## 3 Results and Discussion
### ICEO around a Janus Micropillar Free from External Wiring
#### 3.2.1 Effect of Metal-Phase Angle \(\alpha\) on Horizontal Pumping by ICEO around a Janus Post  
In ICEO, an externally applied electric field acts on its own induced diffuse screening charge within bipolar IDL, leading to time-averaged electro-osmotic streaming on ideally polarizable metal surfaces, even under zero-mean AC fields (Figure 2). In most situations, nonlinear diffuse-charge dynamics on insulating structures can be safely disregarded due to their high electric impedance in front of the bipolar IDL. This suggests that we should enable symmetry breaking in ICEO eddies along the horizontal direction by changing the volumetric fractions of the dielectric and metallic phases of Janus pillars. To explicitly express this kind of adjustable material composition, we must define the interface-included angle or metal-phase angle \(\alpha\). As shown in Figure 1B, for a typical neutral configuration of symmetric material polarizability, \(\alpha\) takes the value of \(180^{\circ}\). Otherwise, if the metal phase becomes less (more) dominant than the dielectric counterpart, \(\alpha\) is smaller (larger) than \(180^{\circ}\), as shown in Figures 3B and 3D, respectively. The volumetric fraction of the metal phase of the Janus pillar \(\rm Vf_{m}\) can then be given by the ratio of \(\alpha\) to \(360^{\circ}\):  
\[\rm Vf_{m}=\frac{\alpha}{360^{\circ}}\times 100\% \tag{35}\]  
With the precise definition of \(\alpha\), a set of numerical simulations are then conducted to clarify the importance of metal-phase volumetric fraction in inducing a net ICEO pump's fluid motion along the channel length. For the special case of \(\alpha=0^{\circ}\), i.e., the pure dielectric limit of the Janus pillar, in the application of a transversal low-frequency AC voltage difference, ICEO streaming emerges and behaves as four counter-rotating electrokinetic micro-vortices around a dielectric post placed in the field center (Figure 3A). Indeed, this kind of ICEO vortex flow field resembles the familiar quadrupolar whirlpools induced next to a conducting cylinder (Figure 2A). Nevertheless, the strength of sinusoidal slipping profile \(\rm u_{\theta}\) merely has a characteristic magnitude of \(0.1~{}\mu m\cdot s^{-1}\) on the dielectric surface (Figure 3E), which is much smaller than that on the metal surface by nearly 2\(\sim\)3 orders of magnitude. This implies that the uniform dielectric pillar loses the function of a liquid actuator from the perspective of ICEK, although special attention must be paid to the sharp transition region between two adjacent materials of discontinuous polarizability. As the interface-included angle increases to \(\alpha=90^{\circ}\), the metal phase makes up a quarter of the entire volume of the Janus post (Figure 3B). With the slightly enhanced polarizability, two counter-rotating ICEO whirlpools emerge selectively at the metal/electrolyte interface. By contrast, the dielectric surface lacks such rotating electrokinetic eddies. This effectively causes a discernable asymmetry in the ICEO's flow profile perpendicular to the field axis, resulting in a net pump-flow component for the unidirectional delivery of fluid samples downstream. The ICEO's flow velocity has a typical magnitude of \(0.5~{}\rm mm\cdot s^{-1}\) at the conducting end, much larger than \(0.1~{}\mu m\cdot s^{-1}\) on the dielectric side. With a further increase in \(\alpha\) towards a neutral value of \(\alpha=180^{\circ}\), we realized the most classical Janus micropillar seen in experiments, divided by the polar interface into exactly two halves of equal volume (Figure 3C). Since the polar interface was parallel to the applied electric field, the dielectric and metal phases were symmetrically located on the left and right sides, respectively. Once again, a pair of strong ICEO vortices with reversed streaming directions was produced on the metal end, resembling the flow profile induced under \(\alpha=90^{\circ}\).