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Elastic tensor-derived properties of composition-dependent disordered refractory binary alloys using DFPT

Surya T. Bijjala, Susan R. Atlas, Pankaj Kumar

TL;DR

This work addresses the challenge of predicting elastic properties in compositionally disordered refractory binaries by leveraging density functional perturbation theory (DFPT) with special quasi-random structures (SQS). It computes both rigid-ion and relaxed-ion elastic tensors and decomposes nuclear-relaxation contributions across Mo–Nb–Ta–W compositions, yielding $C_{ij}^{rigid}$, $C_{ij}^{relaxed}$, and derived moduli $B$, $G$, $E$ along with ductility indicators like $B/G$ and Cauchy pressure. Key findings show excellent agreement with experimental data, Vegard-like linear trends for same-group binaries and nonlinear behavior for different-group binaries, and near-isotropic elasticity with site-resolved nuclear-relaxation maps revealing lattice heterogeneity. The approach provides a rigorous, quantum-mechanical route to composition-dependent elasticity in disordered alloys and supplies microscopic relaxation fields suitable for data-driven materials design and extended multi-component systems.

Abstract

The elastic tensor provides valuable insight into the mechanical behavior of a material with lattice strain, such as disordered binary alloys. Traditional stress-strain methods have made it possible to compute elastic constants for ordered structures and individually tailored alloy compositions. However, this approach depends on predetermined or iteratively-chosen strain tensors. This poses a significant challenge for systematic, composition-dependent studies of disordered materials with low symmetry. DFPT provides a compelling alternative to stress-strain methods: it allows for an unbiased determination of the elastic tensor, as well as access to local field data derived from the underlying general response function framework. Despite its intrinsic flexibility and efficiency, DFPT has seen limited application to the study of disordered systems. At the same time, there is a growing need for expanded quantum mechanical data to improve predictive modeling of complex disordered material properties. Here we present results for the rigid-ion and relaxed-ion elastic tensors computed using DFPT, for a comprehensive set of structural refractory BCC binary alloys of Mo, Nb, Ta, and W. We map the quantum-driven heterogeneity in elastic properties, and associated relaxation fields at each disordered structure lattice site, by computing the force response internal strain tensor and displacement response internal strain tensors. Derived properties -- the bulk modulus ($B$), shear modulus ($G$), Young's modulus ($E$), Poisson's ratio ($ν$), Pugh's ratio ($B/G$), Cauchy pressure and elastic anisotropy -- are reported as a function of composition for all refractory binaries. The DFPT-computed mechanical properties data for the refractory binary alloys at systematically-varied Mo, Nb, Ta, and W compositions are in excellent agreement with available experimental data.

Elastic tensor-derived properties of composition-dependent disordered refractory binary alloys using DFPT

TL;DR

This work addresses the challenge of predicting elastic properties in compositionally disordered refractory binaries by leveraging density functional perturbation theory (DFPT) with special quasi-random structures (SQS). It computes both rigid-ion and relaxed-ion elastic tensors and decomposes nuclear-relaxation contributions across Mo–Nb–Ta–W compositions, yielding , , and derived moduli , , along with ductility indicators like and Cauchy pressure. Key findings show excellent agreement with experimental data, Vegard-like linear trends for same-group binaries and nonlinear behavior for different-group binaries, and near-isotropic elasticity with site-resolved nuclear-relaxation maps revealing lattice heterogeneity. The approach provides a rigorous, quantum-mechanical route to composition-dependent elasticity in disordered alloys and supplies microscopic relaxation fields suitable for data-driven materials design and extended multi-component systems.

Abstract

The elastic tensor provides valuable insight into the mechanical behavior of a material with lattice strain, such as disordered binary alloys. Traditional stress-strain methods have made it possible to compute elastic constants for ordered structures and individually tailored alloy compositions. However, this approach depends on predetermined or iteratively-chosen strain tensors. This poses a significant challenge for systematic, composition-dependent studies of disordered materials with low symmetry. DFPT provides a compelling alternative to stress-strain methods: it allows for an unbiased determination of the elastic tensor, as well as access to local field data derived from the underlying general response function framework. Despite its intrinsic flexibility and efficiency, DFPT has seen limited application to the study of disordered systems. At the same time, there is a growing need for expanded quantum mechanical data to improve predictive modeling of complex disordered material properties. Here we present results for the rigid-ion and relaxed-ion elastic tensors computed using DFPT, for a comprehensive set of structural refractory BCC binary alloys of Mo, Nb, Ta, and W. We map the quantum-driven heterogeneity in elastic properties, and associated relaxation fields at each disordered structure lattice site, by computing the force response internal strain tensor and displacement response internal strain tensors. Derived properties -- the bulk modulus (), shear modulus (), Young's modulus (), Poisson's ratio (), Pugh's ratio (), Cauchy pressure and elastic anisotropy -- are reported as a function of composition for all refractory binaries. The DFPT-computed mechanical properties data for the refractory binary alloys at systematically-varied Mo, Nb, Ta, and W compositions are in excellent agreement with available experimental data.
Paper Structure (21 sections, 49 equations, 22 figures, 4 tables)

