## Materials and Methods
### Mathematical Model of Active Sample Transport and Stirring via Janus AC-FFET  
* (a) AC electric field  
The entire calculation domain can be divided into four correlated subregions, including the liquid volume, the dielectric and metallic phase of the Janus pillar, and the bipolar IDL formed at the polarizable solid/electrolyte interface. The first three are considered bulk zones, while the latter is treated equivalently by interfacial conjugating conditions of an abrupt voltage drop across Debye screening layer according to conservation of charge flux. In the quasi-electrostatic limit, local electroneutrality approximation (LEA) can be safely invoked for a lossy dielectric medium containing sufficient charge carriers, so the electrical potential field \(\widetilde{\phi}\) within all the three bulk subdomains is governed by electric current continuity in a sinusoidal steady state:  
\[\
abla\cdot\big{(}(\sigma_{\text{i}}+j\omega\varepsilon_{i})\
abla\widetilde{ \phi}_{i}\big{)}=0\text{ for i }=\text{ f, d, m }\in\Omega_{\text{f}}\cup\Omega_{\text{d}}\cup\Omega_{\text{m}} \tag{1}\]  
where the subscripts i = f, d, and m stand for the working fluid, dielectric phase, and metal phase of the Janus micropillar of inhomogeneous polarizability (Figure 1F). The \(\sigma\) and \(\varepsilon\) are the electric conductivity and real permittivity, respectively, of corresponding medium bulk. Under the assumption of uniform electrical properties in every subdomain, control equations of the AC electric fields are reduced to Laplace equations with zero induced space charge:  
\[\
abla^{2}\widetilde{\phi}_{i}=0\text{ for i }=\text{ f, d, m }\in\Omega_{\text{f}}\cup\Omega_{\text{d}}\cup\Omega_{\text{m}} \tag{2}\]  
At the structural interface between aqueous electrolyte and dielectric phase of the Janus cylinder, the IZP across the IDL is tightly related to the normal field intensity in the bulk fluid immediately outside the thin boundary layer according to charge conservation:  
\[\Big{(}\sigma_{f}+j\omega\varepsilon_{f}\Big{)}\
abla\widetilde{\phi}_{f} \cdot\textbf{n}=j\omega\text{C}_{0}\Big{(}\widetilde{\phi}_{f}-\widetilde{ \phi}_{d}\Big{)}\in\text{ T}_{\text{FGH}}\cup\text{T}_{\text{KLM}} \tag{3}\]  
\[\varepsilon_{d}\
abla\widetilde{\phi}_{d}\cdot\textbf{n}=\text{C}_{0}\Big{(} \widetilde{\phi}_{f}-\widetilde{\phi}_{d}\Big{)}\in\text{ T}_{\text{FGH}}\cup\text{T}_{\text{KLM}} \tag{4}\]  
where we apply the core idea of classical linear RC-circuit theory of ICEK on ideally polarizable conducting solid surfaces to describe field-induced Debye screening on weakly polarizable dielectric object, simply by replacing the original Ohmic current with the displacement current flowing across the dielectric matrix. This is correct on the macroscopic scale under the Debye-Huckel limit, as demonstrated by our previous investigation on the effect of sharp corners of dielectric obstacles on resulting swirling ICEO streamlines induced along channel sidewalls. The \(C_{0}=C_{D}/(1+\delta)\) denotes the total IDL capacitance, which comprises the diffuse layer capacity \(C_{D}=\varepsilon_{\text{f}}/\lambda_{D}\) of mobile counterions and the Stern layer capacity \(C_{S}\) of immobile ions connected in series, in terms of the double-layer thickness \(\lambda_{D}=(D\varepsilon_{1}/\sigma_{1})^{1/2}\) and surface capacitive impedance ratio of \(\delta=C_{D}/C_{S}\).