## 3 Results and Discussion
### Developing Microfluidic Pump and Mixer Driven by Bipolar Janus AC-FFET
#### 3.4.1 Geometry Dependence of the Integrated Microfluidic Pump and Mixer  
The values of various geometry and voltage parameters were chosen carefully for the preliminary numerical study of the active microfluidic pump and mixer embedding an integrated Janus pillar array: the channel width, \(W_{C}=100~{}\mu\)m; the channel length, \(L_{C}=1000~{}\mu\)m; the radius of the pillar, \(R=5~{}\mu\)m; the inter-pillar separation, \(L_{G}=80~{}\mu\)m; the signal frequency, \(f=200\) Hz; and the driving voltage \(V_{D}=2\) V. The gate voltages for the odd-the numbered and even-numbered posts were \(V_{G\text{-odd}}=1.8\) V and \(V_{G\text{-even}}=0.2\) V, respectively, together with an equal but opposite gate-offset ratio, \(\beta_{G\text{-odd}}=-\beta_{G\text{-even}}=0.8\). We are primarily interested in how the different geometry factors affect the device's performance. For this reason, we must directly compare the unbiased powering mode and the biased counterpart. In the no-bias version in a symmetric powering scheme, the metal phases of all of the Janus pillars are free from external wiring, i.e., \(V_{G\text{-odd}}=V_{G\text{-even}}=0.5V_{D}\) = 1 V with \(\beta_{G\text{-odd}}=\beta_{G\text{-even}}=0\); namely, they are naturally polarized by the driving voltage applied across the two opposing DE plates in the absence of gate-voltage supply. In the biased mode in asymmetric powering conditions, the metal phase of the adjacent units in the pillar array is exposed to two external gate terminals of equal but opposite gate-voltage offsets, i.e., \(V_{G\text{-odd}}=V_{D}-V_{G\text{-even}}=1.8\) V with \(\beta_{G\text{-odd}}=-\beta_{G\text{-even}}=0.8\) for a given \(V_{D}=2\) V, in the presence of bipolar AC field-effect flow control. * (a) Effect of the radius of Janus cylinder As shown in the analysis in Section 3.3.2, a change in the radius of the Janus pillar R can simultaneously affect the ICEO fluid dynamics in terms of changes in \(u_{x}\), \(u_{y}\), and \(u_{y}/u_{x}\), all of which are of great importance in the resulting device performance. Thus, in this subsection, we focus on the influence of R on the desired dual functionality of the proposed microfluidic device. We first show the basic physics of the fluids underlying the improvement in the sample mixing, as the AC's powering condition is switched from the conventional unbiased mode to the more advanced biased mode (Figure 8A-D). As shown in Figure 8A, without any gate-voltage bias, since all the elements in the post array float in electrical potential, two symmetrical eddies appear selectively on the conducting side of each Janus pillar, leading to a net pump-flow component along the channel length, from left to right. Since there is no net laterally rotating fluid motion, although the diffusion interface between the two stratified streams is periodically split and recombined when passing each array unit, the contribution of this almost unidirectional electroconvection behavior to the mixing performance is quite limited, and the molecular-diffusion effect serves as the chief mechanism for interfacial mass exchange across a sharp gradient of the sample's concentration, resulting in a quite low mixing performance, M\({}_{\text{S}}\) = 35.5%, by the unbiased mode (Figure 8B). Upon the application of oppositely biased gate voltages to the sequentially placed pillar unit with \(\beta_{\text{G-odd}}=-\beta_{\text{G-even}}=0.8\), however, strong flow circulation appears around each post and rotates counterclockwise (clockwise) in front of the odd-numbered (even-numbered) array units, respectively, physically originating in the bipolar AC field-effect flow control (Figure 8C). The diffusion-phase interface can then be stretched well by these chaotic ICEO eddies after undergoing sequential rotating motion in alternating directions (Figure 8D).