## 3 Results and Discussion
### ICEO around a Janus Micropillar Free from External Wiring
#### 3.2.2 Effect of the Radius of Janus Pillar on Frequency-Dependent ICEO Pumping  
Since the paired DE plates and the metal and dielectric phases of the Janus cylinder all undergo electrochemical polarization, which is strongly influenced by the double-layer charge relaxation in the AC fields, the frequency dependency of the ICEO's flow speed in the current microsystem must be intricate. After obtaining the optimal symmetric structure of the Janus pillar, we clarified how the resulting ICEO's pump-flow motion responds to the AC field frequency when the central cylinder has different sizes. In the DC limit, with the signal frequency \(\mathrm{f}\ll\mathrm{f}_{\mathrm{RC}}^{\mathrm{DE}}=(1+\delta)\lambda_{\mathrm{D}} \sigma_{\mathrm{f}}/\pi\epsilon_{\mathrm{f}}\mathrm{W}_{c}\), as most of the imposed AC potential drops across the IDL at the DE/liquid interface due to complete Debye screening, almost no electric field is present in the fluid bulk to force the mobile counterions into ICEO streaming. As the field frequency is raised well beyond the characteristic RC charge-relaxation frequency of the blocking metal (dielectric) surface of the Janus pillar, i.e., \(\mathrm{f}\gg\mathrm{f}_{\mathrm{RC}}^{\mathrm{m}}=(1+\delta)\lambda_{ \mathrm{D}}\sigma_{\mathrm{f}}/2\pi\epsilon_{\mathrm{f}}\mathrm{R}(\mathrm{f} \gg\mathrm{f}_{\mathrm{RC}}^{\mathrm{d}}=\sigma_{\mathrm{f}}/2\pi\epsilon_{ \mathrm{d}})\), there is insufficient time for the bipolar diffuse charge to accumulate within the IDL at the metal/electrolyte (dielectric/electrolyte) interface within each half cycle of sinusoidal actuation, resulting in ineffective ICEO streaming, once again originating in incomplete in situ Debye screening. As a result, the quickest ICEO fluid flow occurs in the intermediate frequency range of \(f_{RC}^{\mathrm{DE}}<f_{ideal}^{\mathrm{m}}<f_{RC}^{\mathrm{m}}\) and \(f_{RC}^{\mathrm{DE}}<f_{ideal}^{\mathrm{d}}<f_{RC}^{\mathrm{d}}\) for the metal and dielectric surfaces of the Janus micropillar, respectively. Given the simulation results shown in Figure 4G, at a given post radius R = 5 \(\upmu\)m, the magnitude of ICEO slipping on both conducting and dielectric surfaces of the central post tends to vanish when the field frequency approaches the low-frequency or high-frequency limit and exhibits a single relaxation peak in the middle of the frequency spectrum. The ideal working frequency of ICEO is in the order of \(\mathrm{f}_{\mathrm{RC}}^{\mathrm{d}}\sim 10^{5}\) Hz for the dielectric phase, which is at least 100 times that of \(\mathrm{f}_{\mathrm{RC}}^{\mathrm{m}}\sim 1000\) Hz for the metal phase (Figure 4G), so our numerical solution of ICEO flow field around an individual metal-dielectric Janus pillar coincides well with the analytical scale of their corresponding frequency ratio \(f_{RC}^{\mathrm{d}}/f_{RC}^{\mathrm{m}}\sim\lambda_{D}/R>100\), based on the typical RC circuit theory given above. According to Equation (25), the size of the Janus post R influences the ICEO's pumping motion from two perspectives. The first is the maximum pump-flow rate achievable by ICEO growing linearly with R. The second is the frequency dependence of ICEO streaming, with its high cutoff frequency \(\mathrm{f}_{\mathrm{RC}}^{\mathrm{m}}=(1+\delta)\lambda_{\mathrm{D}}\sigma_{ \mathrm{f}}/2\pi\epsilon_{\mathrm{f}}\mathrm{R}\) inversely proportional to R. As shown by the calculation results in Figure 4E,F, as R increases from 2.5 \(\upmu\)m to 10 \(\upmu\)m, the peak flow frequency shifts from 1000 Hz to 500 Hz, more to the left side of frequency axis, together with an enhanced maximum pump velocity from 139 \(\upmu\)m\(\cdot\)s\({}^{-1}\) to 440 \(\upmu\)m\(\cdot\)s\({}^{-1}\), which is in qualitative agreement with the approximate scaling analysis in Section 2.4.