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The Second Gibbs Paradox

Daan Frenkel

TL;DR

The paper addresses the subtle question of whether the chemical potential inside a crystal nucleus in coexistence with a fluid matches the fluid's chemical potential. Using a particle-based Mullins route and contrasting fixed unit-cell-volume and fixed-number-of-lattice-sites frameworks, it incorporates vacancies and surface-stress effects to resolve the paradox. The key result is that, in equilibrium, the nucleus interior shares the fluid chemical potential μ_fluid, provided point defects are allowed; naive bulk-solid μ calculations mislead due to curvature terms and the distinction between surface free-energy σ and surface stress t. The work clarifies how the nucleation barrier can be consistently described via a surface-tension framework while accounting for curvature corrections, and it discusses simulations and pore-based experimental setups to test these predictions, including measurements of Laplace pressure from lattice spacings.

Abstract

In his Equilibrium of Heterogeneous Substances Gibbs seems to suggest that the chemical potential of a crystal nucleus need not be equal to that of the coexisting fluid. In the field, Gibbs's statement has been something of a hot potato. I argue that a consistent treatment of point defects in the critical nucleus is essential for clarifying the meaning of the chemical potential of the nucleus. In the end -- as always -- Gibbs was right.

The Second Gibbs Paradox

TL;DR

The paper addresses the subtle question of whether the chemical potential inside a crystal nucleus in coexistence with a fluid matches the fluid's chemical potential. Using a particle-based Mullins route and contrasting fixed unit-cell-volume and fixed-number-of-lattice-sites frameworks, it incorporates vacancies and surface-stress effects to resolve the paradox. The key result is that, in equilibrium, the nucleus interior shares the fluid chemical potential μ_fluid, provided point defects are allowed; naive bulk-solid μ calculations mislead due to curvature terms and the distinction between surface free-energy σ and surface stress t. The work clarifies how the nucleation barrier can be consistently described via a surface-tension framework while accounting for curvature corrections, and it discusses simulations and pore-based experimental setups to test these predictions, including measurements of Laplace pressure from lattice spacings.

Abstract

In his Equilibrium of Heterogeneous Substances Gibbs seems to suggest that the chemical potential of a crystal nucleus need not be equal to that of the coexisting fluid. In the field, Gibbs's statement has been something of a hot potato. I argue that a consistent treatment of point defects in the critical nucleus is essential for clarifying the meaning of the chemical potential of the nucleus. In the end -- as always -- Gibbs was right.

Paper Structure

This paper contains 13 sections, 56 equations, 1 figure.

Table of Contents

  1. Preamble
  2. The Chemical Potential
  3. Introduction
  4. The critical crystal nucleus
  5. Accretion/dissolution and deformation
  6. Virtual volume change
  7. Mullins route
  8. Fixed unit-cell volume
  9. Fixed number of lattice sites
  10. Vacancies and the chemical potential inside a nucleus
  11. Measuring the effect of surface stress
  12. Possible experiments?
  13. Conclusions

Figures (1)

  • Figure 1: A crystallite (A) can expand either through expansion (B) or through accretion (C). In the case of expansion, the number of lattice sites remains the same, but the lattice spacing changes. In the case of accretion, new lattice sites are added.