Study on the use of dielectrophoresis and electrothermal forces to produce on-chip micromixers and microconcentrators

Abstract

The present study uses the dielectrophoresis (DEP) and electrothermal (ET) forces to develop on-chip micromixers and microconcentrators. A microchannel with rectangular array of microelectrodes, patterned either on its bottom surface only or on both the top and the bottom surfaces, is considered for the analysis. A mathematical model to compute electrical field, temperature field, the fluid velocity, and the concentration distributions is developed. Both analytical and numerical solutions of standing wave DEP (SWDEP), traveling wave DEP (TWDEP), standing wave ET (SWET), and traveling wave ET (TWET) forces along the length and the height of the channel are compared. The effects of electrode size and their placement in the microsystem on micromixing and microconcentrating performance are studied and compared to velocity and concentration profiles. SWDEP forces can be used to collect the particles at different locations in the microchannel. Under positive and negative DEP effect, the particles are collected at electrode edges and away from the electrodes, respectively, irrespective of the position, size, and number of electrodes. The location of the concentration region can be shifted by changing the electrode position. SWET and TWET forces are used for mixing and producing concentration regions by circulating the fluid at a given location. The effect of forces can be changed with the applied voltage. The TWDEP method is the better method for mixing along the length of the channels among the four options explored in the present work. If two layers of particle suspension are placed side by side in the channel, triangular electrode configuration can be used to mix the suspensions. Triangular and rectangular electrode configurations can efficiently mix two layers of particle suspension placed side-by-side and one-atop-the-other, respectively. Hence, SWDEP forces can be successfully used to create microconcentrators, whereas TWDEP, SWET, and TWET can be used to produce efficient micromixers in a microfluidic chip.

INTRODUCTION

AC electrokinetics is the study of particle manipulation arising due to the interaction of dielectric particles with nonuniform AC electric field.¹ This technique has been used in various biomedical applications for manipulating, separating, sorting, and mixing of bioparticles such as cells, bacteria, viruses, DNA, and proteins.^{2, 3, 4, 5, 6, 7, 8, 9} Dielectrophoresis (DEP) is a frequently used AC electrokinetic technique for manipulating suspended particles in a fluid medium. Usually, this technique is implemented by placing a suspension on a designed, planar microelectrode structure. In some cases, the high electric fields used for manipulating particles generates significant amount of power density in the vicinity of the electrodes due to conduction in the solution, causing localized Joule heating. The resulting temperature distribution surrounding the electrode changes the local conductivity and permittivity, causing electrothermal (ET) fluid flow.^{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} ET fluid flow has been used previously for mixing particle suspensions.¹⁵

Taking the advantage of manipulating the fluid by ET forces and the particles by DEP forces, the present study applies these techniques for developing micromixers and microconcentrators. Micromixer or microconcentrator is one of the important components in lab-on-a-chip (LOC) devices or micro-total analysis systems ($\mu \text{TAS}$). It is used for mixing the chemical reagents and biological analytes in chemical and biomedical analysis.^{21, 22} In macroscale, mixing can be achieved by turbulence,^{21, 22, 23} whereas in microscale, mixing can be achieved by diffusion due to laminar flow in microfluidic systems.^{21, 22} The application of DEP and ET enhances the mixing performance at microscale by enhancing diffusive transport. In the present study, the effects of DEP and ET on the performance of mixing and concentrating of particle suspensions are studied. Both standing wave (SW) and traveling wave (TW) types of DEP and ET effects are studied. Numerical and analytical solutions for SWDEP, SWET, TWDEP, and TWET forces are calculated and the forces along the length and height of the channel are compared. A microchannel with rectangular array of microelectrodes, patterned either on its bottom surface only or on both the top and the bottom surfaces of the channel, is considered for the analysis. The terms DEP and SWDEP are synonymous and same goes for ET and SWET. Henceforth, in this paper, SWDEP and SWET will be used to describe dielectrophoresis and electrothermal effect, respectively.

A few studies are available in literature on micromixers based on DEP technique. Deval et al.²⁴ and Lee et al.²⁵ demonstrated chaotic dielectrophoretic micromixing in a simple microchannel geometry with integrated microelectrodes at the bottom surface. Frequency dependent DEP force was used for mixing the aqueous suspension of polystyrene particles. The combination of electrical actuation and channel geometry variation was used to create chaotic advection. Due to the interaction of electric field with velocity field both in time and space, saddle point regions were generated. At saddle point regions, the particles undergo a sequence of stretching and folding, leading to an exponential stretching rate. Chaotic trajectories, generated due to exponential stretching rate, induce the particles into fast and efficient mixing in the particle suspension. Recently, Gunda et al.²⁶ presented a numerical study on the dielectrophoretic micromixing with novel electrode geometry. They used a microchannel housing array of right angled triangular microelectrodes at the bottom surface. Mathematical model for mixing of two different types of colloidal suspensions was developed based on Laplace, Stokes, and convection-diffusion-migration equations. The effects of electrode pairs and mixing length were studied on the microfluidic system, and a mixing efficiency of $\sim 97\%$ was calculated for four pairs of electrodes.

Ramos et al.^{10, 11, 12} described the various forces that are present in the particle suspension under nonuniform AC electric fields. They provided calculation of DEP and ET forces extensively. Later, the same research group studied the ET fluid flow for pumping and mixing; DEP for manipulation, separation, and sorting of micro- or nano-scale particles.^{27, 28, 29, 30, 31, 32} Sigurdson et al.¹⁴ used the ET effects for binding of antigen in the fluid to antibody that is immobilized on the short length of the microchannel, to develop an immunosensor. Their numerical simulations showed the improvement of binding rate by a factor of 2 to 8. Later, Feldman et al.¹⁸ developed a biotin-streptavidin heterogeneous assay using ET forces. They experimentally investigated the binding of a fluorescently labelled streptavidin, which was suspended in a high conductivity buffer, to biotin, which was immobilized on the channel surface. Their experiments and simulations demonstrated the binding rate of factor 9 at $10V_{\mathit{r}\mathit{m}\mathit{s}}$ applied voltage. They also observed a difference of 1.5 orders of magnitude between the experimental and numerical velocity patterns.

