Simulating high-entropy alloys at finite temperatures: An uncertainty-based approach

A general method is presented for modeling high-entropy alloys as ensembles of randomly sampled, ordered configurations on a given lattice. Statistical mechanics is applied post hoc to derive the ensemble properties as a function of composition and temperature, including the free energy of mixing and local structure. Random sampling is employed to address the high computational costs needed to model alloys with a large number of components. Doing so also provides rigorous convergence criteria, including the quantification of noise due to random sampling, and an estimation of the number of additional samples required to lower this noise to the desired levels. Binary to five-component alloys of the group-IV chalcogenides are used as case examples, for which the predicted miscibility shows excellent agreement with experiment. This method is well-suited for calculating the configurational thermodynamics, local structure, and ensemble properties of complex alloys, and it is attractive for materials with temperature-dependent, short-range order.

Figure 1

Alloys are represented as an ensemble of configurational states within the independent cell approximation. Post hoc application of statistical mechanics then predicts the alloy's local structure, thermodynamic stability, and material properties.

Figure 2

The methodology developed herein begins with specifying the initial conditions; random configurations are then generated within those constraints. The $\mathrm{\Delta }{H}_{\text{mix},i}$ and the temperature-dependent probability of each configuration are calculated. The properties of interest are then determined using ensemble averaging. Additionally, $\mathrm{\Delta }{G}_{\text{mix}}$ is calculated to determine stability with respect to the free-energy convex hull. The central limit theorem is then applied to estimate the uncertainty in the predictions (${\sigma }_{\overline{X}}$) and the number of new configurations that may be needed to further reduce the uncertainty below the convergence criteria, ${\sigma }_{\text{max}}$.

Figure 3

Within a given ensemble, the variation in $\mathrm{\Delta }{H}_{\text{mix},i}$ leads to significant temperature evolution of the probability for each configuration. In the high-temperature limit, the probabilities converge to $n^{-1}$. Here, an ensemble of 50 ${\mathrm{PbSe}}_{0.5}{\mathrm{Te}}_{0.5}$ configurations is considered, with each configuration represented by a line colored by its $\mathrm{\Delta }{H}_{\text{mix},i}$.

Figure 4

To illustrate the CLT, we start with the composition ${\mathrm{PbSe}}_{0.5}{\mathrm{Te}}_{0.5}$ and highlight in (a) that the distribution of $\mathrm{\Delta }{\overline{H}}_n$ values for any given $n$ forms a Gaussian centered around the true value, $\mathrm{\Delta }{\overline{H}}_{\text{tot}}$. The standard deviation of each Gaussian, ${\sigma }_{\mathrm{\Delta }\overline{H},n}$, decreases with increasing $n$. Each Gaussian arises from calculating $\mathrm{\Delta }{\overline{H}}_n$ of 10 000 ensembles. Each point in (b) is determined by taking the standard deviation in $\mathrm{\Delta }\overline{H}$ from 10 000 ensembles, including the Gaussians in (a). In (b), this decrease in ${\sigma }_{\mathrm{\Delta }\overline{H},n}$ can be modeled using the central limit theorem (black line). By simply knowing the ensemble size and ${\sigma }_{\mathrm{\Delta }H,\text{tot}}, {\sigma }_{\mathrm{\Delta }\overline{H},n}$ can be determined. The procedure is repeated for ${\mathrm{PbS}}_{0.5}{\mathrm{Te}}_{0.5}$ to illustrate the importance of chemical composition. Since ${\sigma }_{H,\text{tot}}$ is larger, ${\mathrm{PbS}}_{0.5}{\mathrm{Te}}_{0.5}$ requires more configurations to reach the same level of uncertainty.

Figure 5

The distribution of $\mathrm{\Delta }{H}_{\text{mix},i}$ is shown in the left panel, made from an ensemble of 50 ${\mathrm{PbSe}}_{0.5}{\mathrm{Te}}_{0.5}$ configurations. $\mathrm{\Delta }{\overline{H}}_{50}$ denotes the average $\mathrm{\Delta }{H}_{\text{mix},i}$, and $\mathrm{\Delta }{H}_{\text{min}}$ shows the ground-state configuration. In the right panel, the ensemble $\mathrm{\Delta }{H}_{\text{mix}}$ gradually moves towards $\mathrm{\Delta }{\overline{H}}_{50}$ as the temperature increases. The entropic contributions ($T\mathrm{\Delta }{S}_{\text{mix}}$) and the resulting free energy $\mathrm{\Delta }{G}_{\text{mix}}$ are also shown. For temperatures at which $\mathrm{\Delta }{G}_{\text{mix}}$ is below zero, the alloy is stable against decomposition to its parent compounds (i.e., PbSe and PbTe).

Figure 6

Given the complete ensemble, there is a standard deviation in the values of $\mathrm{\Delta }{H}_{\text{mix}}$ (${\sigma }_{H,\text{tot}}$) denoted by the horizontal gray dashed lines. With increasing ensemble size, the mean standard deviation ${\overline{\sigma }}_{H,n}$ rapidly approaches ${\sigma }_{H,\text{tot}}$. However, the ${\sigma }_{H,n}$ of a single ensemble shows greater variation as denoted by the error bars, corresponding to $\pm$ one standard deviation from the mean. The small variation from ${\sigma }_{H,\text{tot}}$ highlights the accuracy of the approximation of Eq. (10). The larger variation in ${\sigma }_{H,n}$ for ${\mathrm{PbS}}_{0.5}{\mathrm{Te}}_{0.5}$ arises from a broader distribution of $\mathrm{\Delta }{H}_{\text{mix}}$ values in the complete ensemble.

