1 Introduction

Dielectrophoresis (DEP) is a phenomenon in which the electrically neutral but polarizable micro particles suspended in a medium move towards either the maxima or minima of a non-uniform electric field according to the electric properties of the particles and medium^[1-5]. In recent years, DEP has become one of the most popular techniques to manipulate neutral particles or cells in biomedical diagnostic, chemical analysis, and electronics industries, owing to its label-free nature^[6-7], highly selective, and sensitive analysis^[8-9].

The continuous particle separation can be achieved by combining dielectrophoretic and hydrodynamic techniques, such as dielectrophoresis field-flow fractionation (DEP-FFF) and insulating dielectrophoresis (iDEP), which have become the most promising technique for continuous particle separation^[10-16]. Yang et al.^[17] separated the mixtures of the major human leukocyte subpopulations by DEP-FFF. The separation was conducted in a thin chamber equipped with an array of microfabricated interdigitated electrodes on the bottom surface, and the separation performance was characterized by on-line flow cytometry. Piacentini et al.^[18] presented a microfluidic device capable of separating platelets from other blood cells in a continuous flow using DEP-FFF. The hydrodynamic force in combination with the dielectrophoretic force allows the separation of platelets from red blood cells due to their size difference. Srivastava et al.^[19] achieved continuous sorting of polystyrene microparticles utilizing direct current (DC) iDEP at lower voltages. Three different sized particles were sorted into four channel outlets in a microchannel containing a rectangular insulating obstacle. Lewpiriyawong and Yang^[20] presented a new application of the modified H-filter with insulating rectangular blocks using negative and positive DEP for separation of multiple particles in a continuous pressure-driven flow. Mohammadi et al.^[21] continuously separated red blood cells infected in vitro by plasmodium falciparum human-malaria parasites called Pf-iRBCs from healthy red blood cells (h-RBCs) by DC-iDEP using an innovative microfluidic device, which is suitable for clinical diagnosis as well as biological and epidemiological research.

Moreover, the dielectric surfaces of the channel wall acquire an electrical charge when exposed to an electrolyte solution in a microchannel. The nearby ions in the solution are influenced, and the so-called electric double layer (EDL) is formed. By applying an external driving electric field along the channel wall, the net charge in the EDL can be driven by the Coulomb force, and thus the surrounding fluid moves due to the viscous drag force, which is called the electroosmotic flow (EOF)^[22-23]. The EOF is often used for microfluidic pumping and particle manipulation combined with DEP^[24-28]. Zhou and Tilton^[24] patterned and separated colloidal particles and biological cells laterally adjacent to a micropatterned electrode array by coupling dielectrophoretic forces and electroosmostic drag forces. Melvin et al.^[25] proposed a particle collection device capable of concentrating micro-sized particles in a predetermined area by combining the EOF and DEP. An equal and quadrilateral flow directed towards a stagnant region was created by the EOF, which is capable of rapid collection of dense particle. Gencoglu et al.^[26] explored the potential of low-frequency alternating current (AC) insulator-based DEP for separation of polystyrene microparticles and yeast cells. A microchannel with diamond-shaped insulating posts was adopted to generate an EOF gradient by employing an asymmetrical and low frequency electric signal, and the particles can be effectively concentrated and selectively released. Three different sized particles were successfully separated including polystyrene particles and biological particles. Chung et al.^[27] presented a method to fabricate controllable microscale wave structures on the top of regular interdigitated electrode (IDE) arrays using electrically-assisted lithography techniques. Size-dependent particle separation over the wave structure was demonstrated by the joint effects of drag and dielectrophoretic forces when being pumped to flow via the AC EOF. Rezanoor and Dutta^[28] presented a method to rotate dielectrophoretically trapped microparticles using a stationary AC electric field. A non-uniform electric field was employed for DEP trapping of microparticles and EOF generating in the vicinity of the electrodes, resulting in rotation of microparticles in a microfluidic device.

