Design I.
Acknowledgement
The work was supported in part by the National Science Council of Taiwan
under Grant No. NSC-96-2628-E-006-162-MY3.
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Chaotic Mixing in a Microfluidic Device

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