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

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 vertical bar theta - 90 degrees vertical bar decreases, and it changes from 0.5 at vertical bar theta - 90 degrees&VERBAR = 70 degrees to 0.67 at vertical bar theta - 90 degrees&VERBAR = 0 degrees . A wide range of contact angles (60 degrees -107.5 degrees ) 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.