INTRODUCTION
Microfluidic devices have found widespread application in clinical practice and throughout the food and chemical industries. The microscale flow channels incorporated in such devices increase the surfacetovolume ratio of the fluid streams and are therefore advantageous in many applications, including DNA restriction, multiple sample injection, sample extraction, controlled fraction mixing and so forth. However, the fluid flow within the microchannel is constrained to very low Reynolds numbers (typically <10) and thus homogenization of the species solutions takes place through diffusion processes alone. Consequently, obtaining a satisfactory mixing performance requires the use of extended mixing channels and longer mixing times, which diminish the practical benefits of such devices.
The diffusion of two sample streams can be improved by increasing the interfacial
contact area between them. Previous researchers have proposed various passive
microfluidic devices which exploit this phenomenon to create an enhanced mixing
effect. Feeding the different samples through discrete via holes (Miyake et
al., 1993), utilizing cantilever plate valves (Voldman et al., 2000),
configuring the device with multiple channels (Yang et al., 2005) and
so forth. The feasibility of achieving species mixing via an alternate stretching
and folding of the sample flows has also been demonstrated. Branerbjerg et
al. (1996) reported an effective mixing performance within mixing lengths
as short as 100300 ms. Mengeaud et al. (2002) presented a series of
Yform micromixers with various zigzag mixing channel configurations. Luo et
al. (2005) and Chen et al. (2006) investigated the elestroosmotic
flow driven by DC or AC electric fields in a curved microchannel. It is found
some circulations induced in the secondary flows can be used to enhance the
microfluidic mixing efficiency. Passive micromixers utilizing surface hetereogeneities
to stir the mixing species have also been demonstrated. For example, Ajdari
(1995) investigated the complex elestroosmotic flows induced by the application
of nonuniform, timeindependent and timedependent potentials along the conduit
walls. Similarly, Qian and Bau (2002) performed a theoretical investigation
into twodimensional, timeindependent and timedependent elestroosmotic flows
driven by a uniform electric field in a conduit with a nonuniform potential
distribution. The results showed that timewise periodic alternations of the
potentials induced a chaotic advection effect within the fluid. Erickson and
Li (2002) performed threedimensional numerical simulations to study the effects
of surface electrokinetic heterogeneities on the elestroosmotic flow induced
by a uniform electric field and then exploited these effects to enhance the
mixing efficiency of a Tshaped micromixer. Biddiss et al. (2004) employed
an experimental visualization technique to investigate the effects of surface
charge patterning on species mixing and presented an optimized electrokinetic
micromixer applicable to species mixing in the low Reynolds number regime. Chang
and Yang (2004) used a numerical method to simulate the mixing mechanisms of
elestroosmotic flow induced by a DC electric field in a microchannel with oppositely
charged surface heterogeneities. The results showed that local flow circulation
regions promoted species mixing in the microchannel. Luo (2006) and Luo et
al. (2007a) demonstrated the mixing of twodimensional, timedependent elestroosmotic
flows via the application of an AC electric field to a microchannel patterned
with surface heterogeneities. The results showed that the interaction of the
AC electric field with the patchwise surface heterogeneities created vortex
structures within the fluid streams, which yielded a significant reduction in
both the mixing channel length and the retention time required to attain a homogeneous
solution. Luo et al. (2008) utilized electrical field intensity perturbations
to stir the electrokinetic instability in order to enhance the microfluidic
mixing in a Ttype microchannel. Compared to the passive and active schemes
presented in the literature, such EKIbased schemes have the advantages of a
simpler microchannel design, a more straightforward fabrication process and
a rudimentary voltage control scheme.
In addition to the passive mixing schemes, described above, researchers have also proposed many active mixing schemes, in which species mixing is achieved via the application of an external force to the fluid flow. For example, Rife et al. (2000) demonstrated the use of embedded PZT ultrasonic transducers to generate acoustic waves to stir the samples within the mixing channel. Meanwhile, Lu et al. (2002) proposed a sophisticated micromixer incorporating micromagnetic stirrers fabricated using surface micromachining techniques. Lee et al. (2001) developed a microfluidic device featuring embedded microelectrodes to induce the dielectrophoretic stretching and folding of sample fluids. Oddy et al. (2001) demonstrated the feasibility of using electrokinetic instability effects to achieve the rapid mixing of micro and nanoliter volume solutions for bioanalytical applications. Luo et al. (2007b) presented a Tform micromixer which utilized a rectified AC electric field and a DC electric field to induced elestroosmotic flows for sample proportional mixing.
Electrokinetic flow has emerged as the method of choice for transporting species in microfluidic devices designed to carry out immunoassays, DNA hybridization and general cellmolecule interaction applications. In general, such applications require the rapid and efficient mixing of two or more species. Accordingly, the present study proposes a Tshaped micromixer in which an enhanced mixing performance is obtained by applying AC and DC electric fields at the upper and lower inlets, respectively, while grounding the outlet. The characteristics of the proposed micromixer are investigated by performing a series of simulations based upon the backwardsEuler timestepping method. The simulations focus specifically on the respective effects on the mixing performance of the difference in intensity of the AC and DC electric fields, the frequency of the AC electric field and the width of the mixing channel. In general, the results confirm the feasibility of the proposed approach in accomplishing the proportional mixing of microfluidic flows within a finite time and mixing distance.
FORMULATION
The Tshaped microchannel considered in the present study (Fig.
1) has a nominal width and height of 60 mm and is filled with an incompressible
Newtonian electrolyte of uniform dielectric constant, ε and viscosity,
μ. Since the characteristic height of the microchannel is in the order
of magnitude of 10 mm, the interaction between the fluid and the walls is significant
and must be taken into account in the theoretical model. A review of the literature
suggests that a theoretical model comprising the PoissonBoltzmann equation,
the Laplace equation and the NavierStokes equation with body force terms from
the GuoyChapman model provides a reasonable description of the elestroosmotic
flow in the current microchannel. Furthermore, the distributions of the electrical
double layer potential and the applied electric field can be described using
the PoissonBoltzmann equation and the Laplace equation, respectively.

