Geometric view of the thermodynamics of adsorption at a line of three-phase contact

Abstract

We consider three fluid phases meeting at a line of common contact and study the linear excesses per unit length of the contact line (the linear adsorptions Lambda_i) of the fluid's components. In any plane perpendicular to the contact line, the locus of choices for the otherwise arbitrary location of that line that makes one of the linear adsorptions, say Lambda_2, vanish, is a rectangular hyperbola. Two of the adsorptions, Lambda_2 and Lambda_3, then both vanish when the contact line is chosen to pass through any of the intersections of the two corresponding hyperbolas Lambda_2 = 0 and Lambda_3 = 0. There may be two or four such real intersections. It is required, and is confirmed by numerical examples, that a certain expression containing \Lambda_{1(2,3)}, the adsorption of component 1 in a frame of reference in which the adsorptions Lambda_2 and Lambda_3 are both 0, is independent of which of the two or four intersections of Lambda_2 = 0 and Lambda_3 = 0 is chosen for the location of the contact line. That is not true of Lambda_{1(2,3)} by itself; while the adsorptions and the line tension together satisfy a linear analog of the Gibbs adsorption equation, there are additional, not previously anticipated terms in the relation that are required by the line tension's invariance to the arbitrary choice of location of the contact line. The presence of the additional terms is confirmed and their origin clarified in a mean-field density-functional model. The additional terms vanish at a wetting transition, where one of the contact angles goes to 0.