Feng et al.^{33, 34} presented the analytical and numerical solutions of ET flow in microfluidic systems. They applied this technique for mixing fluids and cleaning the contaminants trapped in microcavities. Their simulations showed that in the case of ET based mixing, the mixing time was reduced by three orders of magnitude compared to the diffusion based mixing. They also observed the chaotic behaviour of fluid flow under ET effects. They demonstrated the fast and efficient cleaning of trapped contaminants in microsystems using the combined effect of pressure driven and ET flows. Perch-Nielsen et al.³⁵ studied the TWET effects on fluid flow numerically. Their model suggested that the temperature distribution and velocity of the fluid flow depends on conductivity, applied voltage, geometry of the system, and the type of substrate. They studied the effect of substrate materials (glass and silicon) on fluid flow and observed the opposite flow direction for the two materials in case of TWET flow. They also studied the effect of external applied temperature gradients in fluid flow and observed the reversal of the fluid flow under certain conditions. Molla and Bhattacharjee^{36, 37, 38} have demonstrated the successful use of DEP forces to prevent fouling in membranes.

The present paper focuses on the effect of electrode position, electrode size, number of electrode pairs, and applied voltage on the micromixing due to ET fluid flow and DEP. A detailed description of microchannel considered for the analysis is provided in the Sec. 2. The section also provides the governing equations with boundary conditions. In Sec. 3, comparisons of numerical and analytical solution of DEP and ET forces and comparison of velocity and concentration profiles for SWDEP, SWET, TWDEP, and TWET are discussed. Finally, the paper concludes by discussing the important aspects of this study.

MATHEMATICAL MODELING

Forces acting on particles and fluid

This section describes the various forces acting on the particles and fluid when the particle suspension is under the influence of an AC electric field.^{10, 11, 12}

The interaction between particles and the medium influences the movement of suspended particles in the solution under AC nonuniform electric fields. The effect of gravity, Brownian motion, DEP, and particle-particle interactions influences the motion of the sub-micron particles. Gravity force is one of the main external influencing forces that acts due to the difference in the mass density of the fluid and particles. Brownian motion is another important force acting on the sub-micron particles due to thermal randomization. The dielectrophoretic force is the interaction of a nonuniform electric field with the induced dipole moment of a particle. Electrostatic interactions and van der Waals interactions are important particle-particle interaction forces acting on sub-micron size particles. The electrical interaction force occurs between electrically charged particles, which can be described using Coulomb's law. van der Waals interaction can be either attractive or repulsive in nature between particles depending on the interaction medium. If the solution has finite conductivity, the effects of electrical forces such as AC electroosmosis (ACEO) and ET forces on fluids can be dominant. ACEO occurs due to the interaction of the tangential components of the electric field with an electrical double layer, which forms on the electrode surface because of applied AC signal. ET forces act on the fluid due to the heat generated by AC electrical fields. In brief, the suspended particle suspension under AC electrical fields will undergo the effects of gravity, Brownian motion, DEP, and particle-particle interaction forces on particles, and the effects of ACEO and ET forces on fluids.

Since the present study considers the neutrally buoyant, uncharged, non-interacting particles of $\sim 1-10$μm size at low concentrations, the effects of gravity, Brownian motion, and particle-particle interaction forces are neglected. ACEO is applicable only when there is significantly large induced electric double layer. Castellanos et al.^{6, 39} predicted that AC electroosmosis is not observable for frequencies greater than 100 kHz when medium electrical conductivity is 0.002 S/m. Similarly, Studer et al.⁴⁰ also estimated that ACEO is not important at any frequency for fluids with conductivity greater than 0.140 S/m. Castellanos et al.⁶ presented fluid flow and particle flow maps to see which forces are dominant in microfluidic systems and confirm the limitation of ACEO to relatively low conductivity solutions. Hence, it shows that ACEO is important for low conductivity solutions at low applied field frequencies. The medium conductivity used in this work is considerably high, 0.0575 S/m, the estimated frequency where ACEO is negligible for frequencies greater than $\sim 10\hspace{0.17em}\text{kHz}$. The study will be mostly at higher frequencies (15 kHz and 30 MHz) where the electrical double layer (EDL) formation can be assumed to be small, hence the ACEO is neglected. The effects of the remaining two forces, i.e., DEP and ET forces, are dominant in such cases and studied in the present work. The remainder of this section discusses these two forces in detail.

Dielectrophoretic force

DEP is a phenomenon in which a force is exerted on uncharged particles because of the polarization effects that occur in nonuniform electric fields. Usually this force arises on all types of particles, charged or uncharged. However, the strength of the force depends mainly on electrical properties of the particles; medium, shape, and size of the particles, magnitude, frequency, and phase of the applied AC signal. The typical case is the induced dipole in a lossy dielectric homogeneous spherical particle. Under the influence of sinusoidal varying AC nonuniform electric field with an angular frequency ω, the time averaged DEP force (N) acting on the particle (where the particle is much smaller than the electric field nonuniformities) is given by^{10, 11, 12, 41, 42, 43, 44}

