#### Dr. Thomas Hammerschmidt

Atomistic Modelling and Simulation at ICAMS

Ruhr-Universität Bochum

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**Performant implementation of the atomic cluster expansion (PACE) and application to copper and silicon**

Lysogorskiy, Y. and Oord, C. and Bochkarev, A. and Menon, S. and Rinaldi, M. and Hammerschmidt, T. and Mrovec, M. and Thompson, A. and Csányi, G. and Ortner, C. and Drautz, R.*npj Computational Materials*7 (2021)The atomic cluster expansion is a general polynomial expansion of the atomic energy in multi-atom basis functions. Here we implement the atomic cluster expansion in the performant C++ code PACE that is suitable for use in large-scale atomistic simulations. We briefly review the atomic cluster expansion and give detailed expressions for energies and forces as well as efficient algorithms for their evaluation. We demonstrate that the atomic cluster expansion as implemented in PACE shifts a previously established Pareto front for machine learning interatomic potentials toward faster and more accurate calculations. Moreover, general purpose parameterizations are presented for copper and silicon and evaluated in detail. We show that the Cu and Si potentials significantly improve on the best available potentials for highly accurate large-scale atomistic simulations. © 2021, The Author(s).view abstract 10.1038/s41524-021-00559-9 **Tight-binding bond parameters for dimers across the periodic table from density-functional theory**

Jenke, J. and Ladines, A.N. and Hammerschmidt, T. and Pettifor, D.G. and Drautz, R.*Physical Review Materials*5 (2021)We obtain parameters for nonorthogonal and orthogonal tight-binding (TB) models from two-atomic molecules for all combinations of elements of period 1 to 6 and group 3 to 18 of the periodic table. The TB bond parameters for 1711 homoatomic and heteroatomic dimers show clear chemical trends. In particular, using our parameters we compare to the rectangular d-band model, the reduced sp TB model, as well as canonical TB models for sp- and d-valent systems, which have long been used to gain qualitative insight into the interatomic bond. The transferability of our dimer-based TB bond parameters to bulk systems is discussed exemplarily for the bulk ground-state structures of Mo and Si. Our dimer-based TB bond parameters provide a well-defined and promising starting point for developing refined TB parametrizations and for making the insight of TB available for guiding materials design across the periodic table. © 2021 American Physical Society.view abstract 10.1103/PhysRevMaterials.5.023801 **Understanding creep of a single-crystalline Co-Al-W-Ta superalloy by studying the deformation mechanism, segregation tendency and stacking fault energy**

Volz, N. and Xue, F. and Zenk, C.H. and Bezold, A. and Gabel, S. and Subramanyam, A.P.A. and Drautz, R. and Hammerschmidt, T. and Makineni, S.K. and Gault, B. and Göken, M. and Neumeier, S.*Acta Materialia*214 (2021)A systematic study of the compression creep properties of a single-crystalline Co-base superalloy (Co-9Al-7.5W-2Ta) was conducted at 950, 975 and 1000°C to reveal the influence of temperature and the resulting diffusion velocity of solutes like Al, W and Ta on the deformation mechanisms. Two creep rate minima are observed at all temperatures indicating that the deformation mechanisms causing these minima are quite similar. Atom-probe tomography analysis reveals elemental segregation to stacking faults, which had formed in the γ′ phase during creep. Density-functional-theory calculations indicate segregation of W and Ta to the stacking fault and an associated considerable reduction of the stacking fault energy. Since solutes diffuse faster at a higher temperature, segregation can take place more quickly. This results in a significantly faster softening of the alloy, since cutting of the γ′ precipitate phase by partial dislocations is facilitated through segregation already during the early stages of creep. This is confirmed by transmission electron microscopy analysis. Therefore, not only the smaller precipitate fraction at higher temperatures is responsible for the worse creep properties, but also faster diffusion-assisted shearing of the γ′ phase by partial dislocations. The understanding of these mechanisms will help in future alloy development by offering new design criteria. © 2021view abstract 10.1016/j.actamat.2021.117019 **Atomic scale configuration of planar defects in the Nb-rich C14 Laves phase NbFe2**

