#### Prof. Dr. Ralf Drautz

Atomistic Modelling and Simulation at ICAMS

Ruhr-Universität Bochum

##### Contact

- ralf.drautz@rub.de
- +49 234 3229308
- personal website

##### Hub

**Atomic cluster expansion of scalar, vectorial, and tensorial properties including magnetism and charge transfer**

Drautz, R.*Physical Review B*102 (2020)The atomic cluster expansion [R. Drautz, Phys. Rev. B 99, 014104 (2019)2469-995010.1103/PhysRevB.99.014104] is extended in two ways, the modeling of vectorial and tensorial atomic properties and the inclusion of atomic degrees of freedom in addition to the positions of the atoms. In particular, atomic species, magnetic moments, and charges are attached to the atomic positions, and an atomic cluster expansion that includes the different degrees of freedom on equal footing is derived. Expressions for the efficient evaluation of forces and torques are given. © 2020 American Physical Society.view abstract 10.1103/PhysRevB.102.024104 **Atomistic description of self-diffusion in molybdenum: A comparative theoretical study of non-Arrhenius behavior**

Smirnova, D. and Starikov, S. and Leines, G.D. and Liang, Y. and Wang, N. and Popov, M.N. and Abrikosov, I.A. and Sangiovanni, D.G. and Drautz, R. and Mrovec, M.*Physical Review Materials*4 (2020)According to experimental observations, the temperature dependence of self-diffusion coefficient in most body-centered cubic metals (bcc) exhibits non-Arrhenius behavior. The origin of this behavior is likely related to anharmonic vibrational effects at elevated temperatures. However, it is still debated whether anharmonicity affects more the formation or migration of monovacancies, which are known to govern the self-diffusion. In this extensive atomistic simulation study we investigated thermodynamic properties of monovacancies in bcc molybdenum, here taken as a representative model system, from zero temperature to the melting point. We combined first-principles calculations and classical simulations based on three widely used interatomic potentials for Mo. In our analysis we employ static and dynamic atomistic calculations as well as statistical sampling techniques and thermodynamic integration to achieve thorough information about temperature variations of vacancy formation and migration free energies and diffusivities. In addition, we carry out large-scale molecular dynamics simulations that enable direct observation of high-temperature self-diffusion at the atomic scale. By scrutinizing the results obtained by different models and methods, we conclude that the peculiar self-diffusion behavior is likely caused by strong temperature dependence of the vacancy formation energy. © 2020 American Physical Society.view abstract 10.1103/PhysRevMaterials.4.013605 **BOPcat software package for the construction and testing of tight-binding models and bond-order potentials**

Ladines, A.N. and Hammerschmidt, T. and Drautz, R.*Computational Materials Science*173 (2020)Atomistic models like tight-binding (TB), bond-order potentials (BOP) and classical potentials describe the interatomic interaction in terms of mathematical functions with parameters that need to be adjusted for a particular material. The procedures for constructing TB/BOP models differ from the ones for classical potentials. We developed the BOPcat software package as a modular python code for the construction and testing of TB/BOP parameterizations. It makes use of atomic energies, forces and stresses obtained by TB/BOP calculations with the BOPfox software package. It provides a graphical user interface and flexible control of raw reference data, of derived reference data like defect energies, of automated construction and testing protocols, and of parallel execution in queuing systems. We demonstrate the concepts and usage of the BOPcat software and illustrate its key capabilities by exemplary constructing and testing a parameterization of a magnetic BOP for Fe. We provide a parameterization protocol with a successively increasing set of reference data that leads to good transferability to a variety of properties of the ferromagnetic bcc groundstate and to crystal structures which were not part of the training set. © 2019 Elsevier B.V.view abstract 10.1016/j.commatsci.2019.109455 **Experimental and Theoretical Investigation on Phase Formation and Mechanical Properties in Cr-Co-Ni Alloys Processed Using a Novel Thin-Film Quenching Technique**

Naujoks, D. and Schneider, M. and Salomon, S. and Pfetzing-Micklich, J. and Subramanyam, A.P.A. and Hammerschmidt, T. and Drautz, R. and Frenzel, J. and Kostka, A. and Eggeler, G. and Laplanche, G. and Ludwig, Al.*ACS Combinatorial Science*22 (2020)The Cr-Co-Ni system was studied by combining experimental and computational methods to investigate phase stability and mechanical properties. Thin-film materials libraries were prepared and quenched from high temperatures up to 700 °C using a novel quenching technique. It could be shown that a wide A1 solid solution region exists in the Cr-Co-Ni system. To validate the results obtained using thin-film materials libraries, bulk samples of selected compositions were prepared by arc melting, and the experimental data were additionally compared to results from DFT calculations. The computational results are in good agreement with the measured lattice parameters and elastic moduli. The lattice parameters increase with the addition of Co and Cr, with a more pronounced effect for the latter. The addition of ∼20 atom % Cr results in a similar hardening effect to that of the addition of ∼40 atom % Co. Copyright © 2020 American Chemical Society.view abstract 10.1021/acscombsci.9b00170 **Fast diffusion mechanism in Li4P2S6: Via a concerted process of interstitial Li ions**

Stamminger, A.R. and Ziebarth, B. and Mrovec, M. and Hammerschmidt, T. and Drautz, R.*RSC Advances*10 (2020)The synthesis of Li superionic conductor Li7P3S11 may be accompanied by the formation of a detrimental Li4P2S6 phase due to a high mixing sensitivity of precursor materials. This phase exhibits a poor ionic conductivity whose origins are not fully understood. Recently Dietrich et al. investigated the energetics of Li ion migration in Li4P2S6 with nudged elastic band (NEB) calculations. The observed large migration barrier of 0.51 eV for purely interstitial diffusion leads to an interpretation of the low ionic conductivity by kinetic limitations. Based on ab initio molecular dynamics simulations (AIMD) we propose a new and energetically much more favorable diffusion path available to interstitial Li ion charge carriers that has not been considered so far. It consists of a concerted process in which a second lithium atom is pushed out from its equilibrium lattice position by the diffusing lithium ion. A detailed analysis with NEB calculations shows that the energy barrier for this concerted diffusion is only 0.08 eV, i.e. an order of magnitude lower than the previously reported value for purely interstitial diffusion. Therefore, the observed low ionic conductivity of Li4P2S6 is likely not originating from kinetic limitations due to high diffusion barriers but rather from thermodynamic reasons associated with a low concentration of free charge carriers. We therefore expect that increasing the charge carrier concentration by doping is a viable design route to optimize the ionic conductivity of this material. © 2020 The Royal Society of Chemistry.view abstract 10.1039/d0ra00932f **Identification of a multi-dimensional reaction coordinate for crystal nucleation in Ni3Al**

Liang, Y. and Díaz Leines, G. and Drautz, R. and Rogal, J.*The Journal of chemical physics*152 (2020)Nucleation during solidification in multi-component alloys is a complex process that comprises competition between different crystalline phases as well as chemical composition and ordering. Here, we combine transition interface sampling with an extensive committor analysis to investigate the atomistic mechanisms during the initial stages of nucleation in Ni3Al. The formation and growth of crystalline clusters from the melt are strongly influenced by the interplay between three descriptors: the size, crystallinity, and chemical short-range order of the emerging nuclei. We demonstrate that it is essential to include all three features in a multi-dimensional reaction coordinate to correctly describe the nucleation mechanism, where, in particular, the chemical short-range order plays a crucial role in the stability of small clusters. The necessity of identifying multi-dimensional reaction coordinates is expected to be of key importance for the atomistic characterization of nucleation processes in complex, multi-component systems.view abstract 10.1063/5.0010074 **Role of pre-ordered liquid in the selection mechanism of crystal polymorphs during nucleation**

Menon, S. and Díaz Leines, G. and Drautz, R. and Rogal, J.*Journal of Chemical Physics*153 (2020)We investigate the atomistic mechanism of homogeneous nucleation during solidification in molybdenum employing transition path sampling. The mechanism is characterized by the formation of a pre-structured region of high bond-orientational order in the supercooled liquid followed by the emergence of the crystalline bulk phase within the center of the growing solid cluster. This precursor plays a crucial role in the process as it provides a diffusive interface between the liquid and crystalline core, which lowers the interfacial free energy and facilitates the formation of the bulk phase. Furthermore, the structural features of the pre-ordered regions are distinct from the liquid and solid phases and preselect the specific polymorph that nucleates. The similarity in the nucleation mechanism of Mo with that of metals that exhibit different crystalline bulk phases indicates that the formation of a precursor is a general feature observed in these materials. The strong influence of the structural characteristics of the precursors on the final crystalline bulk phase demonstrates that for the investigated system, polymorph selection takes place in the very early stages of nucleation. © 2020 Author(s).view abstract 10.1063/5.0017575 **Study of grain boundary self-diffusion in iron with different atomistic models**

Starikov, S. and Mrovec, M. and Drautz, R.*Acta Materialia*188 (2020)We studied grain boundary (GB) self-diffusion in body-centered cubic iron using ab initio calculations and molecular dynamics simulations with various interatomic potentials. A combination of different models allowed us to determine the principal characteristics of self-diffusion along different types of GBs. In particular, we found that atomic self-diffusion in symmetric tilt GBs is mostly driven by self-interstitial atoms. In contrast, in general GBs atoms diffuse predominantly via an exchange mechanism that does not involve a particular defect but is similar to diffusion in a liquid. Most observed mechanisms lead to a significant enhancement of self-diffusion along GBs as compared to diffusion in the bulk. The results of simulations are verified by comparison with available experimental data. © 2020 Acta Materialia Inc.view abstract 10.1016/j.actamat.2020.02.027 **Accelerating spin-space sampling by auxiliary spin dynamics and temperature-dependent spin-cluster expansion**

Wang, N. and Hammerschmidt, T. and Rogal, J. and Drautz, R.*Physical Review B*99 (2019)Atomistic simulations of the thermodynamic properties of magnetic materials rely on an accurate modeling of magnetic interactions and an efficient sampling of the high-dimensional spin space. Recent years have seen significant progress with a clear trend from model systems to material-specific simulations that are usually based on electronic-structure methods. Here we develop a Hamiltonian Monte Carlo framework that makes use of auxiliary spin dynamics and an auxiliary effective model, the temperature-dependent spin-cluster expansion, in order to efficiently sample the spin space. Our method does not require a specific form of the model and is suitable for simulations based on electronic-structure methods. We demonstrate fast warm-up and a reasonably small dynamical critical exponent of our sampler for the classical Heisenberg model. We further present an application of our method to the magnetic phase transition in bcc iron using magnetic bond-order potentials. © 2019 American Physical Society.view abstract 10.1103/PhysRevB.99.094402 **Additive manufacturing of CMSX-4 Ni-base superalloy by selective laser melting: Influence of processing parameters and heat treatment**

Lopez-Galilea, I. and Ruttert, B. and He, J. and Hammerschmidt, T. and Drautz, R. and Gault, B. and Theisen, W.*Additive Manufacturing*30 (2019)Selective laser melting (SLM) provides an economic approach to manufacturing Ni-base superalloy components for high-pressure gas turbines as well as repairing damaged blade sections during operation. In this study, two advanced processing routes are combined: SLM, to fabricate small specimens of the nonweldable CMSX-4, and hot isostatic pressing (HIP) with a rapid cooling rate as post-processing to heal defects while the target γ/γ´ microstructure is developed. An initial parametric study is carried out to investigate the influence of the SLM process parameters on the microstructure and defects occurring during SLM. Special emphasis is placed on understanding and characterizing the as-built SLM microstructures by means of high-resolution characterization techniques. The post-processing heat treatment is then optimized with respect to segregation and the γ/γ´ microstructure. © 2019 Elsevier B.V.view abstract 10.1016/j.addma.2019.100874 **Atomic cluster expansion for accurate and transferable interatomic potentials**

Drautz, R.*Physical Review B*99 (2019)The atomic cluster expansion is developed as a complete descriptor of the local atomic environment, including multicomponent materials, and its relation to a number of other descriptors and potentials is discussed. The effort for evaluating the atomic cluster expansion is shown to scale linearly with the number of neighbors, irrespective of the order of the expansion. Application to small Cu clusters demonstrates smooth convergence of the atomic cluster expansion to meV accuracy. By introducing nonlinear functions of the atomic cluster expansion an interatomic potential is obtained that is comparable in accuracy to state-of-the-art machine learning potentials. Because of the efficient convergence of the atomic cluster expansion relevant subspaces can be sampled uniformly and exhaustively. This is demonstrated by testing against a large database of density functional theory calculations for copper. © 2019 American Physical Society.view abstract 10.1103/PhysRevB.99.014104 **BOPfox program for tight-binding and analytic bond-order potential calculations**

Hammerschmidt, T. and Seiser, B. and Ford, M.E. and Ladines, A.N. and Schreiber, S. and Wang, N. and Jenke, J. and Lysogorskiy, Y. and Teijeiro, C. and Mrovec, M. and Cak, M. and Margine, E.R. and Pettifor, D.G. and Drautz, R.*Computer Physics Communications*235 (2019)Bond-order potentials (BOPs) provide a local and physically transparent description of the interatomic interaction. Here we describe the efficient implementation of analytic BOPs in the BOPfox program and library. We discuss the integration of the underlying non-magnetic, collinear-magnetic and noncollinear-magnetic tight-binding models that are evaluated by the analytic BOPs. We summarise the flow of an analytic BOP calculation including the determination of self-returning paths for computing the moments, the self-consistency cycle, the estimation of the band-width from the recursion coefficients, and the termination of the BOP expansion. We discuss the implementation of the calculations of forces, stresses and magnetic torques with analytic BOPs. We show the scaling of analytic BOP calculations with the number of atoms and moments, present options for speeding up the calculations and outline different concepts of parallelisation. In the appendix we compile the implemented equations of the analytic BOP methodology and comments on the implementation. This description should be relevant for other implementations and further developments of analytic bond-order potentials. © 2018 Elsevier B.V.view abstract 10.1016/j.cpc.2018.08.013 **Discovery of ω -free high-temperature Ti-Ta- X shape memory alloys from first-principles calculations**