$${\mathbf{F}}_{\mathit{D}\mathit{E}\mathit{P}}=\frac{1}{4}v{\epsilon }_m\left[{({\tilde{f}}_{\mathit{C}\mathit{M}})}_{\mathcal{R}}\nabla {\left|\tilde{\mathbf{E}}\right|}^2-2{({\tilde{f}}_{\mathit{C}\mathit{M}})}_{\mathcal{I}}(\nabla \times ({\mathbf{E}}_R\times {\mathbf{E}}_{\mathcal{I}}))\right],$$ (1)

where υ is the volume of particle $(=4\pi a_p^3/3)$ and $a_p$ is the radius of particle. $\tilde{\mathbf{E}}$ is the applied electric field $(\tilde{\mathbf{E}}=-\nabla \tilde{\varphi }=-(\nabla {\varphi }_R+i\nabla {\varphi }_I)),\hspace{0.17em}{\epsilon }_m$ is the permittivity of medium, and ${\tilde{f}}_{\mathit{C}\mathit{M}}$ is Clausius-Mossotti (CM) factor, subscripts R and I indicate the real part and imaginary part of the component, respectively, which is given as

$${\tilde{f}}_{\mathit{C}\mathit{M}}=\frac{{\tilde{\epsilon }}_p-{\tilde{\epsilon }}_m}{{\tilde{\epsilon }}_p+2{\tilde{\epsilon }}_m},$$ (2)

where $\tilde{\epsilon }$ is complex permittivity and subscripts p and m represent the particle and medium, respectively. The complex permittivity is given as

$$\tilde{\epsilon }=\epsilon -i\frac{\sigma }{\omega },$$ (3)

where σ is electrical conductivity, ω is angular frequency of the applied electric field, ε is the permittivity, and $i=\sqrt{-1}$. For spatially varying field magnitude (${180}^{°}$ phase difference between adjacent electrodes), the first part of Eq. 1 will be non-zero, exhibiting SWDEP. For spatially varying phase (${90}^{°}$ phase difference between adjacent electrodes), the second part Eq. 1 will be non-zero, indicating TWDEP.

Electrothermal force

ET force is the body force exhibited on the fluid medium due to AC electric fields. The AC nonuniform electrical fields in the system generate heat (Joule heating) at the electrodes, which diffuses rapidly in the fluid medium, leading to the variations in the temperature of medium and concurrent variations in the conductivity and permittivity of medium. The variations in the permittivity and conductivity exert an electrical body force on the medium. The time averaged ET force on unit volume of fluid ${(\mathrm{N}/\mathrm{m}}^3)$ is given as^{10, 11, 12, 35}

$$\begin{array}{c}{\mathbf{f}}_{\mathit{E}\mathit{T}}=\frac{1}{2}[\left\{\frac{{\epsilon }_m(\alpha -\beta )}{1+{(\omega {\epsilon }_m/{\sigma }_m)}^2}\left[(\nabla T·{\mathbf{E}}_{\mathcal{R}}){\mathbf{E}}_{\mathcal{R}}+(\nabla T·{\mathbf{E}}_{\mathcal{I}}){\mathbf{E}}_{\mathcal{I}}\right]\right\}+\left\{\frac{\omega {\epsilon }_m^2(\alpha -\beta )}{{\sigma }_m\left(1+{(\omega {\epsilon }_m/{\sigma }_m)}^2\right)}[(\nabla T·{\mathbf{E}}_{\mathcal{R}}){\mathbf{E}}_{\mathcal{I}}-(\nabla T·{\mathbf{E}}_{\mathcal{I}}){\mathbf{E}}_{\mathcal{R}}]\right\}]\\ -\frac{1}{4}{\epsilon }_m\alpha \nabla T\left[{\left|{\mathbf{E}}_{\mathcal{R}}\right|}^2+{\left|{\mathbf{E}}_{\mathcal{I}}\right|}^2\right],\end{array}$$ (4)

where T is the temperature, $\alpha =(\partial {\epsilon }_m/\partial T)/{\epsilon }_m$ and $\beta =(\partial {\sigma }_m/\partial T)/{\sigma }_m$. For spatially varying field magnitude (${180}^{°}$ phase difference between adjacent electrodes), the imaginary part of Eq. 1 will be zero, exhibiting SWET. For spatially varying phase (${90}^{°}$ phase difference between adjacent electrodes), the imaginary part will be non-zero, indicating TWET.

Governing equations

Governing equations required for solving the combined effects of DEP and ET are provided in this section.

Laplace equation

The phasor notation is used to describe the temporal electric field varying in a sinusoidal manner with angular frequency, ω. The potential distribution in the system created by the AC signal is given by the Laplace equation^{45, 46}

$${\nabla }^2\tilde{\varphi }=0,$$ (5)

where $\tilde{\varphi }={\varphi }_{\mathcal{R}}+i{\varphi }_{\mathcal{I}}$ is the potential phasor. Using Eq. 5, the electric field distribution can be calculated as

$$\tilde{\mathbf{E}}=-\nabla \tilde{\varphi },$$ (6)

where $\tilde{\mathbf{E}}={\mathbf{E}}_{\mathcal{R}}+i{\mathbf{E}}_{\mathcal{I}}=-\nabla {\varphi }_{\mathcal{R}}-i\nabla {\varphi }_{\mathcal{I}}$. This electric field phasor is used to calculate the time averaged dielectrophoretic force, ${\mathbf{F}}_{\mathit{D}\mathit{E}\mathit{P}}$ and electrothermal force, ${\mathbf{F}}_{\mathit{E}\mathit{T}}$ provided in Eqs. 1, 4, respectively.