By applying modulating electric fields in a direction perpendicular to the channel wall, the EDL is strikingly enhanced, a much stronger EOF is obtained, and micro-vortexes are generated around the modulating electrodes, which is called the field-modulated EOF^[29-30]. With an optimal design of discrete modulating electrodes, the magnitude, direction, and flow profiles of the EOF in the microchannel can be effectively controlled. These discrete electrodes create non-uniform or discontinuous wall surface charge. When a particle is positioned near the electrodes, it will experience a strong dielectrophoretic force, which is also strongly dependent on the electrode configuration. Thus, it is more effective to use the joint effects of dielectrophoretic force and the hydrodynamic effect of dielectrophoretic for particle manipulation. In this paper, the micro-vortexes generated by the EOF and dielectrophoretic forces are combined to achieve the separation and trap two different sized particles in the continuous flow of a straight microchannel without complicated channel design. Micro-vortices and dielectrophoretic forces are realized in the microchannel by applying modulating electric fields with various electrode configurations.

2 Mathematical model

Due to the large cross-sectional aspect ratios in practical applications, the fluid flow in such a flat microchannel can be realistically modeled as a two-dimensional (2D) flow between parallel plates. In this work, the EOF is created in a straight finite-length channel positioned between two liquid reservoirs, which are equipped with electrodes to produce an electric field tangential to the EDL, and a pair of symmetrical modulating electrodes are equipped on the channel walls to enhance the EDL and generate micro-vortexes. The fluid is pumped by the EOF to transport the particles from upstream to downstream, and the vortex region is the working space for particle trapping and separation. The geometry of the 2D model is shown in Fig. 1.

2.1 Governing equations of the EOF

An external electric field φ is created by the voltage difference across the electrodes in the reservoirs, as shown in Fig. 1. The voltages on the two electrodes are, respectively, φ₁ and 0. The distribution of φ can be obtained by the Laplace equation,

(1)

The boundary conditions on the rest walls except the electrodes are specified as electrically insulating condition

Introduce

where H is the characteristic length, E₀=φ₁/H is the characteristic electric field strength, and "^*" denotes the dimensionless variables. The equation can be normalized as

(2)

The corresponding boundary conditions are normalized as

An induced electric field ψ in the field-modulated EDL is related to the net charge density ρ_e via the Poisson equation as follows^[31]:

(3)

where ε_f is the permittivity of the electrolyte, and ρ_e is given by

(4)

where e is the elementary charge, z_i is the valence of the ions, and n_i is the ionic number concentration of the electrolyte solution, which obeys the Boltzmann distribution when the EDL is not disturbed by the external fields, boundaries, and EDLs^[31],

(5)

where n₀=CN_A is the ion number concentration of the electrolyte, C is the molar concentration of ions, N_A=6.022×10²³ is Avogadro's number, k_b=1.380×10^-23 J/K is Boltzmann constant, and T is the absolute temperature. For a symmetrical electrolyte, in which the cations and the anions have the same charge valence, that is,

the net charge density and the ionic number concentration can be written as

(6)

(7)

The boundary conditions are specified as ψ_e=ξ for the modulating electrodes and for the rest walls. Introduce

where

is the characteristic electric field in the EDL. Then,

is the dimensionless net charge density. The equations of the modulated electric field can be normalized as

(8)

(9)

(10)

where k=KH is referred to as the double-layer thickness parameter, in which

is called the Debye-Huckel parameter and represents the reciprocal of the characteristic EDL parameter. The corresponding boundary conditions are ψ^*=ξ/ψ₀ for the modulating electrodes and for the rest walls.

The Reynolds number of the fluid flow in this study is very small. Thus, the fluid inertia can be neglected. The continuity equation and the Navier-Stokes equation for the incompressible viscous fluid are given by

(11)

(12)

where V is the velocity vector of the fluid flow, p is the fluid pressure, ρ is the fluid density, η is the fluid viscosity, ρ_e is the net charge density, and E is the electric field composed of two types of electric field as follows:

(13)

The boundary conditions are specified as p=0 for the inlet and outlet and V=0 for the channel walls. Introduce

where V₀=ε_fE₀ψ₀/η is the characteristic velocity, t₀=H/V₀ is the characteristic time, and p₀=ρV₀² is the characteristic fluid pressure. The equations above can be normalized as

(14)

(15)

(16)

where

are dimensionless parameters, and Re=ρV₀H/η is the Reynolds number. The corresponding boundary conditions are p^*=0 for the inlet and outlet and V^*=0 for the channel walls.