Fig. 1: 
Schematic illustration of Tshaped microchannel showing principal
dimensions and fluid flows 
Distributions of electric field and electrical double layer potential: The nondimensional quantities (denoted by an asterisk) are defined as following:
The distribution of the applied electric field is governed by the Laplace equation,
i.e.,
According to electrostatics theory, the distribution of the electric potential
in the EDL is governed by the PoissonBoltzmann equation (Hunter, 1986). The
dimensionless nonlinear PoissonBoltzmann distribution equation can then be
expressed as:
where, ψ is the nondimensional EDL potential, κ = A x K is the nondimensional
EDL thickness, K = (2n_{0}z_{2}e_{2}/εε_{0}k_{b}T)^{1/2}
is the DebyeHuckel parameter, where ε is the dielectric constant of the
fluid, z is the valence, e is the charge carried by an electron, n_{o}
is the bulk electrolyte concentration, k_{b} is the Boltzmann constant
and T is the temperature and A is the height of the microchannel (Note that
the asterisks are omitted here for convenience) (Table 1).
Electroosmotic flow field: When an external electric field is applied
to the microchannel, the resulting liquid flow induced by elestroosmosis is
governed by the momentum equation (Dutta and Beskok, 2001; Yang and Li, 1998),
i.e.,
Assuming that the gravity effect is very small and can be neglected, the only
force (F) acting on the fluid is that produced by the interaction beween the
applied electric field and the free ions within the EDL. This body force induces
a bulk fluid motion generally referred to as elestroosmotic flow. The following
nondimensional quantities (denoted by asterisks) can be introduced: nondimensional
velocity u* = u/(υ/A) or v* = v/(υ/A), nondimensional time t* = t/(A^{2}/υ),
nondimensional pressure p* = (pp_{ref})/(ρυ^{2}/A^{2})
and nondimensional angular velocity Λ* = Λ/(υ/A^{2}),
where υ is the kinetic viscosity of the electrolyte, t is time and Λ
is the angular velocity of the applied electric field, i.e., Λ = 2πf.
Hence, the momentum equation given in Eq. 3 can be rewritten
as:
where, Gx = 2n_{0}k_{b}T/(ρV^{2}/A^{2})
and φ is the applied electric potential. The Reynolds number is defined
as uA/υ. Hence, according to the definition of the nondimensional velocity
quantity, the Reynolds number in the current Tshaped microchannel has a value
of one, as indicated by Eq. 4b and c.
Concentration field: The species mixing of elestroosmotic flows can
be described by the following dimensionless concentration equation:
where, C is the nondimensional concentration of the species and Pe = U_{ref}A/D_{i},
where D_{i} is the diffusion coefficient of the species. The mixing
efficiency at any point in the mixing channel can be evaluated by:
where, C is the nondimensional species concentration profile across the width of the mixing channel and C_{0} and C_{ref} are the solution concentrations in the completely unmixed and completely mixed states, respectively.
Boundary conditions: In the current simulations, the microchannel surfaces
are assumed to be homogeneous and to have a zeta potential of 75 mV, equivalent
to a dimensionless value of ψ = 2.92. The boundary conditions at the walls,
inlets and outlet of the microchannel are given as follows:
• 
At the walls, with noslip conditions: 
Note that n denotes the normal unit vector to the microchannel walls.
• 
At the inlets: 