Fourier's heat conduction equation

According to Ramos et al.,¹⁰ the typical diffusion time for temperature front in the present system can be estimated as ($t_{\mathit{d}\mathit{i}\mathit{f}\mathit{f}}={\rho }_m{c}_p{l}^2/k$) 6 ms. For the frequencies (10 kHz and 20 MHz) used in the system, the temperature change ($\Delta T/T\approx 1/(2\omega t_{\mathit{d}\mathit{i}\mathit{f}\mathit{f}}$) is negligible. Hence, transient term is neglected for the heat equation. For the present system, ${\rho }_m{c}_p{u}_{\mathit{a}\mathit{v}}l/k\approx 2\times {10}^{-2}$,¹⁰ so convective term is neglected in the heat equation. The temperature distribution generated in the fluid medium due to AC electrical fields can be calculated from the steady state conduction equation as

$$k{\nabla }^{2}T+{\sigma }_m{\left|\tilde{E}\right|}^2=0,$$ (7)

where k is the thermal conductivity and T is the temperature of the suspension affected due to the electric field.

Stokes equation

For steady state incompressible laminar flow, Stokes equation with electrothermal body force is given by

$$\nabla p=\mu {\nabla }^2\mathbf{u}+{\mathbf{f}}_{\mathit{E}\mathit{T}},$$ (8)

where p is the pressure and u is the velocity vector of the fluid medium.

Convection-diffusion-migration equation

The steady state concentration distribution of particles in the system with no chemical reactions can be given by the convection-diffusion-migration equation considering DEP force as the migration term^{10, 38}

$$\nabla ·(\mathbf{u}c)=\nabla ·({\mathcal{D}}_{\infty }\nabla c)-\nabla ·\left(\frac{{\mathcal{D}}_{\infty }{\mathbf{F}}_{\mathit{D}\mathit{E}\mathit{P}}}{k_B{T}_{\mathit{a}\mathit{m}\mathit{b}}}c\right),$$ (9)

where c is the concentration of the particles, ${\mathcal{D}}_{\infty }$ is the diffusion constant of the particle, $T_{\mathit{a}\mathit{m}\mathit{b}}$ is the ambient temperature, and $k_B$ is the Boltzmann constant ($=1.381\times {10}^{-23}\hspace{0.17em}\mathrm{J}/\mathrm{K}$).

Computational geometry

Figure 1 depicts the two-dimensional (2D) schematic view of the microchannel considered for the study. Here W is the width of the electrode, G is the gap between electrodes, L is the length of the channel, and H is the height of the channel. Appropriate boundary conditions required to solve the problem for SWDEP, SWET, TWDEP, and TWET are shown in the figure, where q is the heat flux entering/exiting through suspension into the channel (considered same at both the inlet and outlet of the channel), and $u_{\mathit{a}\mathit{v}}$ is the average velocity of the suspension entering the channel. Three types of electrode configurations are used in the study—electrodes at bottom, symmetric, and asymmetric electrodes.

Figure 1.

Two-dimensional schematic diagram considered for the analysis. (a) Electrodes at bottom surface only; (b) symmetric electrodes on both top and bottom surfaces at $0^{°}$ phase; and (c) asymmetric electrodes on both top and bottom surfaces at $180°$ phase difference.

Solution methodology

A 2D computational domain is analyzed as shown in Fig. 1. The finite element method is used to solve the above governing equations inside the microchannel. The procedure for solving the concentration distribution under combined DEP and ET effects is divided into four steps: (i) evaluating the potential distribution in the system using Eq. 5 and electric field distribution in the system using Eq. 6; (ii) evaluating the temperature distribution in the system using Eq. 7; (iii) evaluating the flow field velocity distribution in the system using Eq. 8; and (iv) evaluating the particle spatial concentration distribution using Eq. 9. Steps (ii) and (iii) are coupled because the change in fluid velocity due to body force changes the temperature distribution in the system. The solution obtained in step (i) is used to find the DEP force and ET force at the initial condition. Thereafter, incorporating the ET force in Eq. 8 and solving both Eqs. 7, 8 simultaneously, the temperature and velocity distribution can be calculated. Incorporating DEP force and velocity in Eq. 9, spatial concentration distribution of particles can be calculated next. Flow chart for solution methodology is given in Fig. 2.

Figure 2.

Flow chart for combined DEP and ET solution methodology.

The problem is implemented within a commercial based finite element analysis software, comsol multiphysics (comsol Version 3.5a, COMSOL, Inc., Burlington, MA, USA). The computational model is discretized with Lagrange-quadratic quadrilateral elements (using mapped mesh scheme). The electrode edges are locally refined to capture the effect of high intensity electric field. A mesh independent solution is achieved in this case with the domain finally consisting of 9600 elements. The simultaneous linear equations produced by the finite element method are solved using direct elimination solver (pardiso). Analytical solution is also calculated for the computation domain shown in Fig. 1a. SWDEP and TWDEP forces are calculated based on Fourier series method using the electrical field expressions given by Morgan et al.⁴⁴ in Eq. 1. SWET and TWET forces are calculated by incorporating the electric field expressions given by Morgan et al.⁴⁴ based on Fourier series method and temperature distribution expressions given by Ramos et al.¹⁰ and Feng et al.³⁴ into Eq. 4.

