#### Prof. Dr. Ingo Steinbach

Scale Bridging Thermodynamic and Kinetic Simulation at ICAMS

Ruhr-Universität Bochum

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**Path to single-crystalline repair and manufacture of Ni-based superalloy using directional annealing**

Kalfhaus, T. and Schaar, H. and Thaler, F. and Ruttert, B. and Sebold, D. and Frenzel, J. and Steinbach, I. and Theisen, W. and Guillon, O. and Clyne, T.W. and Vassen, R.*Surface and Coatings Technology*405 (2021)Advanced methods for the repair of single-crystalline (SX) Ni-based superalloys are of special interest for the gas turbine industry. Polycrystalline repair approaches show promising results, while the repair of SX materials is still challenging. Directional annealing experiments resulted in large columnar grains by imposing thermal gradients at the abnormal grain growth temperature of a specific Ni-based superalloy. A numerical model of the Bridgman process is applied to provide an insight into the temperature evolution during zone annealing of the Vacuum-Plasma-Spray (VPS) repair coatings with the aim of promoting grain growth from the SX substrate. The results presented here suggest that this is a promising approach to repair or manufacture SX turbine blades. © 2020 Elsevier B.V.view abstract 10.1016/j.surfcoat.2020.126494 **45-degree rafting in Ni-based superalloys: A combined phase-field and strain gradient crystal plasticity study**

Ali, M.A. and Amin, W. and Shchyglo, O. and Steinbach, I.*International Journal of Plasticity*128 (2020)45° rafting of Ni-based superalloys has been investigated with the help of creep test simulations applying a strain gradient crystal plasticity model coupled to the multi-phase field method. This mode of rafting lies in between P- A nd N-type rafting modes. The model parameters are calibrated against experimental data for N-type rafting under high temperature and low stress creep condition. By increasing the stress level, the mixed-mode rafting of precipitates with a clear tendency toward formation of 45° rafts is observed. We show that the key factor for the occurrence of this type of rafting is the generation of highly localized creep strain of more than 10% due to non-homogeneous creep deformation in the form of slip bands. We have successfully captured the evolution of microstructure under high stress leading to production of localized shear bands. © 2020 Elsevier Ltd.view abstract 10.1016/j.ijplas.2020.102659 **Columnar-equiaxed transition in continuous casting based on a micro-macro solidification model with long-range solutal mixing**

Uddagiri, M. and Spee, J. and Hubig, S. and Steinbach, I.*IOP Conference Series: Materials Science and Engineering*861 (2020)A novel model is proposed to describe the columnar to equiaxed transition (CET) in continuous casting. The model bases on Rappaz and Thevoz's solute diffusion model for equiaxed dendritic growth, combined with a 1-Dimensional solidification model normal to the slab surface. The unique feature of the proposed model is the combination with a mixing term between interdendritic and extradendritic melt, representing long-range solutal mixing by convection. The model can also be applied to predict equiaxed to columnar transition (ECT), i.e. the chill zone thickness. The model consists of modules such as nucleation, growth kinetics, solute and heat balance, and a solute mixing module. Nucleation is considered with a fixed nucleation undercooling. The growth kinetics of the dendrites are treated according to the LGK model. A finite difference scheme is employed for solving 1-Dimensional heat transfer equations and finally, volume averaged solute balance equations are solved in a staggered scheme. Mixing of inter- and extradendritic liquid is, as a first step, treated ideally fast. When applied to Fe-C binary system with the thermo-physical properties obtained from literature and CALPHAD simulations, the model successfully predicts recalescence, phase fraction evolution, and concentration profiles in different phases. Realistic boundary conditions of the continuous casting process are obtained from macroscopic FEM simulation. © Published under licence by IOP Publishing Ltd.view abstract 10.1088/1757-899X/861/1/012014 **Controlling bubble coalescence in metallic foams: A simple phase field-based approach**

Vakili, S. and Steinbach, I. and Varnik, F.*Computational Materials Science*173 (2020)The phase-field method is used as a basis to develop a strictly mass conserving, yet simple, model for simulation of two-phase flow. The model is aimed to be applied for the study of structure evolution in metallic foams. In this regard, the critical issue is to control the rate of bubble coalescence compared to concurrent processes such as their rearrangement due to fluid motion. In the present model, this is achieved by tuning the interface energy as a free parameter. The model is validated by a number of benchmark tests. First, stability of a two dimensional bubble is investigated by the Young-Laplace law for different values of the interface energy. Then, the coalescence of two bubbles is simulated until the system reaches equilibrium with a circular shape. To address the major capability of the present model for the formation of foam structure, the bubble coalescence is simulated for various values of interface energy in order to slow down the merging process. These simulations are repeated in the presence of a rotational flow to highlight the fact that the model allows to suppress the coalescence process compared to the motion of bubbles relative to each other. Moreover, since density is treated as an auxiliary variable “slaved” to the volume occupied by a given phase, the present model allows realization of arbitrarily large liquid-gas density ratios. This property is demonstrated by simulation of a system with ρl/ρg=10,000. © 2019 Elsevier B.V.view abstract 10.1016/j.commatsci.2019.109437 **Effect of γ′ precipitate size on hardness and creep properties of Ni-base single crystal superalloys: Experiment and simulation**

