Abstract
Highentropy alloys are random alloys with five or more components, often near equicomposition, that often exhibit excellent mechanical properties. Guiding the design of new materials across the wide composition space requires an ability to compute necessary underlying material parameters via ab initio methods. Here, density functional theory is used to compute the elemental misfit volumes, alloy lattice constant, elastic constants, and stable stacking fault energy in the fcc noble metal RhIrPdPtNiCu. These properties are then used in a recent theory for the temperature and strainrate dependent yield strength. The parameterfree prediction of 583 MPa is in excellent agreement with the measured value of 527 MPa. This quantitative connection between alloy composition and yield strength, without any experimental input, motivates this general density functional theorybased methodological path for exploring new potential highstrength highentropy alloys, in this and other alloy classes, with the chemical accuracy of firstprinciples methods.
Introduction
Highentropy alloys (HEAs), also called multiple principal element alloys, nominally consist of five or more elements at nearequal compositions in a single crystalline phase.^{1} Many HEAs possess high yield strengths, ultimate strengths, and/or ductility, making them a broad new class of candidates for structural applications.^{2,3,4} As a result, new materials are emerging rapidly. In the noble metal family, Sohn et al.^{5} reported a singlefccphase RhIrPdPtNiCu system with compressive yield strength σ_{y} = 527 MPa, ultimate strength of 1839 MPa, and strain to failure of 32.4%, the latter two properties rivaling the strongest and toughest steels.^{6} However, HEAs must not only match currently available structural alloys but exceed them in one or more properties. Since there is no physical restriction to nearequicomposition random alloys, the available composition space for multiproperty optimization is vast. Searching through that space can be greatly facilitated by theory and modeling for both property prediction and phase stability. Accurate predictions require both accurate theories and the chemically accurate inputs to those models, and the latter leads to the application of firstprinciples methods.
Strength and ductility depend on the motion and interaction of dislocations through the random alloy. Firstprinciples modeling of dislocations in pure elemental metals alone is challenging. Studying the required properties/behaviors of dislocations in complex HEAs is even more formidable. Thus, theories are needed to predict the desired properties based on computationally accessible material inputs. The first property of interest is the initial yield strength. This is the stress at which preexisting dislocations can first start to move through the crystalline lattice and generate the ongoing plastic strain at the macroscopic imposed strainrate and temperature. A general theory has been developed to predict the temperature and strainrate dependent yield strength in random fcc alloys.^{7,8} The theory envisions the HEA as an “effectivemedium matrix”, and each elemental atom in the alloy acts as a solute in the effective matrix. With an additional assumption that the solute/dislocation interaction energies are governed by elasticity, the model then requires only the average material properties of the alloy matrix and the properties of the elemental solutes in that average matrix.
Here, we use firstprinciples density functional theory (DFT) to compute all the necessary alloy properties that enter the theory, for the 6component RhIrPdPtNiCu alloy studied recently. We present a new and general method to compute the required solute misfit volumes in the multicomponent random alloys, which then enables parameterfree and experimentfree prediction of the yield strength. Good agreement with experiment is achieved, establishing the overall methodology as a framework for computationally guided design of new fcc HEAs.