RESULTS AND DISCUSSIONS

Results for the study of dielectrophoresis and electrothermal effects on micromixing and microconcentration are presented in this section. For the simulations, length and height of the channels are taken as $L=240\hspace{0.17em}\mu \mathrm{m}$ and $H=40\mu \mathrm{m}$, respectively. The channel length depends on the number of electrodes in that array. The width of electrode and gap between electrodes are taken as $W=30\mu \mathrm{m}$ and $G=30\mu \mathrm{m}$, respectively. The thickness of electrodes is assumed to be very small as compared to the height of channel and hence neglected in the simulations. The dielectric properties of the colloidal particle suspensions considered for the analysis are given as ${\epsilon }_p=3{\epsilon }_o,\hspace{0.17em}{\epsilon }_m=80.2{\epsilon }_o,\hspace{0.17em}{\sigma }_p=0.1\hspace{0.17em}\mathrm{S}/\mathrm{m}$, and ${\sigma }_m=0.0575\hspace{0.17em}\mathrm{S}/\mathrm{m}$. Here ${\epsilon }_0=8.854\times {10}^{-12}\hspace{0.17em}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$ is the permittivity of free space. The properties of the fluid taken for the analysis are given as thermal conductivity k = 0.598 W/(mK); dynamic viscosity $\mu =1.08\times {10}^{-3}\hspace{0.17em}\text{Pas}$; and density $\rho =1000\hspace{0.17em}{\text{kg}/\mathrm{m}}^3$. The particle radius $a_p$ is taken as $1.5\hspace{0.17em}\mu \mathrm{m}$. The particle suspensions are maintained at the ambient temperature $T_{\mathit{a}\mathit{m}\mathit{b}}=298\hspace{0.17em}\mathrm{K}$. The simulations are conducted in the frequency of 15 kHz for positive DEP and ET effects and 20 MHz for negative DEP effects. All the figures shown for SWDEP and TWDEP are taken under 15 V applied voltage on electrodes. SWET and TWET effects are simulated under 30 V applied voltage on electrodes. The list of physical and chemical properties used in this study can be found in Table TABLE I.. All the results shown for the case when the width of the electrode is equal to gap between the electrodes (W = G). Similar results are observed for other electrode configurations ($W<G$ and $W>G$), which are not described here for the sake of brevity.

TABLE I.

Physical and chemical properties of the system studied.

Property	Value
System	Latex particles in water
Particle radius ($a_p$)	$1.5\times {10}^{-6}\hspace{0.17em}\mu \mathrm{m}$
Density of particle (${\rho }_p$)	$1000\hspace{0.17em}{\text{kg}/\mathrm{m}}^3$
Density of water (${\rho }_m$)	$1000\hspace{0.17em}{\text{kg}/\mathrm{m}}^3$
Temperature ($T_{\mathit{a}\mathit{m}\mathit{b}}$)	$273\hspace{0.17em}\mathrm{K}$
Boltzmann constant ($k_B$)	$1.38\times {10}^{-23}\hspace{0.17em}{\text{kg}/\mathrm{m}}^3$
Dynamic viscosity of water (μ)	$1.0\times {10}^{-3}\hspace{0.17em}{\text{Ns}/\mathrm{m}}^2$
Thermal conductivity (k)	$0.598\hspace{0.17em}\mathrm{W}/\text{mK}$
Conductivity of water (${\sigma }_m$)	$0.0575\hspace{0.17em}\mathrm{S}/\mathrm{m}$
Permittivity of water (${\epsilon }_m$)	$80.2{\epsilon }_0\hspace{0.17em}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$
Conductivity of particle (${\sigma }_p$)	$0.1\hspace{0.17em}\mathrm{S}/\mathrm{m}$
Permittivity of particle (${\epsilon }_p$)	$3{\epsilon }_0\hspace{0.17em}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$
Permittivity of free space (${\epsilon }_0$)	$8.854\times {10}^{-12}\hspace{0.17em}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$
Diffusion constant of the particle (${\mathcal{D}}_{\infty }$)	$1\times {10}^{-11}\hspace{0.17em}{\mathrm{m}}^2/\mathrm{s}$
Inlet concentration (c)	$1\hspace{0.17em}\mu {\text{mol}/\mathrm{m}}^3$
Electrode properties
Electrode width (W)	$30\hspace{0.17em}\mu \mathrm{m}$
Gap between electrodes (G)	$30\hspace{0.17em}\mu \mathrm{m}$
Voltage range (ϕ)	15–30 V
Reference voltage (${\varphi }_o$)	1 V
Frequency (F)	10–10 000 kHz
Channel dimensions
Channel length (L)	$240-720\hspace{0.17em}\mu \mathrm{m}$
Channel height (H)	$40\hspace{0.17em}\mu \mathrm{m}$
Average velocity of fluid ($u_{\mathit{a}\mathit{v}}$)	0.1–0.5 mm/s

Comparison of analytical and numerical solutions

Along channel length

Figure 3 illustrates the comparison of forces with respect to channel length near the electrode plane for both analytical and numerical solutions. This figure shows four plots. The horizontal axis of each plot in this figure shows the non-dimensional length of the channel ($x^{*}$) and vertical axis shows the non-dimensional force term. In Fig. 3, the actual DEP force is non-dimensionalized as ${\mathbf{F}}_{\mathit{D}\mathit{E}\mathit{P}}/{\epsilon }_o{{\varphi }_o}^2$, whereas ET body force is non-dimensionalized as ${\mathbf{f}}_{\mathit{E}\mathit{T}}W/{\epsilon }_o{{\varphi }_o}^2$. Analytical solution is indicated using the circle symbol with red color, whereas numerical solution is indicated using the square symbol with blue color. Figure 3a shows the non-dimensional SWDEP force variation along the length of the channel near electrodes. This plot shows that the DEP force is maximum at the edges of electrodes and minimum elsewhere. SWDEP force has a low value at the mid-point of the electrode compared to mid-point of the gap between electrodes. The attraction or repulsion of particles at the edges of electrodes can be achieved substantially due to a large electric field. Excellent agreement between Fourier series based analytical solution and finite element based numerical solution is observed. Figure 3b represents the non-dimensional SWET force profile along the length of the channel near electrodes. SWET force is also maximum at the edges of electrodes like the SWDEP force. But the SWET force is different than the SWDEP force at the mid-point of electrode and at the mid-point of the gap between electrodes. In this case, SWET force is minimum at the mid-point of the gap between electrodes compared to the mid-point of electrodes. A margin of deviation is observed between analytical and numerical solutions for SWET force at the mid-point of the electrode. Non-dimensional SWDEP force is the force acting on the particles and the value is in the orders of 0.02 to 900, whereas non-dimensional ET force is the force acting on the fluid medium and the value is in the orders of 0.01 to 1000. Figure 3c shows the variation of non-dimensional TWDEP force near the electrode plane along the length of the channel. Like SWDEP force, TWDEP force is maximum at the edges of the electrode, but its variation is different. The difference between the TWDEP force at the mid-point of the electrode and mid-point of the gap between the electrodes is very less as compared to DEP force. The TWDEP force values shown in the plot are three orders of magnitude less than the DEP force. Slight difference in the analytical and numerical solutions is observed for the force on the electrodes, but a significant difference of upto 79% is identified along the electrode gap and at the electrode edges. Analytical solution has captured the maximum TWDEP force at the electrode edges compared to numerical solution. Figure 3d illustrates the change in non-dimensional TWET force along the length of the channel near the electrodes. Profiles for both SWET and TWET are similar, but the TWET force values are reduced by half compared to the ET force. Difference of upto 63% is observed between the analytical and numerical solution of TWET force along the electrode, but negligible difference exists along the gap.

