## 3 Results and Discussion
### Field-Effect ICEO of a Janus micropillar Subjected to a Controllable Gate Voltage
#### 3.3.1 Effect of AC Gate-Voltage Bias on ICEO Mixing- and Pumping-Flow Components
Given the calculation results in Figure 5G, in the absence of field-effect flow control with \(\beta=0\), the ICEO's slipping profile is symmetric in the lateral direction, and the negative and positive \(\upup_{\theta}\) cancel each other out, leading to the absence of net rotating motion along the channel's width (Figure 4B). By invoking AC field-effect flow control, however, the original transversal flow symmetry is destroyed more severely by imposing a biased gate voltage on the metal phase of the central post, and the negative (positive) \(\upup_{\theta}\) becomes dominant over the positive (negative) counterpart with a decrease (increase) in the bias ratio \(\beta\) from 0 to \(-0.9\) (0.9), which corresponds well with the net downward (upward) mixing-flow component \(\upup_{y}\) in the negatively (positively) biased gate terminal shown in Figure 5A,D. In addition, according to the IZP profile in Figure 5H, the gate-voltage bias also indirectly breaks the transversal flow symmetry by selectively superposing a global offset to the value of IZP on the conducting surface, which is coupled actively with the regulation of the local electric-field intensity to produce the rather sophisticated physical mechanism underlying the Janus AC-FFET. 3.3.2 Effect of Pillar Size on Frequency-Dependent Flow Responses of \(\upup_{x}\) and \(\upup_{y}\)  
Similar to the transversally symmetric ICEO circulations passing a Janus pillar free from external wiring, in which the horizontal pump motion \(\upup_{x}\) demonstrates a strong frequency dependence, it is interesting to observe how the asymmetry eddies under AC field-effect flow control respond to the excitation frequency when the central Janus post of varying radii R is subjected to a gate-voltage bias in terms of the pumping \(\upup_{x}\) and mixing-flow behavior \(\upup_{y}\), as well as the velocity ratio \(\chi\) of the latter to the former, at a fixed gate-voltage-offset ratio of \(\beta=0.8\) (Figure 6). The mathematical definition of \(\chi\) is as follows:  
\[\chi=\upup_{y}/\upup_{x}=\upmu_{\text{mix}}/\upmu_{\text{pump}} \tag{38}\]  
As shown in Figure 6A,B, although both the axial pump \(\upup_{x}\) and the transversal mixing velocity \(\upup_{y}\) exhibit a single relaxation peak in a middle-frequency range under different post radii, the IR's ideal exciting frequencies, including \(\text{f}_{\text{ideal-ux}}\) and \(\text{f}_{\text{ideal-uy}}\), are different. Indeed, both \(\text{f}_{\text{ideal-ux}}\) for sample transport and \(\text{f}_{\text{ideal-uy}}\) for analyte stirring under AC field-effect flow control decrease in line with the radius of the Janus micropillar, which is identical to the size-dependent ideal working frequency \(\text{f}_{\text{ideal-ux}}\) of the axial fluid conveyed in the unbiased powering mode (Figure 4E). However, the fidel-uy is always slightly smaller than the fidel-ux, and even the highest fidel-uy, 600 Hz for R = 2.5 \(\upmu\)m, cannot match the lowest fidel-ux, 1000 Hz for R = 10 \(\upmu\)m. This evident difference in ideal actuating frequency implies that the two flow components have completely different physical essences; namely, the horizontal pump motion u\({}_{\mathrm{x}}\) originates in the inherent anisotropic polarizability of metal-dielectric Janus micropillars, while the transversal rotating movement u\({}_{\mathrm{y}}\) stems from the symmetry breaking in the AC powering scheme driven by AC field-effect flow control. Moreover, the peak flow velocity of u\({}_{\mathrm{y}}\) decreases monotonically with R, in good agreement with the scale given by Equation (26), while that of u\({}_{\mathrm{x}}\) first decreases and then increases when R is elevated from 2.5 \(\upmu\)m to 10 \(\upmu\)m, which also demonstrates their distinct driving mechanisms from another perspective. Considering a lower fidel-uy than fidel-ux, the pump/mixing-velocity ratio \(\chi\) = u\({}_{\mathrm{y}}\)/u\({}_{\mathrm{x}}\) invariably decays from a stable low-frequency plateau to zero in the high-frequency range with varying post radii (Figure 6C).