Explicit anharmonicity, defined as the vibrational contribution beyond the quasiharmonic approximation, is qualitatively different between the group V and group VI bcc refractory elements. Group V elements show a small and mostly negative anharmonic entropy, whereas group VI elements have a large positive anharmonic entropy, strongly increasing with temperature. Here, we explain this difference utilizing highly accurate anharmonic free energies and entropies from ab initio calculations for Nb and Ta (group V), and Mo and W (group VI). The numerically calculated entropies are in agreement with prior experimental data. The difference in behavior between the two sets of elements arises not from their high-temperature behavior but rather from the 0K quasiharmonic reference state. We understand this by analyzing the 0K and the high-temperature phonon density of states and the electronic density of states. The qualitative difference disappears when the anharmonicity is instead referenced with a high-temperature effective harmonic potential. However, even for an optimized effective harmonic reference, the remaining effective anharmonicity is significant. The reason is that the anharmonicity in the bcc systems - carried by asymmetric distributions in the nearest neighbors - can never be accounted for by a harmonically restricted potential.

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Solid-state batteries can be built based on thiophosphate electrolytes such as β-Li₃PS₄. For the preparation of these solid electrolytes, various solvent-based routes have been reported. For recycling of end-of-life solid-state batteries based on such thiophosphates, we consider the development of dissolution and recrystallization strategies for the recovery of the model compound β-Li₃PS₄. We show that recrystallization can only be performed in polar, slightly protic solvents such as N-methylformamide (NMF). The recrystallization is comprehensively studied, showing that it proceeds via an intermediate phase with composition Li₃PS₄·2NMF, which is structurally characterized. This phase has a high resistivity for the transport of lithium ions and must be removed in order to obtain a recrystallized product with a conductivity similar to the pristine material. Moreover, the recrystallization from solution results in an increase of the amorphous phase fraction next to crystalline β-Li₃PS₄

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We present the magnetic Moment Tensor Potentials (mMTPs), a class of machine-learning interatomic potentials, accurately reproducing both vibrational and magnetic degrees of freedom as provided, e.g., from first-principles calculations. The accuracy is achieved by a two-step minimization scheme that coarse-grains the atomic and the spin space. The performance of the mMTPs is demonstrated for the prototype magnetic system bcc iron, with applications to phonon calculations for different magnetic states, and molecular-dynamics simulations with fluctuating magnetic moments.

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Multi-principal-component alloys have attracted great interest as a novel paradigm in alloy design, with often unique properties and a vast compositional space auspicious for materials discovery. High entropy alloys (HEAs) belong to this class and are being investigated for prospective nuclear applications with reported superior mechanical properties including high-temperature strength and stability compared to conventional alloys. Computational materials design has the potential to play a key role in screening such alloys, yet for high-temperature properties, challenges remain in finding an appropriate balance between accuracy and computational cost. Here we develop an approach based on density-functional theory (DFT) and thermodynamic integration aided by machine learning based interatomic potential models to address this challenge. We systematically evaluate and compare the efficiency of computing the full free energy surface and thermodynamic properties up to the melting point at different stages of the thermodynamic integration scheme. Our new approach provides a ×4 speed-up with respect to comparable free energy approaches at the level of DFT, with errors on high-temperature free energy predictions less than 1 meV/atom. Calculations are performed on an equiatomic HEA, TaVCrW - a low-activation composition and therefore of potential interest for next generation fission and fusion reactors.

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An active learning approach to train machine-learning interatomic potentials (moment tensor potentials) for multicomponent alloys to ab initio data is presented. Employing this approach, the disordered body-centered cubic (bcc) TiZrHfTax system with varying Ta concentration is investigated via molecular dynamics simulations. Our results show a strong interplay between elastic properties and the structural ω phase stability, strongly affecting the mechanical properties. Based on these insights we systematically screen composition space for regimes where elastic constants show little or no temperature dependence (elinvar effect).