Figure 3.

Comparison of analytical and numerical solution of force terms along the length of the channel near electrodes for the case where electrodes at bottom surface only. (a) Non-dimensional SWDEP Force; (b) non-dimensional SWET force; (c) non-dimensional TWDEP force; and (d) non-dimensional TWET force. The horizontal axis in this plot is non-dimensionalized with respect to electrode width, which is $30\mu \mathrm{m}$ in this case. Gap between electrodes is equal to electrode width. Applied voltage $\varphi =15\hspace{0.17em}\mathrm{V}$ and frequency is 15 kHz. Here, DEP force is non-dimensionalized as ${\mathbf{F}}_{\mathit{D}\mathit{E}\mathit{P}}/{\epsilon }_o{{\varphi }_o}^2$, whereas ET body force is non-dimensionalized as ${\mathbf{f}}_{\mathit{E}\mathit{T}}W/{\epsilon }_o{{\varphi }_o}^2$.

Along the channel height

Figure 4 depicts the comparison of analytical and numerical solution of different force terms along the channel height at electrode edge, mid point of electrode, and mid point of gap between the electrodes. The horizontal axis in this figure shows the non-dimensional height of the channel ($y^{*}$), whereas vertical axis shows the non-dimensional force term in logarithmic scale. In Fig. 4, the DEP force is non-dimensionalized as ${\mathbf{F}}_{\mathit{D}\mathit{E}\mathit{P}}/{\epsilon }_o{{\varphi }_o}^2$, whereas ET body force is non-dimensionalized as ${\mathbf{f}}_{\mathit{E}\mathit{T}}W/{\epsilon }_o{{\varphi }_o}^2$. At the edges of the electrodes, all force terms are exponentially decayed along the height of the channel. In Fig. 4a, non-dimensional SWDEP force is monotonically increased up to $y^{*}=0.92$ and then linearly decreased along the height of channel at the mid-point of the electrode. The linear decaying of SWDEP force is observed at the mid point of gap between the electrodes. At the height $y^{*}=2$ to $y^{*}=3$, the SWDEP force is constant throughout the channel length and its magnitude is linearly decreased along the height of the channel. This decides the effectiveness of SWDEP force in the channel. Excellent agreement between the analytical and numerical solutions for SWDEP force is observed in all cases, i.e., mid-point of electrode, edge of electrode, and mid-point of gap. Some slight discrepancy in solutions for SWDEP force along the height of channel up to $y^{*}=0.5$ at the mid-point of electrodes is observed. Figure 4b shows the variation profiles of the non-dimensional SWET force along the height of the channel at different locations. Similar trend between SWDEP force at the mid-point of the electrode and SWET force at the mid-point of the gap is observed. Analytical and numerical solutions of SWET force are in agreement up to $y^{*}=1.8$, beyond this some discrepancies are observed. Figure 4c illustrates the non-dimensional TWDEP force variation along the height of the channel at different locations. A difference in the analytical and numerical solutions for the TWDEP force along the height of the channel is observed. Similar profile variations at the electrode edge and at the mid-point of the gap for both SWDEP and TWDEP forces are observed, but significant difference in the profile at the mid-point of the electrode is observed. The TWDEP force values are three orders of magnitude lower than the SWDEP force. The TWDEP force at the mid of the electrode decreases along the height of the channel up to or around $y^{*}=1.35$ to $y^{*}=1.4$, then increases up to $y^{*}=2.5$, and then remains constant throughout the height of the channel. Figure 4d shows the non-dimensional TWET variation along the height of the channel at different locations. Some difference between the analytical and numerical solutions is observed for TWET force. Profiles for the SWET and TWET forces are similar at the mid of the electrode and the electrode edge, but significant difference in the profile variation at the mid of the gap is observed.

Figure 4.

Comparison of analytical and numerical solution of force terms along the height of the channel at edge of the electrode, mid point of the electrode, and mid point of gap between electrodes for the case where electrodes at bottom surface only. (a) Non-dimensional SWDEP force; (b) non-dimensional SWET force; (c) Non-dimensional TWDEP force; (d) Non-dimensional TWET force. The horizontal axis in this plot is non-dimensionalized with respect to electrode width, which is $30\mu \mathrm{m}$ in this case. Gap between electrodes is equal to electrode width. Applied voltage $\varphi =15\hspace{0.17em}\mathrm{V}$ and frequency is 15 kHz. Here, DEP force is non-dimensionalized as ${\mathbf{F}}_{\mathit{D}\mathit{E}\mathit{P}}/{\epsilon }_o{{\varphi }_o}^2$, whereas ET body force is non-dimensionalized as ${\mathbf{f}}_{\mathit{E}\mathit{T}}W/{\epsilon }_o{{\varphi }_o}^2$.