Figure 7

(a) To assess the minimum required supercell size, we increase the supercell size for ${\mathrm{PbSe}}_{0.5}{\mathrm{Te}}_{0.5}$ until the distribution converges and the $\mathrm{\Delta }{\overline{H}}_{\text{tot}}$ of the distribution (red) has come into agreement with the $\mathrm{\Delta }{H}_{\text{mix}}$ of a 128-atom SQS supercell (blue). The convergence of the distributions can also be tracked via the ${\sigma }_{\mathrm{\Delta }H,\text{tot}}$ values (gray). Here, each distribution is built from complete sampling of ${\mathrm{PbSe}}_{0.5}{\mathrm{Te}}_{0.5}$ for a given supercell size. (b) The importance of using larger supercells is even more visibly important for a material property like the bulk modulus. For supercell sizes of 12 and 16 atoms, the distributions are significantly different from both the properly converged 20- and 24-atom distributions and the SQS-128.

Figure 8

At low temperature, ${\mathrm{PbSe}}_{0.5}{\mathrm{Te}}_{0.5}$ has distinct Pb-Te and Pb-Se bond lengths and strong clustering of bond angles. The deviation from ${90}^{\circ }$ even at low temperature for these rocksalt structures highlights the internal strains present in these alloys. With increasing temperature, the distribution of bond lengths and angles broadens significantly. The $x$-axis of the 300 K panel has a scale that is four times larger than the 1000 and 10 000 K panels in order to accommodate the large, narrow peaks. Here, the colored distributions are for a random ensemble for 50 configurations, and the dashed lines show the complete sampling for bond lengths. Despite the low number of configurations, these structures still involve 3600 unique bonds and 7200 angles.

Figure 9

The $\mathrm{\Delta }{G}_{\text{mix}}$ of ${\mathrm{PbSe}}_{1-x}{\mathrm{Te}}_x$ gradually becomes more concave with increasing temperature, reaching full convexity at 700 K (within a tolerance of $2{\sigma }_{\overline{H},n}$). Here, the alloy compositions are sampled at 1/12 increments.

Figure 10

Bond-length distributions in ${\mathrm{PbSe}}_{1-x}{\mathrm{Te}}_x$ smear as we move towards $x$-values of 6/12, highlighting the increased structural disorder in these alloys. The peak of each distribution corresponds well to Vegard's law. All distributions are shown in the high-temperature limit, and are generated from 50 configurations.

Figure 11

(a) The calculated enthalpy of mixing ($\mathrm{\Delta }{H}_{\text{mix}}$) for the ${\mathrm{PbS}}_x{\mathrm{Se}}_y{\mathrm{Te}}_{1-x-y}$ space is shown. $\mathrm{\Delta }{H}_{\text{mix}}$ is particularly high for PbS-PbTe-rich compositions. (b) The free energy above the convex hull ($\mathrm{\Delta }{G}_{\text{hull}}$) finds that compositions rich in both PbS and PbTe are at energies above the hull. They are thus subject to decomposition to PbS and PbTe alloys. Gray circles denote being on the hull, and gray triangles signify that the composition is on the hull within the uncertainty from random sampling (i.e., $\mathrm{\Delta }{G}_{\text{hull}}<2{\sigma }_{\mathrm{\Delta }H}$). The simulated results are corroborated by experimental findings, although there is a temperature offset. Both $\mathrm{\Delta }{H}_{\text{mix}}$ and $\mathrm{\Delta }{G}_{\text{hull}}$ were calculated at 600 K. The gray dashed line is an experimental result showing the region of immiscibility at 773 K [52].

Figure 12

(a) The (${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x$)(${\mathrm{Te}}_y{\mathrm{Se}}_z{\mathrm{S}}_{1-y-z}$) phase space is visualized as a prism with cation composition varying along the axis. Pseudoternaries (b)–(d) show three slices of this prism with discrete sampling; the color scheme denotes the free energy above the convex hull. Hexagons with a silver point inside indicate that they are on the hull; hexagons with a triangle denote compositions whose distance from the hull is less than their uncertainty ($2{\sigma }_{\overline{E},n}$, as determined by the CLT). Calculating the convex hull at 600 K using the coarse sampling of (b)–(d) fills the interior of the prism with four-phase tetrahedra. At this temperature, the $x=0.5$ compositions are not on the hull, and the resulting tetrahedra of (a) span from $x=0$ to 1 compositions. (e)–(h) The lower row shows the results at 700 K. At this temperature, far more compositions are on the hull, including in the interior of the prism (g), where many of the compositions exhibit significant entropy stabilization.

Figure 13

Local distortion resulting in alloy bond lengths deviating from their parent bond lengths. Further complications arise when mixing parent compounds with different structures. Average bond lengths, broken up by type, are shown for various compositions. The bars denote the standard deviation in the bond lengths for a given type and composition. For reference, the parent bond lengths are included as well. We are invoking the high-temperature limit by taking simple averages. All averages are generated from 50 configurations.