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

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 $Λi)$ 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 $Λ2,$ vanish, is a rectangular hyperbola. Two of the adsorptions $Λ2$ and $Λ3$ then both vanish when the contact line is chosen to pass through any of the intersections of the two corresponding hyperbolas $Λ2=0$ and $Λ3=0.$ There may be two or four such real intersections. It is found most surprisingly, and confirmed in a numerical example, that $Λ1(2,3),$ the adsorption of component 1 in a frame of reference in which the adsorptions $Λ2$ and $Λ3$ are both 0, depends on which intersection of the hyperbolas $Λ2=0$ and $Λ3=0$ is chosen for the location of the contact line. This implies that what had long been taken to be the line analog of the Gibbs adsorption equation is incomplete; there must be additional, previously unanticipated terms in the relation, consistent with the invariance of the line tension to choice of location of the contact line. It is then not $Λ1(2,3)$ by itself but a related expression containing it that must be invariant, and this invariance is also confirmed in the numerical example. The presence of the additional terms in the adsorption equation is further confirmed and their origin clarified in a mean-field density-functional model. The supplemental terms vanish at a wetting transition, where one of the contact angles goes to 0.