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Abstract

Alternating current electrothermal flow has been extensively used for mixing of fluids in microfluidic systems. This work presents a novel approach to mixing two laminar flow streams in microchannels. The micromixer consists of a pair of asymmetric planar alternating current electrodes disposed along a fluidic channel and a thin film resistive heater below the electrode. Multiphysics (fluid flow, temperature, electrical, and concentration fields) in the micromixer is studied by numerical simulation method. The results indicate that external heating by the thin film resister leads to flow body forces that are proportional to the temperature gradient and chaotic stretching and folding of fluid material lines. Enhancement of mixing is demonstrated by numerical simulation results. Effects of size and location of the thin film resistive heater on mixing efficiency are also investigated.

Keywords

Microfluidics mixing alternating current electrothermal flow resistive heater multiphysics

Introduction

Chip-scale micromixing is essential in many microfluidic devices, which requires rapid mixing of reactants in biochemical processes.¹ However, mixing of fluid streams in microfluidic devices is limited by low value of the mass diffusion coefficient, as well as the low Reynolds number laminar flow and the absence of turbulence to assist mixing in microchannels. To achieve more efficient mixing in a short time and flow length, micromixers based on different actuation methods have been developed, including static,² ultrasonic,³ magnetic,⁴ and electrokinetic^5–8 micromixers. Among these, electrokinetic micromixers have drawn particular interests due to the ease of operation and improved mixing efficiency.

In general, electrokinetic phenomena involve the motion of liquids or suspended particles under a direct current (DC) or alternating current (AC) electric field. AC electrokinetics has been an attractive approach to flow generation and particle manipulation for several reasons. The most important reason is that it uses low AC voltages (1–30 V_pp), which reduces Faradaic currents and the related generation of bubbles. This is in contrast to DC electrokinetic flows which require hundreds to thousands of volts for actuation. AC electrokinetics can be classified as AC electroosmosis, dielectrophoresis, and alternating current electrothermal (ACET) flow;⁹ among which ACET flow has been found particularly useful in the application of fluid mixing in microchannels because ACET flow is the only AC electrokinetic method found viable for actuation of biofluids with high electric conductivities (above 0.1 S/m).¹⁰ In ACET flow, joule heating from the asymmetric AC electrodes induces temperature gradients which produce permittivity and conductivity gradients of the fluid, and net electrothermal body forces are created which result in fluid motion. A number of ACET micromixers^11–14 have been proposed for mixing of fluids in microchannels. The ACET mixer in Sin et al.’s¹¹ work contained annular electrodes, which was capable of generating three-dimensional (3D) vortices, and experimental results demonstrated that ACET mixing can be used to improve the DNA hybridization process and reduce the assay time. Cao et al.¹² proposed an electrothermal vortex enhanced micromixer with electrodes at the two opposite side walls, which utilized ACET flow to enhance mixing efficiency. Such vortex flows have also been shown important in the design and analysis of ultraprecision aerostatic bearings.¹³ Ng et al.¹⁴ proposed an ACET micromixer driven by DC-biased AC voltage from a pair of parallel rectangular electrodes along the channel length, which was efficient and suitable for mixing reagents in diverse microfluidic applications.

Recently, Yuan and Wu¹⁰ showed improvement in ACET pumping effects with imposition of external thermal gradients by copper foils which are connected to AC electrodes with hot and cold plates. Wu et al.¹⁵ improved the efficiency of ACET pumping by placing an electrode heater on top of the microchannel. Williams and colleagues^16,17 used thin film resistive heaters to enhance electrothermal pumping in microfluidic systems, and their work showed that the heater location and electrode spacing can influence the flow rate of pumping.¹⁷

In this work, we present a novel micromixer based on ACET effects with a thin film resistive heater. The thin film resistive heater is embedded in the glass substrate, separated from the electrodes by an electric insulation. Numerical simulation is performed to demonstrate the benefits of the film resistive heater design to promote mixing of fluids in microchannels, and methodology for the design and analysis of such microfluidic systems is provided.

Description of the micromixer

The sketch of the ACET micromixer with thin film resistive heaters is shown in Figure 1. It consists of a pair of parallel rectangular AC electrodes along the channel length (analogous to the design in Ng et al.¹⁴) and a thin film resistive heater, which is embedded in the glass substrate, below one of the AC electrodes with a glass insulation in between. As shown in Figure 1(a), fluid A and fluid B are co-injected into the microchannel at the inlet. Asymmetric AC voltage signals are supplied to the two AC electrodes, and the thin film heater can be independently controlled to create a heat source, as well as a nonuniform temperature field in the microchannel.

