Machine learning methods have become increasingly central in the development of a large variety of versatile tools for molecular simulations, many of which have the potential to advance significantly the fields of computational chemistry and physics. In this chapter, we present a framework for combining machine learning for local structure classification with the definition of a global classifier space as a basis for enhanced sampling of structural transformations in condensed phase systems. The transformation is represented by a path in classifier space, and the associated path collective variable is used to drive the process derived from changes in local structural motifs. Enhanced sampling along this type of path collective variable yields insight into the physical mechanism as well as corresponding free energy barriers of the transition. The idea is generally applicable, and the approach, as outlined here, can be adapted to a wide range of systems. © The Royal Society of Chemistry 2022.

Abstract: Molecular simulations are a powerful tool in the study of crystallization and polymorphic transitions yielding detailed information of transformation mechanisms with high spatiotemporal resolution. However, characterizing various crystalline and amorphous phases as well as sampling nucleation events and structural transitions remain extremely challenging tasks. The integration of machine learning with molecular simulations has the potential of unprecedented advancement in the area of crystal nucleation and growth. In this article, we discuss recent progress in the analysis and sampling of structural transformations aided by machine learning and the resulting potential future directions opening in this area. Graphical Abstract: [Figure not available: see fulltext.]. © 2022, The Author(s), under exclusive License to the Materials Research Society.

Path sampling approaches have become invaluable tools to explore the mechanisms and dynamics of the so-called rare events that are characterized by transitions between metastable states separated by sizable free energy barriers. Their practical application, in particular to ever more complex molecular systems, is, however, not entirely trivial. Focusing on replica exchange transition interface sampling (RETIS) and forward flux sampling (FFS), we discuss a range of analysis tools that can be used to assess the quality and convergence of such simulations, which is crucial to obtain reliable results. The basic ideas of a step-wise evaluation are exemplified for the study of nucleation in several systems with different complexities, providing a general guide for the critical assessment of RETIS and FFS simulations. © 2022 Author(s).

We present an atomistic study of heterogeneous nucleation in Ni employing transition path sampling, which reveals a template precursor-mediated mechanism of crystallization. Most notably, we find that the ability of tiny templates to modify the structural features of the liquid and promote the formation of precursor regions with enhanced bond-orientational order is key to determining their nucleation efficiency and the polymorphs that crystallize. Our results reveal an intrinsic link between structural liquid heterogeneity and the nucleating ability of templates, which significantly advances our understanding toward the control of nucleation efficiency and polymorph selection. © 2022 American Physical Society.

We devise automated workflows for the calculation of Helmholtz and Gibbs free energies and their temperature and pressure dependence and provide the corresponding computational tools. We employ nonequilibrium thermodynamics for evaluating the free energy of solid and liquid phases at a given temperature and reversible scaling for computing free energies over a wide range of temperatures, including the direct integration of P-T coexistence lines. By changing the chemistry and the interatomic potential, alchemical and upscaling free energy calculations are possible. Several examples illustrate the accuracy and efficiency of our implementation. ©2021 American Physical Society.

Gaining a fundamental understanding of crystal nucleation processes in metal alloys is crucial for the development and design of high-performance materials with targeted properties. Yet, crystallization is a complex non-equilibrium process and, despite having been studied for decades, the microscopic aspects that govern the crystallization mechanism of a material remain elusive to date. Recent evidence shows that the spatial heterogeneity in the supercooled liquid, characterised by extended regions with distinctive mobility and order, may be a key microscopic factor that determines the mechanism of crystal nucleation. These findings have advanced our view of the fundamental nature of crystallization, as most research has assumed that crystal clusters nucleate from random fluctuations in a ‘homogeneous’ liquid. Here, by analysing transition path sampling trajectories, we show that dynamical heterogeneity plays a key role in the mechanism of crystal nucleation in an elemental metal, nickel. Our results demonstrate that crystallization occurs preferentially in regions of low mobility in the supercooled liquid, evidencing the collective dynamical nature of crystal nucleation in Ni. In addition, our results show that low mobility regions form before and spatially overlap with pre-ordered domains that act as precursors to the crystal phase that subsequently emerges. Our results show a clear link between dynamical and structural heterogeneity in the supercooled liquid and its impact on the nucleation mechanism, revealing microscopic descriptors that could pave a novel way to control crystallization processes in metals. © 2022 The Royal Society of Chemistry.

Abstract: In molecular simulations, the identification of suitable reaction coordinates is central to both the analysis and sampling of transitions between metastable states in complex systems. If sufficient simulation data are available, a number of methods have been developed to reduce the vast amount of high-dimensional data to a small number of essential degrees of freedom representing the reaction coordinate. Likewise, if the reaction coordinate is known, a variety of approaches have been proposed to enhance the sampling along the important degrees of freedom. Often, however, neither one nor the other is available. One of the key questions is therefore, how to construct reaction coordinates and evaluate their validity. Another challenges arises from the physical interpretation of reaction coordinates, which is often addressed by correlating physically meaningful parameters with conceptually well-defined but abstract reaction coordinates. Furthermore, machine learning based methods are becoming more and more applicable also to the reaction coordinate problem. This perspective highlights central aspects in the identification and evaluation of reaction coordinates and discusses recent ideas regarding automated computational frameworks to combine the optimization of reaction coordinates and enhanced sampling. Graphic abstract: [Figure not available: see fulltext.] © 2021, The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature.

