The second Gibbs paradox

Gibbs's monumental article on the equilibrium of heterogeneous substances contains a paradoxical sentence stating that, for a crystallite in equilibrium with a fluid, the chemical potential of the solid will not be equal to that of the fluid if the surface free-energy density differs from the mechanical surface-tension. How can this be? After all, in chemical equilibrium, the chemical potential of any species should be the same throughout the system. This "second Gibbs paradox" has intrigued many authors. In the present paper, I sketch my interpretation of the approach of Gibbs and that of Mullins [J. Chem. Phys. 81, 1436-1442 (1984)], which accounts for the possibility of vacancies and interstitials. I argue that a consistent treatment of point defects in a critical nucleus is essential for clarifying the meaning of the chemical potential of the nucleus. My paper lacks the rigor of Gibbs or Mullins but will hopefully be more accessible for scientists who think primarily in terms of atoms and molecules. In my attempt, I am motivated by a quote that is sometimes attributed to Paul Valéry: "The glass must be absolutely transparent for one to perceive the mud at the bottom."