A novel three-dimensional electroosmotic micromixer based on the Koch fractal principle

Micromixers are extensively used in the ﬁ eld of biochemistry. In this study, based on the Koch fractal principle, a new type of micromixer is designed. To improve the mixing performance of the micromixer, under the action of fractal and electric ﬁ eld, irregular alternating eddy currents were generated in the microchannel to improve the mixing of ﬂ uids. Using COMSOL Multiphysics 5.4, by changing the fractal structure, direct current (DC) voltage size, electrode spacing, electrode con ﬁ guration, and numerical calculations were determined, and a micromixer with excellent mixing e ﬀ ect was obtained. In particular, when the electroosmotic micromixer has three groups of alternately placed electrode pairs and Re ¼ 0.01, the mixing e ﬃ ciency can reach 99%. velocity. The conclusion is that by increasing the applied voltage and reducing the inlet velocity, the mixing performance is improved. The experimental results are in good agreement with numerical results in terms of quality. 23 There are many studies on the e ﬀ ect of fractal structure and electric  eld to improve the mixing e ﬃ ciency of micromixer. In this study, a  er combining the two factors, we developed


Introduction
In recent years, the integration of micro-scale sample preparation, separation and detection into a micro-scale chip has attracted researchers attention. 1-4 Among them, the micro scale uid mixing has always been a hot issue in the eld of scientic research; therefore, a micromixer has become an important component of the chip laboratory system. [5][6][7][8] Therefore, to improve the mixing efficiency of the uid in the micromixer, researchers usually change the geometry of the micromixer or add external conditions to improve the mixing efficiency of the working uid. [9][10][11] Generally, micromixers are divided into two types according to their working principles: passive and active. 12 The passive micromixer has the advantages of simple structure, low manufacturing difficulty and easy integration. Because it is not affected by external forces, the mixing process primarily depends on molecular diffusion. 13 The mixing of samples in the microchannel is generally realized using the inuence of chaotic convection, although the passive micromixer has the advantages of simple structure, low manufacturing difficulty, and easy integration. 14 However, because of its large physical size and poor controllability for uids, it has not been extensively used in the research process. 15 In contrast, the active micromixer has a greater application prospect. Active micromixers usually use some additional active components to mix different uids, such as magnetic eld, electric eld and pressure eld, to achieve effective mixing of different uid samples. [16][17][18] In the active micromixer, electroosmotic micromixer is widely used because of its simple structure and easy integration. 19,20 Jalili et al. conducted a numerical study on an active micromixer driven by osmotic pressure. The optimal value of the effective parameters of the original micromixer is obtained, and it is applied to the micromixer with different obstacle shapes in the mixing chamber. The results show that the mixing quality largely depends on the inlet velocity of the uid, the phase lag of the electrode, frequency and voltage amplitude. Moreover, when the optimal values of these parameters are used, the mixing quality does not depend on the shape of the obstacle. 21 Shamloo et al. proposed a new type of Ttype electroosmotic micromixer and studied the inuence of different parameters on the mixing process. The inuence of the aspect ratio of the channel cross-section and the radius and depth of obstacles was also discussed. Note that the triangular cavity with circular conductive barrier is the best geometric conguration; moreover, please check its time transient performance. The results show that the use of these structures can not only increase the mixing index but also reduce the mixing time. 22 Usean et al. proposed a new type of electroosmotic micromixer in the presence of alternating current (AC) and DC electric elds. PDMS is used to make microchips, and gold nanoparticles are used to make electrodes. The results show that, for both DC and AC electric elds, the mixing performance can be controlled by the applied voltage value and the uid inlet velocity. The conclusion is that by increasing the applied voltage and reducing the inlet velocity, the mixing performance is improved. The experimental results are in good agreement with numerical results in terms of quality. 23 There are many studies on the effect of fractal structure and electric eld to improve the mixing efficiency of micromixer. In this study, aer combining the two factors, we developed a new type of electroosmotic micromixer with the Koch fractal structure. The effects of fractal structure, voltage size, electrode spacing and electrode structure on the mixing performance have also been studied. The simulation results show that the best mixing efficiency of micromixer can reach 99%, which meets our expectation.

Micromixer designs
Fractal obstacles are set up on the basis of the Koch fractal. The steps are as follows: (1) Take a line segment with a length of 0.4 mm.
(2) Divide the line segment into four equal parts, keep the quarter line segments at both ends unchanged, then move the second line segment down by 0.1 mm, move the third line up by 0.1 mm, and nally in sequence connect the four segments.
The fractal structure is shown in Fig. 1 and 2 shows the micromixer with fractal structure.

