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### ARC system design and material property characterization

A design schematic of our ARC self-cleaning system is shown in Fig. 1a. Hydrophilic curved rungs are patterned on the hydrophobic background on the substrate (silicon wafer or soda-lime glass). Different material combinations with hydrophobic/hydrophilic behaviors can be applied. In our work, we used a perfluoro-octyltrichlorosilane (FOTS)–trimethylsilanol (TMS) self-assembled monolayer (SAM) system and a Cytop–TMS spin-coated thin-film system. Table 1 summarizes the material properties, including static CA, dynamic CA hysteresis, sliding CA on an inclined surface and coating thickness. The SAM is created by chemisorption of the trichlorosilane “headgroups” to the hydroxyl group on the substrate, forming a stable covalent bond. The functional “tail group” can be altered, providing different surface energies to create hydrophobic/hydrophilic contrast. The FOTS “tail group” is highly fluorinated; thus, the surface energy is reduced after treatment to provide a hydrophobic surface finish. The deposition of the SAM can be either in the vapor phase or in solution. The SAM coating is only molecular-level thick, making the coating transparent and optically flat.

Cytop is an amorphous fluoropolymer with good transparency over the visible and UV wavelength ranges, good solubility to coat various surface designs, and excellent water repellent properties. The refractive index of Cytop is 1.34, which allows the material to serve as an anti-reflective coating on glass substrates. Micropatterning on hydrophobic surfaces (such as Teflon and Cytop) is difficult using standard photolithography due to poor adhesion between the photoresist and Cytop. Methods have been proposed using a metal buffer layer^{24,25} or plasma pretreatment of the surface, but the original hydrophobic surface properties will be damaged after treatment with decreasing water droplet CA.

Poly(*p*-xylylene) polymers are a special series of polymers that produce uniform pinhole-free films in a chemical vapor deposition process. Parylene can be etched with oxygen plasma, making it compatible with standard lithography processes. It has been used to create biomolecular stencil arrays^{26} and patterns on soft substrates^{27}. By adopting parylene-C as a stencil mask, we created hydrophilic patterns on top of the Cytop surface without degrading its original surface properties. The characterization results of the surface properties of the coating materials used in this paper are shown in Table 1. SEM images of the patterned Cytop are shown in Supplementary S1.

### Optical transmittance and solar module output

It is important to understand the optical performance of the ARC patterned coating. We performed optical transmission measurements, as shown in Fig. 1c,d. For the FOTS-TMS system, the light transmission was degraded due to the added coating of monolayers, but within a range of less than 1%. The glass after FOTS-TMS treatment was transparent and optically flat. Furthermore, the Cytop-TMS system improved the transmission with an enhancement of 2.5%~3.5% over the visible wavelength range even with an added coating on top of the glass. The reason was that the refractive index of Cytop is ~1.34, which is between those of air (*n*_{air} = 1.0) and the glass substrate (*n*_{glass} = 1.5), providing a refractive index match. Similar to Rayleigh’s film, a portion of the incoming light reflects both at the interface of air/Cytop and Cytop/glass but has less reflection than the single reflection at the air/glass interface with a larger refractive index mismatch. Figure 1d shows the I-V curve measurements for assembled PV modules. The ARC structure was patterned over a 5 cm by 5 cm solar cell surface. The Cytop-TMS coating generated higher optical output power than bare glass and FOTS-TMS surface treatment, in accordance with the light transmission measurements. The optical performance demonstrated that our coating systems were compatible with solar module cover glass and can even have anti-reflective properties to improve solar module power output efficiency.

### Droplet transport characterization

The test wafer with ARC patterns was mounted on a vibration stage. A 10 μL (2.84 mm in diameter) water droplet was pipetted on the surface, and the droplet silhouette was monitored via a high-speed camera with a frame rate of 1000 fps. We tested both FOTS-TMS and Cytop-TMS systems, with the design parameters of *R* = 1000 μm, *P* = 100 μm, *w* = 10 μm, and *W* = 1.8 mm. Figure 2a, b shows a typical droplet leading and trailing position change, CA change and line speed with time as the substrate vibrated orthogonally. The droplet transport speed was 7.5 mm/s on the FOTS-TMS ARC surface and 27 mm/s on the Cytop-TMS ARC surface. In this design, we translated substrate orthogonal vibration into droplet lateral expansion and recession and thus moved the droplet by the anisotropic forces at the leading and trailing edge where the solid–liquid–gas three-phase contact line resided. A detailed theoretical derivation of the anisotropic force can be found in reference^{28}. With the aid of the ARC, the droplet can overcome the force of gravity and climb on inclined surfaces under orthogonal vibration. Our experiment showed that the droplet can climb uphill at up to a 15° inclination of the surface. With higher inclination angles, the droplet tended to be “shaken off ” the surface.

