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We study the phase behaviour of the Al$_x$CrFeCoNi high-entropy alloy. Our approach is based on a perturbative analysis of the internal energy of the paramagnetic solid solution as evaluated within the Korringa-Kohn-Rostoker formulation of density functional theory, using the coherent potential approximation to average over disorder. Via application of a Landau-type linear response theory, we infer preferential chemical orderings directly. In addition, we recover a pairwise form of the alloy internal energy suitable for study via atomistic simulations, which in this work are performed using the nested sampling algorithm, which is well-suited for studying complex potential energy surfaces. When the underlying lattice is fcc, at low concentrations of Al, depending on the value of $x$, we predict either an $\mathrm{L}1_2$ or $\mathrm{D}0_{22}$ ordering emerging below approximately 1000 K. On the other hand, when the underlying lattice is bcc, consistent with experimental observations, we predict $\mathrm{B}2$ ordering temperatures higher than the melting temperature of the alloy, confirming that this ordered phase forms directly from the melt. For both fcc and bcc systems, chemical orderings are dominated by Al moving to one sublattice, Ni and Co the other, while Cr and Fe remain comparatively disordered. On the bcc lattice, our atomistic modelling suggests eventual decomposition into $\mathrm{B}2$ NiAl and Cr-rich phases. These results shed light on the fundamental physical origins of atomic ordering tendencies in these intriguing materials., Comment: 17 pages, 11 figures, 1 table

Refractory high-entropy alloys are under consideration for applications where materials are subjected to high temperatures and levels of radiation, such as in the fusion power sector. However, at present, their scope is limited because they are highly brittle at room temperature. One suggested route to mitigate this issue is by alloying with Ti. In this theoretical study, using a computationally efficient linear-response theory based on density functional theory calculations of the electronic structure of the disordered alloys, we study the nature of atomic short-range order in these multi-component materials, as well as assessing their overall phase stability. Our analysis enables direct inference of phase transitions in addition to the extraction of an atomistic, pairwise model of the internal energy of an alloy suitable for study via, e.g. Monte Carlo simulations. Once Ti is added into either the NbMoTaW or VNbMoTaW system, we find that there is competition between chemical phase ordering and segregation. These results shed light on observed chemical inhomogeneity in experimental samples, as well as providing fundamental insight into the physics of these complex systems., Comment: 11 pages, 6 figures, 2 tables

We describe an integrated modelling approach to accelerate the search for novel, single-phase, multicomponent materials with high magnetocrystalline anisotropy (MCA). For a given system we predict the nature of atomic ordering, its dependence on the magnetic state, and then proceed to describe the consequent MCA, magnetisation, and magnetic critical temperature (Curie temperature). Crucially, within our modelling framework, the same ab initio description of the material's electronic structure determines all aspects. We demonstrate this holistic method by studying the effects of alloying additions in FeNi, examining systems with the general stoichiometries Fe$_4$Ni$_3X$ and Fe$_3$Ni$_4X$, for additives including X= Pt, Pd, Al, and Co. The atomic ordering behaviour predicted on adding these elements, fundamental for determining a material's MCA, is rich and varied. Equiatomic FeNi has been reported to require ferromagnetic order to establish the tetragonal L10 order suited for significant MCA. Our results show that when alloying additions are included in this material, annealing in an applied magnetic field and/or below a material's Curie temperature may also promote tetragonal order, along with an appreciable effect on the predicted hard magnetic properties., Comment: 19 pages, 6 figures, 6 tables

This article is intended as a guide for new graduate students in the field of computational science. With the increasing influx of students from diverse backgrounds joining the ever-popular field, this short guide aims to help students navigate through the various computational techniques that they are likely to encounter during their studies. These techniques span from Bash scripting and scientific programming to machine learning, among other areas. This paper is divided into ten sections, each introducing a different computational method. To enhance readability, we have adopted a casual and instructive tone, and included code snippets where relevant. Please note that due to the introductory nature of this article, it is not intended to be exhaustive; instead, we direct readers to a list of references to expand their knowledge of the techniques discussed within the paper. It is likely that this article will continue to evolve with time, and as such, we advise readers to seek the latest version. Finally, readers should note this article serves as an extension to our student-led seminar series, with additional resources and videos available at \url{https://computationaltoolkit.github.io/} for reference., Comment: 45 pages, 3 figures

We have developed a new machine learned interatomic potential for the prototypical austenitic steel Fe$_{7}$Cr$_{2}$Ni, using the Gaussian approximation potential (GAP) framework. This new GAP can model the alloy's properties with close to density functional theory (DFT) accuracy, while at the same time allowing us to access larger length and time scales than expensive first-principles methods. We also extended the GAP input descriptors to approximate the effects of collinear spins (Spin GAP), and demonstrate how this extended model successfully predicts structural distortions due to antiferromagnetic and paramagnetic spin states. We demonstrate the application of the Spin GAP model for bulk properties and vacancies and validate against DFT. These results are a step towards modelling the atomistic origins of ageing in austenitic steels with higher accuracy.

