Long-range spontaneous droplet self-propulsion on wettability gradient surfaces

The directional and long-range droplet transportation is of great importance in microfluidic systems. However, it usually requires external energy input. Here we designed a wettability gradient surface that can drive droplet motion by structural topography. The surface has a wettability gradient range of over 150° from superhydrophobic to hydrophilic, which was achieved by etching silicon nanopillars and adjusting the area of hydrophilic silicon dioxide plane. We conducted force analysis to further reveal the mechanism for droplet self-propulsion, and found that the nanostructures are critical to providing a large driving force and small resistance force. Theoretical calculation has been used to analyze the maximal self-propulsion displacement on different gradient surfaces with different volumes of droplets. On this basis, we designed several surfaces with arbitrary paths, which achieved directional and long-range transportation of droplet. These results clarify a driving mechanism for droplet self-propulsion on wettability gradient surfaces, and open up new opportunities for long-range and directional droplet transportation in microfluidic system.


Driving force and hysteresis resistance force
During the whole process, the difference of the Gibbs surface energy between its two sides acts as the driving force. Figure S1. Schematic drawing shows a spherical segment water droplet on the wettability gradient stripes from the front view (a) and the top view (b).
As shown in the Fig. S1, if we take the ribbons with width dy into consideration, the driving force for the displacement dx can be expressed as 1 :  With θ oa and θ or being the advancing and receding contact angle at the center of droplet, θ d being the dynamic contact angle which can be calculated as: Here θ a and θ r are the droplet advancing and receding CAs on a wettability gradient surfaces.
where F d and F h are the driving force and hysteresis resistance force, LV γ represents the surface tension of the droplet, R b is the base radius of the droplet in contact with the solid. We assume that the water droplet on gradient surfaces can be considered as a spherical segment and the contact area is a circle with radius R b , the initial volume of water droplet is V o , thus the base radius can be calculated as: As the wetting profile of gradient surface is discrete, the forces acting on the dynamic droplets were simplified and defined points at middle positions of two (four) adjacent wettability gradient regions were used to calculate the driving and hysteresis forces (Fig. S2).
The droplet just overlaps two wettability regions when moving towards hydrophobic regions ( Fig. S2a). Therefore, the wettability gradient dcosθ d is the gradient between wettability regions S i+1 and S i+2 (the red and yellow contact line). And for the hydrophilic regions, droplet may overlap four wettability regions (Fig. S2b). The wettability gradient dcosθ d is divided into two parts: one is the gradient between S i+1 and S i+2 (the red and yellow contact line), and the other is the gradient between S i and S i+3 (the blue contact line).

Viscous resistance force
The viscous resistance force can be calculated as  From flow field analysis, we can get: Where η is the viscosity of water, ( ) x V z shows the velocity as a function of z at position x (please refer to the yellow curved in Fig. S2). According to lubrication approximation Where P is the pressure of Poiseuille flow, P x ∂ ∂ is the pressure gradient along x coordinate.
Combing with boundary condition: ( ) Here, V is the average velocity of water droplet along z-axis, x h is the height of droplet at point x (see Fig. S2). Thus, Thus, the drag force per unit length can be expressed as: If we assume that the flow field is uniform at solid/liquid interface, we can get the overall viscous resistance force: , L 1 =90 µm, h=2 µm. Figure S4. Schematic of the silicon nanopillars and SiO 2 plane

Pattern density and static CA
The droplet static contact angle on the composite pattern surface is provided by the Cassie-Baxter model. The measured and calculated results as table S1 shows. Figure S5. Droplet motion displacement (a) and velocity (b) on different wettability gradient surfaces.

Reproducibility of the wettability gradient surfaces.
We demonstrate the sample reproducibility on the radial path surfaces and the snapshots are shown as follow. Figure S6. Droplet self-motion reproducibility of the wettability gradient surfaces.