Results
Theory of yield strength in multicomponent random alloys
We consider a general fcc alloy containing N_{elem} types of alloying elements at concentrations {c_{n}} (with \(\mathop {\sum}\nolimits_{n = 1}^{N_{{\mathrm{elem}}}} {\kern 1pt} c_n = 1\)), with all alloying elements distributed randomly on the fcc lattice sites. There is thus a probability c_{n} that any given lattice site will be occupied by a typen atom. The strengthening of the random alloy arises through solute/dislocation interactions. The interaction of the dislocation with the random fluctuations in the local solute concentrations induces the dislocation to become wavy over some characteristic wavelength 4ζ_{c} and amplitude w_{c} that minimize the total energy of the system. The wavy dislocation is then locally trapped in lowenergy regions. Dislocation motion, and plastic strain, thus occurs by stressassisted thermal activation across the adjacent highenergy regions along the dislocation glide plane and into the next lowenergy regions. The theory computes the fundamental zerostress energy barrier ΔE_{b} and zerotemperature flow stress τ_{y0} from which the yield stress at finite temperature T and finite strainrate \(\dot \epsilon\) are obtained. The theory is well described in the recent literature^{8} and has been validated against molecular dynamics simulations in model alloys using semiempirical interatomic potentials.^{7}
To make the theory computationally accessible, it has been simplified through the use of elasticity theory.^{7} In this limit, the solute/dislocation interaction energy of a typen solute in the alloy is modeled as −p(x, y)ΔV_{n} where p(x, y) is the elastic pressure field at solute position (x, y) due to a dislocation centered at the origin and ΔV_{n} is the average misfit volume of the typen solute in the alloy. With this simplification, the theory becomes analytic for fcc alloys. Specifically, the zerostress activation barrier ΔE_{b} and the zerotemperature shear yield stress τ_{y0} (stress at which the energy barrier is zero) are
where b is the dislocation Burgers vector, and
is the wellknown δparameter that describes the collective effect of the misfit volumes of all the elements in the alloy. In addition,
are the dislocation line tension and shear modulus for fcc slip on the {111} plane in the 〈110〉 direction expressed in terms of the standard cubic elastic constants, respectively. The dimensionless line tension parameter α = 1/8 is accurate for several fcc metals. The last quantity in Eq. (1) is an elastic coefficient P = P(C_{ij}) associated with the anisotropic dislocation pressure field. For fcc metals, P can be written in terms of appropriate averaged isotropic elastic constants μ^{ave} and ν^{ave} as
Previous applications of the model used experimentally measured isotropic elastic constants of the alloy, μ^{expt} and ν^{expt}.^{7,9,10} A forthcoming full anisotropic analysis will demonstrate that the use of the Voigtaveraged elastic constants
where B is the bulk modulus agrees with the full anisotropic model to within ~3% over a wide range of elastic anisotropy. We will thus use the Voigtaveraged values.
The resulting strength versus temperature T and strainrate \(\dot \epsilon\) is
where \(\dot \epsilon _0\) = 10^{4} s^{−1} is a reference strainrate determined by the dislocation density and other factors. The properties entering τ_{y0}(T) and ΔE_{b}(T) should in principle be computed at the temperature of interest, especially the elastic constants. We discuss this further in Section “Yield strength prediction.” Experiments are typically performed in uniaxial tension on random untextured polycrystalline specimens. The measured uniaxial yield stress is related to the shear yield stress as σ_{y} = Mτ_{y} where M = 3.06 is the Taylor factor for random fcc polycrystals.
Dislocations in fcc metals dissociate into two partial dislocations separated by a stacking fault. The two partials are separated by a distance d and each partial dislocation also spreads across a few atoms with the spreading characterized by a Gaussian function with standard deviation σ. This complex dislocation structure should thus influence the strengthening, since the solutes interact with the actual dislocation structure. Full analysis of the elastic theory as a function of d reveals that the strength is nearly independent of d for d ≳ 6.5b. The spreading parameter σ has been examined in several atomistic simulations using interatomic potentials and the typical values are 1.5 ≤ σ/b ≤ 2.0. The analytic model above (Eq. (1)) with the numerical coefficients given corresponds to a sufficiently large partial separation d ≳ 6.5b and σ/b = 1.5. The use of these values has yielded very good predictions in comparison with experiments.^{7,9,10}
From the above summary, we see that the required inputs to the analytic theory are the fcc lattice constant (a_{0} with \(b = a_0{\mathrm{/}}\sqrt 2\)), the elastic constants (C_{11}, C_{12}, C_{44}), and the solute misfit volumes (ΔV_{n}) at the alloy composition {c_{n}}. The partial separation must also be d ≳ 6.5b, which can be wellestimated in anisotropic elasticity as d = K/γ_{ssf} where γ_{ssf} is the stable stacking fault energy of the alloy and K is an anisotropic elastic parameter computed exactly in terms of dislocation character, elastic constants, and Burgers vector.^{11,12,13,14,15} All of the above quantities can be computed by firstprinciples using the methods described in Section “DFT methodology”, leading to yield strength predictions given in Section “Yield strength prediction.”
