The line adsorption equation: the one-dimensional counterpart of the Gibbs adsorption equation

The thermodynamic development for multiphase contact lines is analogous to that for surfaces or interfaces. However, for one of the most important equations in surface thermodynamics, the Gibbs adsorption equation, the one-dimensional analogue is missing. This paper derives such an analogue, the line adsorption equation. Similarly to the Gibbs adsorption equation, the line adsorption equation is derived from Gibbsian thermodynamics. For a three-phase, three-component contact line system (e.g. an oil lens on the surface of an aqueous solution), the line concentrations (excesses) of two immiscible solvents can be made vanish by appropriately placing the dividing line. Consequently, the line concentration of the solute can be evaluated through the line tension change with the volume concentration of the solute. Such an evaluation provides information about molecular adsorption at the contact line, which is important in physical chemistry of lines, but difficult to obtain by any other means.