This paper contains 21 sections, 49 equations, 22 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Computational Methods
  3. Construction of disordered binary alloy supercells and structure optimization
  4. Density functional perturbation theory
  5. Elastic properties
  6. Nuclear-relaxation fields
  7. Results
  8. Rigid-ion vs. relaxed-ion elastic constants
  9. Rigid-ion elastic constants
  10. Comparison with experiment
  11. Derived mechanical properties
  12. Elastic anisotropy and elastic heterogeneity as a function of composition
  13. Nuclear-relaxation contributions to elastic properties
  14. Discussion
  15. Elastic tensor variations with distortions in lattice
  16. ...and 6 more sections

Figures (22)

  • Figure 1: Composition-dependent Voigt-averaged rigid-ion elastic constants $\bar{C}_{11}^{rigid}$, $\bar{C}_{12}^{rigid}$ and $\bar{C}_{44}^{rigid}$ (solid lines, Redlich-Kister polynomial fit to computed values (filled diamonds)), compared with experimental elastic constants (stars) for (a) MoNb hubbell1972elastic; (b) MoTa armstrong1970influence; (c) MoW; (d) NbTa fisher1981effects; (e) WNb frey1978elastic; and (f) WTa anderson1980single. Computed error bars correspond to standard deviation with respect to Voigt averages (Eq. (\ref{['eq:voigt-averages']})).
  • Figure 2: Composition-dependent elastic moduli $B$, $G$ and $E$ computed using the Voigt method (Eqs. (\ref{['eq:B_voigt']})--(\ref{['eq:E_voigt']})), as a function of composition, compared with experimental $E$ (circles) for (a) MoNb mccoy1964mechanicalcampbell2008elementsWebElements-website; (b) MoTa mccoy1964mechanicalcampbell2008elementsWebElements-websitekhakurel2021machine; (c) MoW mccoy1964mechanicalcampbell2008elementsWebElements-website; (d) NbTa mccoy1964mechanicalcampbell2008elementsWebElements-websitekhakurel2021machine; (e) WNb mccoy1964mechanicalcampbell2008elementsWebElements-website; and (f) WTa mccoy1964mechanicalcampbell2008elementsWebElements-websitekhakurel2021machine. Error bars correspond to the standard deviation about averaged experimental values.
  • Figure 3: Composition-dependent Pugh's ratio ($B/G$; green) and Cauchy pressure (($C_{12} - C_{44}$); blue) for (a) MoNb; (b) MoTa; (c) MoW; (d) NbTa; (e) WNb; and (f) WTa. Horizontal lines indicate maximum Cauchy pressure or Pugh's ratio of the constituent elements.
  • Figure 4: Anisotropy plots of $G$ (GPa), $E$ (GPa), and $\nu$ for: (a)-(c) $\rm{Mo_{0.25}Nb_{0.75}}$; (d)-(f) $\rm{Mo_{0.5}Nb_{0.5}}$; and (g)-(h) $\rm{Mo_{0.75}Nb_{0.25}}$. Plots generated using the ELATE tool gaillac2016elate. The maximum and minimum values of $G$, $E$ and $\nu$ are represented as transparent blue and solid green surfaces respectively.
  • Figure 5: Anisotropy plots of $G$ (in GPa), $E$ (in GPa), and $\nu$ for: (a)-(c) $\rm{W_{0.25}Ta_{0.75}}$; (d)-(f) $\rm{W_{0.5}Ta_{0.5}}$; and (g)-(h) $\rm{W_{0.75}Ta_{0.25}}$. Plots generated using the ELATE tool gaillac2016elate. The maximum and minimum values of $G$, $E$ and $\nu$ are represented as transparent blue and solid green surfaces respectively.
  • ...and 17 more figures