Chemomechanics in alloy phase stability

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Chemomechanics in alloy phase stability

Sesha Sai Behara, John C. Thomas, Brian Puchala, and Anton Van der Ven

Phys. Rev. Materials 8, 033801 – Published 12 March 2024

Abstract

We describe a first-principles statistical mechanics method to calculate the free energies of crystalline alloys that depend on temperature, composition, and strain. The approach relies on an extension of the alloy cluster expansion to include an explicit dependence on homogeneous strain in addition to site occupation variables that track the degree of chemical ordering. The method is applied to the Si-Ge binary alloy and is used to calculate free energies that describe phase stability under arbitrary epitaxial constraints. We find that while the incoherent phase diagram (in which coexisting phases are not affected by coherency constraints) hosts a miscibility gap, coherent phase equilibrium predicts ordering and negative enthalpies of mixing. Instead of chemical instability, the chemomechanical free energy exhibits instabilities along directions that couple the composition of the alloy with a volumetric strain order parameter. This has fundamental implications for phase field models of spinodal decomposition as it indicates the importance of gradient energy coefficients that couple gradients in composition with gradients in strain.

Received 31 October 2023
Revised 17 January 2024
Accepted 13 February 2024

DOI:https://doi.org/10.1103/PhysRevMaterials.8.033801

Physics Subject Headings (PhySH)

Order parameters Phase diagrams Phase separation Thermodynamics of computation Thermodynamics of mixing

Statistical Physics & Thermodynamics

Authors & Affiliations

Sesha Sai Behara ¹, John C. Thomas¹, Brian Puchala², and Anton Van der Ven^1,*

¹Materials Department, University of California, Santa Barbara, California 93106, USA
²Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

^*avdv@ucsb.edu

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Vol. 8, Iss. 3 — March 2024

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Images

Figure 1
(a) Formation energies of fully relaxed Si-Ge configurations. Si and Ge in the fully relaxed diamond parent crystal structure were used as reference states. (b) $e_{1}$ strain order parameter for fully relaxed Si-Ge configurations with pure Si as the reference state.
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Figure 2
Formation energies of various Si-Ge configurations at fixed equilibrium volumes of (a) a Si-Ge alloy at $x = 0.5,$ (b) pure Si, and (c) pure Ge. The dotted black line represents the thermodynamic convex hull whereas the orange squares represent the ground-state configurations that fall on the convex hull. The formation energies in all three plots are calculated relative to the energies of pure Si and Ge at their equilibrium volumes. Formation energies calculated at fixed volume are always higher than the fully relaxed formation energy. The two formation energies for a given structure are only equal when the imposed strain constraints (including volume) are equal to the fully relaxed strains.
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Figure 3
(a) The root mean square error (rmse) upon the incremental addition of cluster basis functions to a truncated configuration plus strain cluster expansion. (b) Same as (a), but within an rmse range of 0 to 10 meV/unit cell. (c) rmse upon the incremental addition of cluster basis functions for a configuration only cluster expansion.
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Figure 4
Helmholtz free energy as a function of concentration $x$ and volumetric strain $e_{1}$ calculated at 298 K. (a) A three-dimensional depiction of the free-energy surface and (b) a contour plot. The solid orange line in (b) traces the minimum free-energy path in $x − e_{1}$ space, while the minimum free energy at each composition is plotted as a function of concentration in orange in (c). (c) Free-energy slices at different values of $e_{1}$ as a function of composition (d) Free-energy slices at different values of composition as a function of $e_{1}$ .
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Figure 5
The calculated compressibility of the ${Ge}_{x} {Si}_{1 - x}$ alloy as a function of concentration at $P = 0$ .
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Figure 6
(a) Schematic illustration of epitaxial growth. (b) Helmholtz free energy at room temperature calculated as a function of composition and $E_{z z}$ under epitaxial strain conditions (fixed $E_{x x}$ and $E_{y y}$ ). (c) Free-energy slices at $x = 0.35, 0.5, 0.65$ as a function of $E_{z z}$ . (d) The solid-orange line shows the epitaxially constrained free energy coinciding with $σ_{z z} = 0$ . The solid-blue line corresponds to the equilibrium free energy when $P = 0$ . The dotted-orange line shows the tangent to the epitaxially constrained free energy at $x = 0.35$ . The black arrows indicate how the chemical potentials of Si and Ge change upon going from an unconstrained state to an epitaxially strained state at fixed $E_{x x}$ and $E_{y y}$ and $σ_{z z} = 0$ .
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