## 3 Results and Discussion
### Developing Microfluidic Pump and Mixer Driven by Bipolar Janus AC-FFET
#### 3.4.2 Effect of the Applied AC Voltage on the Integrated Device's Performance  
Once the device is fabricated with a fixed structure, it is not possible to change the channel geometry any further, and the regulation of the externally imposed AC voltage signal serves as a convenient way to achieve the flexible adjustment of the pumping and mixing performances based on a given kind of analyte dispersed in a standard working fluid. For this reason, it is intriguing to clarify how the dual functionality of the proposed microdevice responds to changes in the applied AC signal in terms of the field frequency, voltage amplitude, and gate-voltage-offset ratio \(\beta\). * (a) Frequency dependence  
As revealed by the basic fluid-mechanics study in Section 3.3, since the conducting DE plates and the metal phases of all the Janus pillars undergo non-negligible electrode polarization, which is potently affected by electrochemical ion relaxation in harmonic AC forcing, the frequency dependence of the two orthogonal ICEO flow-velocity components under AC field-effect control is quite complex, and the same is true for the mixing efficiency, as shown in Figure 11.  
Figure 11: Simulation results of the AC-field-frequency effect on the resulting electrokinetic pumping and mixing performances of the advanced device design driven by bipolar Janus AC-FFET for given values of \(\mathrm{V_{D}}=2\), \(\mathrm{V}\), \(\mathrm{\beta_{odd}}=-\mathrm{\beta_{even}}=0.8\), \(\mathrm{R}=5\)\(\mathrm{\upmu}\), \(\mathrm{L_{G}}=30\)\(\mathrm{\upmu}\), and \(\mathrm{n}=12\). (**A**,**B**) A direct comparisonof the analyte-concentration field in terms of a surface plot between the unbiased (**A**) and biased AC powering modes (**B**) at f = 150 Hz. (**C**,**D**) Counterparts of (**A**,**B**) at f = 10000 Hz. (**E**) Numerical quantification of the mixing efficiency in the unbiased and biased modes and the relative mixing improvement as a function of field frequency. (**F**) A quantitative comparison of the horizontal ICEO pump-flow velocity between unbiased and biased powering modes as a function of field frequency. (**G**) A quantitative comparison of the transversal ICEO mixing-flow velocity between unbiased and biased powering modes as a function of field frequency. (**H**) A quantitative comparison of the frequency-dependent ICEO mixing/pumping-velocity ratio between unbiased and biased powering modes. According to Figure 11F,G, both \(u_{\text{x}}\) and \(u_{\text{y}}\) exhibit a single relaxation peak at a middle frequency fidel. Because the fidel-ux for \(u_{\text{x}}\) is larger than the fidel-uy for \(u_{\text{y}}\), the velocity ratio \(\chi=u_{\text{y}}/u_{\text{x}}\) decreases from a stable low-frequency plateau to zero within the high-frequency limit (Figure 11H). This leads to a reduced mixing index at a higher frequency for the biased powering mode from an overall perspective (the red line in Figure 11E). Moreover, the unbiased mode performs the worst in terms of its mixing function in an intermediate frequency range (the black line in Figure 11E) on account of the shortest residence time tmix from an in situ peak ICEO pump motion (Figure 11F). From this subtle distinction, we can infer that the sample mixing is governed by lateral ICEO flow rotation \(u_{\text{y}}\) and diffusive mass transfer in the biased and non-biased modes, respectively. Consequently, the mixing-index enhancement \(\triangle\)M reaches a maximum of 53.8% at f = 150 Hz (Figure 11E). As the applied field frequency moves away from the critical point of 150 Hz, the \(\triangle\)M decays dramatically. Importantly, the originally positive \(\triangle\)M turns into a negative value for f < 50 Hz and f > 800 Hz (Figure 11E), which indicates that the mixing efficiency in the biased mode is even lower than that in the conventional unbiased mode (Figure 11A-D), which is mainly due to the augmented pump-flow rate along the channel length (Figure 11F).