Misfit volumes
We consider, in general, an alloy containing N_{elem} types of alloying elements at concentrations {c_{n}}. The average misfit volume of a typen atom in the random alloy can be conceptually defined as
where 〈ΔV^{n/m}〉 is the volume change caused by replacing one typem atom with typen atom and with the brackets indicating averaging over many different such replacements where the atom of interest has different surrounding atomic environments.^{16} Performing this actual procedure is computationally costly and unnecessary, but the definition remains useful. Operationally, we consider a macroscopic sample with N^{tot} atoms and total volume \(V_0^{{\mathrm{tot}}}\) at the alloy composition. We then add ΔN_{n} typen atoms by replacing ΔN_{m} = −ΔN_{n}c_{m}/(1 − c_{n}) typem atoms for all m ≠ n at randomly chosen typem atom sites in the sample. The volume of the new random alloy is \(V^{{\mathrm{tot}}} = V_0^{{\mathrm{tot}}} + \mathop {\sum}\nolimits_m \left( {  {\mathrm{\Delta }}N_m} \right)\left\langle {{\mathrm{\Delta }}V^{n/m}} \right\rangle\). The average atomic volume V = V^{tot}/N^{tot} can then be expressed as
After adding the additional typen atoms, the changes of the elemental compositions are (1 − c_{n})x_{s} for the typen atom and −c_{m}x_{s} for all the other element types m ≠ n. The new alloy can thus be considered as a pseudobinary alloy with a formula \({\mathrm{[type  }}n\,{\mathrm{atom]}}_{x_s}\left[ {{\mathrm{matrix}}\left\{ {c_n} \right\}} \right]_{1  x_s}\), where the matrix is the original alloy and the “solutes” are the extra typen atoms introduced into the effective matrix. Similarly, ΔN_{n} typen atoms can be removed (ΔN_{n} < 0) with other atom types added as ΔN_{m} = −ΔN_{n}c_{m}/(1 − c_{n}) for all m ≠ n. The total volume is then \(V^{{\mathrm{tot}}} = V_0^{{\mathrm{tot}}} + \mathop {\sum}\nolimits_m {\kern 1pt} {\mathrm{\Delta }}N_m\left\langle {{\mathrm{\Delta }}V^{m/n}} \right\rangle\), with 〈ΔV^{m/n}〉 = −〈ΔV^{n/m}〉. This procedure leads again to Eq. (8), but with x_{s} being negative. Therefore, measuring the sample volume at different values of \(\left {x_s} \right \ll 1\) for all solute types enables the determination of the desired misfit volumes ΔV_{n} for all solutes in the alloy at the composition of interest.
To compute the average atomic volumes V(x_{s}) of the various pseudobinary random alloys, and the lattice constant and elastic constants at the central composition, there are various methods, such as cluster expansion (CE) method,^{17} small set of ordered structures (SSOS) method,^{18} similar local atomic environment (SLAE) method,^{19} etc. To date, the two most widely used methods are the special quasirandom structures (SQS)^{20} and the coherent potential approximation (CPA).^{21} The supercell method combined with the concept of SQS has the very important advantage that explicit local lattice distortions within the SQS supercell are possible. On the other hand, the CPA method can mimic the perfect chemical disorder within one primitive cell. In this work, we need to compute the atomic volumes, lattice constants, and elastic constants precisely, where the local lattice distortion should be explicitly considered. Previous literature has shown that the local lattice relaxation has a significant effect on the alloy properties, e.g., formation energy.^{22} We also wish to assess accuracy of the computed quantities, which can be accomplished using multiple realizations and sizes of the SQS cells. The SQS method is strongly preferred for these reasons, and is used here.
