Furmidge Equation Revisited

This paper addresses the issue of the discontinuous function that gives rise to the Furmidge equation, a long-standing problem in interfacial science. The force at the contact line of a sliding drop is related to the drop size, the contact angle hysteresis, the surface tension, and a geometrical prefactor <i>k</i> which depends on the distribution of the contact angle about the contact line. The most common expression for the contact line force, called the Furmidge equation, takes <i>k</i> = 1 and is based on a discontinuous contact line, while corrections to this model pose polynomial functional forms for the contact angle, engendering discontinuities in the derivatives. Moreover, experimental findings provide a wide range of <i>k</i> values for different drops on different solids, and this range is yet to be explained in a physical context. Owing to this, the understanding of forces on sliding drops remains lacking. We construct a general model based on a Fourier series, and we further generalize this model by superposing a series of Gaussian curves on our Fourier series. The result of this model is a range of <i>k</i> values, in accordance to the range of experimental values which appear in the literature. Additionally, we fit our functional form for the contact angle to experimental data and find good agreement, as well as good agreement between <i>k</i> values.