Dependence of the contact line roughness exponent on the contact angle on substrates with dilute mesa defects: numerical study

Abstract

We compute the roughness exponent of the averaged contact line width of a liquid on heterogeneous substrates with randomly distributed dilute defects in statics. We study the case of circular “mesa”-type defects placed on homogeneous base. The shape of the liquid meniscus and the contact line are obtained numerically, using the full capillary model when a vertical solid plate, partially dipped in a tank of liquid, is slowly withdrawing from the liquid. The obtained results imply that the contact line roughness exponent depends on the contact angle $\theta$, which the liquid meniscus forms with the solid homogeneous base. The roughness exponent grows when $|\theta - 90^{\circ } |$ decreases, and it changes from 0.5 at $|\theta - 90^{\circ } |= 70^{\circ }$ to 0.67 at $|\theta - 90^{\circ } |= 0^{\circ }$. A wide range of contact angles ($60^{\circ }$–$107.5^{\circ }$) is present, where the roughness exponent is practically constant, equal to previously obtained experimental results on the magnitude of the roughness exponent and its dependence on $\theta$.

Graphic abstract