3 Governing equations of particle motion

The motions of the particles are governed by the kinematic equation as follows:

(17)

where m_p and r_p are the mass and the position vector of particles, respectively. F_dep is the dielectrophoretic force, given by

(18)

where a is the particle radius, Re means the real quantity, and f_cm is the Clausius-Mossotti factor. In the DC electric field, f_cm can be written as

(19)

where σ_p and σ_f are the conductivities of the particles and the fluid, respectively. The conductivity of a solid homogeneous spherical particle can be expressed as^[18-19]

(20)

where σ_{p, bulk} is the bulk conductivity of the particle and is nearly zero for polystyrene dielectric particles, and K_s is the surface conductance ranging from 0.2 nS to 2.1 nS. F_d is the Stokes drag force, given by

(21)

where u and v are the velocities of the fluid and particles, respectively. Introduce

where F₀=ε_fψ₀² is the characteristic force, and m₀=ξ₀t₀η/E₀ is the characteristic mass. The equations can be normalized as

(22)

(23)

(24)

where γ=6πaE₀/ψ₀ is a dimensionless parameter.

4 Results and discussion

The potassium chloride (KCl) aqueous solution at the room temperature T=293 K is used as the working fluid. The fluid permittivity is ε_f=78ε₀, where ε₀=8.85×10^-12 F/m is the vacuum permittivity. The density and viscosity of the fluid are ρ=1 000 kg/m³ and η=10^-3 Pa·s, respectively. The particle density is considered the same as the fluid so that the gravity force can be ignored. In this paper, the separation of particles is achieved based on the particle size. Therefore, a mixture of two types of particles with different radii a₁=10 μm, a₂=5 μm and the same Clausius-Mossotti factor f_cm=-0.45 are used for particle separation, which means that the particles are all negative dielectrophoretic particles and the dielectrophoretic force drives the particles away from the electrodes. In this work, the dimensionless EDL thickness parameter k and the Reynolds number Re are input parameters, from which the height of the channel H and the corresponding characteristic constants are calculated. Pairs of symmetrical modulating electrodes are used to get locally enhanced zeta potentials, as shown in Fig. 1.

The dimensionless EDL thickness parameter is specified as k=399.1. Correspondingly, the dimensions of the system can be obtained as H=k/K=120 μm, L_c=6H=720 μm, L_r=3H=360 μm, and H_r=4H=480 μm. The lengths of the modulating electrodes are specified as L_e=20 μm. Both structural and non-structural grids are adopted in the numerical analysis to keep the numerical solutions independent of the grid size, as shown in Fig. 2. The grid sizes close to the EDLs are smaller than the double-layer thickness to ensure that the net charge density in the EDLs can be accurately calculated. The characteristic velocity and external electric field strength can be obtained once the Reynolds number Re is input by the equations V₀=Reη/(ρH) and E₀=V₀η/(ε_fξ₀), respectively. Thus, the voltage of the external electric field φ₁ is obtained, which is related to the velocity of the EOF and is called the driving voltage.

When the driving voltage is φ₁=0.573 3 V and the modulating voltage is ψ_e=-2ψ₀=-50 mV, the electroosmotic velocity vectors and streamlines are shown in Fig. 3. The velocity field in the flow region between the two symmetrical modulating electrodes is highly non-uniform, due to the local disturbance of the modulating electrical field. Two symmetrical micro-vortexes are generated with opposite directions, which are in essence two electroosmotic pumps in parallel and drive the fluid flow from left to right in the channel.