Ferrari, A. and Paulsen, A. and Langenkämper, D. and Piorunek, D. and Somsen, C. and Frenzel, J. and Rogal, J. and Eggeler, G. and Drautz, R.*Physical Review Materials*3 (2019)The rapid degradation of the functional properties of many Ti-based alloys is due to the precipitation of the ω phase. In the conventional high-temperature shape memory alloy Ti-Ta, the formation of this phase compromises completely the shape memory effect, and high (>100°C) transformation temperatures cannot be maintained during cycling. A solution to this problem is the addition of other elements to form Ti-Ta-X alloys, which often modifies the transformation temperatures; due to the largely unexplored space of possible compositions, very few elements are known to stabilize the shape memory effect without decreasing the transformation temperatures below 100°C. In this study, we use transparent descriptors derived from first-principles calculations to search for new ternary Ti-Ta-X alloys that combine stability and high temperatures. We suggest four alloys with these properties, namely Ti-Ta-Sb, Ti-Ta-Bi, Ti-Ta-In, and Ti-Ta-Sc. Our predictions for the most promising of these alloys, Ti-Ta-Sc, are subsequently fully validated by experimental investigations, the alloy Ti-Ta-Sc showing no traces of ω phase after cycling. Our computational strategy is transferable to other materials and may contribute to suppress ω phase formation in a large class of alloys. ©2019 American Physical Society.view abstract 10.1103/PhysRevMaterials.3.103605 **Erratum: Atomic cluster expansion for accurate and transferable interatomic potentials (Physical Review B (2019) 99 (014104) DOI: 10.1103/PhysRevB.100.249901)**

Drautz, R.*Physical Review B*10 (2019)In Eq. (C6), the relative weight of the different J couplings was omitted. The symbol (Formula Persented). depends on the relative contribution of the different couplings J. The correct form of Eq. (C6) reads.(Formula Persented). with constants cJ that determine the relative contributions of the different couplings J and (Formula Persented). . The coefficients Formula Persented). c J , in general, differ for different basis functions indexed by.(Formula Persented). In analogy, the correct expression for Eq. (C8) is given by(Formula Persented).None of the conclusions or results of the paper are affected by these typing errors. I am indebted to Dr. A. Shapeev from Skoltech, Russia for pointing out the mistake in Eq. (C6). © 2019 American Physical Society.view abstract 10.1103/PhysRevB.100.249901 **First-principles characterization of reversible martensitic transformations**

Ferrari, A. and Sangiovanni, D.G. and Rogal, J. and Drautz, R.*Physical Review B*99 (2019)Reversible martensitic transformations (MTs) are the origin of many fascinating phenomena, including the famous shape memory effect. In this work, we present a fully ab initio procedure to characterize MTs in alloys and to assess their reversibility. Specifically, we employ ab initio molecular dynamics data to parametrize a Landau expansion for the free energy of the MT. This analytical expansion makes it possible to determine the stability of the high- and low-temperature phases, to obtain the Ehrenfest order of the MT, and to quantify its free energy barrier and latent heat. We apply our model to the high-temperature shape memory alloy Ti-Ta, for which we observe remarkably small values for the metastability region (the interval of temperatures in which the high- and low-temperature phases are metastable) and for the barrier: these small values are necessary conditions for the reversibility of MTs and distinguish shape memory alloys from other materials. © 2019 American Physical Society.view abstract 10.1103/PhysRevB.99.094107 **Imaging individual solute atoms at crystalline imperfections in metals**

Katnagallu, S. and Stephenson, L.T. and Mouton, I. and Freysoldt, C. and Subramanyam, A.P.A. and Jenke, J. and Ladines, A.N. and Neumeier, S. and Hammerschmidt, T. and Drautz, R. and Neugebauer, J. and Vurpillot, F. and Raabe, D. and Gault, B.*New Journal of Physics*21 (2019)Directly imaging all atoms constituting a material and, maybe more importantly, crystalline defects that dictate materials' properties, remains a formidable challenge. Here, we propose a new approach to chemistry-sensitive field-ion microscopy (FIM) combining FIM with time-of-flight mass-spectrometry (tof-ms). Elemental identification and correlation to FIM images enabled by data mining of combined tof-ms delivers a truly analytical-FIM (A-FIM). Contrast variations due to different chemistries is also interpreted from density-functional theory (DFT). A-FIM has true atomic resolution and we demonstrate how the technique can reveal the presence of individual solute atoms at specific positions in the microstructure. The performance of this new technique is showcased in revealing individual Re atoms at crystalline defects formed in Ni-Re binary alloy during creep deformation. The atomistic details offered by A-FIM allowed us to directly compare our results with simulations, and to tackle a long-standing question of how Re extends lifetime of Ni-based superalloys in service at high-temperature. © 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/ab5cc4 **Influence of excess volumes induced by Re and W on dislocation motion and creep in ni-base single crystal superalloys: A 3D discrete dislocation dynamics study**

Gao, S. and Yang, Z. and Grabowski, M. and Rogal, J. and Drautz, R. and Hartmaier, A.*Metals*9 (2019)A comprehensive 3D discrete dislocation dynamics model for Ni-base single crystal superalloys was used to investigate the influence of excess volumes induced by solute atoms Re and W on dislocation motion and creep under different tensile loads at 850 °C. The solute atoms were distributed homogeneously only in g matrix channels. Their excess volumes due to the size difference from the host Ni were calculated by density functional theory. The excess volume affected dislocation glide more strongly than dislocation climb. The relative positions of dislocations and solute atoms determined the magnitude of back stresses on the dislocation motion. Without diffusion of solute atoms, it was found that W with a larger excess volume had a stronger strengthening effect than Re. With increasing concentration of solute atoms, the creep resistance increased. However, a low external stress reduced the influence of different excess volumes and different concentrations on creep. © 2019 by the authors. Licensee MDPI, Basel, Switzerland.view abstract 10.3390/met9060637 **Ionic Conductivity and Its Dependence on Structural Disorder in Halogenated Argyrodites Li6PS5X (X = Br, Cl, I)**

Stamminger, A.R. and Ziebarth, B. and Mrovec, M. and Hammerschmidt, T. and Drautz, R.*Chemistry of Materials*(2019)Halogenated argyrodites Li6PS5Br, Li6PS5Cl, and Li6PS5I exhibit large differences in the measured Li ionic conductivities. Crystallographic analysis has shown that these differences may be related to occupations of specific Wyckoff sites in different argyrodite types, but detailed understanding of the relationship between the atomic structure and operating diffusion mechanisms is still lacking. In this work, we employed ab initio molecular dynamics simulations to calculate the Li diffusivity for different argyrodite structure types. Our calculations show that the Li diffusivity does not depend implicitly on the type of halogen but is rather governed by the degree of structural disorder. Assuming disordered structures to arise naturally from the ordered structure type by thermally activated antisite defects, we are able to explain the degree of disorder found for the different types of halogens from the calculated defect formation energies. Comparing the calculated formation energies to the ionic radii of the halogen atoms, we find a strong correlation between the radii and energies required for introducing the antisite defects. © 2019 American Chemical Society.view abstract 10.1021/acs.chemmater.9b02047 **Optimized parallel simulations of analytic bond-order potentials on hybrid shared/distributed memory with MPI and OpenMP**

Teijeiro, C. and Hammerschmidt, T. and Drautz, R. and Sutmann, G.*International Journal of High Performance Computing Applications*33 (2019)Analytic bond-order potentials (BOPs) allow to obtain a highly accurate description of interatomic interactions at a reasonable computational cost. However, for simulations with very large systems, the high memory demands require the use of a parallel implementation, which at the same time also optimizes the use of computational resources. The calculations of analytic BOPs are performed for a restricted volume around every atom and therefore have shown to be well suited for a message passing interface (MPI)-based parallelization based on a domain decomposition scheme, in which one process manages one big domain using the entire memory of a compute node. On the basis of this approach, the present work focuses on the analysis and enhancement of its performance on shared memory by using OpenMP threads on each MPI process, in order to use many cores per node to speed up computations and minimize memory bottlenecks. Different algorithms are described and their corresponding performance results are presented, showing significant performance gains for highly parallel systems with hybrid MPI/OpenMP simulations up to several thousands of threads. © The Author(s) 2017.view abstract 10.1177/1094342017727060 **Phase transitions in titanium with an analytic bond-order potential**

Ferrari, A. and Schröder, M. and Lysogorskiy, Y. and Rogal, J. and Mrovec, M. and Drautz, R.*Modelling and Simulation in Materials Science and Engineering*27 (2019)Titanium is the base material for a number of technologically important alloys for energy conversion and structural applications. Atomic-scale studies of Ti-based metals employing first-principles methods, such as density functional theory, are limited to ensembles of a few hundred atoms. To perform large-scale and/or finite temperature simulations, computationally more efficient interatomic potentials are required. In this work, we coarse grain the tight-binding (TB) approximation to the electronic structure and develop an analytic bond-order potential (BOP) for Ti by fitting to the energies and forces of elementary deformations of simple structures. The BOP predicts the structural properties of the stable and defective phases of Ti with a quality comparable to previous TB parameterizations at a much lower computational cost. The predictive power of the model is demonstrated for simulations of martensitic transformations. © 2019 IOP Publishing Ltd.view abstract 10.1088/1361-651X/ab471d **pyiron: An integrated development environment for computational materials science**

Janssen, J. and Surendralal, S. and Lysogorskiy, Y. and Todorova, M. and Hickel, T. and Drautz, R. and Neugebauer, J.*Computational Materials Science*163 (2019)To support and accelerate the development of simulation protocols in atomistic modelling, we introduce an integrated development environment (IDE) for computational materials science called pyiron (http://pyiron.org). The pyiron IDE combines a web based source code editor, a job management system for build automation, and a hierarchical data management solution. The core components of the pyiron IDE are pyiron objects based on an abstract class, which links application structures such as atomistic structures, projects, jobs, simulation protocols and computing resources with persistent storage and an interactive user environment. The simulation protocols within the pyiron IDE are constructed using the Python programming language. To highlight key concepts of this tool as well as to demonstrate its ability to simplify the implementation and testing of simulation protocols we discuss two applications. In these examples we show how pyiron supports the whole life cycle of a typical simulation, seamlessly combines ab initio with empirical potential calculations, and how complex feedback loops can be implemented. While originally developed with focus on ab initio thermodynamics simulations, the concepts and implementation of pyiron are general thus allowing to employ it for a wide range of simulation topics. © 2019 The Authorsview abstract 10.1016/j.commatsci.2018.07.043 **Reconciling Experimental and Theoretical Data in the Structural Analysis of Ti–Ta Shape-Memory Alloys**

Ferrari, A. and Kadletz, P.M. and Chakraborty, T. and Liao, K. and Langenkämper, D. and Motemani, Y. and Paulsen, A. and Lysogorskiy, Y. and Frenzel, J. and Rogal, J. and Ludwig, Al. and Somsen, C. and Drautz, R. and Schmahl, W.W.*Shape Memory and Superelasticity*5 (2019)The structural characterization of the various phases that occur in Ti–Ta-based high-temperature shape-memory alloys is complicated by the presence of many competing phases as a function of composition. In this study, we resolve apparent inconsistencies between experimental data and theoretical calculations by suggesting that phase separation and segregation of undesired phases are not negligible in these alloys, and that finite temperature effects should be taken into account in the modeling of these materials. Specifically, we propose that the formation of the ω phase at low Ta content and of the σ phase at high Ta content implies a difference between the nominal alloy composition and the actual composition of the martensitic and austenitic phases. In addition, we show that temperature affects strongly the calculated values of the order parameters of the martensitic transformation occurring in Ti–Ta. © 2018, ASM International.view abstract 10.1007/s40830-018-00201-6 **Repair of Ni-based single-crystal superalloys using vacuum plasma spray**

Kalfhaus, T. and Schneider, M. and Ruttert, B. and Sebold, D. and Hammerschmidt, T. and Frenzel, J. and Drautz, R. and Theisen, W. and Eggeler, G. and Guillon, O. and Vassen, R.*Materials and Design*168 (2019)Turbine blades in aviation engines and land based gas-turbines are exposed to extreme environments. They suffer damage accumulation associated with creep, oxidation and fatigue loading. Therefore, advanced repair methods are of special interest for the gas-turbine industry. In this study, CMSX-4 powder is sprayed by Vacuum Plasma Spray (VPS) on single-crystalline substrates with similar compositions. The influence of the substrate temperature is investigated altering the temperature of the heating stage between 850 °C to 1000 °C. Different spray parameters were explored to identify their influence on the microstructure. Hot isostatic pressing (HIP) featuring fast quenching rates was used to minimize porosity and to allow for well-defined heat-treatments of the coatings. The microstructure was analysed by orientation imaging scanning electron microscopy (SEM), using electron backscatter diffraction (EBSD). The effects of different processing parameters were analysed regarding their influence on porosity and grain size. The results show that optimized HIP heat-treatments can lead to dense coatings with optimum γ/γ′ microstructure. The interface between the coating and the substrate is oxide free and shows good mechanical integrity. The formation of fine crystalline regions as a result of fast cooling was observed at the single-crystal surface, which resulted in grain growth during heat-treatment in orientations determined by the crystallography of the substrate. © 2019view abstract 10.1016/j.matdes.2019.107656 **Stress-induced formation of TCP phases during high temperature low cycle fatigue loading of the single-crystal Ni-base superalloy ERBO/1**

Meid, C. and Eggeler, M. and Watermeyer, P. and Kostka, A. and Hammerschmidt, T. and Drautz, R. and Eggeler, G. and Bartsch, M.*Acta Materialia*168 (2019)The microstructural evolution in the single crystal Ni-base superalloy ERBO/1 (CMSX 4 type) is investigated after load controlled low cycle fatigue (LCF) at 950 °C (load-ratio: 0.6, tensile stress range: 420–740 MPa, test frequency: 0.25 Hz, fatigue rupture life: about 1000 - 3000 cycles). Bulk topologically close packed (TCP) phase particles precipitated and were analyzed by three-dimensional focus ion beam slice and view imaging and analytical transmission electron microscopy. The particles did not precipitate homogenously but at locations with enhanced levels of local stresses/strains, such as isolated γ-channels subjected to cross channel stresses, shear bands and in front of micro cracks. The influence of stress/strain is furthermore apparent in the spatial arrangement and the shape of the TCP phase particles. Only μ-phase TCP particles were found by electron diffraction. Results of a structure-map analysis suggest that most of these TCP particles observed after LCF testing would not precipitate in thermodynamic equilibrium. In order to rationalize this effect, the atomic volume was analyzed that transition-metal (TM) elements take in unary fcc and in unary μ-phase crystal structures and found that all TM elements except Zr and V take a larger volume in a unary μ phase than in a unary fcc phase. This trend is in line with the observed localized precipitation of TCP phases that are rich in Ni and other late TM elements. The experimental and theoretical findings suggest consistently that formation of TCP particles in LCF tests is considerably influenced by the local tensile stress/strain states. © 2019 Acta Materialia Inc.view abstract 10.1016/j.actamat.2019.02.022 **Transferability of interatomic potentials for molybdenum and silicon**