Šlapáková, M. and Zendegani, A. and Liebscher, C.H. and Hickel, T. and Neugebauer, J. and Hammerschmidt, T. and Ormeci, A. and Grin, J. and Dehm, G. and Kumar, K.S. and Stein, F.*Acta Materialia*183 (2020)Laves phases belong to the group of tetrahedrally close-packed intermetallic phases, and their crystal structure can be described by discrete layer arrangements. They often possess extended homogeneity ranges and the general notion is that deviations from stoichiometry are accommodated by anti-site atoms or vacancies. The present work shows that excess Nb atoms in a Nb-rich NbFe2 C14 Laves phase can also be incorporated in various types of planar defects. Aberration-corrected scanning transmission electron microscopy and density functional theory calculations are employed to characterize the atomic configuration of these defects and to establish stability criteria for them. The planar defects can be categorized as extended or confined ones. The extended defects lie parallel to the basal plane of the surrounding C14 Laves phase and are fully coherent. They contain the characteristic Zr4Al3-type (O) units found in the neighboring Nb6Fe7 µ phase. An analysis of the chemical bonding reveals that the local reduction of the charge transfer is a possible reason for the preference of this atomic arrangement. However, the overall layer stacking deviates from that of the perfect µ phase. The ab initio calculations establish why these exceptionally layered defects can be more stable configurations than coherent nano-precipitates of the perfect µ phase. The confined defects are observed with pyramidal and basal habit planes. The pyramidal defect is only ~1 nm thick and resembles the perfect µ phase. In contrast, the confined basal defect can be regarded as only one single O unit and it appears as if the stacking sequence is disrupted. This configuration is confirmed by ab initio calculations to be metastable. © 2019view abstract 10.1016/j.actamat.2019.11.004 **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 **Dft-cef approach for the thermodynamic properties and volume of stable and metastable al–ni compounds**

Tumminello, S. and Palumbo, M. and Koßmann, J. and Hammerschmidt, T. and Alonso, P.R. and Sommadossi, S. and Fries, S.G.*Metals*10 (2020)The Al–Ni system has been intensively studied both experimentally and theoretically. Previous first-principles calculations based on density-functional theory (DFT) typically investigate the stable phases of this system in their experimental stoichiometry. In this work, we present DFT calculations for the Al–Ni system that cover stable and metastable phases across the whole composition range for each phase. The considered metastable phases are relevant for applications as they are observed in engineering alloys based on Al–Ni. To model the Gibbs energies of solid phases of the Al–Ni system, we combine our DFT calculations with the compound energy formalism (CEF) that takes the Bragg–Williams–Gorsky approximation for the configurational entropy. Our results indicate that the majority of the investigated configurations have negative energy of formation with respect to Al fcc and Ni fcc. The calculated molar volumes for all investigated phases show negative deviations from Zen’s law. The thermodynamic properties at finite temperatures of individual phases allow one to predict the configurational contributions to the Gibbs energy. By applying a fully predictive approach without excess parameters, an acceptable topology of the DFT-based equilibrium phase diagram is obtained at low and intermediate temperatures. Further contributions can be added to improve the predictability of the method, such as phonons or going beyond the Bragg–Williams–Gorsky approximation that overestimates the stability range of the ordered phases. This is clearly demonstrated in the fcc order/disorder predicted metastable phase diagram. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.view abstract 10.3390/met10091142 **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 **Structural stability of Co–V intermetallic phases and thermodynamic description of the Co–V system**