Figure 1.

Sketch of the micromixer: (a) 3D sketch and (b) cross section.

Numerical model

In our multiphysics numerical model, fluid flow in the micromixer is determined by the body force $f_{E}$ generated by ACET effects and can be described by the steady-state incompressible Navier–Stokes equations

\nabla \cdot v = 0

(1)

ρ (v \cdot \nabla) v = - \nabla p + μ \nabla^{2} v + f_{E}

(2)

where $v$ is the fluid velocity, $ρ$ is the fluid density, p is the pressure, and $μ$ is the dynamic viscosity of the fluid. Other body forces on the fluid, such as the buoyancy force from uneven temperature distribution, can be neglected when compared with the ACET force.¹⁸ AC electroosmosis flow is also ignored when conductivity of the fluid is large enough (≥0.08 S/m),¹⁹ which is the case for many biofluids. No slip boundary condition is prescribed on the channel walls, and a uniform velocity is assumed at the inlet. A zero pressure boundary condition is also imposed on the outlet.

In ACET flows, temperature gradients cause change in the solution conductivity and dielectric constant gradients, resulting in changes in electric field forces exerted on the fluid. The relationships between the dielectric constant and conductivity and temperature are written as²⁰

α = (\frac{1}{ε}) (\frac{\nabla ε}{\nabla T}) = - 0.4 % K^{- 1}

(3)

β = (\frac{1}{σ}) (\frac{\nabla σ}{\nabla T}) = 2 % K^{- 1}

(4)

where $ε$ is the dielectric constant and $σ$ is the electric conductivity. The ACET body force is related to the electric conductivity and dielectric constant gradients and given by

\begin{array}{l} f_{E} = - 0.5 [(\frac{\nabla σ}{σ} - \frac{\nabla ε}{ε}) E \frac{ε E}{1 + (ω τ)} + 0.5 {| E |}^{2} \nabla ε] \\ = 0.5 \cdot ε \cdot \nabla T \cdot E^{2} Π (ω) \end{array}

(5)

Π (ω) = \frac{α - β}{1 + (ω τ)} - \frac{α}{2}

(6)

where $E$ is the electric field intensity vector, $ω$ is the angular frequency of AC potentials, $Π$ is a dimensionless function of frequency, and $τ$ is the charge relaxation time ( $ε / σ$ ). The electric field is governed by the Laplace equation

\nabla^{2} V = 0

(7)

E = - \nabla V

(8)

where V is the electric potential.

For equation (8), insulation boundary conditions are prescribed on channel walls, and an asymmetric pair of AC voltages, $V_{1} = - V_{rms} / 2$ and $V_{2} = V_{rms} / 2$ , is specified on the two rectangular electrodes, where $V_{rms}$ is the root mean square of the AC voltage.

The temperature simulation space includes the complete chip (fluid in the channel, glass substrate, glass coverslip, and insulation), and the temperature field is governed by

k \nabla^{2} T + σ E^{2} = 0

(9)

where T is the temperature and k the thermal conductivity, and convective effects have been ignored.²⁰ Constant temperature (room temperature) boundary conditions are prescribed on the outermost horizontal boundaries.¹⁶ Thermal insulation boundary conditions are imposed on the side walls of the simulation space. A thin film heater is embedded in the glass substrate, separated from the AC electrodes with a distance of 1 µm. A fixed value of heat flux is prescribed on the surface of the heater to model its heating effect.¹⁶ Fluid temperature at the inlet is assumed to be a constant (room temperature), and an outflow boundary condition of zero normal temperature gradients is imposed at the outlet which is far enough from the AC electrodes.

The transport equation of concentration in the microchannel is written as

v \cdot \nabla C = D \nabla^{2} C

(10)

where C is the concentration and D is the mass diffusion coefficient. At the inlet, concentration of fluid A is set as 1, and concentration of fluid B is set as 0. The outlet boundary condition of concentration is $\vec{n} \cdot (- D \nabla C) = 0$ , where $\vec{n}$ is the outward unit normal vector.

The above coupled equations are solved numerically using the finite element software COMSOL Multiphysics. First, equations (1)–(9) are solved simultaneously to obtain the temperature distribution, electric field, and flow field in the microchannel. Then, the flow velocity is substituted into equation (10) to solve for the concentration field. The computational mesh is shown in Figure 2, which consists of 99,215 tetrahedral elements. Grid dependence tests have been performed until successive refinement of the gird results in insignificant changes of the numerical solution. Relative tolerance has been specified to be 0.01 as the convergence criterion.

Figure 2.

Computational mesh.

