Contact-Angle Hysteresis Caused by a Random Distribution of Weak Heterogeneities on a Solid Surface

Abstract

A model according to which contact-angle hysteresis arises as the result of a random distribution of irregularities on the solid surface is investigated on the basis of probability theory. An estimate is obtained of the mathematical expectation of the number of stable equilibria when the effective angle between the liquid–gas surface and the solid surface with which the liquid is in contact deviates from the value, say θ0, which would obtain if the solid surface were uniform, i.e., free from irregularities. It is found that when the effective contact angle deviates from θ0 by less than a critical value, then the expected number of stable equilibria increases exponentially with the length of the contact line; therefore such a contact angle can occur under static conditions. But if the deviation of the contact angle from θ0 exceeds the critical value, then the expected number of stable equilibria decreases exponentially with the length of the contact line, so a stable equilibrium is not possible for a macroscopic length of the contact line. The method is applicable only if the random deviations of the spreading power (defined as the solid–gas surface tension minus the sum of the liquid–gas and liquid–solid surface tensions) from its average are sufficiently small. It is found that the critical deviation of the contact angle from θ0 is, apart from a slowly varying logarithmic factor, proportional to H2ρs, where H is a measure of the amplitude of the surface irregularities and ρs is the surface density (i.e., number per unit area) of the irregularities. This qualitative feature agrees with the results previously obtained by several other authors, and, moreover, there is a surprisingly close agreement of the proportionality factor with the results of some earlier work in which the method of statistical analysis was much less elaborate than here. The effect of the logarithmic factor is to make the critical deviation of the contact angle increase more slowly than the first power of H2ρs, and this is also in qualitative agreement with some earlier work.