Reply to comment on 'Surface thermodynamics and surface stress for deformable bodies'

TL;DR

This paper clarifies the complex thermodynamics of deformable surfaces by introducing an 'ideal transformation' concept, deriving exact expressions for surface stress and energy variations, and resolving misunderstandings about surface quantities.

Contribution

The paper introduces a new 'ideal transformation' concept to accurately determine local thermodynamic variables and surface stress for deformable bodies, clarifying previous misconceptions.

Findings

Derived exact relations between surface grand potential, stress, and strain.

Established the equivalence of Eulerian and Lagrangian surface energy variations.

Provided explicit formulas for surface deformation work and surface stress.

Abstract

The above comment http://dx.doi.org/10.1088/0953-8984/22/42/428001 and a previous letter by the same author reveal a great misunderstanding of what Eulerian and Lagrangian quantities are, and a confusion between the deformation of an element of a surface and the creation of a new element of a surface. Surface thermodynamics is complex because the surface quantities are not 'intuitive' (as surface excesses on some dividing surface) and the thermodynamic variables of the state of a surface are a priori completely unknown. This is why we introduced a new concept ('ideal transformation') and presented detailed proof, leading to the determination of the 'local' thermodynamic variables of the state of the surface, the exact expression of the work of deformation of the surface, and the definition of surface stress, for any deformable body (Olives 2010 J. Phys.: Condens. Matter 22 085005, arXiv:arXiv:1906.11022). These results are not obvious (despite their similarity with some expressions in volume thermodynamics). We explicitly write the Eulerian forms of (i) the relation between the surface grand potential per unit area, the surface stress and the surface strain, showing its exact equivalence with the Lagrangian form, and (ii) the variation of the surface energy due to both the deformation of an element of the surface and the creation of a new element of the surface.