Application of the above strategy to the equicomposition 6component (ABCDEF) alloy such as RhIrPdPtNiCu alloy is achieved as follows. We start with a supercell \(\left( {{\mathrm{ABCDEF}}} \right)_{N_0}\) (total number of atoms N_{tot} = 6N_{0}). We then add 5N_{1} atoms of type A and remove other atom types equally, leading to the pseudobinary composition \({\mathrm{A}}_{6N_1}\left( {{\mathrm{ABCDEF}}} \right)_{N_0  N_1}\) with x_{s} = 6N_{1}/N_{tot}. Similarly, removing 5N_{1} atoms of type A and adding the other elements equally leads to the pseudobinary composition \({\mathrm{A}}_{  6N_1}\left( {{\mathrm{ABCDEF}}} \right)_{N_0 + N_1}\) and x_{s} = −6N_{1}/N_{tot}. This is repeated for each individual atom type (B, C, …) in the alloy. In fact, there is no actual need to use the same supercell when adding/removing atoms; this is just the clearest way to understand the process. Then, the pseudobinary formula becomes of particularly high value. In practice, only the quantities 6N_{1} and N_{0} − N_{1} (or −6N_{1}, N_{0} + N_{1}) need to be integers. For example, in Table 1, the x_{s} = −3.1 at.% case is achieved using N_{1} = 2/3 and N_{0} = 64/3, corresponding to −6N_{1} = −4, N_{0} + N_{1} = 22 with N_{tot} = 128. The pseudobinary concept thus leads to considerable flexibility in the choice of supercell size for each individual composition.
The choice of the supercell composition formulas is flexible but a few guidelines are helpful. First, N_{tot} should be neither too large nor too small. A large value is computationally costly while a small value might be insufficient for achieving an SQS of high quality. According to the literature, good representations of chemical disorder can usually be achieved using approximately several tens of atoms.^{22} Second, x_{s} should be neither too large nor too small. If too large, the volume change might exceed the domain of linear changes around the central composition. If too small, the uncertainty due to the configurational randomness will greatly increase the uncertainty in the deduced misfit volumes. The supercells and formula units used here are shown in Table 1, and satisfy 100 < N_{tot} < 150 and 0.03 < x_{s} < 0.06.
For each composition, the SQSs are generated independently using the atat code.^{23} For the 6element alloys here, SQS generation at the necessary supercell sizes of 100 ~ 150 atoms is computationally intensive.^{18} Fortunately, the resulting structures can be used multiple times by switching the atom types. In addition, these SQSs can immediately provide candidate SQSs for other alloys having fewer components with compositions in multiples of 1/6, such as A_{2}BCDE, A_{4}B_{2}, etc. For a given 6component SQS, there are many different possible SQS structures for alloys with fewer components. The best SQS from among the possibilities is easily determined by evaluating the SQS quality, e.g., the SQS error using available error estimates.^{24} The present SQSs can therefore be easily used for studying new alloys without any further timeconsuming searches for new SQSs. For this reason, we provide the exact SQSs used here (https://doi.org/10.24435/materialscloud:2018.0019/v1).
With the SQS construction as described above, the equilibrium atomic volumes are computed by minimizing the total energy versus volume with full relaxation of all ions and supercell shape. We perform calculations for multiple samples at the same and/or various x_{s} to improve the overall accuracy of the final misfit volumes. From the entire set of DFTcomputed equilibrium atomic volumes, we then perform a linear regression with one constraint \(\mathop {\sum}\nolimits_n {\kern 1pt} c_n{\mathrm{\Delta }}V_n = 0\) to obtain the misfit volumes ΔV_{n} for all solutes and the atomic volume V_{0} at the desired composition {c_{n}}. Figure 1 shows the computed atomic volumes versus compositions x_{s} for all alloys studied here around the RhIrPdPtNiCu composition. We obtain the lattice constant and misfit volumes shown in Table 2. Importantly, from the totality of the data computed, we can compute the 95% confidence intervals for the lattice constant (±0.00017 Å) and misfit volumes (±0.093 Å^{3}), respectively. The misfit quantity δ in the theory for yield strength is then computed as δ = 3.873%, as shown in Fig. 1 and summarized in Table 2, with 95% confidence interval within ±0.2%.