Governing equations
The uid dynamics analysis uses the following Navier-Stokes (N-S) equation and continuity equation: where the velocity vector is u, the uid density is r, the dynamic viscosity is h, and the pressure is p. Note that eqn (1) is the momentum balance equation, whereas eqn (2) is the continuous equation of incompressible uid.
The convection-diffusion equation can be used to analyze the mixing of uids of different concentrations in the microchannel: where concentration is c and diffusion coefficient is D.
The potential distribution of the electric eld can be expressed by the Laplace equation: In the above formula, V represents a potential (V). Adopting electroosmotic velocity boundary conditions, the local slip velocity given by Helmholtz-Smoluchowski is as follows: Among them, 3 0 , 3 r are the dielectric constant of vacuum and electrolyte solution respectively;x 0 is the zeta potential of the inner wall of the xed wall; and V is the electric potential.
The characteristic dimensionless number of the microchannel and the Reynolds number (Re) corresponding to the ow velocity: where D d is the equivalent diameter, h is the height, w is the width, and n is the kinematic viscosity. The calculation equation of mixing efficiency is as follows: The mixing efficiency is M, N is the total number of sampling points, and c j and c are normalized concentration and the expected normalized concentration, respectively. The mixing efficiency ranges from 0 (0%) to 1 (100%).

Finite element modeling
Based on COMSOL Multiphysics 5.4 soware, the electrolyte solution incompressibly ows, and the uid dynamics boundary conditions do not slip on the wall of the micromixer. The outlet pressure of the microchannel is 0 Pa, the sample concentration of the injection port 1 and the injection port 2 is set to C1 ¼ 1 mol L À1 and C2 ¼ 0 mol L À1 , the electric eld follows the law of conservation of charge, the boundaries of the micromixer are insulated except for the electrode area, zeta  potential is À0.1 V, uid density is 988 kg m À3 , relative dielectric constant is 78, ionic solution conductivity is 0.11845 S m À1 , dynamic viscosity is 10 À3 Pa s, and solution diffusion coefficient is 10 À11 m 2 s À1 . The 3D model of the micromixer was designed using the three-dimensional soware Pro/Engineer 5.0.

Grid independence analysis
The selection of the grid has a great inuence on the simulation results. The correct selection of the grid can improve the accuracy and save the calculation time. Fig. 3 shows the local velocity distribution of the line segment at the exit of the micromixer. The test result increases as the number of grid cells increases. However, the difficulty of simulation will increase as the number of grid cells increases. As per the grid dependency test calculation, the number of grid cells in grid 2 is selected as the best result. According to COMSOL 5.4, the complete grid information of grid 2 selected is counted, and the number of cells is 366 033, the minimum cell mass is 0.001271, and the average element can reach 0.5866, tetrahedron the number of units is 295 869, the pyramid cell number is 1816, the prism cell number is 68 348, and the triangle cell number is 36 260.

The effect of electrodes on mixing efficiency
In this study, we compare the passive micromixer with the electroosmotic micromixer with the Koch fractal. As shown in Fig. 4, three pairs of electrodes with opposite polarity are alternately added on the fractal channel wall, and other conditions are the same. Fig. 5 shows that when the Re values are 0.01, 0.05, 0.1, 0.5 and 1, the application of electrodes in the microchannel can effectively improve the mixing efficiency of the solution. Especially, when Re ¼ 0.01, the mixing efficiency can be increased by 64%. Under the action of non-electric eld, the uid mainly ows in laminar ow. The mixing of solution primarily depends on the fractal structure in the microchannel and the diffusion between molecules. At a low Reynolds number, the fractal structure has little effect on the mixing of uid. When the electric eld is applied, the uid in the microchannel forms a vortex under the action of the electric eld. With the help of the fractal structure, the uid motion path is changed so as to improve the uid mixing.