∂ψ/∂y = 0, φ = φ_{m}sin Λt at the upper
inlet, φ = Const. at the lower inlet. 

u = 0, ∂ψ/∂y = 0, p = 0, C = 0 at the upper inlet, C = 1 at
the lower inlet. 
Note that φ_{m} denotes the maximum electric potential generated
by the AC electric power supply.
Full details of the physical models about the elestroosmotic flow are reported in Luo et al. (2006, 2007) and Luo (2006).
Numerical method: The numerical solution procedure performed in this
study employs the backwardsEuler timestepping method to identify the evolution
of the flow when driven by the AC electric field. The applied electric potential,
φ, can be computed from the Laplace equation given in Eq.
1, while the zeta potential distribution in the EDL can be obtained from
Eq. 2. The transient elestroosmotic flow under the applied
electric field can then be simulated by substituting the electric potential
into Eq. 4b and c and solving the simplified
equation set given in Eq. 4ac. The computational
domain is discretized into 351x701 nonequally spaced grid points in the X
and Ydirections. The calculated solutions were proven to be independent of
both the computational grid points and the time step. Full details of the iteration
algorithm about the backwardsEuler timestepping method are reported in Yang
and Luo (2002) and Luo (2004a, b).
RESULTS AND DISCUSSION
As shown in Fig. 1, this study considers a Tshaped microchannel
in which electrodes are installed at the two inlets and the outlet is connected
to ground. During the mixing operation, a DC electric field is applied to the
lower inlet, while an AC electric field is applied to the upper inlet. The intensity
of the DC electric field is assigned a sufficiently high value to ensure that
the fluid entering the microchannel through the lower inlet has sufficient momentum
to drive the fluid stream entering through the upper inlet into the mixing channel.
Furthermore, halfwave rectification is applied to the AC electric field at
the upper inlet to prevent the injected fluid from returning to the reservoir.
In practice, the relative proportion of the two fluid streams entering the mixing
channel (and hence the final solution concentration obtained in the fullymixed
state) is easily controlled by maintaining a constant AC electric field intensity
at the upper inlet while regulating the value of the DC electric field intensity
applied at the lower inlet.