Elastic constants
The elastic constants of the alloy are computed starting from the equilibrium states of the two SQS structures of the random RhIrPdPtNiCu alloy containing 108 and 144 atoms, respectively. These systems have small initial (Pulay) stresses \(\sigma _i^0\) (i = 1...6) using the Reuss contracted notation. We use the wellestablished stress method.^{25} Six linearly independent strain tensors are applied (nonzero components = 0.002) and the corresponding stress tensors are computed in DFT. The elastic constants are then computed from \(\sigma _i  \sigma _i^0 = C_{ij}\epsilon _j\). In one leastsquares fitting, we calculate all 21 C_{ij} components of C_{ij,108} and C_{ij,144} (in GPa). The elastic constants expressed in the standard 6 × 6 Voigt matrix notation for the two different SQS cells are
The fccsymmetry is broken in these small cells of random alloys. To obtain the fcc constants C_{ij}, we take the averages of all the fccsymmetryequivalent components over both structures. The remaining C_{ij} components are set to zero (the averages being negligible). The resulting values are C_{11} = 289 GPa, C_{12} = 176 GPa, and C_{44} = 112 GPa. Since each elastic constant is an average over six values, we can compute the 95% confidence interval, which results in ~±2% of the mean value. The bulk modulus B = (C_{11} + 2C_{12})/3 = 214 GPa obtained from the averaging is very close to the values of B computed from the energy versus volume for the two structures (215 and 214 GPa, respectively).
This alloy is moderately anisotropic, with Zener anisotropy index A = 2C_{44}/(C_{11} − C_{12}) = 1.98. The Voigtaveraged elastic constants that enter the strength theory are μ^{V} = 90 GPa and ν^{V} = 0.316. For this value of A, the full anisotropic value of P in Eq. (1) is only ~0.5% smaller than the isotropic approximation using these Voigt averages. The shear modulus used in the line tension is computed as μ_{110/111} = 75 GPa.
Note that the measured shear modulus of the polycrystalline material is expected to be closer to the VoigtReussHill average μ^{VRH} = 85 GPa.^{26} This value does not enter the strength theory and indicates the typical error that might be expected when using an experimentally measured shear modulus. All of the various elastic constants (anisotropic and associated isotropic average) for RhIrPdPtNiCu are shown in Table 3.
Stable stacking fault energy and partial separation
The stable stacking fault energy γ_{ssf} is computed using the ANNNI model. In this model, γ_{ssf} is related to the difference between hcp and fcc cohesive energies of the alloy as \(\gamma _{{\mathrm{ssf}}} = 2\left( {E_0^{{\mathrm{hcp}}}  E_0^{{\mathrm{fcc}}}} \right){\mathrm{/}}A_0^{{\mathrm{fcc}}}\), where \(E_0^{{\mathrm{hcp}}}\) and \(E_0^{{\mathrm{fcc}}}\) are the atomic energy for the fully relaxed hcp and fcc structures, respectively. \(A_0^{{\mathrm{fcc}}}\) is the fcc atomic area on the stacking fault plane.
Direct computation of γ_{ssf} in random alloys is deemed likely to have large fluctuations since it is a planar defect energy computed from two bulk energies of 3dimensional random structures.^{27} Furthermore, application of the analytic strengthening theory requires only that γ_{ssf} be sufficient to ensure sufficiently large partial dislocation separation d.
Benchmark DFT computations of γ_{ssf} for the individual elemental metals using the direct tiltedcell method^{28} and ANNNI model show differences (which can be positive or negative) within ±20 mJ/m^{2}. The differences arise, in part, from the two fcc/hcp interfacial energies that are not considered in the ANNNI model.^{29} We can thus anticipate that application of the ANNNI model to the 6component alloy will have a similar level of uncertainty.