Ali, M.A. and López-Galilea, I. and Gao, S. and Ruttert, B. and Amin, W. and Shchyglo, O. and Hartmaier, A. and Theisen, W. and Steinbach, I.*Materialia*12 (2020)The role and effect of γ′ precipitate size on the mechanical properties of Ni-base single crystal superalloy is investigated. The underlying mechanisms are analyzed on the one hand with the help of experiments including hardness and creep tests, and on the other hand with the help of two different simulation approaches by taking the typical γ/γ′ microstructure into account. Simulations, based on the crystal plasticity finite element method (CPFEM) are carried out for the hardness tests, whereas simulations, based on the crystal plasticity coupled phase-field method (CPPFM) are carried out for the creep tests. The hardness test simulation results show that the hardness of material varies inversely with the size of γ′ precipitates for a given γ′ phase volume fraction and it varies directly with the volume fraction of γ′ precipitates for a given precipitate size. These results are qualitatively consistent with the experimental observations. The creep simulation results show that the refinement of γ′ precipitates with a certain volume fraction of precipitates leads to an improvement of creep resistance by delaying the plastic activity in the material. © 2020 Acta Materialia Inc.view abstract 10.1016/j.mtla.2020.100692 **Erratum: Quantum-Phase-Field Concept of Matter: Emergent Gravity in the Dynamic Universe (Zeitschrift für Naturforschung A. (2020) 72:1 (51-58) DOI: 10.1515/zna-2016-0270)**

Steinbach, I.*Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences*75 (2020)The work "Quantum-phase-field concept of matter: Emergent gravity in the dynamic universe", published in [1], outlines a framework to describe physical matter from the solution of a one-dimensional non-linear wave equation. Unfortunately, a central result, the presented analytic form of this solution, (13) in [1] is incorrect. The corrected form can be found on arXiv [2]. In the present Erratum, besides the correct solution, some background information about these types of non-linear wave equations and their solutions is added. We start from a functional fDW, the famous "double-well potential", according to Landau's theory of phase transitions [3] with positive constants r and u, expanded in temperature T around the critical temperature Tc. φ. is the "order parameter" in the original Ginzburg-Landau theory. In this theory the potential function is expanded to the fourth order, keeping only even contributions of the order parameter φ: (Equation Presented) The potential has a minimum at φ = 0 for T > Tc and two minima at φm = ±√r(Tc-T)/u for T < Tc. These minima describe the disordered and ordered state of the system, respectively. Here only the ordered state T < Tc is discussed and to be consistent with conventions in the phase-field literature the order parameter is normalized to (Equation Presented), see Figure 1. (Equation Presented) Also, we will understand φ = φ(s, t) as a field variable in space s and time t and define the Hamiltonian H as an integral over the potential density and the Ginsburg gradient operators accounting for fluctuations: (Equation Presented) U is a constant with dimension of energy, ϵ is a constant with dimension of length and γ an inverse length. The well-known minimum solution δ/δφ G = 0 is for the boundary conditions φ(-∞) = 0 and φ(+∞) = 1: (Equation Presented) with η = √72 ϵ/γ for a traveling wave with speed ν. Note that for ν ≠ 0 a symmetry breaking contribution has to be added to the potential (2), which is omitted here to keep the focus on the type of potential. Details can be found in appendix of [4]. For the boundary conditions φ(-∞) = φ(+∞) = 0 we have, besides the trivial solution φ ≡ 0: (Equation Presented) where the waves, peaked at s1 and s2 at t = 0, respectively, s1 < s2, travel with opposite speed, see Figure 2. It is also well established that the special form of the potential (1) or (2) is not fixed from basic principles, besides that, between the two minima there should be a potential barrier to separate the minima with a given activation energy ∝ U. Only close to the critical point, i.e. where the activation energy U → 0, a rigorous renormalization group treatment may be applied to show that higher order contributions will become irrelevant [5]. Aside from the critical point, no argument exists to truncate the Landau expansion of the potential to the fourth order or to select the given form. In the "multiphase-field theory" [6] there arises, however, an additional constraint for the potential. In this theory a set of fields φI, I = 1 ⋯ N is defined which form junctions by the condition ΣN I=1 φI = 1. The functional is replaced by H = ΣN I=1 HI where each term HI has the special form (3). In the center of the junction, where φI = φJ = 1/N for all I, J, the maximum of the potential fDW m can be evaluated: (Equation Presented) i.e. for N > 3 the energy of the junction decreases with the order N and approaches 0 for large N. This must [Figure Presented] be termed "unphysical", as junctions between objects loose their penalty and the system would return to the disordered state. To remedy this problem, the so-called "double-obstacle potential" is introduced [6]: (Equation Presented) It has the same topology as (2) (see Fig. 1) but a maximum power of 2. Further on it has the advantage that it defies a linear wave aside from the breakpoints. We calculate the maximum potential of the junction fDO m: (Equation Presented) i.e. the energy of the junction increases with the order N and approaches a constant ∝ γ for large N, as it should. The main drawback of this potential is the non-analytical form with the absolute signs. The "non-linearity" of (2) is hidden in the breakpoints at φ = 0 and φ = 1. Only a piece-wise solution is possible for the boundary conditions φ(-∞) = 0 and φ(+∞) = 1, η = π √ϵ.γ, (Equation Presented) For φ(-∞) = φ(+∞) = 0, one finds (see Fig. 2): (Equation Presented) The last (10) replaces (13) in [1]. © 2019 De Gruyter. All rights reserved.view abstract 10.1515/zna-2019-0326 **Multi-phase-field simulation of microstructure evolution in metallic foams**