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Recent experiments show that the chemical composition of body-centered cubic (bcc) refractory high entropy alloys (HEAs) can be tuned to enable transformation-induced plasticity (TRIP), which significantly improves the ductility of these alloys. This calls for an accurate and efficient method to map the structural stability as a function of composition. A key challenge for atomistic simulations is to separate the structural transformation between the bcc and the ω phases from the intrinsic local lattice distortions in such chemically disordered alloys. To solve this issue, we develop a method that utilizes a symmetry analysis to detect differences in the crystal structures. Utilizing this method in combination with ab initio calculations, we demonstrate that local lattice distortions largely affect the phase stability of Ti–Zr–Hf–Ta and Ti–Zr–Nb–Hf–Ta HEAs. If relaxation effects are properly taken into account, the predicted compositions near the bcc–hcp energetic equilibrium are close to the experimental compositions, for which good strength and ductility due to the TRIP effect are observed.

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The interface method is a well established approach for predicting melting points of materials using interatomic potentials. However, applying the interface method is tedious and involves significant human intervention. The whole procedure involves several successive tasks: estimate a rough melting point, set up the interface structure, run molecular dynamic calculations and analyze the data. Loop calculations are necessary if the predicted melting point is different from the estimated one by more than a certain convergence criterion, or if full melting/solidification occurs. In this case monitoring the solid–liquid phase transition in the interface structure becomes critical. As different initial random seeds for the molecular dynamic simulations within the interface method induce slightly different melting points, a few ten or hundred interface method calculations with different random seeds are necessary for performing a statistical analysis on these melting points. Considering all these technical details, the work load for manually executing and combining the various involved scripts and programs quickly becomes prohibitive. To simplify and automatize the whole procedure, we have implemented the interface method into pyiron (http://pyiron.org). Our fully automatized procedure allows to efficiently and precisely predict melting points of stable unaries represented by arbitrary potentials with only two user-specified parameters (interatomic potential file and element). For metastable or dynamically unstable unary phases, the crystal structure needs to be provided as an additional parameter. We have applied our automatized approach on fcc Al, Ni, dynamically unstable bcc Ti and hcp Mg and employed a large set of available interatomic potentials. Melting points for classical interatomic potentials of these metals have been obtained with a numerical precision well below 1 K.

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We apply the efficient two-optimized references thermodynamic integration using Langevin dynamics method [Phys. Rev. B 96, 224202 (2017)2469-995010.1103/PhysRevB.96.224202] to calculate highly accurate melting properties of Al and magnetic Ni from first principles. For Ni we carefully investigate the impact of magnetism on the liquid and solid free energies including longitudinal spin fluctuations and the reverse influence of atomic vibrations on magnetic properties. We show that magnetic fluctuations are effectively canceling out for both phases and are thus not altering the predicted melting temperature. For both elements, the generalized gradient approximation (GGA) and the local-density approximation (LDA) are used for the exchange-correlation functional revealing a reliable ab initio confidence interval capturing the respective experimental melting point, enthalpy of fusion, and entropy of fusion.

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Local lattice distortions in a series of body-centered cubic alloys, including refractory high-entropy alloys, are investigated by means of atomic volumes, atomic charges, and atomic stresses defined by the Bader charge analysis based on first-principles calculations. Analyzing the extensive data sets, we find large distributions of these atomic properties for each element in each alloy, indicating a large impact of the varying local chemical environments. We show that these local-environment effects can be well understood and captured already by the first and the second nearest neighbor shells. Based on this insight, we employ linear regression models up to the second nearest neighbor shell to accurately predict these atomic properties. Finally, we find that the elementwise-averaged values of the atomic properties correlate linearly with the averaged valence-electron concentration of the considered alloys.

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The field of atomistic simulations of multicomponent materials and high entropy alloys is progressing rapidly, with challenging problems stimulating new creative solutions. In this Perspective, we present three topics that emerged very recently and that we anticipate will determine the future direction of research of high entropy alloys: the usage of machine-learning potentials for very accurate thermodynamics, the exploration of short-range order and its impact on macroscopic properties, and the more extensive exploitation of interstitial alloying and high entropy alloy surfaces for new technological applications. For each of these topics, we briefly summarize the key achievements, point out the aspects that still need to be addressed, and discuss possible future improvements and promising directions.

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High entropy alloys (HEAs) in the hexagonal close-packed (hcp) phase usually show poor mechanical properties. We demonstrate here, by use of ab initio simulations and detailed experimental investigations, that the mechanical properties can be improved by optimizing the microstructure. In particular we design a dual-phase HEA consisting of a body-centered cubic (bcc) matrix and hcp laths, with nanoprecipitates of the ω phase in the Sc-Ti-Zr-Hf-Re system, by controlling the Re content. This dedicated microstructure reveals, already in the as-cast state, high compressive strength and good ductility of 1910 MPa and 8%, respectively. Our study lifts the hcp-based HEAs onto a competitive, technological level.

