MIXING ENHANCEMENT IN ELECTROOSMOTIC MICROMIXERS

Micromixers have important applications in various pharmaceutical and medical fields. In the present study, the enhancement of mixing index in electroosmotic micromixer with different geometries is investigated. The commercial software COMSOL Multiphysics 5.4 is employed to solve the mathematical models. The SIMPLEC algorithm is employed for coupling the velocity and pressure fields. A second-order upwind scheme is used to reduce the artificial diffusivity. The results show a remarkable effect of the electric field on the mixing efficiency. The optimum geometry is the one with no obstacle in the mixing chamber. For the optimum geometry, it is demonstrated that the mixing efficiency increases with the voltage, however there are optimum values for frequency and inlet velocity in which the micromixer exhibits its best performance. The optimum values of frequency and inlet velocity are 8 Hz and 0.1 mm/s, respectively. It is revealed that the micromixer with no obstacle can reach the mixing efficiency of about 97%.


INTRODUCTION
Micromixers are widely used in chemical reactions, medical industries, and biomechanics [1]. In the biological industries, rapid mixing of large molecules such as proteins, nucleic acids and various biofluids has attracted many attentions [2]. Micromixers are divided into two categories: active and passive micromixers. In active micromixers, mixing is performed using external forces such as electric [3,4], magnetic [5], acoustic [6], thermal [7], etc. Passive micromixers do not require external forces and work based on their geometries [8]. Among these external forces, the electrokinetic force is divided into four types: electroosmotic, electrophoretic, streaming potential and sedimentation potential [9]. An applied voltage can drive a liquid flow in a capillary tube. If slip boundary condition is imposed on the tube walls, the electrochemical double layer results in a streaming profile. This is a principle of the electroosmosis. Electroosmosis is one of the main mechanisms of electrokinetic that works based on the induced charge in the fluid phase due to contact with the dielectric surface [10,11]. Since the Reynolds number in micromixers is of order of 10, the flow regime is laminar and mixing is mainly due to the molecular diffusion. Thus, external energy sources can be used to perturb the flow field and improve the mixing performance.
Many researchers have investigated the mixing performance of electroosmotic devices [12][13][14][15][16]. Bhattacharyya and Bera [12] investigated the mixing index in a long microchannel considering pressure-driven electroosmotic flow. They placed a rectangular block on a wall of channel and demonstrated that the vortical flow generated at the rear of the block depends on the Debye length. It was revealed that as the vortex becomes stronger, the mixing efficiency enhances. Peng and Li [13] showed that the uniformity of flow velocity and electric field in a microchannel depends on the value of ionic concentration of the electrolyte solution. They demonstrated that mixing efficiency is low for two solutions when one of them has very high ionic concentration and another has low concentration. Zhang et al. [14] used planar floating electrodes to enhance the mixing index and reached the mixing efficiency of 94.7% for flow rate of 1500 μm/s. they showed that the mixing index increases with the voltage and decreases with the inlet velocity and fluid viscosity. It was concluded that there is an optimal value for the mixing index when frequency is changed. Wu and Chen [15] used fractal structure to study the influence of electrodes on the mixing performance. They employed an alternating voltage of 5 V and reached the mixing index of 95.2% compared to the electrodeless micromixer with the efficiency of 50% at the same volume flow rate. Usefian and Bayareh [16] proposed a novel electroosmotic micromixer with a convergent chamber connected to a straight microchannel. They 48 employed AC and DC electric fields and revealed that the strength of the vortices created by DC electric field is higher than that of AC one. It was shown that the mixing efficiency enhances with the voltage and decreases with the inlet velocity.
According to the previous researches, micromixer geometry and arrangement of electrodes have significant influences on the mixing performance of electroosmotic micromixers. In the present study, the effect of four obstacles placed in a hexagonal chamber on mixing performance of a micromixer is compared with a no-obstacle one to evaluate the formation of vortices within the channel. The optimal geometry is realized and then the influence of voltage, inlet velocity and frequency on its mixing quality is investigated.

GOVERNING EQUATIONS
The governing equations for mixing of Newtonian incompressible fluids are continuity and momentum equations [15]: where ⃗ is the velocity vector, p the pressure, ρ the density and μ the dynamic viscosity. Since electroosmotic slip velocity is employed for micromixer walls, ⃗ is ignored in the momentum equation. By applying this boundary condition, electroosmotic flow is generated, leading to that the liquid moves in the direction of the electric field with the following velocity [16]: where = −0.1 V indicates zeta potential, ⃗ is the applied electric field, = 80.2 is dielectric constant, and = 8.854 × 10 −12 C/Vm is the permittivity of vacuum.
The electric potential applied in the liquid is calculated by the Laplace equation: where ⃗ = −∇ . Boundary condition is imposed on the electrodes are presented in Eq. 5 for AC electric field [16]: where is the maximum AC potential and is the frequency. Mixing index is calculated as follows [16]: where ̅ is the average concentration, is the concentration at each point and N is the number of nodes in the considered area.

VERIFICATION
In order to verify the present results, the results of Keshavarzian et al. [17] who considered the effect of electrical potential on a micromixer, are used. Figure 1 shows a schematic of their micromixer and Table 1 presents Journal of Thermal Engineering, Research Article, Vol. 7, No. 2, Special Issue 13, pp. 47-57, February, 2021 49 its parameters. They mounted 8 electrodes on mixing chamber wall and inner circular obstacle symmetrically. Figure  3 shows the concentration as a function of channel width obtained from the present work and those reported by Keshavarzian et al. [17]. This figure illustrates that there is good agreement between the present results and those of Keshavarzian et al. [17] with the maximum error of 1.28%.

