On the chemical potential and grand potential density of solids under non-hydrostatic stress

Non-hydrostatic stress has a peculiar effect on the phase equilibrium between solids and liquids. This was already pointed out by Gibbs. Gibbs derived his formulation of the condition for liquid–solid coexistence applying a surface accretion process without imposing chemical equilibrium between liquid and solid. Adding particles to the bulk of a solid was not possible in his view at the time. Chemical potentials for solids were later introduced by material scientists. This required extending chemical and mechanical equilibrium with a third condition involving a relation between grand potential densities controlling the migration of the interface. These issues are investigated using a nonlinear elastic continuum model (technically an open compressible neo-Hookean material) developed in a previous publication (M. Sprik, J. Chem. Phys. 155, 244701 (2021)). In common with a liquid, the grand potential density of the model is equal to minus the mean pressure even if the stress is non-hydrostatic. Applying isothermal compression normal to a liquid–solid interface initially in hydrostatic equilibrium drives the system away from coexistence. We derive the Gibbs–Thomson correction to the pressure of the liquid required to restore phase equilibrium. We find that the coupling between chemical potential of the solid and shear stress is a purely nonlinear effect.