Model simplification
The intricate nature of solving differential equations involves strongly coupled multi-physics phenomena in numerical simulations of microfluidics. This model often entails challenging convergence and computational difficulties. We choose to judiciously simplify the microchannel model before embarking on the numerical simulations.
The electric field and potential generated by the bipolar electrode reaction is shown in Fig. 3. The electric field varies the most at the ends of the bipolar electrode and changes relatively slowly at other positions of the bipolar electrode47. The potential in Fig. 3 shows that the overpotential is highest at positions adjacent to the bipolar electrodes.
The electric field and current density distribution within the microchannel are depicted in Fig. 4. These results show that the electric field is highest at positions adjacent to the bipolar electrodes, and the current density induced by the reaction is also maximized corresponding to the locations. It can be seen that the oxidationâreduction reaction occurs most intensely at the ends of the bipolar electrode from these results. Therefore, we model the ends of the bipolar electrode to replace the entire bipolar electrode, which can reduce model complexity.
The structure of the separation channel is shown in Fig. 5. The OHâ generated by the cathodic reaction increases the conductivity near the cathode. The electric field gradient is changed by increased conductivity, which alters the magnitude and direction of the electric migration. As a result, the plastic particles flow out of the upper channel to achieve the separation effect. The influence of parameters such as the applied voltage, the separation channel angle, and the distance of the bipolar electrode on the separation efficiency of plastic microbeads is investigated.
Boundary conditions
Concerning the schematic diagram of the bifurcated channel in Fig. 5, the main boundary conditions for this system are the surfaces of the bipolar electrodes. The oxidation reactions occur at the anode of the bipolar electrode producing H+. The electrode reaction equation is shown below:
$${2} {\text{H}}_{{2}} {\text{O}} \rightleftharpoons {4} {\text{H}}^{ + } {\text{ + O}}_{{2}} {\text{ + 4e}}^{ – }$$
The current density generated at the anode reaction obeys the ButlerâVolmer equation shown below48:
$$i_{loc} = i_{0,a} \left[ {\exp \left( {\frac{{\alpha \eta_{a} }}{RT}} \right) – \exp \left( {\frac{{ – \beta \eta_{a} }}{RT}} \right)} \right]$$
(10)
$$\eta_{a} = \phi_{s} – \phi_{l} – E_{eq,a}$$
(11)
In Eqs. (10) and (11), iloc represents the local current density for the anodic reaction; i0 denotes the exchange current density for the anodic reaction; α and β are the transfer coefficients for the anodic and cathodic reactions, respectively; R is the universal gas constant; T stands for temperature; ηa denotes the overpotential at the anode of the bipolar electrode; Ïs represents the potential of electrode and Ïl is the potential of the solution near the bipolar electrode; Eeq,a represents the electrode potential at the temperature T solved by Nernst equation. The equation is shown below:
$$E_{eq,a} = E_{0,a} – \frac{RT}{{nF}}\ln \left( {\frac{{C_{{[{\text{H}}^{ + } ]}} }}{{C_{{0,[{\text{H}}^{ + } ]}} }}} \right)$$
(12)
In Eq. (12), E0,a represents the standard electrode potential for the anodic reaction; F is the Faraday constant; n is the number of electrons involved in the reaction transfer; \({C}_{[\text{H}^{+}]}\) is the surface H+ concentration at the bipolar electrode, and \({C}_{0,{[\text{H}^{+}]}}\) is the H+ concentration in the bulk solution.
The rate of chemical reaction of the H+ follows Faraday’s law:
$$R_{{[{\text{H}}^{ + } ]}} = \frac{{v_{{[{\text{H}}^{ + } ]}} i_{loc} }}{nF}$$
(13)
In Eq. (13), \({v}_{{[\text{H}^{+}]}}\) represents the stoichiometric number of H+ in the anodic reaction.