Fig. 2: 
Evolutions of electric field and flow field, respectively,
under application of AC electric field of intensity 165 V/cm and frequency
4 Hz at upper inlet and constant DC electric field intensity of 105 V/cm
at lower inlet 

Fig. 3: 
Evolution of concentration under application of AC electric
field of amplitude 165 V/cm and frequency 4 Hz applied at upper inlet and
constant DC electric field intensity of 105 V/cm applied at lower inlet 
As shown in Fig. 2a, when the amplitude of the AC electric
field intensity is greater than that of the DC electric field, most of the fluid
in the main mixing channel originates from the upper inlet. Conversely, when
the amplitude of the AC electric field intensity is less than that of the DC
electric field, the fluid in the mixing channel originates primarily from the
lower inlet, as shown in Fig. 2b and c.
Overall, the results presented in Fig. 2 indicate that the
relative proportion of the two flow streams entering the mixing channel is governed
by the relative intensities of the AC and DC electric fields.
Figure 3 presents the temporal evolution of the concentration
field within the microchannel over the course of a single period of the AC electric
field. Note that the AC electric field has an amplitude of 165 V/cm and a frequency
of 4 Hz, while the DC electric field has an amplitude of 105 V/cm.

Fig. 4: 
Variation of mixing efficiency along main channel under application
of AC electric field of amplitude 165 V/cm and frequency 2, 4, 6, 12 Hz
at upper inlet and constant DC electric field intensity of 105 V/cm at lower
inlet when t = 5 sec 
As shown, the applied electric fields cause the two species with different
concentrations (i.e., zero in the upper inlet and one in the lower inlet) to
converge in the junction region of the microchannel. Due to the difference in
intensity of the AC and DC electric fields, a momentum imbalance arises at the
interface of the two fluid flows, resulting in the formation of a periodic waveform,
which propagates in the downstream direction at the velocity of the elestroosmotic
flow. The wavelike interface increases the contact area of the two species
and therefore enhances the convective mixing effect between them. Figure
4 shows the corresponding variations of the mixing efficiency along the
length of the main mixing channel with different frequency of the applied AC
electric field. At low frequency (2 Hz), the amount of the samples to be injected
into the mixing channel during a period is greater than those at high frequency
and the injected samples from a series of patches staggered along the main mixing
channel. Therefore, the distribution of the corresponding mixing efficiency
along the mixing channel appears with high amplitude. At high frequency (6 and
12 Hz), the samples can be effectively mixed at the entrance of the mixing channel
through the induced high frequency waveform flow. However, the amplitudes of
these induced high frequency waveform flows are much smaller and these high
frequency waveform flows can not be propagated far in the downstream. Therefore,
the effective mixing efficiency can not be sustained in the downstream of the
mixing channel. From inspection, it is found that for the case with frequency
of 4 Hz, a mixing efficiency as high as 92% can be achieved within a mixing
distance of 0.6 mm.

Fig. 5: 
Variation of mixing efficiency along main channel under application
of AC electric field of amplitude 124, 165, 206, 247 V/cm and frequency
4 Hz at upper inlet and constant DC electric field intensity of 105 V/cm
at lower inlet when t = 4.6 sec 
The performance of the 4 Hz case is the best among the cases. For the case
with high amplitude of the applied AC electric field intensity (206 and 247
V/cm), the amplitude of the induced waveform flow is also greater and the distribution
of the corresponding mixing efficiency also appears with a wavy form with greater
amplitude (Fig. 5). For the case with low amplitude of the
applied AC electric field intensity (124 V/cm), the amplitude of the induced
waveform flow becomes smaller. The induced waveform flows can not be propagated
far in the downstream and the effective mixing efficiency can not be sustained
in the downstream of the mixing channel. In the Fig. 5, it
can be found the mixing performance of the case with the amplitude of 165 V/cm
is the best among the cases. For the case with the application of the AC electric
field with the amplitude of 165 V/cm and the frequency of 4 Hz, a mixing efficiency
as high as 90% can be achieved at t = 2.6 sec and the fullymixed cab be achieved
at t = 4.6 sec. However, for the case with the application of the AC electric
field with the amplitude of 124 V/cm and the frequency of 4 Hz, the driving
electric force is smaller (Fig. 6). A time of 3.65 sec is
required to achieve a mixing efficiency as high as 90%. Therefore, the above
results confirm the effectiveness of the combined AC/DC potential mode in enhancing
the microfluidic mixing efficiency within the Tshaped microchannel and the
mixing performance of the case with the applied AC electric field with an intensity
of 165 V/cm and a frequency of 4 Hz is the best among the cases.
The wavy pattern at the interface not only can enhance the mixing efficiency
of the two streams, but the position of the interface at the junction of the
Ttype microchannel can determine the relative amount of each fluid stream which
enters the mixing channel.