Lysogorskiy, Y. and Hammerschmidt, T. and Janssen, J. and Neugebauer, J. and Drautz, R.*Modelling and Simulation in Materials Science and Engineering*27 (2019)Interatomic potentials are widely used in computational materials science, in particular for simulations that are too computationally expensive for density functional theory (DFT). Most interatomic potentials have a limited application range and often there is very limited information available regarding their performance for specific simulations. We carried out high-throughput calculations for molybdenum and silicon with DFT and a number of interatomic potentials. We compare the DFT reference calculations and experimental data to the predictions of the interatomic potentials. We focus on a large number of basic materials properties, including the cohesive energy, atomic volume, elastic coefficients, vibrational properties, thermodynamic properties, surface energies and vacancy formation energies, which enables a detailed discussion of the performance of the different potentials. We further analyze correlations between properties as obtained from DFT calculations and how interatomic potentials reproduce these correlations, and suggest a general measure for quantifying the accuracy and transferability of an interatomic potential. From our analysis we do not establish a clearcut ranking of the potentials as each potential has its strengths and weaknesses. It is therefore essential to assess the properties of a potential carefully before application of the potential in a specific simulation. The data presented here will be useful for selecting a potential for simulations of Mo or Si. © 2019 IOP Publishing Ltd.view abstract 10.1088/1361-651X/aafd13 **Development of Single-Crystal Ni-Base Superalloys Based on Multi-criteria Numerical Optimization and Efficient Use of Refractory Elements**

Markl, M. and Müller, A. and Ritter, N. and Hofmeister, M. and Naujoks, D. and Schaar, H. and Abrahams, K. and Frenzel, J. and Subramanyam, A.P.A. and Ludwig, Al. and Pfetzing-Micklich, J. and Hammerschmidt, T. and Drautz, R. and Steinbach, I. and Rettig, R. and Singer, R.F. and Körner, C.*Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science*49 (2018)The development of new Ni-base superalloys with a complex composition consisting of eight or more alloying elements is a challenging task. The experimental state-of-the-art development cycle is based on the adaption of already existing compositions. Although new alloy compositions with potentially improved material properties are expected to be similar to already known superalloys, this procedure impedes efficiently finding these compositions in the large multi-dimensional design-space of all alloying elements. Modern alloy development combines numerical optimization methods with experimental validation to guide the development towards promising compositions. In this work, an improved numerical multi-criteria optimization tool using CALPHAD calculations and semi-empirical models for alloy development is presented. The model improvements to its predecessor are described and the successful application for the development of rhenium-free single-crystal Ni-base superalloys ERBO/13 and ERBO/15 is revisited. The optimization tool is described and the designed alloys are discussed regarding phase stability. Finally, a possible phase stability model extending the optimization tool and improving the alloy composition predictions is presented. © 2018, The Author(s).view abstract 10.1007/s11661-018-4759-0 **Electronic structure based descriptor for characterizing local atomic environments**

Jenke, J. and Subramanyam, A.P.A. and Densow, M. and Hammerschmidt, T. and Pettifor, D.G. and Drautz, R.*Physical Review B*98 (2018)A quantitative descriptor of local atomic environments is often required for the analysis of atomistic data. Descriptors of the local atomic environment ideally provide physically and chemically intuitive insight. This requires descriptors that are low-dimensional representations of the interplay between atomic geometry and electronic bond formation. The moments of the local density of states relate the atomic structure to the electronic structure and bond chemistry. This makes it possible to construct electronic structure based descriptors of the local atomic environment that have an immediate relation to the binding energy. We show that a low-dimensional moments-descriptor is sufficient as the lowest moments, calculated from the closest atomic neighborhood, carry the largest contributions to the local bond energy. Here, we construct moments-descriptors that project the space of local atomic environments on a two-dimensional map. We discuss in detail the separation of various atomic environments and their connections in the map. The distances in the map may be related to energy differences between local atomic environments as we show by analytic considerations based on analytic bond-order potentials and by numerical assessment using tight-binding and density-functional theory calculations. Possible applications of the proposed moments-descriptors include the classification of local atomic environments in molecular-dynamic simulations, the selection of structure sets for developing and testing interatomic potentials, as well as the construction of descriptors for machine-learning applications. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.98.144102 **Finite-temperature property-maps of Li-Mn-Ni-O cathode materials from: Ab initio calculations**

Albina, J.-M. and Marusczyk, A. and Hammerschmidt, T. and Eckl, T. and Drautz, R.*Journal of Materials Chemistry A*6 (2018)We report first-principles calculations for determining the phase relationships in multi-component cathode materials. We investigate the effect of delithiation on the phase stability, chemical potential, and open circuit voltage for a selection of cathode materials based on Li-Mn-Ni oxides at various temperatures. Entropic contributions are included by calculating the phonon frequencies in the harmonic approximation. The open circuit voltage in multi-component systems is estimated by a convex hull approach. We confirm that spinel-like phases are predominant during the charging process of layered Li-Mn-O cathode materials and that the addition of Ni reduces the spinel content. The analysis of phase stability upon delithiation suggests that the Li2MnO3 component in the Li2MnO3·Li(Mn,Ni)O2 electrode material should not exceed 60% and that the amount of Ni in the LiMnO2 component should be above 40 at% for minimizing spinel-type phase formation and minimizing oxygen formation. Using the computed structural stability at room temperature, we derive property maps for the design of Li-Mn-Ni-O cathode materials. © The Royal Society of Chemistry 2018.view abstract 10.1039/c7ta07221j **Grain boundaries in bcc-Fe: A density-functional theory and tight-binding study**

Wang, J. and Madsen, G.K.H. and Drautz, R.*Modelling and Simulation in Materials Science and Engineering*26 (2018)Grain boundaries (GBs) have a significant influence on material properties. In the present paper, we calculate the energies of eleven low-σ (σ > 13) symmetrical tilt GBs and two twist GBs in ferromagnetic bcc iron using firstprinciples density functional theory (DFT) calculations. The results demonstrate the importance of a sufficient sampling of initial rigid body translations in all three directions. We show that the relative GB energies can be xplained by the miscoordination of atoms at the GB region. While the main features of the studied GB structures were captured by previous empirical interatomic potential calculations, it is shown that the absolute values of GB energies calculated were substantially underestimated. Based on DFT-calculated GB structures and energies, we construct a new d-band orthogonal tight-binding (TB) model for bcc iron. The TB model is validated by its predictive power on all the studied GBs. We apply the TB model to block boundaries in lath martensite and demonstrate that the experimentally observed GB character distribution can be explained from the viewpoint of interface energy. © 2018 IOP Publishing Ltd.view abstract 10.1088/1361-651X/aa9f81 **Unusual composition dependence of transformation temperatures in Ti-Ta-X shape memory alloys**

Ferrari, A. and Paulsen, A. and Frenzel, J. and Rogal, J. and Eggeler, G. and Drautz, R.*Physical Review Materials*2 (2018)Ti-Ta-X (X = Al, Sn, Zr) compounds are emerging candidates as high-temperature shape memory alloys (HTSMAs). The stability of the one-way shape memory effect (1WE), the exploitable pseudoelastic (PE) strain intervals, as well as the transformation temperature in these alloys depend strongly on composition, resulting in a trade-off between a stable shape memory effect and a high transformation temperature. In this work, experimental measurements and first-principles calculations are combined to rationalize the effect of alloying a third component to Ti-Ta-based HTSMAs. Most notably, an increase in the transformation temperature with increasing Al content is detected experimentally in Ti-Ta-Al for low Ta concentrations, in contrast to the generally observed dependence of the transformation temperature on composition in Ti-Ta-X. This inversion of trend is confirmed by the ab initio calculations. Furthermore, a simple analytical model based on the ab initio data is derived. The model can not only explain the unusual composition dependence of the transformation temperature in Ti-Ta-Al but also provide a fast and elegant tool for a qualitative evaluation of other ternary systems. This is exemplified by predicting the trend of the transformation temperature of Ti-Ta-Sn and Ti-Ta-Zr alloys, yielding a remarkable agreement with available experimental data. © 2018 American Physical Society.view abstract 10.1103/PhysRevMaterials.2.073609 **{110} planar faults in strained bcc metals: Origins and implications of a commonly observed artifact of classical potentials**

Möller, J.J. and Mrovec, M. and Bleskov, I. and Neugebauer, J. and Hammerschmidt, T. and Drautz, R. and Elsässer, C. and Hickel, T. and Bitzek, E.*Physical Review Materials*2 (2018)Large-scale atomistic simulations with classical potentials can provide valuable insights into microscopic deformation mechanisms and defect-defect interactions in materials. Unfortunately, these assets often come with the uncertainty of whether the observed mechanisms are based on realistic physical phenomena or whether they are artifacts of the employed material models. One such example is the often reported occurrence of stable planar faults (PFs) in body-centered cubic (bcc) metals subjected to high strains, e.g., at crack tips or in strained nano-objects. In this paper, we study the strain dependence of the generalized stacking fault energy (GSFE) of {110} planes in various bcc metals with material models of increasing sophistication, i.e., (modified) embedded atom method, angular-dependent, Tersoff, and bond-order potentials as well as density functional theory. We show that under applied tensile strains the GSFE curves of many classical potentials exhibit a local minimum which gives rise to the formation of stable PFs. These PFs do not appear when more sophisticated material models are used and have thus to be regarded as artifacts of the potentials. We demonstrate that the local GSFE minimum is not formed for reasons of symmetry and we recommend including the determination of the strain-dependent (110) GSFE as a benchmark for newly developed potentials. © 2018 American Physical Society.view abstract 10.1103/PhysRevMaterials.2.093606 **A kinetic Monte Carlo approach to diffusion-controlled thermal desorption spectroscopy**

Schablitzki, T. and Rogal, J. and Drautz, R.*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*375 (2017)Atomistic simulations of thermal desorption spectra for effusion from bulk materials to characterize binding or trapping sites are a challenging task as large system sizes as well as extended time scales are required. Here, we introduce an approach where we combine kinetic Monte Carlo with an analytic approximation of the superbasins within the framework of absorbing Markov chains. We apply our approach to the effusion of hydrogen from BCC iron, where the diffusion within bulk grains is coarse grained using absorbingMarkov chains, which provide an exact solution of the dynamics within a superbasin. Our analytic approximation to the superbasin is transferable with respect to grain size and elliptical shapes and can be applied in simulations with constant temperature as well as constant heating rate. The resulting thermal desorption spectra are in close agreement with direct kinetic Monte Carlo simulations, but the calculations are computationally much more efficient. Our approach is thus applicable to much larger system sizes and provides a first step towards an atomistic understanding of the influence of structural features on the position and shape of peaks in thermal desorption spectra. © 2017 The Author(s) Published by the Royal Society. All rights reserved.view abstract 10.1098/rsta.2016.0404 **Ab-initio study of C and N point defects in the C14-Fe2Nb phase**

Ladines, A.N. and Drautz, R. and Hammerschmidt, T.*Journal of Alloys and Compounds*693 (2017)Nb-alloying of steels can lead to the formation of topologically close-packed (TCP) phases, particularly Fe2Nb Laves and Fe7Nb6 μ phases. The stability of these TCP phases is strongly affected by the presence of light elements like C and N. We calculate the solution energy of C and N in C14-Fe2Nb using density functional theory. N shows a strong preference to dissolve in larger interstitial voids while C shows a strong tendency to bind with a neighbouring Nb atom. The computed solution energies suggest N incorporation into Fe2Nb Laves phases while C is hardly soluble. The N-N interaction in Fe2Nb is strongly attractive and twice as strong as that of C-C. A comparison to C interstitials in the μ-Fe7Nb6 phase shows similar dependence of the solution energy on the atomic environment. In order to aid future work, we additionally provide the coordinates of interstitial sites in all TCP phases (A15, Zr4Al3, C14, C15, C36, χ, μ, σ, M, P, δ and R.) in the supplementary material. © 2016 Elsevier B.V.view abstract 10.1016/j.jallcom.2016.10.030 **Atomistic insight into the non-classical nucleation mechanism during solidification in Ni**

Díaz Leines, G. and Drautz, R. and Rogal, J.*Journal of Chemical Physics*146 (2017)Nucleation is a key step during crystallization, but a complete understanding of the fundamental atomistic processes remains elusive. We investigate the mechanism of nucleation during solidification in nickel for various undercoolings using transition path sampling simulations. The temperature dependence of the free energy barriers and rate constants that we obtain is consistent with the predictions of classical nucleation theory and experiments. However, our analysis of the transition path ensemble reveals a mechanism that deviates from the classical picture of nucleation: the growing solid clusters have predominantly non-spherical shapes and consist of face-centered-cubic and random hexagonal-close-packed coordinated atoms surrounded by a cloud of prestructured liquid. The nucleation initiates in regions of supercooled liquid that are characterized by a high orientational order with structural features that predetermine the polymorph selection. These results provide atomistic insight not only into the nucleation mechanism of nickel but also into the role of the preordered liquid regions as precursors for crystallization. © 2017 Author(s).view abstract 10.1063/1.4980082 **Identification of a ternary μ-phase in the Co-Ti-W system – An advanced correlative thin-film and bulk combinatorial materials investigation**