The chemical reaction equation for producing OHâ at the cathode of the bipolar electrode is as follows:
$$2 {\text{H}}_{2} {\text{O + 2}} {\text{e}}^{ – } \rightleftharpoons 2 {\text{OH}}^{ – } + {\text{H}}_{2}$$
The cathodic reaction at the bipolar electrode also follows the Butler-Volmer equation and Faraday’s law:
$$i_{loc} = i_{0,c} \left[ {\exp \left( {\frac{{\alpha \eta_{c} }}{RT}} \right) – \exp \left( {\frac{{ – \beta \eta_{c} }}{RT}} \right)} \right]$$
(14)
$$\eta_{c} = \phi_{s} – \phi_{l} – E_{eq,c}$$
(15)
$$E_{eq,c} = E_{0,c} – \frac{RT}{{nF}}\ln \left( {\frac{{C_{{[{\text{OH}}^{ – } ]}} }}{{C_{{0,[{\text{OH}}^{ – } ]}} }}} \right)$$
(16)
$$R_{{[{\text{OH}}^{ – } ]}} = \frac{{v_{{[{\text{OH}}^{ – } ]}} i_{loc} }}{nF}$$
(17)
where i0,c represents the exchange current density for the cathodic reaction, ηc is the overpotential for the cathodic reaction; Eeq,c is the electrode potential for the cathodic reaction at the temperature T; E0,c is the standard electrode potential for the cathodic reaction; \(C_{{[{\text{OH}}^{ – } ]}}\) and \(C_{{0,[{\text{OH}}^{ – } ]}}\) represent the OHâ concentration near the cathodic electrode and in the bulk solution, respectively; n is the stoichiometric number of OHâ.
The boundary conditions applied at the inlet of the microchannel: (1) specified ion concentration c0,i; (2) specified voltage V0; and (3) pressure is set to 0:
$$c = c_{0,i} ,\;\;\; V = V_{0} ,\;\;\; p = 0$$
(18)
The boundary conditions applied at the outlet of the microchannel: (1) zero diffusion flux, (2) zero voltage, and (3) pressure is set to 0:
$${\mathbf{n}} \cdot D_{i} \nabla c_{i} = 0,\;\;\; V = 0,\;\;\; p = 0$$
(19)
The boundary conditions imposed at the microchannel walls: (1) zero flux, (2) electrical insulation, and (3) the velocity of electroosmotic flow within the microchannel follows the HelmholtzâSmoluchowski equation49:
$$- {\mathbf{n}} \cdot {\mathbf{J}}_{{\mathbf{i}}} = 0,\;\;\; {\mathbf{n}} \cdot {\mathbf{D}} = 0,\;\;\; {\mathbf{u}} = – \frac{{\zeta \varepsilon {\mathbf{E}}}}{\eta }$$
(20)
where ζ represents the potential difference between the diffusion layer and the bulk solution, E is the electrical field strength inside the microchannel, Eâ=ââ âV, and V is the applied potential within the solution.
Model verification
We use the straight channel model commonly used in the field of microfluidics to verify the feasibility of the simulation method to ensure that the simulation results are real and valid. The results of the simulations are compared with the experimental results of other researchers to prove that the simulation method is feasible.
Ionic current density is depicted in Fig. 6. Because of the electrode reaction of the model, the result of ionic current density can vary along the length of the channel. These variables changed most significantly around the bipolar electrodes. Therefore, the values of current density near the three bipolar electrodes are extracted and compared with the experimental data. The results of the simulation show an almost equal effect to those of other researchers46, which means a correct simulation model.
The ionic current density is measured at three positions in the experiments of other scientific researchers, which are 500 μm of the electrode cathode (upstream of BPE), 1500 μm downstream of the electrode cathode (between BPE poles), and 1000 μm downstream of electrode anode (downstream of BPE). These results are shown in Table 1. The results show that the simulation results are not much different from the experimental data within the allowable range of error, and the results obtained by the simulation model are valid.
Particle separation in the channel
The separation microchannel structure is shown in Fig. 5. A solution of KCl with a concentration of 5 mol/m3 (\({z}_{{\text{K}^{+}}}\) =â1, â\({z}_{{\text{cl}^{-}}}\) =ââ 1, â\({z}_{{\text{H}^{+}}}\) =â1, â\({z}_{{\text{OH}^{-}}}\) =ââ 1, â\({D}_{{\text{K}^{+}}}\) =â1.97âÃâ10â9 m2/s, â\({D}_{{\text{cl}^{-}}}\) =â2.033âÃâ10â9 m2/s, â\({D}_{{\text{H}^{+}}}\) =â9.103âÃâ10â9 m2/s, â\({D}_{{\text{OH}^{-}}}\) =â5.28âÃâ10â9 m2/s) is used at the inlet of the channel. In order to ensure the validity of the model and to be able to implement the simulation function, we assume microplastics are evenly distributed in solution. A polystyrene microplastic particle is used and its concentration is 3âÃâ10â12 mol/m3. Although environmental factors lead to different charges of microplastic particles, we choose a special kind of polystyrene microplastic, whose charge is â 2 (zBeadâ=ââ 2). Microbeads can be affected by diffusion in this simulation, so the diffusion coefficient is considered, which is 7.85âÃâ10â8 m2/s (DBeadâ=â7.85âÃâ10â8 m2/s). Additionally, the diameter of particles is 0.99 μm. The relative permittivity of the solution is 80, and the zeta potential is â 80 mV. The governing Eqs. (2), (3), (5), (6) and (7) are solved using the commercial finite element software Comsol Multiphysics V6.0. By varying the applied voltage V0, separation channel angle θ, and position of the bipolar electrode d, the microfluidics separation mechanism and the impact on the efficiency of microchannel separation are investigated.