Fig. 6: 
Variation of mixing efficiency with time along the cross section
at the outlet under application of AC electric field of amplitude 124, 165
and frequency 4 Hz at upper inlet and constant DC electric field intensity
of 105 V/cm at lower inlet 
Therefore, the position of the interface governs the solution concentration
in the fullymixed state. In practice, the interface position is easily controlled
by assigning a constant AC electric field intensity at the upper inlet and regulating
the intensity of the DC electric field applied at the lower inlet. Note that
the mean intensity of the AC electric field is 105 V/cm. When the DC electric
field has an intensity of 132 V/cm, a significant difference exists in the intensities
of the two electric fields (Fig. 7). The momentum of the stream
introduced through the lower inlet (C = 1) is much greater than that of the
stream introduced from the upper inlet (C = 0) and thus the interface between
the two streams is located within the upper inlet channel. Consequently, the
main mixing channel is filled almost entirely with fluid from the lower inlet
and hence the fullymixed dimensionless concentration has a value of 0.99. When
the DC electric field intensity is reduced to 115 V/cm, the difference in the
momentum of the two streams is reduced and thus the interface between the fluid
streams is located just beneath the egress of the upper inlet into the junction
region of the microchannel. Under these conditions, the bulk of the fluid entering
the main mixing channel still originates from the lower inlet, but a greater
amount of fluid from the upper inlet also enters the mixing channel.
As a result, the fullymixed dimensionless concentration at the outlet reduces
to 0.744. When the intensity of the DC electric field is further reduced to
105 V/cm, the momentums of the two streams are approximately equal and hence
the fluid interface is located near the center of the junction region. As a
result, the two streams flow into the microchannel in approximately equal proportions
and thus the fullymixed dimensionless concentration is 0.49. Finally, when
the DC electric field intensity is reduced to 91 V/cm, the momentum of the stream
entering from the upper inlet is greater than that of the stream entering from
the lower inlet and thus the interface between them is located in the lower
inlet. As a result, the mixing channel is filled almost entirely with fluid
from the upper inlet and hence the fullymixed dimensionless concentration has
a value of 0.09 (Fig. 7ad).
Note that in interpreting the xaxis, a negative (positive) value indicates
that the applied DC field potential is lower (greater) than the mean value of
the AC electric field potential (Fig. 8). As shown in Fig.
7, for a given AC electric field intensity, the value of the dimensionless
concentration at the outlet of the microchannel is governed by the intensity
of the DC electric field. Figure 8 shows that when the DC
electric potential exceeds the mean value of the AC electric potential by 3
V or more, the mixing channel is filled with fluid from the lower inlet and
the fullymixed dimensionless concentration has a value close to one. However,
when the DC electric potential is 2 V or more lower than the mean AC electric
potential, the fluid from the upper inlet fills the main mixing channel and
the value of the fullymixed dimensionless concentration is close to zero. When
the difference in the electric potentials applied at the two inlet channels
falls within these two threshold values, i.e., 2 V~3 V, the fullymixed concentration
varies approximately linearly with the potential difference. Overall, the results
presented in Fig. 8 confirm the ability of the combined AC/DC
potential mode proposed in this study to achieve the proportional mixing of
microfluidic elestroosmotic flows.
As described earlier, the AC electric field applied at the upper inlet of the
current Tshaped microchannel undergoes a process of halfwave rectification
to prevent a backflow of the injected fluid into the reservoir (Fig.
9). For a sinusoidal electric field with an intensity of 165 V/cm and a
frequency of 2 Hz, the amplitude remains at 165 V/cm following the rectification
process, but the frequency increases to 4 Hz. The mean value of the rectified
sinusoidal AC electric potential over the course of a single period is given
by:
where, ω = 2πf, in which f is the frequency, T is the period and
Amp is the amplitude. From inspection, it is apparent that the mean value of
the AC electric field intensity is independent of the frequency.