### The frequency response of the droplet transport

It is important to understand the frequency response of the droplet and to match it with the dynamic behavior of the PV modules^{29}. The water droplet exhibits different resonance modes depending on mass and surface tension. The *n*^{th} resonance mode of the droplet can be expressed as^{30,31,32}

$$f_n = frac{pi }{2}left( {frac{{n^3gamma }}{{24m}}frac{{mathrm{cos}^3theta – 3cos theta + 2}}{{theta ^3}}} right)^{frac{1}{2}}$$

(1)

where *n* = 2, 3, 4,… is the mode number, *γ* is the water surface tension (in N/m), *θ* is the CA (in radians) and *m* is the water mass (in kg). We modeled the droplet on the uniform hydrophobic surface under vibration as a forced mass-spring oscillator system^{33} and then characterized the water drop resonance frequency in low-frequency bandwidth regions (10–100 Hz, at every 5 Hz) by monitoring the droplet width change at different vibration frequencies. The vibration acceleration amplitude was kept constant at 1 g, meaning the droplet was driven by a periodic external force. Figure 3a shows the plot of the relative droplet width change of 5, 10, and 15 μL droplets at different frequencies. Due to the mechanical resonance behavior of the droplet, it was relatively easy to drive the center volume of the droplet on the substrate using frequencies close to its resonance when relatively low orthogonal vibration energy was required to achieve enough droplet sideway expansion amplitude. Figure 3b shows examples of droplet transport with higher modes on ARC surfaces at 50, 200, 300, and 500 Hz. The acceleration required to move the water droplet along the ARC rises as the mode number increases.

To study the required power input to drive the droplet, if we ignore the mass of the substrate, we have the following expression describing a damped oscillator with a harmonic driving force^{34}

$$frac{{mathrm{d}^2}}{{mathrm{d}t^2}}xleft( t right) + {mathrm{Gamma }}frac{mathrm{d}}{{mathrm{d}t}}xleft( t right) + omega _0^2xleft( t right) = frac{{F_0cos left( {omega _dt} right)}}{m}$$

(2)

where *x*(*t*) is droplet position with time, Γ is the damping constant of the electromagnetic vibration exciter (in s^{−1}), *F*_{0} is the driven force, *ω*_{0}/2*π* is the natural frequency of the oscillator and *ω*_{d}/2*π* is the driven frequency. The solution to the equation above is

$$xleft( t right) = Acos left( {omega _dt} right) + Bsin (omega _dt)$$

(3)

$$A = frac{{left( {omega _0^2 – omega _d^2} right)a}}{{(omega _0^2 – omega _d^2)^2 + {mathrm{Gamma }}^2omega _d^2}}$$

(4)

$$B = frac{{{mathrm{Gamma }}omega _da}}{{(omega _0^2 – omega _d^2)^2 + {mathrm{Gamma }}^2omega _d^2}}$$

(5)

where *A* is the elastic amplitude and *B* is the absorptive amplitude. (a = frac{{F_0}}{m}) can be measured from experiments. The average power within any single oscillation period is related to the *B* term

$$P = frac{1}{T}mathop {int }limits_{0}^{T} Ffrac{{mathrm{d}{it{x}}}}{{mathrm{d}t}}mathrm{d}t = frac{1}{2}{it{F}}_{0}omega_{it{d}}{it{B}} = frac{1}{2}{it{m}}aomega_{it{d}}{it{B}}$$

(6)

To estimate the power required to drive the droplet, we measured the minimum acceleration required to drive the droplet over the 20~100 Hz frequency bandwidth, as shown in Fig. 3c. We then calculated the average power based on Eq. (6). All the parameters could be inserted through measurement or datasheet, except for the damping coefficient of the electromagnetic vibration exciter. We estimated a large damping coefficient Γ = 1000 s^{−1} in our calculation based on the fact that the electromagnetic vibration exciter was a highly damped system and the conversion efficiency from electrical to mechanical was low at its maximum load (~0.1%) among the working frequencies^{35}. The power calculations based on experimentally measured data on Cytop-TMS ARC designs are presented in the Discussion section below.

### Surface anisotropic force by SA measurement

To evaluate the anisotropic forces, a slip test was performed with different ARC designs (Fig. 4). The inclination angle is defined when the droplet starts to slide off the tilted surfaces. The gravity of the water droplet overcomes the surface adhesion at the inclination angle. The ARC radius of curvature is pointed either uphill or downhill on inclined surfaces, as indicated in Fig. 4. The anisotropic pinning force can be expressed as

$$F_{mathrm{anis}} = F_{mathrm{slip,uphill}} – F_{mathrm{slip,downhill}} = mgsin delta _{mathrm{uphill}} – mgsin delta _{mathrm{downhill}}$$

(7)

where *δ* is the surface inclination angle. We performed a slip test on the Cytop-TMS ARC surface. For the (a1)–(a3) design group, we changed the center-to-center period width (*P*) to 50, 100, and 200 μm while keeping the other parameters the same. For the (b1)–(b4) design group, we altered the ARC pattern radius of curvature (*R*) with 1000, 1100, 1500 μm, and straight lines. As seen from the results of the change in period (a1)–(a3), the SA for both uphill and downhill decreases as the period gap increases. The anisotropic forces remained similar for different designs. However, we could observe a decrease in the anisotropic force as we increased the ARC radius of curvature from the measurement results of (b1)–(b4). The average anisotropic force on a radius of curvature *R* = 1000, 1100, and 1500 μm was 73, 66, and 11 μN, respectively. When there were only straight hydrophilic lines, no SA differences were observed for uphill and downhill measurements, and the anisotropic forces were close to zero. To achieve a better ratcheting performance, we chose the ARC radius of curvature *R* = 1000 μm to drive the droplet to move.

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