The geometry of the used microfluidic channel is depicted in Fig. 7. Upon applying a voltage V0 on both sides of the separation channel, the bipolar electrodes at the bottom of the channel become activated. Microplastic particles flow out from the top of the channel, while purified water flows out from the bottom of the channel.
Effect of applied voltage on separation efficiency
When the applied voltage V0 is varied, the magnitude of the overpotential on the bipolar electrode is changed, which affects the oxidationâreduction reaction rate. Finally, the reaction rate impacts the separation efficiency of microparticles.
The concentration distribution of microplastic particles is illustrated in a partial model of the bifurcation channel in Fig. 8, indicated by the red dashed box in Fig. 7. cm,avg is the average concentration of microplastics in the top channel. The electrochemical reaction rate near the cathode accelerates as the voltage V0 increases, which leads to a change in the nearby electric field strength. This alteration impacts the force experienced by microplastic particles in the solution and results in a decrease in particle concentration in the bottom channel and an increase in concentration in the top channel.
We simulated the distribution of flow and electric fields to consider the convection and electromigration of microplastics. The flow rate of electroosmotic flow is accelerating with the increase of the applied voltage V0, which is depicted in Fig. 9a. Electroosmotic flow is predominant in the motion of particles away from the bipolar electrode. Additionally, the properties of fluid flow become complex near the electrode for the changing electric field.
We can also observe the changing electric field in Fig. 9b. The reaction of the electrode affects the electric field in the bipolar electrode cathode region, which generates a large electric gradient. the direction and magnitude of electroosmotic flow and electromigration in this region, therefore, they can affect the motion trajectory of microplastics.
To study the factors affecting the electric field, the simulation was conducted to model the ion concentrations at different voltages. Figure 10 illustrates the distribution of OHâ ion concentration along the centerline of the bottom channel as indicated by the red dashed line in the schematic of Fig. 7. With the increase in voltage, the concentration of hydroxide ions near the cathodic increases. The increased concentration indicates a faster chemical reaction rate at the bipolar electrode.
The distribution of the electric field at the same position is shown in Fig. 11. The change in the chemical reaction rate results in a greater variation of the electric field near the cathode. The varying electric field alters the trajectory of microplastic particles.
The curves represent the electric field strength at xâ=â1900 μm in Fig. 12, as indicated by the black dashed line in the schematic of Fig. 7. With the increase in the applied voltage V0, the overall electric field near the cathode rises. This phenomenon enhances the electric migration effect experienced by microplastic. The initial equilibrium is disrupted because of the increasing electric migration effect. The microplastic particles move toward the lower electric field direction.
Figure 13 depicts the distribution of microplastic concentration at the same location. Due to the alteration in electric field strength, the forces acting on microplastic particles are modified and influence the concentration of microparticles near the cathode. Lastly, the force affects the trajectory of the particles.
To investigate the influence of different voltages on the separation factor of plastic, we define the separation factor r as the ratio of the concentration of the tiny plastic in the top channel to that in the bottom channel.
$$r = \frac{{c_{m,up} }}{{c_{m,bottom} }}$$
(21)
cm,up and cm,bottom represent the microplastic particle concentration in the top channel and the bottom channel, respectively. A higher value of r implies a greater proportion of microplastic particles exiting from the top channel, which indicates a more effective separation. Conversely, a lower r implies poorer separation.
The variation of the separation factor under different voltages is illustrated in Fig. 14. As the voltage V0 increased from 30 to 50 V, the separation factor increases by up to about 50%. The relationship between voltage and separation factor is as follows:
$$r = 0.62 + 0.081V_{0}$$
(22)
Due to the large electromigration effect of the accelerated reaction electrochemical reaction rate, this phenomenon indicates an improved separation efficiency with the development of voltage. However, with the increase in voltage, more secondary reactions occurred on the bipolar electrodes. These reactions can render the separation of microplastic particles uncontrollable. Simultaneously, escalating the voltage leads to an increase in the temperature within the microchannels because of the generation of more heat. Therefore, it is imperative to comprehensively consider the influence of voltage based on the actual application scenario.
