Cluster expansion and the configurational theory of alloys

Sanchez, Juan M

doi:10.1103/physrevb.81.224202

Cited by 165 publications

(132 citation statements)

References 14 publications

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“…[11,12] These basis sets are, in fact, members of an infinite family of closely related basis sets, which include the one introduced in Ref 5, and that, under a specific definition of the scalar product, are complete and orthogonal in the full configurational space. [13,14] Thus, as we point out in section 2.1, a CE cannot be carried out in the canonical ensemble since completeness of the basis functions, which is central to the theory, holds only when the scalar product is defined in the space with 2 N configurations; i.e. in the grand canonical ensemble.…”

Section: Introductionmentioning

confidence: 99%

“…One such advantage is that we can account in a relatively straightforward manner for the concentration dependence of the equilibrium volume of the alloy, which gives rise to concentration dependent ECIs and to a much faster convergence of the expansion relative to the CW method. In particular, an approach that has been referred to as the Variable Basis Cluster Expansion (VBCE) [14] describes the energies of an alloy with fixed concentration in terms of basis functions defined in such a manner that their expectations values (or averages over the crystal) vanish in the random state with the same concentration. This distinguishes the VBCE from the usual CW method since, in the latter, the expectation values of the basis functions vanish in the disordered state only at the equiatomic concentration (50/50).…”

Section: Introductionmentioning

confidence: 99%

“…without relaxations, and that such projections are done using a specific definition of the scalar product in configurational space. [5,13,14] As mentioned, the obvious advantage of the VBCE over the commonly use CW method is that the VBCE naturally captures the dominant concentration dependence of the equilibrium volume through the concentration dependence of the ECIs. [14] In its simplest form, the VBCE can be implemented by fitting a set of ECIs, at each concentration, to the energies of sets of compounds at their equilibrium volume, all with the same concentration.…”

Section: Introductionmentioning

confidence: 99%

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Foundations and Practical Implementations of the Cluster Expansion

Sanchez

2017

J. Phase Equilib. Diffus.

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Different versions of the cluster expansion are explored using the Mo-Ta system as an example. One of the objectives of this work is to establish a clear distinction between phenomenological expansions that express the energy of an alloy in the form of a generalized Ising model, i.e. with constant pair and many body interactions, and cluster expansions that use a set of complete basis functions in configurational space and define the interactions as projections of the energy onto the basis functions. For the latter case, the interactions are functions of concentration and depend, furthermore, on the full state of order of the system. Such dependence is expected since the configurational energy is shown to be a homogeneous function of degree one in the complete set of configurational variables, or correlation functions, with the interactions being the Euler derivatives of the energy with respect to the correlation functions. For the Mo-Ta system we show that, by including the concentration dependence of the interactions either explicitly or through their dependence on volume, the cluster expansion converges significantly faster than the phenomenological Ising-like models commonly used to represent the energies of disordered alloys.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Foundations and Practical Implementations of the Cluster Expansion

Sanchez

2017

J. Phase Equilib. Diffus.

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“…Detailed illustration of the method can be found in many papers. 18,20,26,[33][34][35][36][37][38][39] Here a brief description of the main aspects is given. In the cluster expansion, the alloy is treated as a lattice problem in which the lattice sites are fixed at those of the underlying Bravais lattice (fcc, bcc, etc.)…”

Section: A Cluster Expansionmentioning

confidence: 99%

First-principles calculation of phase equilibrium of V-Nb, V-Ta, and Nb-Ta alloys

et al. 2012

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In this paper, we report the calculated phase diagrams of V-Nb, V-Ta, and Nb-Ta alloys computed by combining the total energies of 40-50 configurations for each system (obtained using density functional theory) with the cluster expansion and Monte Carlo techniques. For V-Nb alloys, the phase diagram computed with conventional cluster expansion shows a miscibility gap with consolute temperature T c = 1250 K. Including the constituent strain to the cluster expansion Hamiltonian does not alter the consolute temperature significantly, although it appears to influence the solubility of V-and Nb-rich alloys. The phonon contribution to the free energy lowers T c to 950 K (about 25%). Our calculations thus predicts an appreciable miscibility gap for V-Nb alloys. For bcc V-Ta alloy, this calculation predicts a miscibility gap with T c = 1100 K. For this alloy, both the constituent strain and phonon contributions are found to be significant. The constituent strain increases the miscibility gap while the phonon entropy counteracts the effect of the constituent strain. In V-Ta alloys, an ordering transition occurs at 1583 K from bcc solid solution phase to the V 2 Ta Laves phase due to the dominant chemical interaction associated with the relatively large electronegativity difference. Since the current cluster expansion ignores the V 2 Ta phase, the associated chemical interaction appears to manifest in making the solid solution phase remain stable down to 1100 K. For the size-matched Nb-Ta alloys, our calculation predicts complete miscibility in agreement with experiment.

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“…Cluster expansions 12,13 are based on expanding the energy of an alloy configuration using nearest-neighbor lattice clusters. Each term consists of an unspecified prefactor and the product of "spin variables" for sites within a cluster.…”

Section: Introductionmentioning

confidence: 99%

Statistical physics of multicomponent alloys using KKR-CPA

2016

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We apply variational principles from statistical physics and the Landau theory of phase transitions to multicomponent alloys using the multiple-scattering theory of Korringa-Kohn-Rostoker (KKR) and the coherent potential approximation (CPA). This theory is a multicomponent generalization of the S (2) theory of binary alloys developed by G. M. Stocks, J. B. Staunton, D. D. Johnson and others. It is highly relevant to the chemical phase stability of high-entropy alloys as it predicts the kind and size of finite-temperature chemical fluctuations. In doing so it includes effects of rearranging charge and other electronics due to changing site occupancies. When chemical fluctuations grow without bound an absolute instability occurs and a second-order order-disorder phase transition may be inferred. The S (2) theory is predicated on the fluctuation-dissipation theorem; thus we derive the linear response of the CPA medium to perturbations in site-dependent chemical potentials in great detail. The theory lends itself to a natural interpretation in terms of competing effects: entropy driving disorder and favorable pair interactions driving atomic ordering. To further clarify interpretation we present results for representative ternary alloys CuAgAu, NiPdPt, RhPdAg, and CoNiCu within a frozen charge (or band-only) approximation. These results include the so-called Onsager mean field correction that extends the temperature range for which the theory is valid.

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Cluster expansion and the configurational theory of alloys

Cited by 165 publications

References 14 publications

Foundations and Practical Implementations of the Cluster Expansion

Foundations and Practical Implementations of the Cluster Expansion

First-principles calculation of phase equilibrium of V-Nb, V-Ta, and Nb-Ta alloys

Statistical physics of multicomponent alloys using KKR-CPA

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