The chemical potential in thermodynamic systems under non-hydrostatic stresses

It is proved that a chemical potential μ v = u v – Ts v + pv v may be introduced for every chemical component v which may be considered a possible component everywhere in a multiphase system in thermodynamic equilibrium under non-hydrostatic stresses, where —3 p is the trace of the stress tensor. It is a condition of equilibrium that μ v has the same value throughout such a system and it is shown that in a virtual infinitesimal variation d U = T d S + d W + Ʃ v μ v d N v , where U, S are the total energy and entropy of the multi-phase system, and d W is the total mechanical work done on the system. At an interface between phases where a discontinuous displacement is permitted, it is shown also that μ v = u v - Ts v + P n v v , for both phases in contact at the interface, P n being the normal component of the pressure at the interface. In a system in which each phase is under a uniform stress and is connected to at least one other phase by such an interface, all phases at equilibrium must thus have the same value of p , and the normal component of the pressure at every such interface must also be p . An important example of this latter result is that of a fluid-solid system, for which, if p is the fluid pressure, the solid must be under an equal hydrostatic pressure p together with a shear stress whose principal directions are perpendicular to the normal of the interface, this new result representing a considerable restriction on the possible stress in a solid at chemical equilibrium with the fluid. The chemical potential is not assumed to exist but is introduced as an undetermined multiplier in the application of the Gibbs condition of thermodynamic equilibrium, and all its important properties are deduced. The same method may be applied more simply in hydrostatic cases.