Naujoks, D. and Eggeler, Y.M. and Hallensleben, P. and Frenzel, J. and Fries, S.G. and Palumbo, M. and Koßmann, J. and Hammerschmidt, T. and Pfetzing-Micklich, J. and Eggeler, G. and Spiecker, E. and Drautz, R. and Ludwig, Al.*Acta Materialia*138 (2017)The formation of a ternary μ-phase is documented for the system Co-Ti-W. The relevant compositional stability range is identified by high-throughput energy dispersive X-ray spectroscopy, electrical resistance and X-ray diffraction maps from a thin-film materials library (1 μm thickness). Bulk samples of the identified compositions were fabricated to allow for correlative film and bulk studies. Using analytical scanning and transmission electron microscopy, we demonstrate that in both, thin film and bulk samples, the D85 phase (μ-phase) coexists with the C36-phase and the A2-phase at comparable average chemical compositions. Young's moduli and hardness values of the μ-phase and the C36-phase were determined by nanoindentation. The trends of experimentally obtained elastic moduli are consistent with density functional theory (DFT) calculations. DFT analysis also supports the experimental findings, that the μ-phase can solve up to 18 at.% Ti. Based on the experimental and DFT results it is shown that CALPHAD modeling can be modified to account for the new findings. © 2017 Acta Materialia Inc.view abstract 10.1016/j.actamat.2017.07.037 **Metallic NiPS3@NiOOH Core-Shell Heterostructures as Highly Efficient and Stable Electrocatalyst for the Oxygen Evolution Reaction**

Konkena, B. and Masa, J. and Botz, A. J. R. and Sinev, I. and Xia, W. and Kossmann, J. and Drautz, R. and Muhler, M. and Schuhmann, W.*ACS Catalysis*7 (2017)We report metallic NiPS3@NiOOH core shell heterostructures as an efficient and durable electrocatalyst for the oxygen evolution reaction, exhibiting a low onset potential of 1.48 V (vs RHE) and stable performance for over 160 h. The atomically thin NiPS3 nanosheets are obtained by exfoliation of bulk NiPS3 in the presence of an ionic surfactant. The OER mechanism was studied by a combination of SECM, in situ Raman spectroscopy, SEM, and XPS measurements, which enabled direct observation of the formation of a NiPS3@NiOOH core shell heterostructure at the electrode interface. Hence, the active form of the catalyst is represented as NiPS3@NiOOH core shell structure. Moreover, DFT calculations indicate an intrinsic metallic character of the NiPS3 nanosheets with densities of states (DOS) similar to the bulk material. The high OER activity of the NiPS3 nanosheets is attributed to a high density of accessible active metallic-edge and defect sites due to structural disorder, a unique NiPS3@NiOOH core shell heterostructure, where the presence of P and S modulates the rface electronic structure of Ni in NiPS3, thus providing excellent conductive pathway for efficient electron-transport to the NiOOH shell. These findings suggest that good size control during liquid exfoliation may be advantageously used for the formation of electrically conductive NiPS3@ NiOOH core shell electrode materials for the electrochemical water oxidation.view abstract 10.1021/acscatal.6b02203 **Oxygen activity and peroxide formation as charge compensation mechanisms in Li2MnO3**

Marusczyk, A. and Albina, J.-M. and Hammerschmidt, T. and Drautz, R. and Eckl, T. and Henkelman, G.*Journal of Materials Chemistry A*5 (2017)In the search for high energy density battery materials, over-lithiated transition metal oxides have attracted the attention of many researchers worldwide. There is, however, no consensus regarding the underlying mechanisms that give rise to the large capacities and also cause the electrochemical degradation upon cycling. As a key component and prototype phase, Li2MnO3 is investigated using density functional theory. Our calculations show that hole doping into the oxygen bands is the primary charge compensation mechanism in the first stage of delithiation. Upon further delithiation, there is an energetic driving force for peroxide formation with an optimal number of peroxide dimers that is predicted as a function of lithium concentration. Unlike the defect-free phases, the peroxide structures are highly stable, which leads to two competing mechanisms for charge compensation: (i) oxygen loss and densification at the surface and (ii) peroxide formation in the bulk. Our results show that both have a detrimental effect on the electrochemical performance and therefore the stabilization of oxygen in the crystal lattice is vital for the development of high energy cathode materials. The insights into the origin and implications of peroxide formation open the door for a more profound understanding of the degradation mechanism and how to counteract it. © The Royal Society of Chemistry 2017.view abstract 10.1039/c7ta04164k **Structure map including off-stoichiometric and ternary sp-d-valent compounds**

Hammerschmidt, T. and Bialon, A.F. and Drautz, R.*Modelling and Simulation in Materials Science and Engineering*25 (2017)Structure maps predict the crystal structure of a compound from the knowledge of constituent elements and chemical composition. We recently developed a highly predictive, three-dimensional structure map for stoichiometric binary sp- d-valent compounds. Here we show that the descriptors of this structure map are transferable to off-stoichiometric compounds with similar predictive power. We furthermore demonstrate that the descriptors are suitable for ternary prototypes. In particular, we construct a three-dimensional structure map for 129 prototypical crystal structures for ternary compounds. The crystal structure is predicted correctly with a probability of 78%. With a confidence of 95% the correct crystal structure is among the three most likely crystal structures predicted by the structure map. © 2017 IOP Publishing Ltd.view abstract 10.1088/1361-651X/aa83c3 **Atomistically informed extended Gibbs energy description for phase-field simulation of tempering of martensitic steel**

Shchyglo, O. and Hammerschmidt, T. and Čak, M. and Drautz, R. and Steinbach, I.*Materials*9 (2016)In this study we propose a unified multi-scale chemo-mechanical description of the BCT (Body-Centered Tetragonal) to BCC (Body-Centered Cubic) order-disorder transition in martensitic steel by adding the mechanical degrees of freedom to the standard CALPHAD (CALculation of PHAse Diagrams) type Gibbs energy description. The model takes into account external strain, the effect of carbon composition on the lattice parameter and elastic moduli. The carbon composition effect on the lattice parameters and elastic constants is described by a sublattice model with properties obtained from DFT (Density Functional Theory) calculations; the temperature dependence of the elasticity parameters is estimated from available experimental data. This formalism is crucial for studying the kinetics of martensite tempering in realistic microstructures. The obtained extended Gibbs energy description opens the way to phase-field simulations of tempering of martensitic steel comprising microstructure evolution, carbon diffusion and lattice symmetry change due to the ordering/disordering of carbon atoms under multiaxial load. © 2016 by the authors.view abstract 10.3390/ma9080669 **Collective Atomic Displacements during Complex Phase Boundary Migration in Solid-Solid Phase Transformations**

Duncan, J. and Harjunmaa, A. and Terrell, R. and Drautz, R. and Henkelman, G. and Rogal, J.*Physical Review Letters*116 (2016)The A15 to bcc phase transition is simulated at the atomic scale based on an interatomic potential for molybdenum. The migration of the phase boundary proceeds via long-range collective displacements of entire groups of atoms across the interface. To capture the kinetics of these complex atomic rearrangements over extended time scales we use the adaptive kinetic Monte Carlo approach. An effective barrier of 0.5 eV is determined for the formation of each new bcc layer. This barrier is not associated with any particular atomistic process that governs the dynamics of the phase boundary migration. Instead, the effective layer transformation barrier represents a collective property of the complex potential energy surface. © 2016 authors. Published by the American Physical Society.view abstract 10.1103/PhysRevLett.116.035701 **Complexity analysis of simulations with analytic bond-order potentials**

Teijeiro, C. and Hammerschmidt, T. and Seiser, B. and Drautz, R. and Sutmann, G.*Modelling and Simulation in Materials Science and Engineering*24 (2016)The modeling of materials at the atomistic level with interatomic potentials requires a reliable description of different bonding situations and relevant system properties. For this purpose, analytic bond-order potentials (BOPs) provide a systematic and robust approximation to density functional theory (DFT) and tight binding (TB) calculations at reasonable computational cost. This paper presents a formal analysis of the computational complexity of analytic BOP simulations, based on a detailed assessment of the most computationally intensive parts. Different implementation algorithms are presented alongside with optimizations for efficient numerical processing. The theoretical complexity study is complemented by systematic benchmarks of the scalability of the algorithms with increasing system size and accuracy level of the BOP approximation. Both approaches demonstrate that the computation of atomic forces in analytic BOPs can be performed with a similar scaling as the computation of atomic energies. © 2016 IOP Publishing Ltd.view abstract 10.1088/0965-0393/24/2/025008 **Crystal-structure analysis with moments of the density-of-states: Application to intermetallic topologically close-packed phases**

Hammerschmidt, T. and Ladines, A.N. and Koßmann, J. and Drautz, R.*Crystals*6 (2016)The moments of the electronic density-of-states provide a robust and transparent means for the characterization of crystal structures. Using d-valent canonical tight-binding, we compute the moments of the crystal structures of topologically close-packed (TCP) phases as obtained from density-functional theory (DFT) calculations. We apply the moments to establish a measure for the difference between two crystal structures and to characterize volume changes and internal relaxations. The second moment provides access to volume variations of the unit cell and of the atomic coordination polyhedra. Higher moments reveal changes in the longer-ranged coordination shells due to internal relaxations. Normalization of the higher moments leads to constant (A15,C15) or very similar (χ, C14, C36, µ, and σ) higher moments of the DFT-relaxed TCP phases across the 4d and 5d transition-metal series. The identification and analysis of internal relaxations is demonstrated for atomic-size differences in the V-Ta system and for different magnetic orderings in the C14-Fe2Nb Laves phase. © 2016 by the authors; licensee MDPI, Basel, Switzerland.view abstract 10.3390/cryst6020018 **Diffusion of solutes in fcc Cobalt investigated by diffusion couples and first principles kinetic Monte Carlo**

Neumeier, S. and Rehman, H.U. and Neuner, J. and Zenk, C.H. and Michel, S. and Schuwalow, S. and Rogal, J. and Drautz, R. and Göken, M.*Acta Materialia*106 (2016)The interdiffusivity of Al and the transition metal solutes Ti, V, Cr, Mn, Fe, Nb, Mo, Ru, Ta, W, and Re in fcc Co is characterized at 1373 K, 1473 K and 1573 K by binary diffusion couples. The experimental results are complemented by first-principles calculations in combination with kinetic Monte Carlo simulations to investigate the diffusion of Re, W, Mo and Ta in fcc Co. The interdiffusion coefficients of alloying elements in fcc Co are generally smaller than in fcc Ni, but the correlation between interdiffusion coefficients and the atomic number of metal solutes is comparable in Co and Ni. With increasing atomic number and decreasing atomic radii the interdiffusion coefficients of the investigated elements, except for Mn and Fe, decrease strongly. The trends in the diffusivity determined by experiment and simulation are in excellent agreement. Re is the slowest diffusing element in fcc Co among the investigated elements. The electronic structure calculations indicate that this is caused by strong directional bonds between Re and neighboring Co atoms. The overall lower diffusivity of solute atoms in Co as compared to Ni suggests that diffusion controlled processes could be slower in Co-base superalloys. © 2016 Acta Materialia Inc. All rights reserved.view abstract 10.1016/j.actamat.2016.01.028 **Efficient parallelization of analytic bond-order potentials for large-scale atomistic simulations**

Teijeiro, C. and Hammerschmidt, T. and Drautz, R. and Sutmann, G.*Computer Physics Communications*204 (2016)Analytic bond-order potentials (BOPs) provide a way to compute atomistic properties with controllable accuracy. For large-scale computations of heterogeneous compounds at the atomistic level, both the computational efficiency and memory demand of BOP implementations have to be optimized. Since the evaluation of BOPs is a local operation within a finite environment, the parallelization concepts known from short-range interacting particle simulations can be applied to improve the performance of these simulations. In this work, several efficient parallelization methods for BOPs that use three-dimensional domain decomposition schemes are described. The schemes are implemented into the bond-order potential code BOPfox, and their performance is measured in a series of benchmarks. Systems of up to several millions of atoms are simulated on a high performance computing system, and parallel scaling is demonstrated for up to thousands of processors. © 2016 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cpc.2016.03.008 **First-principles study of carbon segregation in bcc iron symmetrical tilt grain boundaries**

Wang, J. and Janisch, R. and Madsen, G.K.H. and Drautz, R.*Acta Materialia*115 (2016)Segregation of light elements can profoundly affect the energies and cohesive properties of grain boundaries. First-principles calculations have been performed to determine the carbon solution energies and cohesive properties of three different grain boundaries in presence of carbon. It is demonstrated that the most stable segregation sites possess the greatest coordination number and maximum Fe-C nearest neighbor distance. Thereby a geometric criterion for predicting the segregation sites is suggested. Open grain boundary structures are shown to be more attractive to C atoms than the compact grain boundary structure, vacancies and dislocations, and C segregation at open grain boundaries decreases the grain boundary energy. The theoretical fracture strength of grain boundaries increases with C concentration and tend to similar values for certain areal concentrations irrespective of the grain boundary structures. This implies that the maximum fracture strength of a grain boundary depends on the maximum C areal concentration it can accommodate. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2016.04.058 **The thermal stability of topologically close-packed phases in the single-crystal Ni-base superalloy ERBO/1**

Lopez-Galilea, I. and Koßmann, J. and Kostka, A. and Drautz, R. and Mujica Roncery, L. and Hammerschmidt, T. and Huth, S. and Theisen, W.*Journal of Materials Science*51 (2016)In Ni-base superalloys, the addition of refractory elements such as Cr, Mo, Co, W, and Re is necessary to increase the creep resistance. Nevertheless, these elements induce the formation of different kinds of intermetallic phases, namely, the topologically close-packed (TCP) phases. This work focuses on intermetallic phases present in the second-generation single-crystal (SX) Ni-base superalloy ERBO/1. In the as-cast condition, the typical γ/γ′ structure is accompanied by undesirable intermetallic phases located in the interdendritic regions. The nature of these precipitates as well as their thermal stability between 800 and 1200 °C has been investigated by isothermal heat treatments. The investigation techniques include DSC, SEM, EDX, and TEM. The experimental information is complemented by (1) comparison with a structure map to link the local chemical composition with phase stability, as well as (2) thermodynamic calculations based on the CALPHAD method to determine the occurrence and composition of phases during solidification and in equilibrium conditions. The TCP phases Laves, µ and σ were identified in various temperature/time ranges. © 2015, Springer Science+Business Media New York.view abstract 10.1007/s10853-015-9579-7 **Thermodynamic and kinetic solid-liquid interface properties from transition path sampling**

