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Gas Adsorption in Metals

Derivation by Bruce Eichinger

This document describes the application of statistical thermodynamics to gas absorption in metals. It reviews the basic tenets of thermodynamics in application to the phase equilibria of interest. In an open system – the metal that is absorbing gas – the chemical potential of the gaseous component must be equal in the gas and in the metal. This establishes a critical link between the well-known properties of the gas and the properties (to be discovered) of the gaseous component atoms in the solid. The equilibrium constant that relates the composition in the solid to the pressure of the gas is determined by the nature of the standard states that are chosen for two phases. These standard states are discussed at length, as they must be thoroughly understood before one can know how to put thermochemical and computational information together to determine the equilibrium constant.

The representation of the free energy of the solid phase is the province of statistical mechanics. The two components of the free energy, the energy and entropy, must be formulated and calculated to determine the relation between composition and chemical potential in the solid phase. The MEDEA-Phonon module in conjunction with MEDEA-VASP can be used to determine the free energy, via Debye theory, of representatives of the MmXn system for various values of m and n. The partition function and free energy of supercells of m metal atoms and n absorbed atoms X in specified atomic arrangements can be determined by this method. What remains is to relate the configurational entropy of the system to the computed free energies of these (m,n) supercells. (The use of the word cluster is purposely avoided here.) It is the goal of this document to relate the configurational entropy, and hence system free energy, to any determined set of (m,n) supercell free energies.

This objective is met via a sequence of models of increasing complexity. At first we will consider the case of absorption into single sites, such that the absorbed atoms are independent of one another. The next most complicated case is absorption into two different sites with different energetics (actually free energies). Following this is a discussion of the quasi-chemical approximation, which provides a second approximation to the configurational entropy by weighting the formation of pairs of adjacent absorbed atoms by the free energy for pair formation. This is one step beyond the independent particle approximation. A model at this level is seen to be one stage in the more general cluster expansion method, which is the final model to be addressed. The theoretical background to the cluster expansion method is however not discussed. The reader is referred to the other sources and some references are provided.

Following the development of these models a short discussion of Widom insertion and the Gibbs Ensemble Method is presented, as these techniques ultimately provide “model independent” methods for determination of phase equilibria. At the end of the document, a few cautionary notes on the limitations of the methods are given.

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