Fig. 7: 
Dimensionless concentration in fullymixed condition under
application of AC electric field of intensity 165 V/cm and frequency 4 Hz
at upper inlet and constant DC electric field intensities of 132, 115, 105
and 91 V/cm at lower inlet. (a) DC voltage 132 V/cm, the dimensionless concentration
at the outlet is 0.99. (b) DC voltage 115 V/cm, the dimensionless concentration
at the outlet is 0.744. (c) DC voltage 105 V/cm, the dimensionless concentration
at the outlet is 0.5 and (d) DC voltage 91 V/cm, the dimensionless concentration
at the outlet is 0.09 
Since the relative proportion of the two species in the main mixing channel
is determined by the difference between the DC and mean AC electric field intensities,
the dimensionless concentration in the fullymixed state is therefore also independent
of the AC electric field frequency, as shown in Fig. 9.

Fig. 8: 
Variation of dimensionless concentration in fullymixed state
with difference between DC and AC electric field voltages 

Fig. 9: 
Variation of fluid concentration in fullymixed state with
difference between DC and AC electric field intensities as function of AC
frequency 
Note that the maximum channel width is limited to 150 μm since this value
represents the effective physical limit beyond which elestroosmotic flow can
not be induced. Note also that in deriving the results presented in the Fig.
10, the mixing channel was extended as required to ensure a fullymixed
condition as the channel width was progressively increased. From inspection,
it is seen that the linear range bounded by the two threshold values of the
DC electric field intensity expands slightly as the width of the microchannel
is increased.

Fig. 10: 
Variation of fluid concentration in fullymixed state with
difference between DC and AC electric field intensities as function of mixing
channel width 
However, the rate of change of the threshold values is significantly lower
than that of the microchannel width, indicating that the threshold values are
relatively insensitive to the microchannel width in the current range.
CONCLUSION
This study has proposed a Tshaped micromixer in which a halfwave rectified AC electric field is applied at the upper inlet, a DC electric field is applied at the lower inlet and the outlet is grounded. The combined effects of the DC and AC electric fields prompt the formation of a periodic waveform at the interface of the two fluid streams which increases the interfacial contact area and therefore improves the mixing efficiency. The numerical results have revealed that for a given AC electric field intensity, the position of the fluid interface within the intersection region of the microchannel can be regulated by adjusting the value of the DC electric potential. In this way, the relative amounts of the two fluid streams entering the mixing channel can be accurately controlled such that a proportional microfluidic mixing capability is obtained. For a constant AC electric potential, two threshold values of the DC electric field intensity have been found, corresponding to fullymixed dimensionless concentrations of zero (upper inlet fluid only) and one (lower inlet fluid only), respectively. The results have shown that at DC electric field intensities between these two threshold values, the dimensionless concentration varies only as a function of the DC electric potential and is independent of the frequency of the AC electric field. Moreover, it has been shown that increasing the microchannel width has little effect on the fullymixed values of the dimensionless concentration provided that the mixing length is adequately extended and results in no more than a minor expansion in the linear range bounded by the two threshold values.
ACKNOWLEDGMENTS
The current authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under Grant No. NSC 952221E167028 and NSC 952622E167010CC3.