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.

Collective interstitial ordering is at the core of martensite formation in Fe–C-based alloys, laying the foundation for high-strength steels. Even though this ordering has been studied extensively for more than a century, some fundamental mechanisms remain elusive. Here, we show the unexpected effects of two correlated phenomena on the ordering mechanism: anharmonicity and segregation. The local anharmonicity in the strain fields induced by interstitials substantially reduces the critical concentration for interstitial ordering, up to a factor of three. Further, the competition between interstitial ordering and segregation results in an effective decrease of interstitial segregation into extended defects for high interstitial concentrations. The mechanism and corresponding impact on interstitial ordering identified here enrich the theory of phase transitions in materials and constitute a crucial step in the design of ultra-high-performance alloys. © 2020, The Author(s), under exclusive licence to Springer Nature Limited.

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).

The martensitic start temperature (M s) is a technologically fundamental characteristic of high-temperature shape memory alloys. We have recently shown [Chakraborty et al 2016 Phys. Rev. B 94 224104] that the two key features in describing the composition dependence of M s are the T = 0 K phase stability and the difference in vibrational entropy which, within the Debye model, is directly linked to the elastic properties. Here, we use density functional theory together with special quasi-random structures to study the elastic properties of disordered martensite and austenite Ti-Ta alloys as a function of composition. We observe a softening in the tetragonal shear elastic constant of the austenite phase at low Ta content and a non-linear behavior in the shear elastic constant of the martensite. A minimum of 12.5% Ta is required to stabilize the austenite phase at T = 0 K. Further, the shear elastic constants and Young's modulus of martensite exhibit a maximum for Ta concentrations close to 30%. Phenomenological, elastic-constant-based criteria suggest that the addition of Ta enhances the strength, but reduces the ductile character of the alloys. In addition, the directional elastic stiffness, calculated for both martensite and austenite, becomes more isotropic with increasing Ta content. The reported trends in elastic properties as a function of composition may serve as a guide in the design of alloys with optimized properties in this interesting class of materials. © 2020 IOP Publishing Ltd.

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.

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.

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.

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.

The investigation of the microscopic processes underlying structural phase transformations in solids is extremely challenging for both simulation and experiment. Atomistic simulations of solid-solid phase transitions require extensive sampling of the corresponding high-dimensional and often rugged energy landscape. Here, we propose a rigorous construction of a 1D path collective variable that is used in combination with enhanced sampling techniques for efficient exploration of the transformation mechanisms. The path collective variable is defined in a space spanned by global classifiers that are derived from local structural units. A reliable identification of the local structural environments is achieved by employing a neural-network-based classification scheme. The proposed path collective variable is generally applicable and enables the investigation of both transformation mechanisms and kinetics. © 2019 American Physical Society.

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.

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.

Understanding the underlying mechanism of crystal nucleation is a fundamental aspect in the prediction and control of materials properties. Classical nucleation theory (CNT) assumes that homogeneous nucleation occurs via random fluctuations within the supercooled liquid, that the structure of the growing clusters resembles the most stable bulk phase, and that the nucleus size is the sole reaction coordinate (RC) of the process. Many materials are, however, known to exhibit multiple steps during crystallization, forming different polymorphs. As a consequence, more complex RCs are often required to capture all relevant information about the process. Here, we employ transition path sampling together with a maximum likelihood analysis of candidate order parameters to identify suitable RCs for the nucleation mechanism during solidification in Ni. In contrast to CNT, the analysis of the reweighted path ensemble shows that a prestructured liquid region that surrounds the crystal cluster is a relevant order parameter that enhances the RC and therefore plays a key role in the description of the nucleus and the interfacial free energy. We demonstrate that prestructured liquid clusters that emerge within the liquid act as precursors of the crystallization in a nonclassical two-step mechanism, which predetermines the coordination of the selected polymorphs. Copyright © 2018 American Chemical Society.

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.

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.

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).

Employing ab initio calculations we demonstrate that the complex structural modulations experimentally observed in ultrathin Fe films on Cu(001) originate from Fe bulk phases that arise under extreme deformations. Specifically, we show that the structural modulations correspond to the motifs observed when transforming fcc Fe to bcc Fe in the Pitsch orientation relationship [(001)fcc||(110)bcc]. The observed structural equivalence between surface and unstable bulk structures naturally explains the experimentally reported magnetic and structural transitions when going from low (two to four MLs) to intermediate (four to ten MLs) film coverages. © 2017 American Physical Society.

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.