Results and discussions

The micromixer studied in this work consists of a Y-shaped microchannel with a pair of coplanar rectangular electrodes at the bottom of the channel, similar to the electrothermal micromixer in Ng et al.¹⁴ The microchannel that we study is 350 µm long, 20 µm high, and 40 µm wide, which is on a glass substrate (1.0 mm thick) and capped with a glass coverslip (80 µm thick). The two AC electrodes are 120 µm long and 15 µm wide and separated by a distance of 10 µm. Asymmetric AC potentials of 5 V(rms) at a frequency of 10⁶ Hz are applied on the two AC electrodes. The dielectric constant and conductivity of the fluid are $7.1 \times 10^{- 10} F / m$ , and 0.1 S/m, respectively. Density and viscosity of the fluid are $997 kg / m^{3}$ and $10^{- 3} N s / m^{2}$ , respectively. The thermal conductivity and diffusion coefficient of the fluid are $0.6 J / m s K$ and $1 0^{- 11} m^{2} / s$ , respectively. The inlet temperature of the fluid is assumed to be 298 K, and the inlet fluid velocity is 200 µm/s. Inspired by Williams and colleagues’^16,17 electrothermal microfluidic pumps using thin film resistive heaters, a rectangular thin film resistive heater with the same size as the AC electrode (120 µm long and 15 µm wide) is embedded in the glass substrate (thermal conductivity of glass is $1.0 J / m s K$ ) 1 µm below one of the AC electrodes. A heat flux of $2.5 \times 1 0^{5} W / m^{2}$ is prescribed on the surface of the thin film resistive heater, which is feasible because a heat flux has also been used as the boundary condition of the heater in Yuan and Wu.¹⁰ We use the above choice of geometric and physical parameters as an example of this novel micromixer, and it should be noted that this specific choice of geometric and physical parameters will not affect the methodology of design and analysis of this micromixer.

The electric field and temperature field

Distributions of the electric potential and log₁₀E in the channel are plotted in Figures 3 and 4, respectively. Temperature distribution on the same cross section above the AC electrodes in the microchannel with and without the resistive heater is compared in Figure 5, where the color represents the magnitude of temperature and arrows indicate both the magnitudes and directions of temperature gradients. As can be seen from Figure 5, film resistive heating plays a dominant role in the resulting temperature gradient distribution. As shown in Figure 5(a), when there is no external resistive heating, distributions of both temperature and its gradients are symmetric, with the maximum temperature appearing at the bottom of the channel between the AC electrodes due to the strongest electric field (Figure 4) between the asymmetric AC electric electrodes. In contrast, when external resistive heating is present (Figure 5(b)), the maximum temperature appears at the left edge of the right AC electrode, which is a result of the asymmetric heating by the resistive heater. It is also seen in Figure 5(b) that temperature gradient distribution becomes nonsymmetric and significantly increased in the transverse direction (normal to the initial fluid interface) along the vertical center line, which, according to equation (5), is proportional to the fluid body force, and it is this transverse temperature gradient that plays an important role in ACET stretching and folding of fluid interfaces.

Figure 3.

Distribution of electric potential (V).

Figure 4.

Distribution of log₁₀E.

Figure 5.

Distribution of temperature (K) on the channel cross section (a) without the heater and (b) with the heater.

The flow field

Figure 6 compares the velocity distribution on a cross section (viewed from upstream) above the AC electrodes, with and without the resistive heater, where the color represents velocity magnitude and the arrows indicate both its magnitude and direction. It can be seen from Figure 6 that flow velocities have been increased significantly by external heating, especially in the transverse direction along the vertical center line of the channel. This can be explained by the increased temperature gradient (Figure 5(b)) and the relation between temperature and fluid body force (equation (5)). The maximum velocity (107 µm/s) in Figure 6(b) is also increased by about 919% by external heating, compared with that (10.5 µm/s) in Figure 6(a). The transverse motion of the fluid results in stretching and folding of material lines, and mixing is enhanced by exponentially increased material contact area and molecular diffusion across the interfaces.

Figure 6.

Velocity (m/s) distribution on the cross section (a) without the heater and (b) with the heater.

The concentration field

Figure 7 plots concentration distribution on different cross sections with and without the resistive heater. It is clearly seen that very limited mixing by molecular diffusion occurs near the vertical center line throughout the channel when there is no resistive heater (Figure 7(a)). However, rapid mixing can be seen when the flow passes the electrodes in the presence of the heater (Figure 7(b)), which is due to the chaotic ACET flow field in Figure 6(b).

Figure 7.

Concentration distribution on different cross sections (a) without the heater and (b) with the heater.