The formation energy of the fcc HEA phase is obtained from the previous DFT calcuations. For all fcc samples, we have the equilibrium energies. Performing a linear regression to the DFT energy data, analogous to what was done for the equilibrium atomic volumes in Fig. 1, gives the formation energy of fcc RhIrPdPtNiCu as \(E_0^{{\mathrm{fcc}}} = 34\) meV/atom, relative to the compositional average of the pure metal atomic energies in the fcc phase. The positive formation energy indicates that the singlephase RhIrPdPtNiCu HEA is either entropically stabilized or metastable. This result is in contrast to a negative formation energy roughly estimated from an empirical literature model.^{5} To calculate \(E_0^{{\mathrm{hcp}}}\), we use the process employed for fcc at the composition of interest. We create the hcp SQSs for the HEA (https://doi.org/10.24435/materialscloud:2018.0019/v1), and perform DFT calculations to obtain the ground state energy on the compositions surrounding the central composition to improve the accuracy, and then perform linear regression to the DFT energy data for hcp structures. The resulting formation energy of RhIrPdPtNiCu in hcp phase is \(E_0^{{\mathrm{hcp}}} = 61\) meV/atom relative to the same compositional average of the pure metal atomic energies in the fcc phase. With \(A_0^{{\mathrm{fcc}}} = (\sqrt 3 {\mathrm{/}}4)a_0^2\), the ANNNI model yields a stable stacking fault energy γ_{ssf} = 138 mJ/m^{2} for RhIrPdPtNiCu. The uncertainty in γ_{ssf} is determined from the 95% confidence intervals of \(E_0^{{\mathrm{fcc}}}\) and \(E_0^{{\mathrm{hcp}}}\), which are ±1 and ±2 meV, respectively. Therefore, the uncertainty in the DFT result itself for γ_{ssf} is about ±15 mJ/m^{2}. Combined with the intrinsic error/uncertainty of the ANNNI model, a conservative estimate of the uncertainty in the stable stacking fault energy is ~±35 mJ/m^{2}.
Finally, using the computed anisotropic elastic constants and lattice constant, the elastic parameter is K = 0.180 eV/Å and the estimated partial separation is then d = K/γ_{ssf} = 7.8b. Considering the uncertainty in γ_{ssf}, the corresponding partial separation ranges from 6.2b to 10.4b. This estimation is an upper bound of the uncertainty since the value ±20 mJ/m^{2} is probably higher than the true value in the HEA. The analysis thus generally satisfies the requirement d ≳ 6.5b for applying the analytic solute strengthening model above.
Yield strength prediction
With all of the necessary material properties computed via DFT as described in the previous section, the yield strength of the alloy can be predicted. The experiments were performed at T = 300 K and strainrate \(\dot \epsilon\) = 10^{−4} s^{−1}. As shown in Table 3, the predicted strength using only the DFTcomputed inputs is σ_{y} = 583 MPa. The uncertainty in this prediction due to the uncertainties in DFTcomputed materials parameters is small. Using the 95% confidence intervals of the lattice constant, misfit volumes, and elastic constants, the 95% confidence interval for the strength prediction is ~±10% of the above mean value. This uncertainty is due to the configurational randomness in the computations, at the SQSs qualities that we achieve.
The average prediction of 583 MPa is in very good agreement with the single experimental value of 527 MPa. While there is only one experimental data point, the yield strength of wellcharacterized alloys usually has small scatter. As analyzed in previous work using approximate methods,^{10} this ascast alloy has two different phases but these phases have very similar lattice constants and estimated strengths. The above experimental value is thus likely close to the “true value” of a singlephase homogenous alloy.
There are a few other aspects that could affect the strength prediction here. First, the DFTcalculated elastic constants must have some (unknown) error, as demonstrated by the deviations between DFT and experiment for the elemental metals. Here, we might expect that the PerdewBurkeErnzerhof (PBE) DFT predictions of the elastic moduli C_{ij} of the HEA are slightly lower than the (unknown) true experimental values due to the underestimate of the moduli of elemental Pd and Pt, as discussed in Section “DFT methodology.”
Second, the experiments are at room temperature and so the roomtemperature elastic constants should be used in the predictions. Roomtemperature calculations are possible but computationally onerous, and so here we make reasonable estimates based on the elemental values. In experiments, the elastic constants C_{ij} of the pure metals all decrease, but by less than 10% from T = 0 K to T = 300 K. Therefore, the yield strength predictions made using the T = 0 K elastic constants are likely <10% too high. Thus, use of the (unknown) T = 300 K elastic constants would bring the predictions closer to the experimental value. For purposes of efficient identification of trends for alloy design, we advocate the use of the easily computed T = 0 K elastic constants, with temperature effects at room temperature estimated from a concentrationweighted average of the elemental elastic constants.
