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Prediction of Phase Diagrams

Computation and Theoretical Methods

  1. Didier de Fontaine

Published Online: 12 OCT 2012

DOI: 10.1002/0471266965.com011.pub2

Characterization of Materials

Characterization of Materials

How to Cite

de Fontaine, D. 2012. Prediction of Phase Diagrams. Characterization of Materials. 1–24.

Author Information

  1. University of California, Berkeley, CA, USA

Publication History

  1. Published Online: 12 OCT 2012

Abstract

A phase diagram is a graphical object, usually determined experimentally, indicating phase relationships in thermodynamic space. Usually, one coordinate axis represents temperature; the others may represent pressure, volume, concentrations of various components, and so on. This unit is concerned only with temperature-concentration diagrams, limited to binary (two-component) and ternary (three-component) systems. Since more than one component is considered, the relevant thermodynamic systems are alloys, by definition, of metallic, ceramic, or semiconductor materials. The emphasis here is placed primarily on metallic alloys.

Phase diagrams can be classified broadly into two main categories: experimentally and theoretically determined. The object of the present unit is the theoretical determination—i.e., the calculation of phase diagrams, meaning ultimately their prediction. But calculation of phase diagrams can mean different things: there are prototype, fitted, and first-principles approaches. Prototype diagrams are calculated under the assumption that energy parameters are known a priori or given arbitrarily. Fitted diagrams are those whose energy parameters are fitted to known, experimentally determined diagrams or to empirical thermodynamic data. First-principles diagrams are calculated on the basis of energy parameters calculated from essentially only the knowledge of the atomic numbers of the constituents, hence by actually solving the relevant Schrödinger equation.

Theory also enters in the experimental determination of phase diagrams, as these diagrams not only indicate the location in thermodynamic space of existing phases but also must conform to rigorous rules of thermodynamic equilibrium (stable or metastable). The fundamental rule of equality of chemical potentials imposes severe constraints on the graphical representation of phase diagrams, while also permitting an extraordinary variety of forms and shapes of phase diagrams to exist, even for binary systems.

That is one of the attractions of the study of phase diagrams, experimental or theoretical: their great topological diversity subject to strict thermodynamic constraints. In addition, phase diagrams provide essential information for the understanding and designing of materials, and so are of vital importance to materials scientists. For theoreticians, first-principles (or ab initio) calculations of phase diagrams provide enormous challenges, requiring the use of advanced techniques of quantum and statistical mechanics.

Keywords:

  • phase diagrams;
  • classical approach;
  • mean-field approach;
  • cluster approach;
  • cluster expansion;
  • electronic structure calculations;
  • ground state;
  • static displacive interactions