Vakili, S. and Steinbach, I. and Varnik, F.*Scientific Reports*10 (2020)This paper represents a model for microstructure formation in metallic foams based on the multi-phase-field approach. The model allows to naturally account for the effect of additives which prevent two gas bubbles from coalescence. By applying a non-merging criterion to the phase fields and at the same time raising the free energy penalty associated with additives, it is possible to completely prevent coalescence of bubbles in the time window of interest and thus focus on the formation of a closed porous microstructure. On the other hand, using a modification of this criterion along with lower free energy barriers we investigate with this model initiation of coalescence and the evolution of open structures. The method is validated and used to simulate foam structure formation both in two and three dimensions. © 2020, The Author(s).view abstract 10.1038/s41598-020-76766-z **Quantum-phase-field: From de broglie-bohm double-solution program to doublon networks**

Kundin, J. and Steinbach, I.*Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences*75 (2020)Different forms of linear and non-linear field equations, so-called 'phase-field' equations, are studied in relation to the de Broglie-Bohm double-solution program. This defines a framework in which elementary particles are described by localised non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schrödinger-type equation. First, we consider the phase-field order parameter as the phase for the linear pilot wave, second as the pilot wave itself, and third as a moving soliton interpreted as a massive particle. In the last case, we introduce the equation for a superwave, the amplitude of which can be considered as a particle moving in accordance to the de Broglie-Bohm theory. Lax pairs for the coupled problems are constructed in order to discover possible non-linear equations that can describe the moving particle and to propose a framework for investigating coupled solutions. Finally, doublons in 1 + 1 dimensions are constructed as self-similar solutions of a non-linear phase-field equation forming a finite space object. Vacuum quantum oscillations within the doublon determine the evolution of the coupled system. Applying a conservation constraint and using general symmetry considerations, the doublons are arranged as a network in 1 + 1 + 2 dimensions, where nodes are interpreted as elementary particles. A canonical procedure is proposed to treat charge and electromagnetic exchange. © 2020 I. Steinbach et al., published by De Gruyter, Berlin/Boston 2020.view abstract 10.1515/zna-2019-0343 **Roadmap on multiscale materials modeling**