The effect of electrode spacing and voltage on mixing efficiency
As shown in Fig. 6, red is high potential and blue is low potential. Two groups of electrode pairs were selected to study. The electrode spacing was 0.15 mm and 0.3 mm. It can be seen from the    calculation results in Fig. 7 that within the scope of study, the mixing efficiency is effectively improved with the decrease of electrode spacing. As shown in Fig. 7, with the increase of voltage, the mixing efficiency of the two micromixers is effectively improved, especially when the voltage increases from 1 to 3 V. When the voltage is increased, the mixing efficiency is not so obvious.
When the electrode spacing is small, the energy efficiency level is higher. With the increase of electrode spacing, the energy efficiency rapidly decreases in the low voltage stage. When the voltage increases, the energy efficiency difference of different electrode spacing will narrow. Because the ions in the diffusion layer of the electric double layer are affected by the horizontal and vertical electric elds, the ions in the diffusion layer will be disturbed, and the viscosity of the uid will be reduced. Moreover, the ions in the diffusion layer will then drive the surrounding uid molecules to form a vertical motion with the wall. Aer reducing the distance between the electrode pairs, the energy efficiency is increased, thus strengthening the convection, which makes the uid stretch and fold repeatedly in the microchannel and improves the mixing efficiency. Therefore, it is feasible to adjust the electrode spacing and voltage value to control the mixing of the micromixer.

The effect of electrode conguration on mixing efficiency
The charge in the microchannel is primarily related to the applied electric eld, which can form eddy currents near the electrodes and improve uid mixing. Therefore, the direction of the electric eld has a signicant impact on the mixing efficiency. This section studies the inuence of different congurations of electrodes on the mixing efficiency. As shown in Fig. 8, the electrodes are divided into three groups. We have studied eight congurations. The two electrodes in each group have opposite potentials, and the external conditions of the eight congurations are exactly the same. The mixing efficiency of the eight conguration outlets is shown in Fig. 9. The best mixing efficiency is the conguration given in Fig. 8, which is about 95%. It can be seen that changing the potential has a great inuence on the performance of the mixer. Because the electrodes placed in the micromixer change the uid from a uniform ow to a non-uniform ow, the uid is mixed. The potential difference in the microchannel is adjusted to allow more adequate mixing of the working uid. When an asymmetric potential difference is applied to the ow path, greater disturbances are generated, resulting in complex uid disturbances. Therefore, the mixing performance of the electroosmotic micromixer can be improved by changing the potential of the microelectrode. Properly placed potential in microchannels can strengthen the uid to stretch and fold in different directions to achieve the purpose of mixing.

Flow analysis in microchannels
To more intuitively study the effect of the electrode on the uid on the fractal micromixer, we have selected three velocity streamline diagrams under different voltages, and the streamline density of the three diagrams is equal. The color of the streamline indicates the concentration distribution of the uid. Fig. 10a shows that when no voltage is applied to the fractal micromixer, the mixing efficiency at this time is not ideal. Because the Reynolds number is low at this time, the mixing of uids is mainly affected by molecular diffusion and residence time. In the whole process, the fractal hinders the uid, and the weak chaotic convection promotes the mixing of the uid. As shown in Fig. 10b, when the voltage is increased to 3 V in the fractal microchannel, the uid undergoes obvious deection and folding under the action of the electric eld and the fractal  structure. From the streamline diagram, due to the action of electric eld, the uid generated a series of chaotic vortices at the fractal, which promoted the mixing of uid in the microchannel. As shown in Fig. 10c, when the voltage in the microchannel is continuously increased, the chaotic motion of the uid becomes more intense; therefore, appropriately adjusting the electrode voltage on the fractal structure will help the uid to mix.

Conclusions
In this study, using the Koch fractal principle and electroosmotic effect to improve the mixing efficiency of the micromixer, it is conrmed that: (1) Alternating electrodes in a micromixer with the Koch fractal structure is a method to generate vortex and enhance mixing.
(2) Because the ions in the diffusion layer of the electric double layer will be affected by the horizontal and vertical electric elds, the ions in the diffusion layer will be disturbed and the viscosity of the uid will decrease. The surrounding uid molecules vertically move with the wall. Aer reducing the distance between the electrode pairs, the energy efficiency is improved, thereby enhancing the convection and making the uid repeatedly stretch and folding in the microchannel as well as improving the mixing efficiency.
(3) Because the electrodes placed in the micromixer change the uid from a uniform ow to an uneven ow, the uids are mixed. When an asymmetric potential difference is applied to the ow path, greater interference will be generated, resulting in complex uid interference. Therefore, the mixing performance of the electroosmotic micromixer can be improved by changing the potential of the microelectrode. To achieve the purpose of mixing, the placed potential difference will cause the uid to stretch and fold in different directions.
(4) Because of the fractal structure, the contact surface of the two uids is increased, the mixing of the uids is improved, and the microelectrodes can be better placed in the microchannels, which is more conducive to the effect of the electric eld, thereby improving the mixing efficiency of the micromixer.

Conflicts of interest
The authors declare no conicts of interest.