The magnetocrystalline anisotropy energy of atomically ordered $\mathrm{L}1_0$ FeNi (the meteoritic mineral tetrataenite) is studied within a first-principles electronic structure framework. Two compositions are examined: equiatomic Fe$_{0.5}$Ni$_{0.5}$ and an Fe-rich composition, Fe$_{0.56}$Ni$_{0.44}$. It is confirmed that, for the single crystals modelled in this work, the leading-order anisotropy coefficient $K_1$ dominates the higher-order coefficients $K_2$ and $K_3$. To enable comparison with experiment, the effects of both imperfect atomic long-range order and finite temperature are included. While our computational results initially appear to undershoot the measured experimental values for this system, careful scrutiny of the original analysis due to N\'{e}el et al. [J. Appl. Phys. 35, 873 (1964)] suggests that our computed value of $K_1$ is, in fact, consistent with experimental values, and that the noted discrepancy has its origins in the nanoscale polycrystalline, multivariant nature of experimental samples, that yields much larger values of $K_2$ and $K_3$ than expected a priori. These results provide fresh insight into the existing discrepancies in the literature regarding the value of tetrataenite's uniaxial magnetocrystalline anisotropy in both natural and synthetic samples., Comment: 9 pages, 3 figures, 2 tables

The impact of magnetism on predicted atomic short-range order in three medium- and high-entropy alloys is studied using a first-principles, all-electron, Landau-type linear response theory, coupled with lattice-based atomistic modelling. We perform two sets of linear-response calculations: one in which the paramagnetic state is modelled within the disordered local moment picture, and one in which systems are modelled in a magnetically ordered state, which is ferrimagnetic for the alloys considered in this work. We show that the treatment of magnetism can have significant impact both on the predicted temperature of atomic ordering and also the nature of atomic order itself. In CrCoNi, we find that the nature of atomic order changes from being $\mathrm{L}1_2$-like when modelled in the paramagnetic state to MoPt$_2$-like when modelled assuming the system has magnetically ordered. In CrFeCoNi, atomic correlations between Fe and the other elements present are dramatically strengthened when we switch from treating the system as magnetically disordered to magnetically ordered. Our results show it is necessary to consider the magnetic state when modelling multicomponent alloys containing mid- to late-$3d$ elements. Further, we suggest that there may be high-entropy alloy compositions containing $3d$ transition metals that will exhibit specific atomic short-range order when thermally treated in an applied magnetic field. This has the potential to provide a route for tuning physical and mechanical properties in this class of materials., Comment: 26 pages, 4 figures, 2 tables

We describe implementation and analysis of a first-principles theory, derived in an earlier work, for the leading terms in an expansion of a Gibbs free energy of a multi-component alloy in terms of order parameters that characterize potential, compositional phases. The theory includes effects of rearranging charge and other electronics from changing atomic occupancies on lattice sites. As well as the rigorous description of atomic short-range order in the homogeneously disordered phase, pairwise interaction parameters suited for atomistic modelling in a multicomponent setting can be calculated. From our study of an indicative series of the Cantor-Wu alloys, NiCo, NiCoCr, NiCoFeCr, and NiCoFeMnCr, we find that the interactions are not approximated well either as pseudobinary or restricted to nearest neighbour range. Our computed order-disorder transition temperatures are low, consistent with experimental observations, and the nature of the ordering is dominated by correlations between Ni, Co, and Cr, while Fe and Mn interact weakly. Further atomistic modelling suggests that there is no true single-phase low-temperature ground state for these multicomponent systems. Instead the single-phase solid solution is kept stable to low temperatures by the large configurational entropy and the Fe, Mn dilution effects. The computationally cost-effectiveness of our method makes it a good candidate for further exploration of the space of multicomponent alloys.

Using an all-electron, first principles, Landau-type theory, we study the nature of short-range order and compositional phase stability in equiatomic refractory high entropy alloys, NbMoTa, NbMoTaW, and VNbMoTaW. We also investigate selected binary subsystems to provide insight into the physical mechanisms driving order. Our approach examines the short-range order of the solid solutions directly, infers disorder/order transitions, and also extracts parameters suitable for atomistic modelling of diffusional phase transformations. We find a hierarchy of relationships between the chemical species in these materials which promote ordering tendencies. The most dominant is a relative atomic size difference between the 3d element, V, and the other 4d and 5d elements which drives a B32-like order. For systems where V is not present, ordering is dominated by the difference in filling of valence states; pairs of elements which are isoelectronic remain weakly correlated to low temperatures, while pairs with a valence difference present B2-like order. Our estimated order-disorder transition temperature in VNbMoTaW is sufficiently high for us to suggest that SRO in this material may be experimentally observable., Comment: 12 pages, 5 figures, 3 tables. v2 contains extension of discussion of linear response theory and inclusion of supplementary material. v3 moves to two-column format and corrects a a typographical error in Table 1