Şopu, D. and Rogal, J. and Drautz, R.*Journal of Chemical Physics*145 (2016)We perform transition path sampling simulations to determine two of the key quantities in solidification, the solid-liquid interface energy and velocity, in a Lennard-Jones system. Our approach is applicable to a wide range of temperature and pressure conditions, at the melting temperature and out-of-equilibrium. We show that small system sizes are sufficient for good values of interface energies and velocities. The transition path sampling method thus offers an attractive and robust alternative for the evaluation of solid-liquid interface properties. © 2016 Author(s).view abstract 10.1063/1.4972583 **Three-Parameter Crystal-Structure Prediction for sp-d-Valent Compounds**

Bialon, A.F. and Hammerschmidt, T. and Drautz, R.*Chemistry of Materials*28 (2016)We present a three-dimensional structure-map based on experimental data for compounds that contain sp-block elements and transition metals. The map predicts the correct crystal structure with a probability of 86% and has a confidence of better than 98% that the correct crystal structure is among three predicted crystal structures. The three descriptors of the structure map are physically intuitive functions of the number of valence electrons, atomic volume, and electronegativity of the constituent elements. We test the structure map against standard density-functional theory calculations for 1:1 sp-d-valent compounds and show that our three-parameter model has a comparable predictive power. We demonstrate the application of the structure map in conjunction with density-functional theory calculations. © 2016 American Chemical Society.view abstract 10.1021/acs.chemmater.5b04299 **Unraveling the composition dependence of the martensitic transformation temperature: A first-principles study of Ti-Ta alloys**

Chakraborty, T. and Rogal, J. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*94 (2016)The martensitic start temperature Ms is one of the key characteristics of shape memory materials. High-temperature shape memory alloys are a special class of materials where transformation temperatures between the martensite and austenite phase above 373 K are desirable. For the design of new high-temperature shape memory alloys it is therefore important to understand and predict the dependence of Ms on the composition of the material. Using density functional theory in combination with the quasiharmonic Debye model, we evaluate the different contributions to the free energy to determine the transition temperature T0 over a wide range of compositions in Ti-Ta alloys. Our approach provides physical insight into the various contributions that explain the strong composition dependence of Ms that is observed experimentally. Based on our calculations, we identify the relative phase stability at T=0 K and the vibrational entropy difference between the involved phases as critical parameters to predict changes in T0. We propose a simple, one-dimensional descriptor to estimate the transition temperature that can be used in the identification of new alloys suitable for high-temperature shape memory applications. © 2016 American Physical Society.view abstract 10.1103/PhysRevB.94.224104 **Bond-order potentials: Derivation and parameterization for refractory elements**

Drautz, R. and Hammerschmidt, T. and Čák, M. and Pettifor, D.G.*Modelling and Simulation in Materials Science and Engineering*23 (2015)The bond-order potentials are derived from density functional theory by a systematic coarse graining of the electronic structure. Within their functional form the bond-order potentials comprise covalent bond formation, charge transfer and magnetism. We review the derivation of the bond-order potentials from density functional theory and discuss their application to the simulation of refractory transition metals. We show that the derived functional form of the bond-order potentials ensures the transferability of the potentials to atomic environments that have not been taken into account in the parameterization. © 2015 IOP Publishing Ltd.view abstract 10.1088/0965-0393/23/7/074004 **Martensitic transformation between competing phases in Ti-Ta alloys: A solid-state nudged elastic band study**

Chakraborty, T. and Rogal, J. and Drautz, R.*Journal of Physics Condensed Matter*27 (2015)A combined density functional theory and solid-state nudged elastic band study is presented to investigate the martensitic transformation between β → (α″, ω) phases in the Ti-Ta system. The minimum energy paths along the transformation are calculated and the transformation mechanisms as well as relative stabilities of the different phases are discussed for various compositions. The analysis of the transformation paths is complemented by calculations of phonon spectra to determine the dynamical stability of the β, α ″, and ω phase. Our theoretical results confirm the experimental findings that with increasing Ta concentration there is a competition between the destabilisation of the α ″ and ω phase and the stabilisation of the high-temperature β phase. © 2015 IOP Publishing Ltd.view abstract 10.1088/0953-8984/27/11/115401 **Microsegregation and precipitates of an as-cast Co-based superalloy—microstructural characterization and phase stability modelling**

Koßmann, J. and Zenk, C.H. and Lopez-Galilea, I. and Neumeier, S. and Kostka, A. and Huth, S. and Theisen, W. and Göken, M. and Drautz, R. and Hammerschmidt, T.*Journal of Materials Science*50 (2015)The demand for increased efficiency of industrial gas turbines and aero engines drives the search for the next generation of materials. Promising candidates for such new materials are Co-based superalloys. We characterize the microsegregation and solidification of a multi-component Co-based superalloy and compare it to a ternary Co–Al–W compound and to two exemplary Ni-based superalloys by combining the experimental characterization of the as-cast microstructures with complementary modelling of phase stability. On the experimental side, we characterize the microstructure and precipitates by electron microscopy and energy-dispersive X-ray spectroscopy and determine the element distributions and microsegregation coefficients by electron probe microanalysis (EPMA). On the modelling side, we carry out solidification simulations and a structure map analysis in order to relate the local chemical composition with phase stability. We find that the microsegregation coefficients for the individual elements are very similar in the investigated Co-based and Ni-based superalloys. By interpreting the local chemical composition from EPMA with the structure map, we effectively unite the set of element distribution maps to compound maps with very good contrast of the dendritic microstructure. The resulting compound maps of the microstructure in terms of average band filling and atomic-size difference explain the formation of topologically close-packed phases in the interdendritic regions. We identify B2, C14, and D0<inf>24</inf> precipitates with chemical compositions that are in line with the structure map. © 2015, Springer Science+Business Media New York.view abstract 10.1007/s10853-015-9177-8 **Non-collinear magnetism with analytic Bond-Order Potentials**

Ford, M.E. and Pettifor, D.G. and Drautz, R.*Journal of Physics Condensed Matter*27 (2015)The theory of analytic Bond-Order Potentials as applied to non-collinear magnetic structures of transition metals is extended to take into account explicit rotations of Hamiltonian and local moment matrix elements between locally and globally defined spin-coordinate systems. Expressions for the gradients of the energy with respect to the Hamiltonian matrix elements, the interatomic forces and the magnetic torques are derived. The method is applied to simulations of the rotation of magnetic moments in α iron, as well as α and β manganese, based on d-valent orthogonal tight-binding parametrizations of the electronic structure. A new weighted-average terminator is introduced to improve the convergence of the Bond-Order Potential energies and torques with respect to tight-binding reference values, although the general behavior is qualitatively correct for low-moment expansions. © 2015 IOP Publishing Ltd.view abstract 10.1088/0953-8984/27/8/086002 **On the effect of alloy composition on martensite start temperatures and latent heats in Ni-Ti-based shape memory alloys**

Frenzel, J. and Wieczorek, A. and Opahle, I. and Maaß, B. and Drautz, R. and Eggeler, G.*Acta Materialia*90 (2015)In the present work we explain the concentration dependence of the martensite start temperature (MS) in Ni-Ti-based shape memory alloys (SMAs). We briefly review the present level of understanding and show that there is a need for further work. We then investigate the strong dependence of MS on alloy composition in binary Ni-Ti, ternary Ni-Ti-X (X = Cr, Cu, Hf, Pd, V, Zr) and quaternary Ni-Ti-Cu-Y (Y = Co, Pd) SMAs. For binary Ni-Ti, we combine differential scanning calorimetry experiments with insight gained through the application of the density functional theory (DFT) to show that heats of transformation ΔH decrease as Ni concentrations increase from 50.0 to 51.2 at.%. This causes a shift in the Gibbs free energy curves of austenite GA(T) and martensite GM(T), which in turn results in a lower MS temperature. Our DFT results suggest that the strong decrease of ΔH is caused by a stabilization of the B2 phase by structural relaxations around Ni antisite atoms, together with a gradual destabilization of B19′. The martensite start temperatures and the latent heats of transformation for binary, ternary and quaternary Ni-Ti-based SMAs are closely related. We observe smaller latent heats when the geometrical differences between the crystal structures of austenite and martensite decrease. © 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2015.02.029 **On the role of Re in the stress and temperature dependence of creep of Ni-base single crystal superalloys**

Wollgramm, P. and Buck, H. and Neuking, K. and Parsa, A.B. and Schuwalow, S. and Rogal, J. and Drautz, R. and Eggeler, G.*Materials Science and Engineering A*628 (2015)In the present study we investigate the creep behavior of a Ni-base single crystal superalloy. We evaluate the stress and temperature dependence of the minimum creep rate, which shows a power law type of stress dependence (characterized by a stress exponent n) and an exponential type of temperature dependence (characterized by an apparent activation energy Qapp). Under conditions of high temperature (1323K) and low stress (160MPa) creep, n and Qapp are determined as 5.3 and 529kJ/mol, respectively. For lower temperatures (1123K) and higher stresses (600MPa) the stress exponent n is higher (8.5) while the apparent activation energy of creep is lower (382kJ/mol). We show that there is a general trend: stress exponents n increase with increasing stress and decreasing temperature, while higher apparent activation energies are observed for lower stresses and higher temperatures. We use density functional theory (DFT) to calculate the activation energy of diffusion for Re in a binary Ni-Re alloy with low Re-concentrations. The resulting energy is almost a factor 2 smaller than the apparent activation energy of creep. We explain why it is not straightforward to rationalize the temperature dependence of creep merely on the basis of the diffusion of one alloying element. We show that the evolution of the microstructure also must be considered. © 2015 Elsevier B.V.view abstract 10.1016/j.msea.2015.01.010 **Reduced tight-binding models for elemental Si and N, and ordered binary Si-N systems**

Gehrmann, J. and Pettifor, D.G. and Kolmogorov, A.N. and Reese, M. and Mrovec, M. and Elsässer, C. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*91 (2015)Silicon nitride is a bulk and a coating material exhibiting excellent mechanical properties. We present a transferable reduced tight-binding (TB) model for the silicon nitride system, developed within the framework of coarse graining the electronic structure from density-functional theory (DFT) to tight binding (TB) to bond-order potentials (BOPs). The TB bond integrals are obtained directly from mixed-basis DFT projections of wave functions onto a minimal basis of atom-centered orbitals. This approach reduces the number of overall parameters to be fitted. Furthermore, applying the reduced TB approximation automatically leads to a single σ bond order that contributes to the bond energy. DFT binding energies of ground state and metastable crystal structures are used as the benchmark to which the TB repulsive parameters are fitted. The quality of the TB models is demonstrated by comparing their predictions for the binding energies, heats of formation, elastic constants, and defect energies with DFT and experimental values. © 2015 American Physical Society.view abstract 10.1103/PhysRevB.91.054109 **Solubility and ordering of Ti, Ta, Mo and W on the Al sublattice in L1**2 -Co3 Al

Koßmann, J. and Hammerschmidt, T. and Maisel, S. and Müller, S. and Drautz, R.*Intermetallics*64 (2015)Co-Al-W-based alloys are promising new materials for high-temperature applications. They owe their high-temperature strength to hardening by ternary L1<inf>2</inf>-Co<inf>3</inf>(Al<inf>1-x</inf>W<inf>x</inf>) precipitates, which may form even though binary Co<inf>3</inf>Al is not stable. In the current work, density functional theory calculations are performed to study the solubility and ordering of the transition metals W, Mo, Ti, and Ta at the Al sublattice in L1<inf>2</inf>-Co<inf>3</inf>Al. The sublattice disorder is modelled with a newly parametrised cluster expansion and compared to results using special quasi-random structures. Our results for W and Mo show that the mixing energy exhibits a minimum at approximately x = 0.7. However, the computed small values of the mixing energies indicate that W and Mo atoms are fully disordered with the Al atoms already at low temperatures. For Ti and Ta we find no sizeable driving force for ordering with the Al atoms. The computed solubilities on the Al sublattice obtained are in the range of 40-80 meV/atom for W and Mo and less than 25 meV/atom for Ti and Ta. © 2015 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.intermet.2015.04.009 **Structural stability of Fe-based topologically close-packed phases**

Ladines, A.N. and Hammerschmidt, T. and Drautz, R.*Intermetallics*59 (2015)Precipitates of topologically close-packed (TCP) phases play an important role in hardening mechanisms of high-performance steels. We analyze the influence of atomic size, electron count, magnetism and external stress on TCP phase stability in Fe-based binary transition metal alloys. Our density-functional theory calculations of structural stability are complemented by an analysis with an empirical structure map for TCP phases. The structural stability and lattice parameters of the Fe-Nb/Mo/V compounds are in good agreement with experiment. The average magnetic moments follow the Slater-Pauling relation to the average number of valence-electrons and can be rationalized in terms of the electronic density of states. The stabilizing effect of the magnetic energy, estimated by additional non-magnetic calculations, increases as the magnetic moment increases with band filling for the binary systems of Fe and early transition metals. For the case of Fe2Nb, we demonstrate that the influence of magnetism and external stress is sufficiently large to alter the energetic ordering of the closely competing Laves phases C14, C15 and C36. We find that the A15 phase is not stabilized by atomic-size differences, while the stability of C14 is increasing with increasing difference in atomic size. © 2014 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.intermet.2014.12.009 **Structural transformations among austenite, ferrite and cementite in Fe-C alloys: A unified theory based on ab initio simulations**

Zhang, X. and Hickel, T. and Rogal, J. and Fähler, S. and Drautz, R. and Neugebauer, J.*Acta Materialia*99 (2015)Structural transformations in Fe-C alloys are decisive for the mechanical properties of steels, but their modeling remains a challenge due to the simultaneous changes in Fe lattice and redistribution of C. With a combination of the orientation relationships between austenite, ferrite and cementite, we identify a metastable intermediate structure (MIS), which can serve as a link between the three phases. Based on this framework, different mechanisms depending on the local conditions (C concentration, strain, magnetism) are revealed from ab initio nudged elastic band simulations, which allow us to construct a unified theory for the structural transformations among austenite, ferrite and cementite. © 2015 Acta Materialia Inc. All rights reserved.view abstract 10.1016/j.actamat.2015.07.075 **Analytic bond-order potentials for the bcc refractory metals Nb, Ta, Mo and W**

