A micromechanical derivation of the differential equation of interfacial statics

Abstract

A novel method is employed to derive the standard equation of interfacial statics from basic thermodynamic principles by employing the general ideas of matched asymptotic expansions. This entails describing the continuum thermodynamic properties of the system (such as pressure, species concentrations and chemical potentials) on two different length scales: (i) a “microscopic” length scale l, from which viewpoint the system is viewed as being continuous over its entire infinite extent. No interface, as such, exists at this level of description. Instead, however, there exists a distribution of short-range external forces (of intermolecular origin) acting on each of the chemical species i in a special direction over an effective length of O( l); (ii) a “macroscopic” or “bulk” length scale L( L ⪢ l). A “sharp” interface [of thickness O( l)] exists at this coarser level of description, separating the thermodynamic system into two continua. However, no external forces are observed at this level. Using singular perturbation theory, thermodynamic properties at the l and L scales are asymptotically matched so as to furnish the same macroscopic stress (i.e., pressure) distribution. The property of interfacial tension then arises naturally to reconcile the apparent difference between the exact l-scale microscopic stress distribution and the coarser L-scale macroscopic stress. In effect, the interfacial tension embodies the macroscopic manifestation of the l-scale intermolecular forces, which forces do not appear explicitly in the L-scale description of the phenomena because of their short-range [i.e., O( l)] nature. These forces do, however, appear implicitly in the L-scale description under the guise of an interfacial tension. The theory, which is rigorous, is shown to be formally equivalent to prior, ad hoc, hybrid, “microscopic” continuum theories which model interfacial phenomena by assuming a priori the existence of a transversely isotropic stress tensor in proximity to the interface. In contrast, the stress tensor is isotropic in our theory.