Wang, P. and Hammerschmidt, T. and Kattner, U.R. and Olson, G.B.*Calphad: Computer Coupling of Phase Diagrams and Thermochemistry*68 (2020)The Co–V system has been reviewed. Density functional theory (DFT) calculations using the generalized gradient approximation (GGA) were used to obtain the energies for the end-members for all three intermediate phases, Co3V, σ and CoV3. Results from DFT calculations considering spin polarization were used to evaluate the CALPHAD (Calculation of phase diagrams) model parameters. The method to evaluate the contribution of the magnetism to the energies of Co-rich compounds that was introduced in our previous work is presented in more detail in the present work. For the description of the σ phase, the magnetic part of the total energy is included in the description of the pure Co end-member compound resulting in a non-linear description of the magnetic contribution over composition. The calculated phase diagram obtained from the present CALPHAD description is in good agreement with the experimental data. The metastable FCC-L12 phase diagram was calculated and compared with experimental data. © 2019 Elsevier Ltdview abstract 10.1016/j.calphad.2019.101729 **Thermodynamic modelling of the Ni–Zr system**

Jana, A. and Sridar, S. and Fries, S.G. and Hammerschmidt, T. and Kumar, K.C.H.*Intermetallics*116 (2020)In this work, we report the thermodynamic modelling of the Ni–Zr system using the Calphad method combined with ab initio calculations. Density functional theory (DFT) is employed to calculate the enthalpy of formation of the intermediate phases. The calculated enthalpies of formation are in close agreement with the experimental data. An approach based on special quasirandom structures (SQS) was used for calculating the enthalpy of mixing of the fcc solid solution. The vibrational contribution to the heat capacities of NiZr, NiZr2, Ni3Zr and Ni7Zr2 phases were calculated using the quasiharmonic approximation (QHA) and the corresponding electronic contribution was obtained using an approach based on Mermin statistics. The total heat capacities for these phases were fitted to appropriate expressions and integrated to obtain the Gibbs energy functions valid down to 0 K. The calculated thermochemical properties along with critically selected experimental constitutional and thermochemical data served as input for the thermodynamic optimisation of the system. The calculated phase equilibria and the thermodynamic properties using the optimised Gibbs energy functions are in good agreement with the input data. The calculated congruent melting points of NiZr and NiZr2 phases are close to the recent experimental data. The Ni10Z7 phase forms by a peritectic reaction, which is also in agreement with the experimental data. © 2019 Elsevier Ltdview abstract 10.1016/j.intermet.2019.106640 **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 **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 **Crowd-sourcing materials-science challenges with the NOMAD 2018 Kaggle competition**

Sutton, C. and Ghiringhelli, L.M. and Yamamoto, T. and Lysogorskiy, Y. and Blumenthal, L. and Hammerschmidt, T. and Golebiowski, J.R. and Liu, X. and Ziletti, A. and Scheffler, M.*npj Computational Materials*5 (2019)A public data-analytics competition was organized by the Novel Materials Discovery (NOMAD) Centre of Excellence and hosted by the online platform Kaggle by using a dataset of 3,000 (AlxGayIn1–x–y)2O3 compounds. Its aim was to identify the best machine-learning (ML) model for the prediction of two key physical properties that are relevant for optoelectronic applications: the electronic bandgap energy and the crystalline formation energy. Here, we present a summary of the top-three ranked ML approaches. The first-place solution was based on a crystal-graph representation that is novel for the ML of properties of materials. The second-place model combined many candidate descriptors from a set of compositional, atomic-environment-based, and average structural properties with the light gradient-boosting machine regression model. The third-place model employed the smooth overlap of atomic position representation with a neural network. The Pearson correlation among the prediction errors of nine ML models (obtained by combining the top-three ranked representations with all three employed regression models) was examined by using the Pearson correlation to gain insight into whether the representation or the regression model determines the overall model performance. Ensembling relatively decorrelated models (based on the Pearson correlation) leads to an even higher prediction accuracy. © 2019, The Author(s).view abstract 10.1038/s41524-019-0239-3 **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 **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 **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 **Thermodynamic assessment of the Co-Ta system**