Third, the lattice constant a_{0} and the δparameter are also temperaturedependent. Applying Vegard’s law (defined below) using the temperaturedependent lattice constants of the pure metals to estimate both the alloy volume and solute misfit volumes at finite temperature, we find that the overall δparameter decreases by <2% with increasing temperature. The thermal expansion effects can thus be safely neglected.
Although there are uncertainties in the DFTcomputed properties and the temperature dependencies, there are no adjustable/fitted parameters in the model. Overall, we thus consider the predictions to be robust and the agreement indicative of the applicability of the theory.
Discussion
The present methodology can be expanded to scan over a wide range of the composition space so as to predict optimized composition(s) with the highest yield strength(s). Suppose we have a database of the atomic volumes over a set of equispaced compositions spanning the entire configurational space. Then one could fit the entire set to some overall function of the compositions, i.e., V({c_{n}}). The misfit volume of typen atom at composition {c_{n}} could be computed by taking the derivative respect to x_{s} in the direction of {(1 − c_{n}) for n, −c_{m} for m ≠ n}, i.e.,
with x_{s} → 0. Here, the challenging task is the computation of the data set. In order to have a quantitative estimate of the size of the database, we assume that the wellspaced compositions consist of elemental concentrations spanning from 0 to 100% by the step of 10%. For senary alloys studied here, the database needs 3003 different compositions for SQS sizes of ~100 atoms; this is not computationally feasible at present. However, for quaternary and ternary alloys, such a database requires 286 and 66 compositions, respectively, and so could be achieved computationally. In any case, the computation of the data set remains demanding but the methodology for the misfit volumes remains useful and will become more feasible as computational power increases.
Given the combinatoric challenge of performing DFT over the huge composition space of a manycomponent alloy, it is very useful to consider other moreapproximate but moreefficient approaches. One such approach is by assuming Vegard’s law and assuming a simple ruleofmixtures for elastic constants.^{10} Then only the lattice constants and elastic constants of the elemental metals are needed in the crystal structure of the HEA. These quantities are computationally trivial to obtain if all the elements are metastable in the HEA crystal structure. Vegard’s law uses the atomic volumes \(V_0^{(n)}\) of the individual elements and predicts the atomic volume of the alloy as \(V_0^{{\mathrm{Vegard}}} = \mathop {\sum}\nolimits_n {\kern 1pt} c_nV_0^{(n)}\). The misfit volumes follow as \({\mathrm{\Delta }}V_n^{{\mathrm{Vegard}}} = V_0^{(n)}  V_0^{{\mathrm{Vegard}}}\). The ruleofmixtures (ROM) estimate for the elastic constants is simply \(C_{ij}^{{\mathrm{ROM}}} = \mathop {\sum}\nolimits_n {\kern 1pt} c_nC_{ij}^{(n)}\) and yields a ruleofmixtures Voigt average for the required μ and ν. With DFT results computed explicitly here for fcc RhIrPdPtNiCu, we can assess the accuracy of such a simple model since all the elements are stable in the fcc structure.
Using the DFTcomputed atomic volumes for the constituents of the RhIrPdPtNiCu, Vegard’s law yields δ = 4.193%. This is ~8% larger than the direct DFT result of δ = 3.873% on the actual random alloy, as compared in Table 2. Similarly, using the DFT values for the elastic constants of the elements, the alloy ROM values are \(C_{11}^{{\mathrm{ROM}}} = 326\) GPa, \(C_{12}^{{\mathrm{ROM}}} = 177\) GPa and \(C_{44}^{{\mathrm{ROM}}} = 131\) GPa. Two of these values are ~15% larger than the DFTcomputed alloy values (Table 3). The resulting Voigt estimate μ^{V} is then ~20% larger (108 GPa vs. 90 GPa). With the approximate estimates for both δ and μ^{V} being larger than the true DFTcomputed properties for the alloy, the predicted strength using the approximate model is 775 MPa, which is rather larger than the full prediction.