Čák, M. and Hammerschmidt, T. and Rogal, J. and Vitek, V. and Drautz, R.*Journal of Physics Condensed Matter*26 (2014)Bond-order potentials (BOPs) are based on the tight-binding approximation for determining the energy of a system of interacting atoms. The bond energy and forces are computed analytically within the formalism of the analytic BOPs. Here we present parametrizations of the analytic BOPs for the bcc refractory metals Nb, Ta, Mo and W. The parametrizations are optimized for the equilibrium bcc structure and tested for atomic environments far from equilibrium that had not been included in the fitting procedure. These tests include structural energy differences for competing crystal structures; tetragonal, trigonal, hexagonal and orthorhombic deformation paths; formation energies of point defects as well as phonon dispersion relations. Our tests show good agreement with available experimental and theoretical data. In practice, we obtain the energetic ordering of vacancy, [1 1 1], [1 1 0], and [1 0 0] self-interstitial atom in agreement with density functional theory calculations. © 2014 IOP Publishing Ltd.view abstract 10.1088/0953-8984/26/19/195501 **Applications of scale-bridging to computational materials design**

Drautz, R. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*22 (2014)view abstract 10.1088/0965-0393/22/3/030201 **Convergence of an analytic bond-order potential for collinear magnetism in Fe**

Ford, M.E. and Drautz, R. and Hammerschmidt, T. and Pettifor, D.G.*Modelling and Simulation in Materials Science and Engineering*22 (2014)Analytic bond-order potentials (BOPs) for magnetic transition metals are applied for pure iron as described by an orthogonal d-valent tight-binding (TB) model. Explicit analytic equations for the gradients of the binding energy with respect to the Hamiltonian on-site levels are presented, and are then used to minimize the energy with respect to the magnetic moments, which is equivalent to a TB self-consistency scheme. These gradients are also used to calculate the exact forces, consistent with the energy, necessary for efficient relaxations and molecular dynamics. The Jackson kernel is used to remove unphysical negative densities of states, and approximations for the asymptotic recursion coefficients are examined. BOP, TB and density functional theory results are compared for a range of bulk and defect magnetic structures. The BOP energies and magnetic moments for bulk structures are shown to converge with increasing numbers of moments, with nine moments sufficient for a quantitative comparison of structural energy differences. The formation energies of simple defects such as the monovacancies and divacancies also converge rapidly. Other physical quantities, such as the position of the high-spin to low-spin transition in ferromagnetic fcc (face centred cubic) iron, surface peaks in the local density of states, the elastic constants and the formation energies of the self-interstitial atom defects, require higher moments for convergence. © 2014 IOP Publishing Ltd.view abstract 10.1088/0965-0393/22/3/034005 **First-principles-based phase diagrams and thermodynamic properties of TCP phases in Re-X systems (X = Ta, V, W)**

Palumbo, M. and Fries, S.G. and Hammerschmidt, T. and Abe, T. and Crivello, J.-C. and Breidi, A.A.H. and Joubert, J.-M. and Drautz, R.*Computational Materials Science*81 (2014)The structural stability of topologically close-packed phases in binary transition metal alloys is investigated with a combination of first-principles calculations based on density-functional theory and the Bragg-Williams-Gorsky approximation for the description of the configurational entropy. For a variety of different (i) exchange-correlation functionals, (ii) pseudopotentials, and (iii) relaxation schemes, for the relevant phases in Re-X (X = Ta, V, W) binary systems, we compare the energy of formation at T = 0 K, as well as the phase diagrams and site occupancies at finite temperatures. We confirm previous findings that the configurational entropy plays a stabilising role for complex phases in these systems at elevated temperatures. Small differences in the calculated energy of formation for different exchange-correlation functionals, pseudopotentials and relaxation schemes are expected, but give rise to qualitatively different phase diagrams. We employ these differences in order to estimate the order of magnitude of the standard deviation necessary in the qualitatively-reliable calculation of phase diagrams and site occupancies. In an attempt to determine the accuracy that is required to assure a qualitatively correct prediction of phase diagrams, we modify our first-principles results numerically by random variations with the determined standard deviation as maximum amplitude. Taking the order of site occupancies and the set of stable phases as simple criteria for a qualitatively correct prediction, we find that the accuracy required for the energy of formation of the individual configurations in these systems is approximately 5 meV/atom (≈0.5 kJ/mol at). © 2013 Elsevier B.V. All rights reserved.view abstract 10.1016/j.commatsci.2013.08.051 **Magnetic analytic bond-order potential for modeling the different phases of Mn at zero Kelvin**

Drain, J.F. and Drautz, R. and Pettifor, D.G.*Physical Review B - Condensed Matter and Materials Physics*89 (2014)It is known that while group VII 4d Tc and 5d Re have hexagonally close-packed (hcp) ground states, 3d Mn adopts a complex χ-phase ground state, exhibiting complex noncollinear magnetic ordering. Density functional theory (DFT) calculations have shown that without magnetism, the χ phase is still the ground state of Mn implying that magnetism and the resultant atomic-size difference between large- and small-moment atoms are not the critical factors, as is commonly believed, in driving the anomalous stability of the χ phase over hcp. Using a canonical tight-binding (TB) model, it is found that for a more than half-filled d band, while harder potentials stabilize close-packed hcp, a softer potential stabilizes the more open χ phase. By analogy with the structural trend from open to close-packed phases down the group IV elements, the anomalous stability of the χ phase in Mn is shown to be due to 3d valent Mn lacking d states in the core which leads to an effectively softer atomic repulsion between the atoms than in 4d Tc and 5d Re. Subsequently, an analytic bond-order potential (BOP) is developed to investigate the structural and magnetic properties of elemental Mn at 0 K. It is derived within BOP theory directly from a new short-ranged orthogonal d-valent TB model of Mn, the parameters of which are fitted to reproduce the DFT binding energy curves of the four experimentally observed phases of Mn, namely, α, β, γ, δ, and ε-Mn. Not only does the BOP reproduce qualitatively the DFT binding energy curves of the five different structure types, it also predicts the complex collinear antiferromagnetic (AFM) ordering in α-Mn, the ferrimagnetic ordering in β-Mn, and the AFM ordering in γ-, δ-, and ε-Mn that are found by DFT. A BOP expansion including 14 moments is sufficiently converged to reproduce most of the properties of the TB model with the exception of the elastic shear constants, which require further moments. The current TB model, however, predicts values of the shear moduli and the vacancy formation energies that are approximately a factor of 2 too small, so that a future more realistic model for MD simulations will require these properties to be included from the outset in the fitting database. © 2014 American Physical Society.view abstract 10.1103/PhysRevB.89.134102 **Parameterized electronic description of carbon cohesion in iron grain boundaries**

Hatcher, N. and Madsen, G.K.H. and Drautz, R.*Journal of Physics Condensed Matter*26 (2014)We employ a recently developed iron-carbon orthogonal tight-binding model in calculations of carbon in iron grain boundaries. We use the model to evaluate the properties of carbon near and on the Σ5 (3 1 0)[0 0 1] symmetric tilt grain boundary (GB) in iron, and calculations show that a carbon atom lowers the GB energy by 0.29 eV/atom in accordance with DFT. Carbon segregation to the GB is analyzed, and we find an energy barrier of 0.92 eV for carbon to segregate to the carbon-free interface while segregation to a fully filled interface is disfavored. Local volume (via Voronoi tessellation), magnetic, and electronic effects are correlated with atomic energy changes, and we isolate two different mechanisms governing carbon's behavior in iron: a volumetric strain which increases the energy of carbon in interstitial α iron and a non-strained local bonding which stabilizes carbon at the GB. © 2014 IOP Publishing Ltd.view abstract 10.1088/0953-8984/26/14/145502 **Vacancy mobility and interaction with transition metal solutes in Ni**

Schuwalow, S. and Rogal, J. and Drautz, R.*Journal of Physics Condensed Matter*26 (2014)Interaction of Re, Ta, W and Mo solutes with vacancies and their diffusion in fcc Ni is investigated by density-functional theory in combination with kinetic Monte Carlo simulations. Interaction energies are calculated for the first six neighbor shells around the solutes and a complete set of diffusion barriers for these shells is provided. Further, diffusion coefficients for the four elements in Ni as well as for vacancies in the presence of these elements are calculated. The calculated solute diffusion coefficients based on our ab initio data are found to compare favorably to experimental values. The mobility of the vacancies as a key factor in dislocation climb is only minimally influenced by the solute atoms within the dilute limit. © 2014 IOP Publishing Ltd.view abstract 10.1088/0953-8984/26/48/485014 **Ab initio study of boron in α-iron: Migration barriers and interaction with point defects**

Bialon, A.F. and Hammerschmidt, T. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*87 (2013)Boron is a common alloying element in modern steels with a significant influence on the mechanical properties already at concentrations of only a few parts per million. The effect of boron depends on its distribution in the microstructure. Here, we characterize the elemental factors that determine the boron distribution in α-iron by density functional theory calculations. Boron as point defect has been considered in substitutional and interstitial sites. The calculated migration barriers for the substitutional and interstitial mechanisms show the first nearest-neighbor hops being preferred over second nearest-neighbor hops. A dissociative mechanism shows boron migrating via an interstitial mechanism to be likely trapped by vacancies. In order to characterize the interaction with other point defects, we determined the distance-dependent interaction energy of a boron defect with a vacancy, a second boron, and with hydrogen, carbon, nitrogen, oxygen, aluminum, silicon, phosphorus, and sulfur atoms. We find that substitutional boron binds strongly to interstitial point defects with dumbbell formation and weaker to substitutional point defects. Interstitial boron tends to repel substitutional and interstitial point defects. We find a similarity of substitutional boron and vacancies regarding their influence on elastic properties and their interaction with point defects in α-iron. © 2013 American Physical Society.view abstract 10.1103/PhysRevB.87.104109 **Analytic bond-order potential expansion of recursion-based methods**

Seiser, B. and Pettifor, D.G. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*87 (2013)We show that the analytic bond-order potentials (BOPs) may be used to reproduce the density of states and energy of recursion-based methods for close-packed atomic configurations. In this way, we demonstrate that the analytic BOPs can efficiently recast the numerical bond-order potentials in a polynomial approximation. By introducing damping factors for the expansion coefficients in analogy to the kernel polynomial method, negative regions of the density of states are removed such that the analytic BOPs may be applied also to open systems with band gaps. By estimating higher moments from the termination of the Lanczos recursion chain, we, furthermore, achieve a faster convergence than the usual kernel polynomial method at a negligible additional computational cost. © 2013 American Physical Society.view abstract 10.1103/PhysRevB.87.094105 **Comparison of analytic and numerical bond-order potentials for W and Mo**

Čák, M. and Hammerschmidt, T. and Drautz, R.*Journal of Physics Condensed Matter*25 (2013)Bond-order potentials (BOPs) are derived from the tight-binding approximation and provide a linearly-scaling computation of the energy and forces for a system of interacting atoms. While the numerical BOPs involve the numerical integration of the response (Green's) function, the expressions for the energy and interatomic forces are analytical within the formalism of the analytic BOPs. In this paper we present a detailed comparison of numerical and analytic BOPs. We use established parametrizations for the bcc refractory metals W and Mo and test structural energy differences; tetragonal, trigonal, hexagonal and orthorhombic deformation paths; formation energies of point defects as well as phonon dispersion relations. We find that the numerical and analytic BOPs generally are in very good agreement for the calculation of energies. Different from the numerical BOPs, the forces in the analytic BOPs correspond exactly to the negative gradients of the energy. This makes it possible to use the analytic BOPs in dynamical simulations and leads to improved predictions of defect energies and phonons as compared to the numerical BOPs. © 2013 IOP Publishing Ltd.view abstract 10.1088/0953-8984/25/26/265002 **Environmental tight-binding modeling of nickel and cobalt clusters**

McEniry, E.J. and Drautz, R. and Madsen, G.K.H.*Journal of Physics Condensed Matter*25 (2013)Tight-binding models derived from density functional theory potentially provide a systematic approach to the development of accurate and transferable models of multicomponent systems. We introduce a systematic methodology for environmental tight binding in which both the overlap and environmental contributions to the electronic structure are included. The parameters of the model are determined directly from ab initio considerations, thus providing a formal conceptual link to density functional approaches. In order to test the validity of the approach, the model is applied to small clusters of Ni and Co, whose electronic structure is largely determined by the interplay of tightly bound d-valent states and the disperse s-states. We numerically illustrate that it is essential to include environmental contributions in the tight-binding approach in order to reliably reproduce the electronic structure of such clusters. © 2013 IOP Publishing Ltd.view abstract 10.1088/0953-8984/25/11/115502 **High-throughput ab initio screening of binary solid solutions in olivine phosphates for Li-ion battery cathodes**

Hajiyani, H.R. and Preiss, U. and Drautz, R. and Hammerschmidt, T.*Modelling and Simulation in Materials Science and Engineering*21 (2013)A promising approach to improving the performance of iron-phosphate FePO4 cathode materials for Li-ion batteries is to partly or fully substitute Fe with other metals. Here, we use high-throughput density-functional theory (DFT) calculations to investigate binary mixtures of metal atoms M and M′ in (Li)MyM'1-yPO4 olivine phosphates. We determine the formation energy for various stoichiometries of different binary combinations of metals for the cases of full lithiation and delithiation. Systematic screening of all combinations of Fe and Mn with elements of the 3d transition-metal (TM) series allows us to identify trends with average band filling and atomic size. We also included compounds that verify the observed relations or that were discussed as cathode materials, particularly Ni-Co, V-Cu and V-Ni, as well as combinations with 4d TMs (Fe-Zr, Fe-Mo, Fe-Ag) and with Mg (Fe-Mg and Ni-Mg). Based on our DFT calculations for each compound, we estimate the volume change during intercalation, the intercalation voltage, the energy density and the thermal stability with respect to reaction with oxygen. Our calculations indicate that the energy density of the binary TM phosphates increases with average band filling while the thermal stability of the compounds decreases. © 2013 IOP Publishing Ltd.view abstract 10.1088/0965-0393/21/7/074004 **High-throughput study of the structural stability and thermoelectric properties of transition metal silicides**

