Sunday, November 27, 2005

Young-Laplace or Gibbs-Thomson equation

"Every idea in the study of phase transformations can be traced to Gibbs" is a favourite quote of Abi. The latest American Journal of Physics carries an article about the Young-Laplace equation and its derivation, which, in metallurgical literature goes by the name of Gibbs-Thomson equation. I found the following paragraph about the merits of local derivation quite interesting.
A more important point is the character of Young's and Laplace's equations. Because they are local conditions that apply to any portion of the surfaces, a local derivation is more appropriate. From the global approach, one might think that the total energy of the system must be minimal for these equations to be valid, which is not true. The condition of minimum energy is restrictive. If the contact line moves slowly enough (see Sec. III E), Young's equation is obeyed even if the rest of the system (far from the contact line) is not in equilibrium. In contrast, a rigorous global derivation must minimize the energy at constant entropy or the Helmholtz free energy at constant total volume. These additional constraints to the minimization problem should be irrelevant for the derivation of a local equation. Although necessary in the global approach, they may be a source of confusion about whether sigmai is the surface Helmholtz or Gibbs' free energy. (In this respect, the paper by Tolman is enlightening.) In the local approach, the surfaces move without any constraint. In particular, the local derivations are valid even for a nonequilibrium state where the temperature is not homogeneous.
But the conclusion was far more interesting:
In spite of the merits of the local approach, Gibbs' derivation (global approach) seems aesthetically more appealing because all laws governing equilibrium (thermal, chemical, and mechanical equilibrium) are deduced at once. We conclude that the global approach is more suitable if one is interested in the equilibrium conditions of a fluid system, whereas a local approach is preferred for the derivation of Laplace's and Young's equations.
A pedagogical article; well worth the efforts of reading.

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