4.1 The effect of driving voltage on the trapping rate and channel flux

A mixture consisting of twenty large particles with a₁=10 μm and twenty small particles with a₂=5 μm is released in the upstream of the fluid orderly, as shown in Fig. 4(a). The particles are primarily driven by the hydrodynamic forces of the fluid and move to downstream of the channel. When they get close to the modulating electrodes, a strong negative dielectrophoretic force arises on the particles due to the locally non-uniform electric field. The particles are driven away from the electrodes and move towards the vortexes in the channel centre. Keep the modulating voltage ψ_e=-50 mV unchanged, which means that the dielectrophoretic forces of the particles stay the same. By adjusting the driving voltage φ₁, the velocity of the EOF is changed. Thus, the hydrodynamic force of the particles can be controlled. Once the dielectrophoretic force is large enough compared with the hydrodynamic force by adjusting φ₁, the particles with a larger dielectrophoretic force are pushed into the recirculation region and trapped by the vortexes, and the others are transported to the downstream by the fluid, as shown in Figs. 4(b)-4(d). For example, when φ₁=1.146 5 V, only one large particle is trapped by the vortex, and the other large particles and all small particles are transported to the downstream (see Fig. 4(b)) due to a strong hydrodynamic force. When the driving voltage is decreased to φ₁=0.286 6 V, the hydrodynamic forces become weaker to drive the particles to the downstream, and as a result thirteen large particles are pushed into the vortexes by the negative dielectrophoretic force, while only seven large particles and twenty small particles are transported to the downstream (see Fig. 4(c)). Moreover, when the driving voltage is further reduced to φ₁=0.114 7 V, only five small particles are transported to the downstream, while the other particles are all trapped by the vortexes (see Fig. 4(d)).

ζ_L is defined as the trapping rate of the large particles and denotes the ratio of the number of large particles trapped by the vortexes to the total large particles released in the upstream. The trapping rate is ζ_L=5% in Fig. 4(b) and ζ_L=65% in Fig. 4(c). If the small particles are also trapped by the vortexes, as shown in Fig. 4(d), the particles in the vortexes are still the mixture of large particles and small particles. Thus, the trapping rate is defined to be ζ_L=0. The variations of the channel flux Q and trapping rate of the large particles ζ_L with the driving voltage φ₁ when ψ_e=-50 mV are shown in Fig. 5. The channel flux Q increases linearly with the driving voltage. When the driving voltage is too large, the hydrodynamic force is dominant compared with the dielectrophoretic force. Both types of particles are transported to the downstream, and no particles are pushed into the vortexes by the dielectrophoretic force. Thus, ζ_L=0. When the driving voltage decreases, the trapping rate increases as a number of the large particles are trapped by the vortexes. When all large particles are trapped by the vortexes and all small particles are transported to the downstream, the trapping rate reaches the optimal value ζ_L=100%. When the driving voltage further decreases, the dielectrophoretic force is dominant compared with the hydrodynamic force, the small particles are also trapped by the vortexes, the large particles in the vortexes cannot be separated from the small particles, and thus ζ_L=0.

4.2 The effect of modulating voltage on the trapping rate and channel flux

When the trapping rate of the large particles reaches the optimal value ζ_L=100% with the modulating voltage ψ_e=-50 mV, the driving voltage is φ₁=0.263 7 V. Keeping the optimal driving voltage unchanged, the variations of the channel flux Q and the trapping rate of large particles ζ_L with the modulating voltage ψ_e are shown in Fig. 6. The channel flux Q increases with the absolute value of the modulating voltage ψ_e linearly when the driving voltage remains φ₁=0.263 7 V. When the absolute value of the modulating voltage is larger than 50 mV, both the dielectrophoretic and hydrodynamic forces of the particles increase, and the trapping rate decreases and does not vary regularly with ψ_e. When the absolute value of the modulating voltage is smaller than 50 mV, both the dielectrophoretic and hydrodynamic forces of the particles decrease, and the trapping rate can keep the optimal value for an interval and also does not vary linearly with ψ_e. By adjusting the modulating voltage, both the dielectrophoretic force and the hydrodynamic force of the particles can be changed. It is hard to estimate which one takes the dominant position in guiding motions of the particles.

The variations of the channel flux Q with the modulating voltage ψ_e when the trapping rate of the large particles keeps the optimal value ζ_L=100% are shown in Fig. 7. It can be seen that, by enhancing the absolute value of the modulating field, the channel flux increases when the trapping rate reaches the optimal value ζ_L=100%, by adjusting the driving voltage φ₁. The particle separation efficiency can be improved by increasing the modulating voltage.