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The unique and unanticipated properties of multiple principal component alloys have reinvigorated the field of alloy design and drawn strong interest across scientific disciplines. The vast compositional parameter space makes these alloys a unique area of exploration by means of computational design. However, as of now a method to compute efficiently, yet with high accuracy the thermodynamic properties of such alloys has been missing. One of the underlying reasons is the lack of accurate and efficient approaches to compute vibrational free energies—including anharmonicity—for these chemically complex multicomponent alloys. In this work, a density-functional-theory based approach to overcome this issue is developed based on a combination of thermodynamic integration and a machine-learning potential. We demonstrate the performance of the approach by computing the anharmonic free energy of the prototypical five-component VNbMoTaW refractory high entropy alloy.

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Multicomponent alloys with multiple principal elements including high entropy alloys (HEAs) and compositionally complex alloys (CCAs) are attracting rapidly growing attention. The endless possibilities to explore new alloys and the hope for better combinations of materials properties have stimulated a remarkable number of research works in the last years. Most of these works have been based on experimental approaches, but ab initio calculations have emerged as a powerful approach that complements experiment and serves as a predictive tool for the identification and characterization of promising alloys. The theoretical ab initio modeling of phase stabilities and mechanical properties of multi-principal element alloys by means of density functional theory (DFT) is reviewed. A general thermodynamic framework is laid down that provides a bridge between the quantities accessible with DFT and the targeted thermodynamic and mechanical properties. It is shown how chemical disorder and various finite-temperature excitations can be modeled with DFT. Different concepts to study crystal and alloy phase stabilities, the impact of lattice distortions (a core effect of HEAs), magnetic transitions, and chemical short-range order are discussed along with specific examples. Strategies to study elastic properties, stacking fault energies, and their dependence on, e.g., temperature or alloy composition are illustrated. Finally, we provide an extensive compilation of multi-principal element alloys and various material properties studied with DFT so far (a set of over 500 alloy-property combinations).

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The temperature-dependent intrinsic stacking fault Gibbs energy is computed based on highly converged density-functional-theory (DFT) calculations for the three prototype face-centered cubic metals Al, Cu, and Ni. All relevant temperature-dependent contributions are considered including electronic, vibrational, magnetic, and explicit anharmonic Gibbs energy contributions as well as coupling terms employing state-of-the-art statistical sampling techniques. Particular emphasis is put on a careful comparison of different theoretical concepts to derive the stacking fault energy such as the axial-next-nearest-neighbor-Ising (ANNNI) model or the vacuum-slab approach. Our theoretical results are compared with an extensive set of previous theoretical and experimental data. Large uncertainties in the experimental data highlight the necessity of complementary parameter-free calculations. Specifically, the temperature dependence is experimentally unknown and poorly described by thermodynamic databases. Whereas calphad derived data shows an increase of the stacking fault energy with temperature for two of the systems (Cu and Ni), our results predict a decrease for all studied systems. For Ni, the temperature induced change is in fact so strong that in the temperature interval relevant for super-alloy applications the stacking fault energy falls below one third of the low temperature value. Such large changes clearly call for a revision of the stacking fault energy when modeling or designing alloys based on such elements.

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Quantum-mechanical calculations are used to determine the temperature dependence of the Gibbs energy of vacancy formation in nickel. Existing data reveal a discrepancy between the high-temperature estimates from experiments and low-temperature approximations from density functional theory. Our finite-temperature calculations - which include the effects of magnetism and fully interacting phonon vibrations - demonstrate that this discrepancy is mostly caused by the previously neglected explicit anharmonic contribution.

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We study the mutual coupling of spin fluctuations and lattice vibrations in paramagnetic CrN by combining atomistic spin dynamics and ab initio molecular dynamics. The two degrees of freedom are dynamically coupled, leading to nonadiabatic effects. Those effects suppress the phonon lifetimes at low temperature compared to an adiabatic approach. The dynamic coupling identified here provides an explanation for the experimentally observed unexpected temperature dependence of the thermal conductivity of magnetic semiconductors above the magnetic ordering temperature.