On high-dimensional and complex potential energy surfaces, the identification of the most likely mechanism for the transition between local minima is a challenging task. Usually the steepest-descent path is used interchangeably with the minimum-energy path and is associated with the most likely path. Here we compare the meaning of the steepest-descent path in complex energy landscapes to the path integral formulation of a trajectory that minimizes the action functional for Brownian dynamics. In particular, for energy landscapes with bifurcation points and multiple minima and saddle points, there can be several steepest-descent paths associated with specific saddles that connect two predetermined states but largely differ from the path of maximum likelihood. The minimum-action path, however, additionally takes into account the scalar work along the trajectory. Minimizing the scalar work can be less ambiguous in the identification of the most likely path in different gradient systems. It can also be used to distinguish between multiple steepest-descent paths that connect reactant and product states. We illustrate that in systems with complex energy landscapes a careful assessment of the steepest-descent path is thus advisable. Here the evaluation of the action can provide valuable information on the analysis and description of the most likely path. © 2016 American Physical Society.

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.

Diffusionless displacive lattice rearrangements, which include martensitic transformations, are in real materials often accompanied by a displacive drag of interstitials. The interplay of both processes leads to a particular atomistic arrangement of the interstitials in the product phase, which is decisive for its performance. An archetype example is the martensitic transformation in Fe-C alloys. One of the puzzles for this system is that the deviation from the cubic symmetry (i.e., the tetragonality) in the martensite resulting from this interplay is lower than what thermodynamics dictates. In our ab initio approach, the relative motion of C in the transforming lattice is studied with the nudged elastic band method. We prove that an atomic shearlike shuffle mechanism of adjacent (112) Fe layers along the ±[111]bcc directions is essential to achieve a redistribution of C atoms during the fcc → bcc transition, which fully explains the abnormal behavior. Furthermore, the good agreement with experiment validates our method to treat a diffusionless redistribution of interstitials and a displacive rearrangement of the host lattice simultaneously. © 2016 American Physical Society.

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).

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.

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.

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.

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.

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.

We review and discuss methods for including the role of point defects in calculations of the free energy, composition and phase stability of elements and compounds. Our principle aim is to explain and to reconcile, with examples, the perspectives on this problem that are often strikingly different between exponents of CALPHAD, and others working in the overlapping fields of physics, chemistry and materials science. Current methodologies described here include the compound energy formalism of CALPHAD, besides the rather different but related canonical and grand-canonical formalisms. We show how the calculation of appropriate defect formation energies should be formulated, how they are included in the different formalisms and in turn how these yield equilibrium defect concentrations and their contribution to free energies and chemical potentials. Furthermore, we briefly review the current state-of-the-art and challenges in determining point defect properties from first-principles calculations as well as from experimental measurements. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The dimer method is a minimum mode following algorithm for finding saddle points on a potential energy surface of atomic systems. Here, the dimer method is extended to include the cell degrees of freedom for periodic solid-state systems. Using this method, reaction pathways of solid-solid phase transitions can be determined without having to specify the final state structure or reaction mechanism. Example calculations include concerted phase transitions between CdSe polymorphs and a nucleation and growth mechanism for the A15 to BCC transition in Mo. © 2014 AIP Publishing LLC.

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.

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.

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.

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.

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.

We present a flexible nonlinear reaction coordinate analysis method for the transition path ensemble based on the likelihood maximization approach developed by Peters and Trout [J. Chem. Phys. 125, 054108 (2006)]. By parametrizing the reaction coordinate by a string of images in a collective variable space, we can optimize the likelihood that the string correctly models the committor data obtained from a path sampling simulation. The collective variable space with the maximum likelihood is considered to contain the best description of the reaction. The use of the reweighted path ensemble [J. Rogal, J. Chem. Phys. 133, 174109 (2010)] allows a complete reaction coordinate description from the initial to the final state. We illustrate the method on a z-shaped two-dimensional potential. While developed for use with path sampling, this analysis method can also be applied to regular molecular dynamics trajectories. © 2010 American Institute of Physics.

Developed for complex systems undergoing rare events involving many (meta)stable states, the multiple state transition path sampling aims to sample from an extended path ensemble including all possible trajectories between any pair of (meta)stable states. The key issue for an efficient sampling of the path space in this extended ensemble is sufficient switching between different types of trajectories. When some transitions are much more likely than others the collective sampling of the different path types can become difficult. Here we introduce a Wang-Landau based biasing approach to improve the sampling. We find that the biasing of the multiple state path ensemble does not influence the switching behavior, but does improve the sampling and thus the quality of the individual path ensembles. © 2010 American Institute of Physics.

We introduce a reweighting scheme for the path ensembles in the transition interface sampling framework. The reweighting allows for the analysis of free energy landscapes and committor projections in any collective variable space. We illustrate the reweighting scheme on a two dimensional potential with a nonlinear reaction coordinate and on a more realistic simulation of the Trp-cage folding process. We suggest that the reweighted path ensemble can be used to optimize possible nonlinear reaction coordinates. © 2010 American Institute of Physics.