Effect of SWDEP force

Figure 5 shows the comparison of DEP velocity under positive DEP effect, in terms of contours and vectors for the following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances. Arrows in figure indicate the direction of particles movement due to DEP force. The length of the arrow does not indicate the magnitude of the DEP velocity. Contours in the figure indicate the constant DEP velocities along the curves. Converging of arrows at electrode edges indicates the collection of particles in that area, which can be used as concentration regions (or collectors). However, the distance traveled by the particle varies in the channel. Figure 6 shows the comparison of DEP velocity under negative DEP effect, in terms of contours and arrows for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances. Arrows in the figure indicate the direction of particles movement due to DEP force. The length of the arrow does not indicate the magnitude of the DEP velocity. Contours in the figure indicate the constant DEP velocities. In this case, the direction of the particle movement has changed and the particles are being repelled from the electrode. As this is a closed channel, the particles are collecting at the top wall of the channel and at a position parallel to the mid-point of the electrode. The region where the particles are collecting is referred to as concentrating regions. Position of electrodes at both top and bottom changes the concentration regions. In Fig. 6b, where the electrodes at both top and bottom of the channel are symmetrically located, the concentration region has shifted to the mid-point of the channel height and parallel to the mid-point of the electrodes. When the electrode position on the top of the channel is shifted by one electrode width, as shown in Fig. 6c, the concentration regions are shifted from the mid-point of the electrode to the edge of the electrodes along the mid-plane of the microchannel.

Figure 5.

Comparison of DEP velocity contour and arrow plots under positive DEP effect for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Figure 6.

Comparison of DEP velocity contour and arrow plots under negative DEP effect for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Effect of SWET force

Figure 7 shows the comparison of fluid velocity under ET forces at applied voltage of $30V_{\mathit{r}\mathit{m}\mathit{s}}$ and angular frequency of 15 kHz in terms of scalar and contour plot for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances. The figure shows the scalar plot of fluid velocity under ET force and contour shows the constant velocity in that curve. It is observed that near (around 0.01% of the height of the channel) the electrode edges, the fluid velocity is less which gradually increases up to a certain height and eventually starts decreasing along the channel height. The contours indicate that the fluid is circulating in the region above the electrode edges without getting transported along the channel. At these circulation regions, the particles also move from the bottom surface to the top surface of the channel, and the particles and fluid in the lower half of the channel can easily mix up with the colloidal suspension in the upper half of channel. Position of the electrodes can change the intensity and number of circulation regions across the channel. The colors in the contour plot indicate the velocity magnitudes. The legend in the right side of each graph indicates the colors used for velocity magnitudes in the contour and scalar plots. The shape of the circular regions also depends on the position of the electrode.

Figure 7.

Comparison of fluid velocity scalar plot with contours under ET force at $30V_{\mathit{r}\mathit{m}\mathit{s}}$ applied voltage for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Figure 8 illustrates the comparison of scalar concentration and ET velocity in terms of stream lines and arrows for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances. Arrows in the figure indicates the direction of fluid medium under ET forces. The length of the arrow does not indicate the magnitude of the velocity. It is observed that the particles in the fluid are moving along with the fluid. In Fig. 8a, arrows and stream lines show the movement of particles and the fluid movement in the channel where electrodes are placed at the bottom of the channel only. The size of circulation region depends on the applied voltage. The arrow directions at the beginning of the electrode and at the end of the electrode are opposite to each other which makes the fluid rotate at the same location instead of moving towards the other stream. If the applied voltage is less, the circulation region will be less and the fluid can easily move to opposite side making some small concentrated circulation region at the edges of electrodes. In Fig. 8b, it is observed that the size of the circulation regions is increased with the addition of electrodes at the top wall of the channel. In certain regions, the suspension is well mixed at the upper half of the channel and in some other regions, the suspension remains confined in the lower and upper halves of the channel, respectively. The effect of applied voltage on velocity of the suspension under ET forces is illustrated in Fig. 9. This figure compares the change in velocity and concentration distribution under ET force with respect to applied voltage of $15V_{\mathit{r}\mathit{m}\mathit{s}}$ and $30V_{\mathit{r}\mathit{m}\mathit{s}}$. The figure also clearly shows the difference in the number of circulation regions and their location in the form of contours and scalar plot of suspension velocity.

Figure 8.

Comparison of scalar concentration and fluid velocity streamline and arrow plots under ET force at $30V_{\mathit{r}\mathit{m}\mathit{s}}$ applied voltage for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Figure 9.

Comparison of change in velocity and concentration distribution under ET force with respect to applied voltage at electrodes for a case of channel having electrodes at both top and bottom surface: (a) Velocity scalar plot with contours at applied voltage of $15V_{\mathit{r}\mathit{m}\mathit{s}}$ and (b) Concentration scalar plot with velocity stream lines and arrows at applied voltage of $15V_{\mathit{r}\mathit{m}\mathit{s}}$.

Effect of TWDEP force

Figure 10 shows the comparison of TWDEP velocity in terms of contours and arrows for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances. Arrows in the figure indicate the direction of particles movement due to TWDEP force. The length of the arrow does not indicate the magnitude of the TWDEP velocity. Contours in the figure indicate the constant TWDEP velocities. This TWDEP can be used for moving the particles from the inlet to the outlet of the channel. Later the movement of the particles has changed. This change in the direction of the particle near the electrode plane and away from the electrode plane can be used for mixing. Figure 10b shows the converging and diverging of arrows along the channel. There are no re-circulation regions in this case. Figure 10c shows the waviness in the direction of arrows from the top to the bottom and then bottom to top along the channel. This TWDEP force on the particles can be used for moving the particles from inlet to outlet as shown in the Fig. 10a and can be advantageously used in mixing of two different layers of the suspensions with electrode configurations shown in Figs. 10b, 10c.

