# Spreading Dynamics of Droplet Impact on a Wedge-Patterned Biphilic Surface
## 1 Introduction  
Droplet impact is of great importance in many technological applications for pesticide spray coatings, drag reduction, water harvesting, heat exchangers, anti-icing and thermal management [1; 2; 3; 4; 5; 6]. In practice, the spreading dynamics of droplet impact on a solid surface is a complex multi-phase flow problem, and yet it is not understood completely. After impacting on a solid surface, a droplet spreads radially outside. Then, the kinetic energy is transferred into surface energy and the dissipation energy is induced by viscosity. Finally, droplet retracts back under the action of surface tension, after reaching the maximum spreading diameter. Eventually, the droplet oscillates off the solid surface, or on it, depending on the initial impact states [7; 8]. The wettability effect on droplet spreading dynamics has been investigated, using a static [9] or dynamic [10] contact angle. One can motivate droplets by using a surface with different wettabilities. There are several ways to build surface tension gradients: coatings, microstructures, temperature and wedged patterns [11]. The wettability of the solid surface described by the contact angle \(\theta\) can be changed by coating different polymer particles with different hydrophilic properties. From the insights of the Wenzel [12] and the Cassie-Baxter model [13], one can change the practical contact area by changing the surface morphology as well. Accordingly, for now, a popular method of regulating thesurface morphology involves manufacturing microstructure arrays onto a solid surface. As the surface tension coefficient is a function that is dependent on temperature \(\gamma(T)\), there is a famous phenomenon \"Marangoni effect\" that is based on this dependence. Changing the temperature is another practical way to control the surface tension. The gradient contact area can be accomplished by designing different wedged patterns (2-D) and cones (3-D), like a cactus spine, along the droplet-spreading direction. Then, the contact length, \(L\), can be controlled. An early experiment for droplet transfer on a wettable gradient surface was performed by [PERSON] [14]. They made a surface with a wettable gradient along the radial direction by chemical coating, and studied the spreading characteristics that occur when a droplet impacts an edge of different wettability. Recently the statics and dynamics of liquid barrels in wedge geometries have been studied theoretically and experimentally [15; 16]. Besides the wetting properties, the impact velocity shows great importance in droplet spreading after impact. Commonly, the Weber number (\(We=\rho V^{2}D/\gamma\)) is applied, to characterize the initial state of impact velocity, as well as the surface tension. A low Weber number means that the surface tension is rather strong, compared with the inertia effect. [PERSON] et al. [17] performed experiments to measure the maximum spreading ratio under low Weber numbers (0.05 to 2) and low Reynolds numbers (\(Re=\rho DV/\eta\), 10-100). In their study, the surface energy and the kinetic energy was comparable. Considering the effects of viscosity and surface tension, [PERSON] et al. [18] performed a series of experiments at different Weber numbers, and promoted a universal scaling law of spreading time. The mesoscopic simulation on spreading diameter was accomplished in [PERSON] and [PERSON]'s work [19]. They simulated the spreading dynamics for when a droplet impacts on a homogeneous superhydrophobic surface. The ideal Cassie state was used, and three linear weight functions were applied in rebuilding the liquid-solid interaction. Considering the energy balance of the initial impact state, and that of the final spread, many models have been promoted [20; 21]. Though the mechanisms of droplet impact have been researched abundantly, many detailed issues still remain unknown. The detailed hydrodynamic model of droplet spreading on a wedge-patterned surface with wettability difference has not been investigated before.