To quantify the effects of mixing, a mixing efficiency defined as in Erickson and Li²¹ is used

η = [1 - \frac{\int \int | C - C_{\infty} | d y d z}{\int \int | C_{0} - C_{\infty} | d y d z}] \times 100 %

(11)

where $C_{\infty} = 0.5$ is the concentration for fully mixed fluids, $C_{0} = 0$ or $C_{0} = 1$ is for unmixed fluids, and the integration is performed at a specified cross sectional plane. With this definition, $η = 0$ is for no mixing at all and $η = 100 %$ is for perfect mixing. With the concentration field calculated above, the mixing efficiency 80 µm downstream from the electrode edge without any ACET flow, that is, due to pure molecular diffusion, is 27.8%, and ACET mixing efficiencies without and with external heating are calculated to be 28.7% and 69.8%, respectively. Improved mixing is, thus, achieved with external heating by the thin film resistive heater.

Validation

A computational model with the same geometric and physical parameters as an ACET micromixer in Ng et al.¹⁴ is first developed. Although this does not have external heating, experimental results in Ng et al.¹⁴ are used to validate our computational model and numerical results. The 3D computational model is 1350 µm long, 100 µm wide, and 44 µm high. The coplanar AC electrodes are 500 µm long and 40 µm wide and separated by a distance of 20 µm at the bottom of the channel. Asymmetric AC potentials of 15 V_pp at the frequency of 500 kHz are applied on the two AC electrodes. The fluid conductivity is 0.5 S/m, and the fluid diffusion coefficient is $1 0^{- 10} m^{2} / s$ . The inlet velocity is set as 7.58 mm/s. Our numerical concentration distribution (top view) is shown in Figure 8(a), which is in good qualitative agreement with the experimental image of the intensity profile in Ng et al.¹⁴ We also calculated the mixing efficiency to be 24.9% 150 µm downstream from the electrode edge, and the mixing coefficiency at the same cross section was experimentally obtained to be 26% in Ng et al.¹⁴ Agreement of the mixing efficiency further confirms the validity of our multiphysics computational model and the numerical results.

Figure 8.

Comparison of our simulation results with experimental results: (a) numerical concentration distribution (top view) and (b) experimental image of the intensity profile from Ng et al.¹⁴

Parametric effects

Effects of resistive heating strength

The heat flux on the resistive heater surface is reduced to $1 \times 1 0^{5} W / m^{2}$ and increased to $5 \times 1 0^{5} W / m^{2}$ to investigate its effects on the mixing efficiency. The results are listed in Table 1, where the mixing efficiencies are calculated at the same cross section. It can be seen from Table 1 that the mixing efficiency increases with increasing resistive heating strength.

Table 1.

Mixing efficiencies with different heat flux values on the heater.

Heat flux ( $W / m^{2}$ )	Mixing efficiency (%)
$1 \times 10^{5}$	43.1
$2.5 \times 1 0^{5}$	69.8
$5 \times 1 0^{5}$	82.6

Effects of resistive heater size

Length of the resistive heater is first varied (symmetrically in both directions) to investigate its effects, and the results are plotted in Figure 9, which shows that the mixing efficiency increases with increasing heater size. This is due to the increased longitudinal length of fluids with temperature gradients above the heaters. However, the mixing efficiency reaches an asymptotic value when the heater is slightly (about 60 µm) longer than the AC electrodes because of the exponentially decaying electric field away from the AC electrodes.

Figure 9.

Variation of the mixing efficiency with length of the resistive heater.

Width of the resistive heater is also varied, with its length being equal to that of the AC electrode, to investigate its effects on the mixing efficiency, and the results are listed in Table 2. From Table 2, it can be concluded that optimal mixing occurs when the resistive heater occupies half width of the channel. This is due to the strongest temperature gradients at the transverse center plane (fluid interface) when the heater has one-half width of the channel (Figure 10(c)).

Table 2.

Mixing efficiencies with different heater widths.

Width of resistive heater (µm)	Mixing efficiency (%)
10	55.4
15^a	69.8
17.5	75.4
20^b	76.6
22.5	75.7
25	71.3

Same width as the alternating current electrode.

Half the width of the channel.

Figure 10.

Temperature gradients (K/m) on a cross section above the AC electrodes with different resistive heater widths:(a) 10 μm wide; (b) 15 μm wide; (c) 20 μm wide and (d) 22.5 μm wide.