Opahle, I. and Parma, A. and McEniry, E.J. and Drautz, R. and Madsen, G.K.H.*New Journal of Physics*15 (2013)The phase stability, electronic structure and transport properties of binary 3d, 4d and 5d transition metal silicides are investigated using high-throughput density functional calculations. An overall good agreement is found between the calculated 0 K phase diagrams and experiment. We introduce descriptors for the phase-stability and thermoelectric properties and hereby identify several candidates with potential for thermoelectric applications. This includes known thermoelectrics like Mn4Si7, β-FeSi2, Ru2Si3 and CrSi2 as well as new potentially meta-stable materials like Rh3Si5, Fe2Si3 and an orthorhombic CrSi2 phase. Analysis of the electronic structure shows that the gap formation in most of the semiconducting transition metal silicides can be understood with simple hybridization models. The transport properties of the Mn4Si 7, Ru2Ge3 and Ir3Si5 structure types and the orthorhombic CrSi2 phase are discussed. The calculated transport properties are in good agreement with available experimental data. It is shown that a better thermoelectric performance may be achieved upon optimal doping. Finally, the high-throughput data are analysed and rationalized using a simple tight-binding model. © IOP Publishing and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/15/10/105010 **Modelling the lattice dynamics in SixGe1-x alloys**

Katre, A. and Drautz, R. and Madsen, G.K.H.*Journal of Physics Condensed Matter*25 (2013)The development of simplified models for the simulation of thermodynamic and thermal transport properties in random alloys is of great importance. In this paper we show how a simple second nearest neighbour model can reliably capture the lattice dynamics of SixGe1-x alloys. The model parameters are extracted from DFT-calculated force constant matrices for pure Si, pure Ge and the Si0.5Ge0.5 ordered alloy. We extract the nearest neighbour contributions directly from density functional theory, whereas effective interactions are obtained for the second nearest neighbour contributions. We demonstrate how the thermal properties, including the expansion coefficient, can be reliably reproduced and that the model is transferable to random SixGe1-x alloys. © 2013 IOP Publishing Ltd.view abstract 10.1088/0953-8984/25/36/365403 **Size versus electronic factors in transition metal carbide and TCP phase stability**

Pettifor, D.G. and Seiser, B. and Margine, E.R. and Kolmogorov, A.N. and Drautz, R.*Philosophical Magazine*93 (2013)The contributions of atomic size and electronic factors to the structural stability of transition metal carbides and topologically close-packed (TCP) phases are investigated. The hard-sphere model that has been used by Cottrell to rationalize the occurrence of the octahedral and trigonal local coordination polyhedra within the transition metal carbides is shown to have limitations in TiC since density functional theory (DFT) predicts that the second most metastable phase closest to the B1 (NaCl) ground state takes the Bk (BN) structure type with 5-atom local coordination polyhedra with very short Ti-C bond lengths. The importance of electronic factors in the TCP phases is demonstrated by DFT predictions that the A15, σ and Χ phases are stabilized between groups VI and VII of the elemental transition metals, whereas the μ and Laves phases are destabilized. The origin of this difference is related to the bimodal shape parameter of the electronic density of states by using the bond-order potential expansion of the structural energy within a canonical tight-binding model. The importance of the size factor in the TCP phases is illustrated by the DFT heats of formation for the binary systems Mo-Re, Mo-Ru, Nb-Re and Nb-Ru which show that the μ and Laves phases become more and more stable compared to A15, σ and Χ as the size factor increases from Mo-Re through to Nb-Ru. © 2013 Taylor & Francis.view abstract 10.1080/14786435.2013.771824 **Topological fingerprints for intermetallic compounds for the automated classification of atomistic simulation data**

Schablitzki, T. and Rogal, J. and Drautz, R.*Modelling and Simulation in Materials Science and Engineering*21 (2013)We introduce a method to determine intermetallic crystal phases by creating topological fingerprints using coordination polyhedra. Many intermetallic crystal phases have complex structures that cannot be determined from the information of their nearest neighbour environment alone, but need information from a further reaching local environment. We obtain the coordination polyhedra of each atom in the structure and use this information in a topological fingerprint to determine the crystal phases in the structure as locally as possible. This allows us to analyse complex crystal phases like the topologically close-packed phases and multi-phase structures. With the information extracted from the coordination polyhedra and topological fingerprint, it is also possible to find and identify point and extended defects. Therefore, our method is able to track interface regions in multi-phase structures, and follow structural changes during phase transformations. © 2013 IOP Publishing Ltd.view abstract 10.1088/0965-0393/21/7/075008 **Topologically close-packed phases in binary transition-metal compounds: Matching high-throughput ab initio calculations to an empirical structure map**

Hammerschmidt, T. and Bialon, A.F. and Pettifor, D.G. and Drautz, R.*New Journal of Physics*15 (2013)In steels and single-crystal superalloys the control of the formation of topologically close-packed (TCP) phases is critical for the performance of the material. The structural stability of TCP phases in multi-component transitionmetal alloys may be rationalized in terms of the average valence-electron count N and the composition-dependent relative volume-difference 1V/V. We elucidate the interplay of these factors by comparing density-functional theory calculations to an empirical structure map based on experimental data. In particular, we calculate the heat of formation for the TCP phases A15, C14, C15, C36, χ, μ and σ for all possible binary occupations of the Wyckoff positions. We discuss the isovalent systems V/Nb-Ta to highlight the role of atomic-size difference and observe the expected stabilization of C14/C15/C36/μ by 1V/V at 1N = 0 in V-Ta. In the systems V/Nb-Re, we focus on the well-known trend of A15! → σ → χ stability with increasing N and show that the influence of 1V/V is too weak to stabilize C14/C15/C36/μ in Nb-Re. As an example for a significant influence of both N and 1V/V, we also consider the systems Cr/Mo-Co. Here the sequence A15 → σ → χ is observed in both systems but in Mo-Co the large size-mismatch stabilizes C14/C15/C36/μ. We also include V/Nb-Co that cover the entire valence range of TCP stability and also show the stabilization of C14/C15/C36/μ. Moreover, the combination of a large volumedifference with a large mismatch in valence-electron count reduces the stability of the A15/σ/χ phases in Nb-Co as compared to V-Co. By comparison to nonmagnetic calculations we also find that magnetism is of minor importance for the structural stability of TCP phases in Cr/Mo-Co and in V/Nb-Co. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/15/11/115016 **A DFT study of formation energies of Fe-Zn-Al intermetallics and solutes**

Klaver, T.P.C. and Madsen, G.K.H. and Drautz, R.*Intermetallics*31 (2012)We report Density Functional Theory results on FeAl and FeZn intermetallics and Fe, Zn and Al solute atoms. The formation energies of fully relaxed intermetallic geometries were determined, as well as solution energies of the three elements in host lattices of the other two elements. Since it is know that the outcome of the magnetic states of some FeAl intermetallics and Fe solutes in Al depends on subtle details of how the calculations are carried out, we have determined many of our results with two different parameterisations, PBE and PBEsol, so see how the parameterisation influences the results. The relaxed intermetallic geometries are in good agreement with experimental results, with PBEsol calculations resulting in slightly smaller geometries than PBE calculations (0.7-2.1%). Intermetallic formation energies fall within ranges of experimental results where available, and are in excellent or reasonable agreement with other DFT results, except for the FeAl 2 phase. For this phase a structure revision was recently suggested and the heat of formation of the newly suggested structure is 0.1 eV/atom lower than for the long-accepted structure. The formation energies of Fe aluminides are an order of magnitude more negative than those of FeZn intermetallics. Most of the calculated magnetic states of the intermetallics are at odds with experimental results. However, the intermetallic formation energies are often not strongly affected by this. Fe/Al solute systems have the most negative solution energies. All other solution energies are positive and smaller in absolute value than the Fe/Al solution energies. Solution energies were all some tenths of eV. Where comparisons could be made, calculated and experimental results differed by some hundredths of eV. The magnetic moment found on an Fe solute in Al is at odds with experimental results. As with FeAl, the outcome of the magnetic state subtly depends on the details of how calculations were performed and has little energetic effect. Lattice relaxation around solute atoms is mostly in agreement with simple atomic size considerations. The slight relaxation of Al neighbours away from a Zn solute is at odds with this pattern, and also with experimental results. © 2012 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.intermet.2012.06.017 **Ab initio study of the interaction of H with substitutional solute atoms in α-Fe: Trends across the transition-metal series**

Psiachos, D. and Hammerschmidt, T. and Drautz, R.*Computational Materials Science*65 (2012)The extent of hydrogen embrittlement in steel depends strongly on the H distribution in the microstructure. Alloying elements might serve to detract hydrogen from regions prone to embrittlement and to distribute it within areas where it causes less damage. We present an ab initio study of the interaction of interstitial hydrogen in α-iron with substitutional transition-metal atoms as alloying elements. We find similar trends for the 3d, 4d, and 5d transition metal elements: the elements in the middle of the transition-metal series repel hydrogen while those on the sides tend to attract hydrogen. The trend is in line with the volume change that the transition-metal solute atom exerts on the iron lattice. The interaction energy decreases rapidly with separation distance with a range of approximately 5 . We use a simple parametrisation in order to estimate finite-size effects in the ab initio data. © 2012 Elsevier B.V. All rights reserved.view abstract 10.1016/j.commatsci.2012.06.020 **DFT-based tight-binding modeling of iron-carbon**

Hatcher, N. and Madsen, G.K.H. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*86 (2012)A coherent transferable tight-binding (TB) parametrization including magnetism has been developed for the Fe-C interaction. We use a downfolding procedure to obtain continuous and transferable Fe-C bond integrals from density functional theory. A TB model is constructed using these bond integrals and a parameterized interatomic repulsion based on simple exponentials fit to the excess energy of interstitial carbon in α, γ, and ε iron. An accurate description of the energy hierarchy of these structures can be achieved with only three fitting parameters. The model is used to calculate bulk properties and energies of a variety of Fe carbides, and good agreement is found for a number of properties. Excellent agreement with the individual elastic constants of cementite is achieved. We obtain a satisfactory migration energy barrier of carbon in α iron and confirm the instability of carbon in the tetrahedral interstitial site. Defect binding energies are calculated for a number of defects and largely agree with density functional theory calculations. This simple model based on physical insights may be used to study systems containing thousands of atoms. Furthermore, it may be employed as the basis for O(N) bond-order potentials. © 2012 American Physical Society.view abstract 10.1103/PhysRevB.86.155115 **Diffusion of hydrogen within idealized grains of bcc Fe: A kinetic Monte Carlo study**

Du, Y.A. and Rogal, J. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*86 (2012)Structural defects in materials such as vacancies, grain boundaries, and dislocations may trap hydrogen and a local accumulation of hydrogen at these defects can lead to the degradation of the materials properties. An important aspect in obtaining insight into hydrogen-induced embrittlement on the atomistic level is to understand the diffusion of hydrogen in these materials. In our study we employ kinetic Monte Carlo (kMC) simulations to investigate hydrogen diffusion in bcc iron within different microstructures. All input data to the kMC model, such as available sites, solution energies, and diffusion barriers, are obtained from first-principles calculations. We find that hydrogen mainly diffuses within the interface region with an overall diffusivity that is lower than in pure bcc Fe bulk. The concentration dependence of the diffusion coefficient is strongly nonlinear and the diffusion coefficient may even decrease with an increasing hydrogen concentration. To describe the macroscopic diffusion coefficient we derive an analytic expression as a function of hydrogen concentrations and temperatures which is in excellent agreement with our numerical results for idealized microstructures. © 2012 American Physical Society.view abstract 10.1103/PhysRevB.86.174110 **High throughput density functional investigations of the stability, electronic structure and thermoelectric properties of binary silicides**

Opahle, I. and Madsen, G.K.H. and Drautz, R.*Physical Chemistry Chemical Physics*14 (2012)The structural stabilities of binary Mg-X (X = Si, Ge, Sn) and 4d transition metal silicides Mo-Si and Ru-Si are investigated. The convex hulls of stable alloys are in overall good agreement with the known experimental phase diagrams. It is shown how the Si-rich Ru-Si structures have band gaps at the Fermi-level and how the Ru 2Si 3 structure is stabilized compared to the corresponding Fe 2Si 3 structure. We discuss the band structure of Ru 2Si 3 and show how the anisotropic band masses lead to favorable calculated thermoelectric properties. © 2012 the Owner Societies.view abstract 10.1039/c2cp41826f **Structural Stability of Topologically Close-Packed Phases: Understanding Experimental Trends in Terms of the Electronic Structure**

Hammerschmidt, T. and Seiser, B. and Cak, M. and Drautz, R. and Pettifor, D.G.*Proceedings of the International Symposium on Superalloys 2012*(2012)Topologically close-packed (TCP) phases in single crystal Ni-based superalloys have a detrimental effect on the mechanical properties. In order to gain a microscopic understanding of the factors that control TCP phase stability, we carry out atomistic calculations based on the electronic structure. In particular, we use a hierarchy of methods that treat the electronic structure at different levels of coarse-graining, i.e. at different levels of computational cost and accuracy. The applied levels of approximation range from density functional theory (DFT) to tight-binding (TB) to bond-order potentials (BOPs). This hierarchy of electronic structure methods allows us to interpret the findings of a recently derived structure map of experimentally observed TCP stability. The TB and BOP calculations are compared to extensive high-throughput DFT calculations for the TCP phases A15, C14, C15, C36, ì, ó, and X of transition-metal elements. These findings are extended to binary systems based on DFT heat-of-formations for TCP phases in the systems V/Nb-Ta, Nb/Mo-Ru, V/Cr/Nb/Mo-Re, V/Cr/Nb/Mo-Co. By pairwise comparisons of selected systems, we illustrate the interplay of the difference in average valenceelectron concentration N and the composition-dependent relative volume difference AV/V. Such an approach could be useful to predict the change of expected TCP phase stability due to changes of the composition for a given multi-component alloy. © 2012 The Minerals, Metals, & Materials Society. All rights reserved.view abstract 10.1002/9781118516430.ch15 **Ab initio study of the modification of elastic properties of α-iron by hydrostatic strain and by hydrogen interstitials**