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Refractory high entropy alloys feature outstanding properties making them a promising materials class for next-generation high-temperature applications. At high temperatures, materials properties are strongly affected by lattice vibrations (phonons). Phonons critically influence thermal stability, thermodynamic and elastic properties, as well as thermal conductivity. In contrast to perfect crystals and ordered alloys, the inherently present mass and force constant fluctuations in multi-component random alloys (high entropy alloys) can induce significant phonon scattering and broadening. Despite their importance, phonon scattering and broadening have so far only scarcely been investigated for high entropy alloys. We tackle this challenge from a theoretical perspective and employ ab initio calculations to systematically study the impact of force constant and mass fluctuations on the phonon spectral functions of 12 body-centered cubic random alloys, from binaries up to 5-component high entropy alloys, addressing the key question of how chemical complexity impacts phonons. We find that it is crucial to include both mass and force constant fluctuations. If one or the other is neglected, qualitatively wrong results can be obtained such as artificial phonon band gaps. We analyze how the results obtained for the phonons translate into thermodynamically integrated quantities, specifically the vibrational entropy. Changes in the vibrational entropy with increasing the number of elements can be as large as changes in the configurational entropy and are thus important for phase stability considerations. The set of studied alloys includes MoTa, MoTaNb, MoTaNbW, MoTaNbWV, VW, VWNb, VWTa, VWNbTa, VTaNbTi, VWNbTaTi, HfZrNb, HfMoTaTiZr.@en

We introduce a new class of high-entropy alloys (HEAs), i.e., quinary (five-component) dual-phase (DP) HEAs revealing transformation-induced plasticity (TRIP), designed by using a quantum mechanically based and experimentally validated approach. Ab initio simulations of thermodynamic phase stabilities of Co₂₀Cr₂₀Fe_40-xMn₂₀Ni_x (x = 0–20 at. %) HEAs were performed to screen for promising compositions showing the TRIP-DP effect. The theoretical predictions reveal several promising alloys, which have been cast and systematically characterized with respect to their room temperature phase constituents, microstructures, element distributions and compositional homogeneity, tensile properties and deformation mechanisms. The study demonstrates the strength of ab initio calculations to predict the behavior of multi-component HEAs on the macroscopic scale from the atomistic level. As a prototype example a non-equiatomic Co₂₀Cr₂₀Fe₃₄Mn₂₀Ni₆ HEA, selected based on our ab initio simulations, reveals the TRIP-DP effect and hence exhibits higher tensile strength and strain-hardening ability compared to the corresponding equiatomic CoCrFeMnNi alloy.

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Free energies of bulk materials are nowadays routinely computed by density functional theory. In particular for metals, electronic excitations can significantly contribute to the free energy. For an ideal static lattice, this contribution can be obtained at low computational cost, e.g., from the electronic density of states derived at T=0 K or by utilizing the Sommerfeld approximation. The error introduced by these approximations at elevated temperatures is rarely known. The error arising from the ideal lattice approximation is likewise unexplored but computationally much more challenging to overcome. In order to shed light on these issues we have computed the electronic free energies for all 3d,4d, and 5d transition elements on the ideal lattices of the bcc, fcc, and hcp structures using finite-temperature density-functional theory. For a subset of elements we have explored the impact of explicit thermal vibrations on the electronic free energies by using ab initio molecular dynamics simulations. We provide an analysis of the observed chemical trends in terms of the electronic density of states and the canonical d band model and quantify the errors in the approximate methods. The electronic contribution to the heat capacities and the corresponding errors due to the different approximations are studied as well.

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Multi-principle element alloys have enormous potential, but their exploration suffers from the tremendously large range of configurations. In the last decade such alloys have been designed with a focus on random solid solutions. Here we apply an experimentally verified, combined thermodynamic and first-principles design strategy to reverse the traditional approach and to generate a new type of hcp Al-Hf-Sc-Ti-Zr high entropy alloy with a hitherto unique structure. A phase diagram analysis narrows down the large compositional space to a well-defined set of candidates. First-principles calculations demonstrate the energetic preference of an ordered superstructure over the competing disordered solid solutions. The chief ingredient is the Al concentration, which can be tuned to achieve a D0₁₉ ordering on the hexagonal lattice. The computationally designed D0₁₉ superstructure is experimentally confirmed by transmission electron microscopy and X-ray studies. Our scheme enables the exploration of a new class of high entropy alloys.

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