Figure 10.

Comparison of TWDEP velocity contour and arrow plots for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Figure 11 shows the effect of TWDEP with the different types of electrode shapes. The electrodes shown in the figures are placed at the bottom of the channels. For this type of configurations, three-dimensional computational domain is necessary to solve the problem. The figure shows the direction of the particle movements in terms of arrows from the inlet to outlet. The waviness of arrows in the flow field shows the mixing of the two layers of suspensions which are introduced side by side in the channel. The effect of particle direction due to the edge effect of the right angled triangular electrodes is shown in Figs. 11a, 11b. The different types of triangular electrodes with blunt corners are shown in the Fig. 11c. In all these cases, the direction of the particles has drastically changed at the slant edge of the electrodes.

Figure 11.

Plot showing the TWDEP velocity vector arrows at the mid of the channel, which has microelectrodes of (a) right angled triangular electrode with small curve at one corner, (b) right angled triangular electrode with large curve at one corner and (c) triangular electrode with curves at all corners. The top view of the microchannel is shown here.

Effect of TWET force

Figure 12 shows the comparison of TWET velocity in terms of scalar and contour plot for the following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances. Figure 13 illustrates the comparison of scalar concentration and TWET velocity in terms of stream lines and arrows for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances. In both these figures, the circulation regions for all cases are similar to the case of SWET force. Here in TWET, just the magnitude of the velocity has been reduced compared to the ET forces.

Figure 12.

Comparison of fluid velocity scalar plot with contours under TWET force for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Figure 13.

Comparison of fluid velocity streamline and arrow plots under TWET force for following cases: (a) channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Comparison of particle velocity and particle vorticity

The variation of normalized total particle velocity due to combined effect of DEP and ET with respect to the change in channel height is shown in Fig. 14 for three different cases studied in this work. It is clearly seen in Fig. 14 that DEP force is dominant at the electrode edges (at $x^{*}=7.5$ and $x^{*}=10.5$), whereas ET is dominant in other cases (at $x^{*}=6$ and $x^{*}=9$). A single peak in normalized particle velocity is observed in the case of channel with electrodes at the bottom surface, whereas double peaks are found for channel with electrodes at both top and bottom surfaces. The particle velocity profiles are similar to those reported by He and Liu.⁴⁷ He and Liu observed that the combined DEP and ET effect enhanced the particle-fluid flow. The change in velocity magnitudes along the channel height as well as along the length of channel makes the particles move in vortex motion. Particles are trapped or concentrated at the points where velocity is zero, i.e., stagnation points. Variation of normalized total particle vorticity due to the combined effect of DEP and ET with respect to the change in channel height is shown in Fig. 15 for three different cases studied in this work. The peaks shown in the variation profiles of vorticity provide the information of closed streamlines which make the particles to concentrate at one location as well as to mix different types of particles in the same location.

Figure 14.

Comparison of normalized particle velocity ($u_{\mathit{t}\mathit{o}\mathit{t}}/{(u_{\mathit{t}\mathit{o}\mathit{t}})}_{\mathit{m}\mathit{a}\mathit{x}}$, Here $u_{\mathit{t}\mathit{o}\mathit{t}}$ is particle velocity due to combined effect of ET and DEP). (a) Channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Figure 15.

Comparison of normalized particle vorticity ($\Omega /{\Omega }_{\mathit{m}\mathit{a}\mathit{x}}$, Here Ω is particle vorticity due to combined effect of ET and DEP). (a) Channel with electrodes at the bottom surface; (b) channel with electrodes at both top and bottom surfaces with the same position as in the previous case; and (c) channel with electrodes at both top and bottom surfaces placed in alternate distances.

Summarizing, the DEP forces can be used to collect the particles at different locations. Under positive DEP effect, the particles are collected at electrode edges, irrespective of the position, size, or number of electrodes. Under negative DEP effect, the particles are collected at the lower electric field strength regions. The location of the concentration regions can be shifted by changing the electrode position in the channel. Hence, DEP forces can be successfully used to create microconcentrators in a microfluidic chip. ET and TWET forces are used for mixing and producing concentration regions by circulating the fluid at a given location. The effect of forces can be changed with the applied voltage. The TWDEP method is the better method for mixing along the length of the channels among the four options explored in the present work. If two layers of particle suspension are placed side by side in the channel, triangular electrode configuration can be used to mix the suspensions. If two layers are placed one on the top of the other, the rectangular electrode configuration with electrodes at the top and bottom will be efficient to mix the two layers of suspensions.

CONCLUSIONS

A mathematical model is developed based on Laplace, heat conduction, Stokes, and convection-diffusion-migration equations to calculate electric field, temperature, velocity, and concentration distributions, respectively. The effects of SWDEP, TWDEP, SWET, and TWET on the performance of micromixers and microconcentrators are studied numerically. Analytical and numerical solutions of DEP and ET forces along the length of the channel and height of the channel are compared. The effects of electrode size and placement on micromixing and microconcentrating performance are studied and the velocity and concentration distributions have been compared. The results indicate that SWDEP forces can be used to collect the particles at different locations. SWET and TWET forces can be used to manipulate the fluid along with the particles. Based on position and size of the electrodes, the size and location of the concentration regions changes by both DEP and ET forces. The movement of particles from the lower half of the channel to the upper half of the channel is observed using TWDEP forces.