The approximate approach above is thus of limited value for quantitative prediction in this family of alloys. However, the approximate approach may be too high mainly due to the moduli estimates, and the moduli estimates may be systematically high for other composition in the configurational space. This can be further validated with the calculations at other {c_{n}}.
The approximate approach may be useful only for rapid relative assessment of the relative strengths of alloy compositions. But even such a relative assessment can then point to a limited set of promising new alloy compositions that could be studied quantitatively using the computationally intensive but chemically accurate DFT calculations proposed in this work.
In summary, we have presented a general approach to DFTlevel computations of the properties necessary to make strength predictions in HEAs and other random alloys using a recent analytic theory. The theory requires solute misfit volumes, alloy lattice constant, elastic constants, and stable stacking fault energy, all of which are accessible within current DFT capabilities. The DFT computational methodology for misfit volumes in multicomponent random alloys is new and general. Executing the methodology on the fcc RhIrPdPtNiCu HEA, we obtain parameterfree predictions of the yield strength with no experimental inputs and find very good agreement with the experimental strength. We are also able to assess uncertainties in the approach and to assess concepts for highly efficient scanning of the huge composition space of HEAs. This work thus connects alloy composition to yield strength, establishing a general firstprinciples methodology for the computationally guided design of highstrength alloys.
Methods
DFT methodology
All firstprinciples calculations are performed using DFT as implemented in the vasp code^{30} within the generalized gradient approximation (GGA) and using the PBE XC functional.^{31} The core electrons are replaced by the projector augmented wave (PAW) pseudopotentials.^{32} The valenceelectron eigenstates are expanded using a spinpolarized plane wave basis set with a cutoff energy 550 eV. A firstorder MethfesselPaxton method^{33} with smearing parameter 0.2 eV is used. In reciprocal space, a Γcentered MonkhorstPack^{34} kmesh is used with line density consistent across all geometries. The interval between the neighboring kpoints along each reciprocal lattice vector b_{j} is 0.02 Å^{−1} (in vasp, a_{i} · b_{j} = δ_{ij}). This kmesh density leads to, for example, 14 × 14 × 14 for Ni and Cu in the fcc cubic unit cell, and 13 × 13 × 13 for the other four elements. Ionic forces are relaxed to less than 1 meV/Å, corresponding to a stress tolerance of ~0.01 GPa in vasp. Benchmark calculations of the lattice constant, elastic constants, and stable stacking fault energy for the pure metal elements are in very good agreement with previous DFT literature.^{35,36,37}
In the benchmark results for the elements, the PBE functional predicts the bulk moduli for Rh, Ir, Ni, and Cu within ±5% of experiment values but underestimates the bulk moduli of Pd and Pt by ~13% (169 GPa vs. 195 GPa for Pd, 249 GPa vs. 288 GPa for Pt; see ref. ^{35} (Table S25.3) and ref. ^{38}). Results using the PBEsol functional^{39} improve the results for Pd and Pt relative to experiments but increase the differences between DFT and experiments for the other four elements in the alloy. These trends are already known in the literature. For this study, we use PBE so as to capture the better agreement with experiment for four of the six alloying elements studied here.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request. The SQSs used in this work are provided in (https://doi.org/10.24435/materialscloud:2018.0019/v1).
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Acknowledgements
This research was supported by the NCCR MARVEL, funded by the Swiss National Science Foundation. We also acknowledge support of highperformance computing provided by Scientific IT and Application Support (SCITAS) at EPFL.
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The initial project idea was from W.A.C., B.Y. and W.A.C. devised the methodology for the DFT computation of misfit volumes in multicomponent random alloys. B.Y. performed all the calculations. B.Y. and W.A.C. analyzed the results and wrote the manuscript.
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Yin, B., Curtin, W.A. Firstprinciplesbased prediction of yield strength in the RhIrPdPtNiCu highentropy alloy. npj Comput Mater 5, 14 (2019). https://doi.org/10.1038/s415240190151x
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DOI: https://doi.org/10.1038/s415240190151x
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