Statistical physics of multicomponent alloys using KKR-CPA

Suffian N. Khan, J. B. Staunton, and G. M. Stocks

Phys. Rev. B 93, 054206 – Published 16 February 2016

Abstract

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.

Received 21 December 2015

DOI:https://doi.org/10.1103/PhysRevB.93.054206

Physics Subject Headings (PhySH)

Order parameters Phase diagrams

Transition metal alloys

Density functional theory Variational approach

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Suffian N. Khan^1,*, J. B. Staunton^2,†, and G. M. Stocks^1,‡

¹Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6114, USA
²Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom

^*khansn@ornl.gov
^†J.B.Staunton@warwick.ac.uk
^‡stocksgm@ornl.gov

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Vol. 93, Iss. 5 — 1 February 2016

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Images

Figure 1
(a) High-temperature fully disordered state of a hypothetical one-dimensional equiatomic alloy “ABC” with unspecified atomic interactions. The filling at each site represents an ensemble average of the occupancy by atoms of type A (red cross hatch), B (blue grid), and C (green north east lines). We are concerned with the free-energy change on imposing an infinitesimal, site-dependent variation in these average occupancies (i.e., concentrations). (b) A finite variation in site concentrations establishing partial to full ordering. This variation may be viewed as the sum of two concentration waves: $c (x) = [1 / 3, 1 / 3, 1 / 3] + η ([- 1 / 2 - \sqrt{3} i / 2, - 1 / 2 + \sqrt{3} i / 2, 1] e^{i 2 π x / 3 a} + c . c .)$ for components A, B, and C, respectively, and lattice constant $a$ .
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Figure 2
Results for a toy one-dimensional binary alloy consisting of twelve sites with periodic boundary conditions. Atoms are of kind A or B and the units of energy are arbitrary. Nearest-neighbor A-A pair energy is 5.0 and second nearest-neighbor A-A pair energy is 2.0. All other pair interactions are zero. (a) Exact and variational grand potential for fixed concentration 0.5. The exact answer is calculated from a grand canonical weighted sum over all $2^{12}$ configurations. Note that the variational upper bound becomes loose at large $β$ . (b) Exact and variational Helmholtz free energy versus concentration of A atoms for $β = 0.2$ (lower curves) and $β = 2.0$ (upper curves). The variational bound becomes tight in the ordered limits $c \to 0$ and $c \to 1$ .
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Figure 3
Variations in concentration space of a three-component alloy with hypothetical $δ F^{(1)}$ contours (solid red) centered about the origin $δ \bar{c} = (0, 0, 0)$ (red dot). The contours have a contrived symmetry so that every vector within the subspace $δ {\bar{c}}_{α_{1}} + δ {\bar{c}}_{α_{2}} + δ {\bar{c}}_{α_{3}} = 0$ is an eigenvector. One orthogonal pair of eigenvectors is shown (centered arrows). The projection of contours and this pair of eigenvectors onto the subspace spanned by $(δ {\bar{c}}_{α_{1}}, δ {\bar{c}}_{α_{2}}, 0)$ is shown below (dashed blue). The inset shows a top-down view of projected contours and eigenvectors. Key here is that projected eigenvectors do not align with projected contours. Therefore care must be taken to define eigenvectors in an unambiguous and consistent manner.
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Figure 4
Short-range order parameters for the same toy model as described in the caption of Fig. 2. One-dimensional short-ranged models are a especially difficult case for mean-field theories. Nevertheless, this toy model serve as a useful illustration of the utility of an Onsager reaction-field correction. (a) The exact short-range order of Eq. (25). From top to bottom (at large $β$ ) curves correspond to on-site, second, fourth, fifth, third, and first neighbors, respectively. Note that the small $β$ ordering does not necessarily reflect the large $β$ ordering. At large $β$ , the system establishes a -A-B-A-B-A-B-A-B-A-B-A-B- pattern. (b) The variational short-range order $\bar{Ψ}$ of Eq. (26) and Onsager corrected short-range order $\hat{Ψ}$ of Sec. 10 in the small $β$ limit. Note that inclusion of an Onsager correction suppresses a divergence in $\bar{Ψ}$ and shows a striking increase in range of validity.
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Figure 5
Eigenvalues of the chemical stability matrix of Eq. (30) along special $k$ directions in the FCC Brillouin zone (solid curves) for (a) CuAgAu at 750 K, (b) NiPdPt at 1100 K, (c) RhPdAg at 5000 K, and (d) CoNiCu at 400 K. The eigenvalues represent the quadratic coefficient of the energy cost of a concentration wave with wave-vector $k$ . Dash-dot curves include an Onsager reaction field. The nature of the eigenvectors is discussed in Sec. 8.
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