Experiments and simulations have shown that a droplet can move spontaneously and directionally on a conical substrate. The driving force originating from the gradient of curvatures is revealed as the self-propulsion mechanism. Theoretical analysis of the driving force is highly desirable; currently, most of them are based on a perturbative theory with assuming a weakly curved substrate. However, this assumption is valid only when the size of the droplet is far smaller than the curvature radius of the substrate. In this paper, we derive a more accurate analytical model for describing the driving force by exploring the geometric characteristics of a spherical droplet on a cylindrical substrate. In contrast to the perturbative solution, our model is valid under a much weaker condition, i.e., the contact region between the droplet and the substrate is small compared with the curvature radius of the substrate. Therefore, we show that for superhydrophobic surfaces, the derived analytical model is applicable even if the droplet is very close to the apex of a conical substrate. Our approach opens an avenue for studying the behavior of droplets on the tip of the conical substrate theoretically and could also provide guidance for the experimental design of curved surfaces to control the directional motion of droplets.

中文翻译：

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液滴在锥形基底上的自发运动：驱动力的理论分析

实验和模拟表明，液滴可以在锥形基底上自发地定向移动。源自曲率梯度的驱动力被揭示为自推进机制。非常需要对驱动力进行理论分析；目前，它们中的大多数是基于微扰理论，假设基板为弱弯曲。然而，只有当液滴的尺寸远小于基板的曲率半径时，这种假设才有效。在本文中，我们通过探索圆柱形基板上球形液滴的几何特征，推导出了一个更准确的分析模型来描述驱动力。与微扰解相比，我们的模型在更弱的条件下是有效的，即 与基板的曲率半径相比，液滴与基板之间的接触区域较小。因此，我们表明，对于超疏水表面，即使液滴非常接近锥形基底的顶点，导出的分析模型也适用。我们的方法为理论上研究锥形基板尖端上的液滴行为开辟了一条途径，也可以为曲面的实验设计以控制液滴的定向运动提供指导。