Describing Droplet Motion on Surface-Textured Ratchet Tracks with an Inverted Double Pendulum Model

We describe the motion of a droplet on a textured ratchet track using a nonlinear resonator model. A textured ratchet track is composed of a semicircular pillar array that induces a net surface tension local gradient on a droplet placed on it. When a vertical vibration is applied, hysteresis is overcome, and the droplet moves toward the local lower energy barrier; however, due to the repetitive structure of texture, it keeps moving until the end of the track. The droplet motion depends on the amplitude and frequency of the vertical oscillation, and this dependence is nonlinear. Therefore, finding a fully analytic solution to represent this motion is not trivial. Consequently, the droplet motion remains poorly understood. In this study, we elaborate on the utility of a double pendulum as a basis for modeling the droplet motion on surfaces inducing asymmetric force. Similar to the droplet motion, resonators, such as a double pendulum, are simple, yet nonlinear systems. Moreover, an inverted double pendulum motion has key characteristics such as the two-phase motion and the double peak motion, which are also observed in the droplet motion. We use various data-processing methods to highlight the similarity between these two systems both qualitatively and quantitatively. After establishing this comparison, we propose a model that utilizes an inverted double pendulum mounted on a moving cart to successfully simulate the motion of a droplet on a ratchet track. This methodology will lead to the development of an accurate droplet-motion modeling approach, and we believe that it will be useful to understand droplet dynamics more deeply.