Adsorption equation for the line of three-phase contact

Abstract

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(ρ₁, ρ₂; b) where ρ₁(r) and ρ₂ (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 dτ/db of the line tension τ with respect to b would have been the same as the limit as R → ∞ of the difference

, 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 Σ is the sum of the three interfacial tensions. We see by explicit numerical calculation that this limiting difference is not dτ/db, thus illustrating that what used to be thought to be the line analogue of the Gibbs adsorption equation is incomplete, as recently surmised.

Acknowledgements

This work was supported by the National Science Foundation and the Cornell Center for Materials Research. The authors wish to thank the computing facility of the Cornell Center for Materials Research for computational resources and support. We also thank Professor Siegfried Dietrich for helpful comments.