Effects of resistive heater location

The resistive heater location in section “Results and discussions” is used as the benchmark and then varied to investigate its effects on the mixing efficiency. First, the heater is moved in the longitudinal direction, and the results are plotted in Figure 11, where positive displacement means moving the heater downstream and negative displacement means moving the heater upstream. From Figure 11, one can see that the mixing efficiency decreases when the heater is moved downstream because of the decreasing overlapping area of temperature gradients and AC electric fields, where the ACET body force helps to improve fluid mixing. When the heater is moved slightly upstream, temperature gradients prior to the leading edge of the AC electrodes help to improve mixing where the AC electric field is still nonnegligible. While at the downstream side above the heater, fluid temperature and concentration have become more uniform after external heating and ACET mixing, and contribution of the AC electric field to ACET mixing is much less significant. When the heater is moved further upstream, most of the cross sections above the heater prior to the AC electrode leading edge experience vanishing AC electric fields, and the mixing efficiency decreases. These explain why the curve in Figure 11 first goes up, reaches a maximum, and then goes down. The optimal mixing efficiency is obtained when the heater is moved slightly upstream (about 7.5 µm).

Figure 11.

Variation of the mixing efficiency with different heater displacements in the longitudinal direction.

Then, the resistive heater is moved in the transverse direction toward the center of the channel, and the results are listed in Table 3. As can be seen from Table 3, transverse location of the resistive heater has negligible effects on the mixing efficiency until the heater edge reaches the center line of the channel, in which case the mixing efficiency starts to decrease. This is more clearly seen from Figure 12, in which temperature gradients normal to the fluid interface at the center start to decrease when the heater is moved past the center line.

Table 3.

Mixing efficiencies with different heater displacements in the transverse direction.

Displacement (µm)	Mixing efficiency (%)
0	69.8
1.5	70.9
2.5	71.2
3.5	71.5
5	72.0
7.5	62.6

Figure 12.

Temperature gradients (K/m) with different resistive heater locations in the transverse direction: (a) 2.5 µm displacement; (b) 5 µm displacement and (c) 7.5 µm displacement.

The distance between the resistive heater and the AC electrode is also varied to investigate its effects, and the results are listed in Table 4. From Table 4, it can be seen that the mixing efficiency decreases with increasing distance between the resistive heater and the AC electrode.

Table 4.

Mixing efficiencies with different distances between the resistive heaters and AC electrodes.

Distance (µm)	Mixing efficiency (%)
1	69.8
2	69.0
3	65.8

AC: alternating current.

Comparison of mixing efficiencies

From the above results, it can be calculated that optimal mixing for the above micromixer with a thin film resistive heater can be achieved when the heater is below one of the AC electrodes, 60 µm longer than the AC electrode with half the width of the channel, and moved 7.5 µm upstream. When the heat flux on the heater surface is $5 \times 1 0^{5} W / m^{2}$ , which corresponds approximately to an externally applied 7 V_pp voltage, this optimal mixing efficiency is calculated to be 88.5%. In comparison, the maximum mixing efficiency in Cao et al.¹² is <50% where two pairs of electrodes were embedded on the opposite side walls of the microchannel. In Ng et al.,¹⁴ a mixing efficiency of 65% was obtained when an AC voltage of 15 V_pp with a 2 V biased voltage was applied. Therefore, it is said that improved mixing can be achieved in the proposed ACET micromixer with a thin film resistive heater.

Conclusion

In this article, a novel ACET micromixer with an external thin film resistive heater has been proposed. To the best of our knowledge, this is the first micromixer that uses thin film resistive heaters to enhance ACET mixing of fluids. After validation of our multiphysics numerical model of the micromixer by quantitative comparison with experimental results in the literature, our numerical results show that rapid mixing of fluids in the microchannel can be achieved by this novel ACET micromixer with thin film resistive heaters. Our numerical results also indicate that optimal mixing can be achieved by a thin film rectangular resistive heater below one of the AC electrodes, slightly (about 60 µm) longer than the AC electrode with half the width of the channel, and moved slightly upstream (about 7.5 µm). Mixing efficiency of the micromixer also increases with increasing heating strength and decreasing distance between the heater and the AC electrode. However, the minimal distance between the heater and the AC electrode is usually limited by fabrication techniques.

Footnotes

Academic Editor: Mark J Jackson

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (numbers 11172111, 11572139, 51175196, 51435006, and 51235005).

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Alternating current electrothermal micromixer with thin film resistive heaters

Abstract

Keywords

Introduction

Description of the micromixer

Numerical model

Results and discussions

The electric field and temperature field

The flow field

The concentration field

Validation

Parametric effects

Effects of resistive heating strength

Effects of resistive heater size

Effects of resistive heater location

Comparison of mixing efficiencies

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References