Effect of separation channel angle on separation efficiency
The varying angle θ of the separation channel alters the flow patterns of the fluid, which causes a shift in the locations where H+ and OHâ accumulate. This change affects the magnitude of the electric and the trajectory of the microplastic particles field within the microchannel. The microplastic concentration and flux near the cathode are depicted in Fig. 15. The average concentration of the upper channel is almost constant, and the flux changes very little at angles between 5° and 30°. When the angle is greater than 30°, the flux of microplastic particles changes significantly.
To further explore the above changes, we simulate the electric field inside the microchannel. The distribution of the electric field is illustrated along the midline of the bottom channel in Fig. 16. The conductivity of the solution changes very little because the overall concentration in the solution barely changes at angles between 5° and 30°. Therefore, the electric field does not change much in this range. However, the local electric field strength changes due to the change in particle flux at more than 30°.
The relationship between the angle and the separation factor is depicted in Fig. 17. The relationship between angle and separation factor is as follows:
$$r{ = }3.017 + 0.001\theta$$
(23)
As the angle θ of the bifurcation channel continuously increases, the electric field strength remains relatively constant. Therefore, the magnitude of the separation factor ranges from 3.02 to 3.06, showing a subtle variation in the separation efficiency. It manifests that the varying angle has little effect on separation efficiency.
Effect of bipolar electrode distance on separation efficiency
The cathode position of the bipolar electrode represents the most dynamically changing region of the electric field. In this simulation, we keep the position of the anode fixed while altering the distance d between the two electrodes to change the cathode position. Altering the cathode position leads to variations in the most dynamically changing region of electric field strength. Consequently, the electro-migration effect experienced by microplastic particles is altered and affects the trajectory of plastic microbead motion.
The concentration distribution of microplastic particles is presented in the channel for different cathode positions of the bipolar electrode in Fig. 18. Figure 18 shows that the concentration of microplastic particles rises in the bottom channel and decreases in the top channel as distance d between the two electrodes increases. The phenomenon indicates a weakening of the separation efficiency.
To further analyze the results of the microplastic concentration distribution, we conduct additional simulations involving electric field and microplastic particle concentration. Figure 19 shows the electric field at the centerline position of the bottom channel.
It is evident that the electric field near the bipolar electrode increases significantly from 7.5 to 7.9 kV/m when the cathode of the bipolar electrode moves towards the positive voltage side. This increase is attributed to an increased potential near the inlet of the runners. The rate of chemistry reaction is increased by elevated potential, which leads to increased solution conductivity. Significant variations in the electric field gradient occur near the bipolar electrode.
However, due to the premature change in the electric field gradient within the microchannel, microplastic particle concentrations become re-mixed after separation. It fails to achieve the desired separation effect. When the bipolar electrode is close to the bifurcation point of the channel (xâ=â2000 μm), the electric field is the highest near the bifurcation point. Microplastic particles experience a substantial electro-migration effect, resulting in more thorough separation.
Figure 20 represents the microparticle concentration at xâ=â1900 μm. It is observable that the curve becomes smoother for the increasing distance d, indicating a poorer separation efficiency. When the distance d is 2900 μm, the ratio of the highest concentration to the lowest concentration can reach 7.95, which shows an effective separation.
Figure 21 illustrates the relationship between the separation factor r and the distance d between the bipolar electrodes. The expression of this relationship is as follows:
$$r \, = \, 6,418,435.891{\text{e}}^{{ – \frac{d}{190.981}}} + 1.44$$
(24)
The separation factor exhibits an exponential decay as distance increases in Fig. 21. The electric field is highest near the bifurcation point in the smallest distance, resulting in the strongest electro-migration effect and better separation efficiency. Conversely, the overall increased field strength is unable to achieve effective separation due to being far from the bifurcation point, which leads to a decrease in the separation factor.
To further confirm the separation effect of the microchannels, the separation factor without bipolar electrode action is calculated in Fig. 21. The results show that the separation factor without bipolar electrodes (BPE inactive) is always 1, which means that there is no significant separation effect without bipolar electrodes. On the contrary, the separation factor with the bipolar electrode (BPE active) is higher than 1. This phenomenon indicates that the concentration of the top channel is higher than that of the bottom channel, which means a good effect of separation.
The outlet concentration of microplastics is measured to clearly illustrate the effect of microchannel separation. These results depicted in Table 2 show that the outlet concentration is consistent under the condition of no bipolar electrode. In addition, the top outlet concentration is greater than that of the bottom outlet concentration under the influence of bipolar electrodes. The concentration of the lower channel is reduced to less than half of the original level when the separation factor reaches more than 2, which means an acceptable range in this study.