## 2 Materials and Methods
### Working Principle  
We perform the transport of the O/W droplet by applying a centrifugal force field perpendicularly to the axis of flow. Convection is induced by designing a structure with different distances between inlet and outlet from the center of rotation, resulting in a net hydrostatic pressure difference. Therefore, we designed a square-section microchannel structure with an Archimedes helix as the track line, and the polar coordinate expression of the helix as follows:  
\[r=r_{0}+a\cdot\theta \tag{1}\]  
where \(r_{0}\) is the initial spiral radius, \(a\) is the change rate of spiral radius, and \(\theta\) is the polar angle in polar coordinates. In this case, the net hydrostatic pressure \(\Delta P\) difference between inlet (\(r_{1}\)) and outlet (\(r_{2}\)) can be expressed as [19]:  
\[\Delta P=\rho_{o}\overline{\sigma}\Delta r\omega^{2} \tag{2}\]  
where \(r\) is the distance between the center of rotation and the mean radial point of the fluid inside the microchannel, \(\Delta r\) is the distance difference between the inlet and outlet relative to the center of rotation, \(\omega\) is the rotational angular velocity, and \(\rho_{o}\) is the density of the oil phase. To study the motion of the oil phase, we consider quasi-equilibrium conditions, where \(\omega\) is a constant, and \(\omega\)\(\varpi\) (\(\varpi\) is the absolute velocity of the fluid) is so small that the influence of Euler force (\(F_{E}=\rho r\frac{d\omega}{dt}\)) and Coriolis force (\(F_{\mathrm{C}}=2\rho\omega v\)) can be ignored. At this point, the flow \(Q\) of oil phase in the microchannel can be given by net hydrostatic pressure difference \(\Delta P\) and hydrodynamic resistance \(R_{hd}\)[20; 21]:  
\[Q=\Delta P/R_{hd} \tag{3}\]  
The hydrodynamic resistance of the square microchannel can be approximately expressed as:  
\[R_{hd}=\frac{8(1+A_{r})^{2}\mu_{o}l}{A_{r}A^{2}} \tag{4}\]  
where \(A\tau\) is the aspect ratio of the microchannel, \(A\) is the cross-sectional area, \(\mu_{o}\) is the viscosity of oil, and \(l\) is the length of the microchannel. Assuming a parabolic flow profile in the continuous phase channel, and the cross-sectional area of all parts in the square microchannel as equal, the expression for the average velocity of the oil phase can be given by:  
\[u_{o}^{*}=\frac{Q}{A}=\frac{\Delta P}{R_{hd}A} \tag{5}\]  
Since the droplet is generally smaller than the microchannel hydraulic diameter in the present tests, the local fluid velocity at the droplet center can be expressed as:  
\[u_{o}=\frac{3}{2}u_{o}^{*}[1-(\frac{d_{h}-D}{d_{h}})^{2}] \tag{6}\]  
where \(D\) is the diameter of droplet and \(d_{h}\) is the continuous phase hydraulic diameter. For a square microchannel, hydraulic diameter is equal to the microchannel diameter. The droplet movement in the oil phase is governed by the interplay of centrifugal force, drag force, and surface tension. The net resultant force of centrifugal and drag force should overcome the obstacles caused by unbalance of surface tension during the movement. The force analysis of droplet can be seen in Figure 3 (a detailed description of the model is shown in Appendix C). We perform a simple theoretical estimate of those three forces acting on the droplet in the oil phase. With the action of the centrifugal field, the centrifugal force applied to the droplet in the rotating microchannel, \(F_{\omega}\), is:  
\[F_{\omega}=\frac{\pi}{6}D^{3}\rho_{w}\omega^{2}r_{i} \tag{7}\]  
where \(\rho_{w}\) is the density of droplet, and \(r_{i}\) is the distance of the droplet from the rotation center. At this time, the component value of the centrifugal force in the direction of velocity can be replaced by (a detailed derivation is shown in Appendix D):  
\[F_{\omega 1}=\frac{\pi}{6}D^{3}\rho_{w}\omega^{2}a \tag{8}\]  
The effect of drag force in the direction of velocity is positive and promotes droplet movement.