Grid Study
In this study, tetrahedral grids are used for the computational domain [18]. Nine grid resolutions (from extremely coarse to extremely fine) are employed to determine the appropriate grid for the case a of Figure 3. Figure  4 demonstrates that further increase in the grid points of finer case does not lead to significant change in the results. Thus, finer grid resolution with 682390 elements is used for further simulations. It should be pointed out that this grid is appropriate for all cases considered in Figure 3.

Determination of Optimal Geometry
To determine the optimal geometry, mixing quality is calculated for five cases considered in Figure 3. AC electric field with the frequency of 8 Hz and electric potential of 3 V is applied to the fluid flow within the micromixers. The inlet velocity is 0.1 mm/s and number of nodes is N = 20 for each cross section. Figure 5 shows the concentration contours for different geometries. It can be observed that the case a has the best mixing performance qualitatively. To quantify the mixing index, Equation 6 is used and the corresponding results are shown in Figure 6. This figure confirms that the geometry with no obstacle has the maximum mixing efficiency than other cases. The mixing quality of geometries a, b, c, d, and e at their outlets is 91.67, 73.92, 71.42, 59.44, and 64.99 respectively. Figure 7 illustrates the streamlines throughout different micromixers for different micromixers at Uo = 0.1 mm/s, v= o.1 V, f = 8 Hz. This figure shows the considerable effect of electric field on the streamlines for all cases. The vortices are created around the obstacles and close to the electrodes. These vortices lead to the circulation of the streams, resulting in an enhancement of mixing quality. It is worth noting that the vortices in the mixing chamber of the micromixer without obstacle circulate more regions of the two flows, while the vortices of other micromixers circulate each stream separately in the regions between the wall and the obstacles. Thus, the mixing rate in the micromixer with no obstacle is higher than those with obstacles.

Voltage Effect
In this section, the effect of voltage on the performance of micromixer is studied. It was known that as the voltage increases, the mixing quality enhances due to higher influence of electroosmotic force than inertial one [16]. The strength of vortices depends on the electroosmotic force. Figure 8 illustrates the variations of mixing quality for different values of the voltage. It can be observed that as the voltage increases, the mixing index is enhanced. It varies from 64.77% for the voltage of 0.05 V to 96.99% for the voltage of 0.7 V. Streamlines are plotted in Figure 9 for these two voltages. This figure demonstrates that the vortices occupy greater area in the chamber as the voltage enhances, leading to higher rate of mixing. As the voltage increases, the two main vortices circulate more parts of the fluid flow at the interface due to stronger electroosmotic force.  Figure 10 shows the values of mixing index for different inlet velocities of 0.05, 0.08, 0.1, 0.15, 0.2 and 0.5 mm/s. This figure demonstrates that inlet velocity of 0.1 mm/s leads to maximum mixing efficiency. In other words, there is an optimal value for inlet flow velocity. As mentioned before, there is a competition between inertial and electroosmotic forces in electroosmotic micromixers. Hence, three cases can be occurred: i) mixing efficiency increases with the inlet velocity, ii) mixing efficiency decreases with the inlet velocity, and iii) mixing efficiency has an optimal value. These conditions depend on the voltage, frequency, phase amplitude and geometrical parameters. In the present study, case (iii) happens, i.e. an optimal value is achieved for different inlet velocities. The value of mixing index is calculated at the channel outlet for different inflow velocities. It is 84. 54 increases, the inertial force becomes stronger. Electroosmotic force overcomes the inertial force for inlet velocity of 0.1, resulting in the formation of stronger vortices ( Figure 11). As shown in Figure 11, for the case of Uo = 0.5 mm/s, inertial force prevents the creation of strong vortices and mixing quality decreases considerably.

Frequency Effect
In this section, the influence of frequency is examined on the mixing performance of the micromixer a. The frequency has the values of 4, 6, 8, 10, 12, 16 Hz. Figure 12 shows the mixing efficiency for v = 0.1 V, Uo = 0.1 mm/s and different frequencies. The figure reveals that there is an optimum value for the frequency (f = 8 Hz), in which the mixing quality has maximum value of 91.69% at channel outlet. The variation of mixing efficiency with respect to the frequency is similar to that of inlet velocity. In other words, the effective parameters such as frequency, voltage, inlet velocity and micromixer geometry determine the existence of optimal value for mixing index. For example, Usefian and Bayareh [16] showed that the mixing efficiency enhances with the frequency monotonically for their proposed micromixer. Figure 13 illustrates the streamlines through the micromixer for the optimal frequency and the one in which the mixing index is minimum. It can be observed that the strength of two main vortices is affected by the amount of the frequency. Electroosmotic force is larger for f = 8 Hz compared to f = 16 Hz. In other words, the deviation of interface is smaller when the frequency increases from 8 Hz to 16 Hz.

CONCLUSION
In the present paper, the effect of different micromixer geometries on the mixing efficiency was investigated. It was found that the one with no obstacle in the mixing chamber has the maximum mixing quality. Then, the effect of the voltage, inlet velocity and frequency on the mixing performance of the optimum micromixer was examined. There is a competition between inertial and electroosmotic forces in electroosmotic micromixers. Hence, when effective parameters change, mixing efficiency may increase, decrease, or has an optimal value. It was demonstrated that the mixing efficiency increases with the voltage, however there are optimum values for frequency and inlet velocity in which the micromixer exhibits its best performance. The optimum values of frequency and inlet velocity were 8 Hz and 0.1 mm/s, respectively. It was revealed that the micromixer with no obstacle can reach the mixing efficiency of about 97%. There are some suggestions for future works: i) investigation of the effect of electrode arrangement, ii) evaluation of mixing for non-Newtonian fluids in the same geometries, and iii) study of higher magnitudes of Reynolds number.