Adsorption equation for the line of three-phase contact

A mean-field density-functional model of three-phase equilibrium that has been much studied to illustrate quantitative properties of line tension is recalled here to address a question about the line analogue of the Gibbs adsorption equation. The local term in the model free-energy density is of the form F(rho(1), rho(2); b) where rho(1)(r) and rho(2)(r) are two densities varying with location r in any plane perpendicular to the three-phase contact line and b is a single independently variable thermodynamic field (temperature or chemical potential). If what had at one time been thought to be the line analogue of the Gibbs adsorption equation had been correct, then in this model the rate of change d tau/db of the line tension tau with respect to b would have been the same as the limit as R --> infinity of tile difference integral(partial derivative F/partial derivative b) da - Rd Sigma/db, where the integral is over the interior of the Neumann triangle whose sides are distant R from the chosen location of the contact line, da is the element of area in the integration and Sigma is the sum of the three interfacial tensions. We see by explicit numerical calculation that this limiting difference is not d tau/db, thus illustrating that what used to be thought to be the line analogue of the Gibbs adsorption equation is incomplete, as recently surmised.