Wang, P. and Koßmann, J. and Kattner, U.R. and Palumbo, M. and Hammerschmidt, T. and Olson, G.B.*Calphad: Computer Coupling of Phase Diagrams and Thermochemistry*64 (2019)The Co-Ta system has been reviewed and the thermodynamic description was re-assessed in the present work. DFT (density functional theory) calculations considering spin polarization were performed to obtain the energies for all end-member configurations of the C14, C15, C36 and μ phases for the evaluation of the Gibbs energies of these phases. The phase diagram calculated with the present description agrees well with the experimental and theoretical data. Considering the DFT results was essential for giving a better description of the μ phase at lower temperatures. © 2018 Elsevier Ltdview abstract 10.1016/j.calphad.2018.12.002 **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 **{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 **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 **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 **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 **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 **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 **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 **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 **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 **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 **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 **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 **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 **Including the effects of pressure and stress in thermodynamic functions**

Hammerschmidt, T. and Abrikosov, I.A. and Alfè, D. and Fries, S.G. and Höglund, L. and Jacobs, M.H.G. and Koßmann, J. and Lu, X.-G. and Paul, G.*Physica Status Solidi (B) Basic Research*251 (2014)Most applications of thermodynamic databases to materials design are limited to ambient pressure. The consideration of elastic contributions to thermodynamic stability is highly desirable but not straight-forward to realise. We present examples of existing physical models for pressure-dependent thermodynamic functions and discuss the requirements for future implementations given the existing results of experiments and first-principles calculations. We briefly summarize the calculation of elastic constants and point out examples of nonlinear variation with pressure, temperature and chemical composition that would need to be accounted for in thermodynamic databases. This is particularly the case if a system melts from different phases at different pressures. Similar relations exist between pressure and magnetism and hence set the need to also include magnetic effects in thermodynamic databases for finite pressure. We present examples to illustrate that the effect of magnetism on stability is strongly coupled to pressure, temperature, and external fields. As a further complication we discuss dynamical instabilities that may appear at finite pressure. While imaginary phonon frequencies may render a structure unstable and destroy a crystal lattice, the anharmonic effects may stabilize it again at finite temperature. Finally, we also outline a possible implementation scheme for strain effects in thermodynamic databases. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/pssb.201350156 **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 **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 **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 **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 **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 **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 **Analytic many-body potential for GaAs(001) homoepitaxy: Bulk and surface properties**

Fichthorn, K.A. and Tiwary, Y. and Hammerschmidt, T. and Kratzer, P. and Scheffler, M.*Physical Review B - Condensed Matter and Materials Physics*83 (2011)We employ atomic-scale simulation methods to investigate bulk and surface properties of an analytic Tersoff-Abell type potential for describing interatomic interactions in GaAs. The potential is a modified form of that proposed by Albe and colleagues [Phys. Rev. BPRBMDO1098-012110.1103/PhysRevB.66. 035205 66, 035205 (2002)] in which the cut-off parameters for the As-As interaction have been shortened. With this modification, many bulk properties predicted by the potential for solid GaAs are the same as those in the original potential, but properties of the GaAs(001) surface better match results from first-principles calculations with density-functional theory (DFT). We tested the ability of the potential to reproduce the phonon dispersion and heat capacity of bulk solid GaAs by comparing it to experiment and the overall agreement is good. In the modified potential, the GaAs(001) β2(2×4) reconstruction is favored under As-rich growth conditions in agreement with DFT calculations. Additionally, the binding energies and diffusion barriers for a Ga adatom on the β2(2×4) reconstruction generally match results from DFT calculations. These studies indicate that the potential is suitable for investigating aspects of GaAs(001) homoepitaxy. © 2011 American Physical Society.view abstract 10.1103/PhysRevB.83.195328 **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 **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 **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 **Erratum: Analytic many-body potential for InAs/GaAs surfaces and nanostructures: Formation energy of InAs quantum dots (Physical Review B (2008) 77 (235303)**

Hammerschmidt, T. and Kratzer, P. and Scheffler, M.*Physical Review B - Condensed Matter and Materials Physics*81 (2010)view abstract 10.1103/PhysRevB.81.159905 **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

#### alloys

#### atomistic simulations

#### bond-order potential

#### density functional theory

#### high-throughput simulations

#### modelling and simulation

#### phase stability

#### structure maps

#### tight binding