Psiachos, D. and Hammerschmidt, T. and Drautz, R.*Acta Materialia*59 (2011)The effect of hydrostatic strain and of interstitial hydrogen on the elastic properties of α-iron is investigated using ab initio density-functional theory calculations. We find that the cubic elastic constants and the polycrystalline elastic moduli to a good approximation decrease linearly with increasing hydrogen concentration. This net strength reduction can be partitioned into a strengthening electronic effect which is overcome by a softening volumetric effect. The calculated hydrogen-dependent elastic constants are used to determine the polycrystalline elastic moduli and anisotropic shear moduli. For the key slip planes in α-iron, [11̄0] and [112̄], we find a shear modulus reduction of approximately 1.6% per at.% H. © 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2011.03.041 **Development of orthogonal tight-binding models for Ti-C and Ti-N systems**

Margine, E.R. and Kolmogorov, A.N. and Reese, M. and Mrovec, M. and Elsässer, C. and Meyer, B. and Drautz, R. and Pettifor, D.G.*Physical Review B - Condensed Matter and Materials Physics*84 (2011)We develop p-d orthogonal tight-binding (OTB) models for the description of TiCx and TiNx compounds in the 1.0>x>0.5 composition range. For the parametrization of bond integrals we use a recently developed method allowing projection of the one-electron wave functions obtained within the density functional theory onto optimized atom-centered orbitals. The performance of the OTB models is investigated for a wide range of properties: binding energy of elements and compounds, density of states, formation energy of vacancy-ordered defects, elastic constants, and phonon dispersions. The models provide a good description of the ground state properties at 1:1 composition and show a fair transferability for various atomic environments in elemental and binary phases. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.84.155120 **First-principles study on the interaction of H interstitials with grain boundaries in α- and γ-Fe**

Du, Y.A. and Ismer, L. and Rogal, J. and Hickel, T. and Neugebauer, J. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*84 (2011)The presence of hydrogen may weaken the bonding of iron atoms at grain boundaries, leading to intergranular embrittlement and thus failure of the bulk material. In this paper, we study the interaction of hydrogen interstitials with close-packed and open grain boundary structures in α- and γ-Fe using density-functional theory. We find that hydrogen accommodation within the grain boundaries strongly depends on the local coordination of the available interstitial sites. Within the open grain boundary structures larger interstitial sites are available, enhancing the solubility as compared to that in the respective bulk phases. The mobility of hydrogen within the investigated grain boundaries is low compared to diffusion in perfect single-crystalline bulk. The grain boundaries do not provide fast diffusion channels for hydrogen, but act as hydrogen traps. Hydrogen that is accumulated within the grain boundaries can lead to a lowering of the critical strain required to fracture the material. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.84.144121 **From electrons to materials**

Hammerschmidt, T. and Madsen, G.K.H. and Rogal, J. and Drautz, R.*Physica Status Solidi (B) Basic Research*248 (2011)In this article, we discuss how microstructural length and time scales may be reached in atomistic simulations. We bridge from electronic properties to properties of materials by employing a systematic coarse graining of the electronic structure to effective interatomic interactions. In combination with extended time scale simulations the elementary processes of microstructural evolution may then be described. We present our approach to the derivation of tight-binding models from density functional theory, the characterization of the interatomic interaction using bond-order potentials and extended time scale simulations based on adaptive kinetic Monte Carlo. Applications to structural stability in iron, internal interfaces in tungsten and hydrogen diffusion in iron are discussed briefly and relate our approach to Manfred Fähnle's work. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/pssb.201147121 **Optimized orthogonal tight-binding basis: Application to iron**

Madsen, G.K.H. and McEniry, E.J. and Drautz, R.*Physical Review B - Condensed Matter and Materials Physics*83 (2011)The formal link between the linear combination of atomic orbitals approach to density functional theory and two-center Slater-Koster tight-binding models is used to derive an orthogonal d-band tight-binding model for iron with only two fitting parameters. The resulting tight-binding model correctly predicts the energetic ordering of the low-energy iron phases, including the ferromagnetic bcc, antiferromagnetic fcc, hcp, and topologically close-packed structures. The energetics of test structures that were not included in the fit are equally well reproduced as those included, thus demonstrating the transferability of the model. The simple model also gives a good description of the vacancy formation energy in the nonmagnetic fcc and ferromagnetic bcc iron lattices. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.83.184119 **Possible routes for synthesis of new boron-rich Fe-B and Fe1-x Crx B4 compounds**

Bialon, A.F. and Hammerschmidt, T. and Drautz, R. and Shah, S. and Margine, E.R. and Kolmogorov, A.N.*Applied Physics Letters*98 (2011)We use ab initio calculations to examine thermodynamic factors that could promote the formation of recently proposed unique oP10-FeB4 and oP12-FeB2 compounds. We demonstrate that these compact boron-rich phases are stabilized further under pressure. We also show that chromium tetraboride is more stable in the new oP10 rather than the reported oI10 structure which opens up the possibility of realizing an oP10-(Fex Cr1-x) B4 pseudobinary material. In addition to exhibiting remarkable electronic features, oP10-FeB4 and oP12-FeB2 are expected to be harder than the known Fe-B compounds commonly used for hard coating applications. © 2011 American Institute of Physics.view abstract 10.1063/1.3556564 **Solubility of carbon in α-iron under volumetric strain and close to the Σ5(3 1 0)[0 0 1] grain boundary: Comparison of DFT and empirical potential methods**

Hristova, E. and Janisch, R. and Drautz, R. and Hartmaier, A.*Computational Materials Science*50 (2011)The solubility of carbon in α-Fe as a function of lattice strain and in the vicinity of the ∑5(310)[001] symmetrical tilt grain boundary is calculated with ab initio methods based on density-functional theory (DFT). The results are compared to four different empirical potentials: the embedded-atom method (EAM) potentials of Lau et al. [1], Ruda et al. [2] and Hepburn et al. [3], and the modified embedded-atom method (MEAM) potential of Lee [4]. The results confirm that the solubility of carbon in body-centered-cubic (bcc) Fe increases under local volume expansion and provide quantitative data for the excess enthalpy to be used in thermodynamic databases. According to our study the excess enthalpy obtained from DFT is more strain-sensitive than the ones obtained from the tested empirical potentials. The comparison of the applied methods furthermore reveals that among the empirical potentials the MEAM is most appropriate to describe the solubility of C in bcc Fe under strain. The differences between the four empirical potentials stem from different parameterizations of the EAM potentials and, in the case of the MEAM, from the altogether different formalism that also includes angular dependent terms in the binding energy. © 2010 Elsevier B.V. All rights reserved.view abstract 10.1016/j.commatsci.2010.11.006 **TCP phase predictions in Ni-based superalloys: Structure maps revisited**

Seiser, B. and Drautz, R. and Pettifor, D.G.*Acta Materialia*59 (2011)The traditional methods for predicting the occurrence of deleterious topologically close-packed (TCP) phases in Ni-based superalloys have been based on the PHACOMP and newPHACOMP methodologies. These schemes use the average number of holes Nh or the centre of gravity of the elemental d-bands Md to predict whether or not a given multicomponent alloy will be prone to TCP formation. However, as both these one-dimensional methodologies are well-known to fail with respect to new generations of alloys, a novel two-dimensional structure map (N,ΔV/V) is introduced where N is the average electron concentration and ΔV/V is a composition-dependent size-factor difference. This map is found to separate the experimental data on the TCP phases of binary A-B transition metal alloys into well-defined but sometimes overlapping regions corresponding to different structure types such as A15, σ, χ, R, P, δ, μ, M and Laves. Detailed investigations of ternary phase diagrams and multicomponent systems show that TCP phases, regardless of the number of constituents, are located in the same regions of the structure map that are favoured by the binary compounds of the same structure type. The structure map is then used in conjunction with CALPHAD computations of σ phase stability to show that the predictive power of newPHACOMP for the seven component Ni-Co-Cr-Ta-W-Re-Al system studied recently by Reed et al. [24] is indeed poor. This supports a growing consensus that robust methods of TCP phase prediction in multicomponent alloys will require the inclusion of reliable first-principles thermodynamic databases within the semi-empirical CALPHAD scheme. © 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2010.10.013 **Theory of structural trends within 4d and 5d transition metal topologically close-packed phases**

Seiser, B. and Hammerschmidt, T. and Kolmogorov, A.N. and Drautz, R. and Pettifor, D.G.*Physical Review B - Condensed Matter and Materials Physics*83 (2011)A combination of electronic-structure methodologies from density functional theory (DFT) through a tight-binding (TB) model to analytic bond-order potentials (BOPs) has been used to investigate structural trends within TCP phases, which we recently discussed using an empirical structure map. First, DFT is used to calculate the structural energy differences across the elemental 4d and 5d transition metal series and the heats of formation of the binary alloys Mo-Re, Mo-Ru, Nb-Re, and Nb-Ru, where we show that the valence electron concentration stabilizes A15, σ, and χ phases but destabilizes μ and Laves phases. Second, a one-parameter canonical d-band TB model in combination with the structural energy difference theorem is found to reproduce the observed elemental DFT structural trends. The structural energy difference theorem is also used to rationalize the influence of the relative size differences on the stability of μ and Laves phases in binary systems. Third, analytic BOP theory using the TB bond integrals as input is shown to converge to the TB structural energy difference curves as the number of moments in the BOP expansion is increased. In order to provide a simple interpretation of these structural energy difference curves in terms of analytic response functions and the differences in the moments of the density of states (DOS), an expression is used for the difference in the band energy that is correct to first order in the Fermi energy differences. We find that the fourth-moment contribution separates the A15, σ, and χ phases from the μ and Laves phases in agreement with the empirical structure map due to difference in the bimodality of the corresponding DOS caused mainly by distortions in their coordination polyhedra from ideal Frank-Kasper polyhedra. Finally, it is shown that at least six moments are needed to predict the structural trend A15→σ→χ. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.83.224116 **Tight-binding simulation of transition-metal alloys**

McEniry, E.J. and Madsen, G.K.H. and Drain, J.F. and Drautz, R.*Journal of Physics Condensed Matter*23 (2011)In order to perform atomistic simulations of steel, it is necessary to have a detailed understanding of the complex interatomic interactions in transition metals and their alloys. The tight-binding approximation provides a computationally efficient, yet accurate, method to investigate such interactions. In the present work, an orthogonal tight-binding model for Fe, Mn and Cr, with the explicit inclusion of magnetism, has been parameterized from ab initio density-functional calculations. © 2011 IOP Publishing Ltd.view abstract 10.1088/0953-8984/23/27/276004 **Valence-dependent analytic bond-order potential for magnetic transition metals**

Drautz, R. and Pettifor, D.G.*Physical Review B - Condensed Matter and Materials Physics*84 (2011)We extend the analytic bond-order potentials for transition metals to include ferro, antiferro, and noncollinear magnetism and charge transfer. This is achieved by first deriving a suitable tight-binding model through the expansion of the spin-density energy functional to second order with respect to magnetic and charge fluctuations. The tight-binding model is then approximated locally by the bond-order potential expansion, where the variational property of the bond-order potential expansion allows us to derive analytic expressions for the forces and torques on the atoms. From the bond-order potentials we then extract a hierarchy of multispin interactions beyond the conventional Heisenberg model. The explicit valence dependence of the bond-order potentials enables us to characterize the magnetic properties of the 3d transition metals and to reproduce the trend from antiferromagnetic spin ordering close to the center of the d band through noncollinear spin configurations to ferromagnetic ordering toward the edges of the d band. The analytic representation of the energy within the bond-order potentials is then further expanded in the form of a Ginzburg-Landau expansion, deriving the prefactors explicitly from tight-binding and bond-order potentials. Thus, in this paper we present a coherent simplification from fundamental to empirical models of magnetism through coarse graining the electronic structure from spin-density functional theory to tight binding to bond-order potentials to the Ginzburg-Landau expansion. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.84.214114 **New superconducting and semiconducting Fe-B compounds predicted with an Ab initio evolutionary search**

Kolmogorov, A.N. and Shah, S. and Margine, E.R. and Bialon, A.F. and Hammerschmidt, T. and Drautz, R.*Physical Review Letters*105 (2010)New candidate ground states at 14, 12, and 11 compositions are identified in the well-known Fe-B system via a combination of ab initio high-throughput and evolutionary searches. We show that the proposed oP12-FeB2 stabilizes by a break up of 2D boron layers into 1D chains while oP10-FeB 4 stabilizes by a distortion of a 3D boron network. The uniqueness of these configurations gives rise to a set of remarkable properties: oP12-FeB2 is expected to be the first semiconducting metal diboride and oP10-FeB4 is shown to have the potential for phonon-mediated superconductivity with a Tc of 15-20 K. © 2010 The American Physical Society.view abstract 10.1103/PhysRevLett.105.217003 **Theoretical investigation of the Pt3Al ground state**

Chauke, H.R. and Minisini, B. and Drautz, R. and Nguyen-Manh, D. and Ngoepe, P.E. and Pettifor, D.G.*Intermetallics*18 (2010)The deleterious low-temperature tetragonal phases in prototypical Pt-based superalloys have variously been reported as taking the tI16-U3Si (DOc), tI16-Ir3Si (DOc′) and tP16-Pt3Ga structure-types in contrast to the high-temperature cubic cP4-Cu3Au (L12) phase. We have investigated the relative stability of these four structure-types at absolute zero by using density functional theory. We find that the ground state of stoichiometric Pt3Al is tP16-Pt3Ga and that the other three lattices are mechanically unstable at absolute zero. Experiments are needed to measure the internal displacement parameters of these three competing tetragonal phases. © 2009 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.intermet.2009.08.016

#### Center for Interface-Dominated High Performance Materials (ZGH)

#### TRR 103: From Atoms to Turbine Blades – Scientific Framework for a New Generation of Single-Crystal Superalloys (Coordination: RUB)

#### FOR 1766: High Temperature Shape Memory Alloys (concluded)

#### SPP 1599: Caloric Effects in Ferroic Materials: New Concepts for Cooling (concluded)

#### Data Driven Materials Development (DDMD) - Datengetriebene Entwicklung neuer Materialien (concluded)

#### Ziel2: ICAMS Start-up Funding (concluded)

#### SFB 459: Shape Memory Technology (Coordination: RUB) (concluded)

#### alloys

#### atomistic simulations

#### bond-order potential

#### electronic structure

#### modelling and simulation

#### multiscale modelling

#### phase stability

#### steel

#### transition metals