4.3 The effect of electrode length on the trapping rate and channel flux

The length of the modulating electrode has significant influence on the non-uniformity of the local modulating field, and thus has a relationship with the dielectrophoretic force and hydrodynamic force of the particles. Figure 8 shows the variation of the channel flux Q with the electrode length when ψ_e=-62.5 mV and the trapping rate of the large particles keeps ζ_L=90% by adjusting the driving voltage φ₁. With the same modulating voltage, when the modulating electrode gets longer, the non-uniformity of the local electric field becomes weaker, which results in a smaller dielectrophoretic force. To get the large particles trapped by the vortexes, the driving voltage should be reduced to weaken the hydrodynamic force of the particles. When the length of the modulating electrode is L_e=25 μm and the trapping rate reaches ζ_L=90%, the driving voltage is φ₁=0.028 7 V and the corresponding channel flux is Q=4.804 1×10^-12 m³/s. When the length of the modulating electrode is reduced to L_e=20 μm and the trapping rate remains ζ_L=90%, the driving voltage is φ₁=0.515 9 V and the corresponding channel flux is Q=6.932 1×10^-11 m³/s, which is much larger. When the length of the modulating electrode is further reduced to L_e=15 μm, the dielectrophoretic force is dramatically increased due to the more strong disturbance, and both large and small particles can be easily trapped by the vortexes. A much larger driving voltage φ₁=2.866 3 V is needed to ensure that all small particles are taken to the downstream, while the large particles are trapped by the vortexes. The corresponding channel flux is Q=2.902 7×10^-10 m³/s. By decreasing the length of the electrode, the channel flux at the same trapping rate ζ_L=90% can be obviously increased, which prominently improves the separation efficiency. However, an electrode shorter than L_e=20 μm brings difficulties in manufacture. Both the shorter electrode and the higher modulating voltage may lead to the thermal effect that cannot be neglected. Thus, the optimization design of the electrode is important in practical applications.

4.4 The effect of channel height on the trapping rate and channel flux

The effect of the height of the microchannel H on the trapping rate of large particles and the channel flux is also studied in this work. Keep the length and the voltage of the modulating electrodes as L_e=20 μm and ψ_e=-50 mV, and the driving voltage is φ₁=0.263 7 V. The variations of channel flux and trapping rate with the channel height are shown in Fig. 9. The channel flux increases linearly with the height channel H. The trapping rate of the large particles is ζ_L=100% when the channel height is H=120 μm according to Fig. 5, and the large particles trapped by the vortexes and all small particles transported to the downstream can be seen in Fig. 10(b). When the channel height is smaller than 120 μm, the dielectrophoretic force increases due to the more non-uniformity of the modulating field, and the vortexes grow stronger to trap particles. Both large and small particles are trapped by the vortexes when H=100 μm, and ζ_L=0, as shown in Fig. 10(a). On the contrary, when the channel height is larger than 120 μm, the dielectrophoretic force is decreased, and the vortexes are weakened to trap particles. All particles are transported to the downstream when H=140 μm and ζ_L=0, as shown in Fig. 10(c).

5 Concluding remarks

A new method is proposed in this paper to continuously separate different sized particles by the combination of dielectrophoretic force and field-modulated electroosmotic vortex. The separation process of two different sized particles and the optimization of the relevant parameters are numerically studied. A pair of symmetrical modulating electrodes are assembled on the microchannel walls to generate two symmetrical vortexes and the dielectrophoretic forces for the particles. The results indicate that the large particles are successfully trapped by the vortexes from the mixture of two different sized particles, while the small particles are transported to the downstream, and the optimal trapping rate ζ_L=100% can be achieved. The channel flux increases linearly with the driving voltage, the modulating voltage, and the channel height. It is more effortless to achieve a higher trapping rate by adjusting the driving voltage rather than the modulating voltage, which affects both the dielectrophoretic force and the hydrodynamic force and brings difficulty in estimating the trapping rate. By increasing the modulating voltage, a higher channel flux can be obtained at the same trapping rate via adjusting the driving voltage, and it makes the separation more efficient. The manufacturing difficulty and the thermal effect must be taken into consideration in the electrode design. The interdigitated electrode array with lower voltage may be applied to avoid the thermal effect. When the channel height gets smaller, the vortexes become stronger to trap the large particles. However, if the channel height is comparable to the particle size, the particle interaction needs to be sufficiently considered, which may have a negative effect on the particle separation.