A general theory of ordering effects on fixed lattices is presented. Effective multiatom interactions are derived from the electronic structure and are shown to account for many observed ordering trends.

Order and Phase Stability in Alloys

Publisher Summary This chapter presents the formalism and applications of cluster expansions to the problem of ab initio calculations of thermodynamic properties of crystalline alloys. The success of the method owes much to the parallel developments of statistical and quantum mechanical methods and, of course, to the availability of powerful computational hardware and software. The technique of orthogonal expansions in cluster functions has revolutionized the treatment of configurationally disordered systems. In addition to offering the most rational and general state-of-order description of alloys, cluster methods provide the rigorous and essential link between quantum and statistical mechanical aspects of first-principles thermodynamic calculations. Cluster functions constitute a complete orthonormal set. This basis set produces a rigorous cluster algebra that is used for systems containing an arbitrary degree of configurational order (or disorder). Cluster methods are, in principle, ideally suited for calculating alloy phase equilibria.

Cluster Approach to Order-Disorder Transformations in Alloys

We present three 16-atom special quasirandom structures (SQS's) for ${\mathrm{A}}_{1\ensuremath{-}x}{\mathrm{B}}_{x}$ bcc substitutional alloys at compositions $x=0.25$, 0.50 and 0.75, respectively. The structures possess local pair and multisite correlation functions that mimic those of the corresponding random bcc alloy. The introduction of these SQS's allows for the possibility of first-principles calculations of bcc solid solutions, even those with significant size-mismatch or atomic relaxation. We have tested our SQS's via first-principles calculations in the Mo--Nb, Ta--W and Cr--Fe systems, in which the bcc solid solution is observed to be stable over the whole composition range. Our first-principles SQS results provide formation enthalpies, equilibrium lattice parameters and magnetic moments of these bcc alloys which agree satisfactorily with most existing experimental data in the literature. In an effort to understand the atomic relaxation behavior in bcc solid solutions, we have also investigated the nearest neighbor bond length distributions in the random bcc alloys. The proposed bcc SQS's are quite general and can be applied to other binary bcc alloys.

http://www.mmm.psu.edu/CJiang2004PRB_BCCSQS.pdf

First-principles study of binary bcc alloys using special quasirandom structures

To predict the ground-state structures and finite-temperature properties of an alloy, the total energies of many different atomic configurations $\mathbf{\ensuremath{\sigma}}\ensuremath{\equiv}{{\ensuremath{\sigma}}_{i};i=1,\dots{},N}$, with $N$ sites $i$ occupied by atom A $({\ensuremath{\sigma}}_{i}=\ensuremath{-}1)$, or B $({\ensuremath{\sigma}}_{i}=+1)$, must be calculated accurately and rapidly. Direct local-density approximation (LDA) calculations provide the required accuracy, but are not practical because they are limited to small cells and only a few of the ${2}^{N}$ possible configurations. The ``mixed-basis cluster expansion'' (MBCE) method allows to parametrize LDA configurational energetics ${E}_{\mathrm{LDA}}[{\ensuremath{\sigma}}_{i};i=1,\dots{},N]$ by an analytic functional ${E}_{\mathrm{MBCE}}[{\ensuremath{\sigma}}_{i};i=1,\dots{},N]$. We extend the method to bcc alloys, describing how to select ${N}_{\ensuremath{\sigma}}$ ordered structures (for which LDA total energies are calculated explicitly) and ${N}_{F}$ pair and multibody interactions, which are fit to the ${N}_{\ensuremath{\sigma}}$ energies to obtain a deterministic MBCE mapping of LDA. We apply the method to bcc Mo-Ta. This system reveals an unexpectedly rich ground-state line, pitting Mo-rich (100) superlattices against Ta-rich complex structures. Predicted finite-$T$ properties such as order-disorder temperatures, solid-solution short-range order and the random alloy enthalpy of mixing are consistent with experiment.

Mixed-basis cluster expansion for thermodynamics of bcc alloys

This unique and comprehensive introduction offers an unrivalled and in-depth understanding of the computational-based thermodynamic approach and how it can be used to guide the design of materials for robust performances, integrating basic fundamental concepts with experimental techniques and practical industrial applications, to provide readers with a thorough grounding in the subject. Topics covered range from the underlying thermodynamic principles, to the theory and methodology of thermodynamic data collecting, analysis, modeling, and verification, with details on free energy, phase equilibrium, phase diagrams, chemical reactions, and electrochemistry. In thermodynamic modelling, the authors focus on the CALPHAD method and first-principles calculations. They also provide guidance for use of YPHON, a mixed-space phonon code developed by the authors for polar materials based on the supercell approach. Including worked examples, case studies, and end-of-chapter problems, this is an essential resource for students, researchers, and practitioners in materials science.

Computational Thermodynamics of Materials

A model tight-binding Hamiltonian including off-diagonal disorder and Hartree-Fock electronic self-consistency is used to calculate the enthalpies of formation of the Mo-Cr, Mo-Nb, and Mo-Ta alloys. The approach combines the coherent-potential approximation, the generalized perturbation method, and the cluster-variation method. The model is also used to calculate the phase diagram and the equilibrium activities for the Mo-Cr system.

M. Kosugi

Papers

Calculation of thermodynamic properties and phase diagrams of binary transition-metal alloys.

The effect of the structural energy of Cr on the NiCr equilibrium phase diagram