Van Der Giessen, E. and Schultz, P.A. and Bertin, N. and Bulatov, V.V. and Cai, W. and Csányi, G. and Foiles, S.M. and Geers, M.G.D. and González, C. and Hütter, M. and Kim, W.K. and Kochmann, D.M. and Llorca, J. and Mattsson, A.E. and Rottler, J. and Shluger, A. and Sills, R.B. and Steinbach, I. and Strachan, A. and Tadmor, E.B.*Modelling and Simulation in Materials Science and Engineering*28 (2020)Modeling and simulation is transforming modern materials science, becoming an important tool for the discovery of new materials and material phenomena, for gaining insight into the processes that govern materials behavior, and, increasingly, for quantitative predictions that can be used as part of a design tool in full partnership with experimental synthesis and characterization. Modeling and simulation is the essential bridge from good science to good engineering, spanning from fundamental understanding of materials behavior to deliberate design of new materials technologies leveraging new properties and processes. This Roadmap presents a broad overview of the extensive impact computational modeling has had in materials science in the past few decades, and offers focused perspectives on where the path forward lies as this rapidly expanding field evolves to meet the challenges of the next few decades. The Roadmap offers perspectives on advances within disciplines as diverse as phase field methods to model mesoscale behavior and molecular dynamics methods to deduce the fundamental atomic-scale dynamical processes governing materials response, to the challenges involved in the interdisciplinary research that tackles complex materials problems where the governing phenomena span different scales of materials behavior requiring multiscale approaches. The shift from understanding fundamental materials behavior to development of quantitative approaches to explain and predict experimental observations requires advances in the methods and practice in simulations for reproducibility and reliability, and interacting with a computational ecosystem that integrates new theory development, innovative applications, and an increasingly integrated software and computational infrastructure that takes advantage of the increasingly powerful computational methods and computing hardware. © 2020 The Author(s). Published by IOP Publishing Ltd.view abstract 10.1088/1361-651X/ab7150 **Role of coherency loss on rafting behavior of Ni-based superalloys**

Ali, M.A. and Görler, J.V. and Steinbach, I.*Computational Materials Science*171 (2020)The role of coherency loss on rafting of superalloys under high temperature low stress creep conditions is investigated by phase-field crystal plasticity simulations. It is demonstrated that coalescence, critically depending on the state of coherency between precipitate and matrix is crucial to understand the rafting behavior of superalloys. An explicit mechanisms is developed predicting coherency loss based on the plastic activity in the matrix. The simulations are verified using experimental creep test results. © 2019 Elsevier B.V.view abstract 10.1016/j.commatsci.2019.109279 **Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth**

Salama, H. and Kundin, J. and Shchyglo, O. and Mohles, V. and Marquardt, K. and Steinbach, I.*Acta Materialia*188 (2020)The role of inclination dependence of grain boundary energy on the microstructure evolution and the orientation distribution of grain boundary planes during grain growth in polycrystalline materials is investigated by three-dimensional phase-field simulations. The anisotropic grain boundary energy model uses the description of the faceted surface structure of the individual crystals and constructs an anisotropic energy of solid-solid interface. The energy minimization occurs by the faceting of the grain boundary due to inclination dependence of the grain boundary energy. The simulation results for a single grain show the development of equilibrium shapes (faceted grain morphologies) with different families of facets which agrees well with the theoretical predictions. The results of grain growth simulations with isotropic and anisotropic grain boundary energy for cubic symmetry show that inclination dependence of grain boundary energy has a significant influence on the grain boundary migration, grain growth kinetics and the grain boundary plane distribution. It has been shown that the model essentially reproduces the experimental studies reported for NaCl and MgO polycrystalline systems where the anisotropic distribution of grain boundary planes has a peak for the low-index {100} type boundaries. © 2020 Acta Materialia Inc.view abstract 10.1016/j.actamat.2020.02.043 **Combined phase-field crystal plasticity simulation of P- and N-type rafting in Co-based superalloys**

Wang, C. and Ali, M.A. and Gao, S. and Goerler, J.V. and Steinbach, I.*Acta Materialia*175 (2019)We combine a phase-field model with a crystal plasticity model to simulate the microstructural evolution during creep in the Co-based superalloy ERBOCo-2Ta. Three-dimensional simulations of tensile and compressive creep tests in [100] direction were performed to study the rafting behavior in Co-based superalloys. The loss of coherency between γ matrix and γ′ precipitate, which is essential for the understanding of rafted structures, is modeled in relation to the dislocation activity in the γ-channels. Special attention is given to the interplay between creep deformation and microstructure stability. Appropriate constitutive modeling is applied to simulate realistic microstructure evolution under creep conditions. Thus, with the removal of the misfit stress, γ′ precipitates lose their cuboidal shape and form rafts. During N-type rafting more γ′ precipitates coalesce than during P-type rafting. The γ′ volume fraction during rafting increases under tensile stress but decreases under compressive stress. The morphological evolution of γ′ precipitates under tensile and compressive stresses in Co-based superalloy is consistent with the rafting characteristics in experimental observations. © 2019 Acta Materialia Inc.view abstract 10.1016/j.actamat.2019.05.063 **Comparative study of different anisotropy and potential formulations of phase-field models for dendritic solidification**

Kundin, J. and Steinbach, I.*Computational Materials Science*170 (2019)Phase-field model formulations with double well and double obstacle potentials, and different anisotropy models are investigated with respect to their potential to simulate (i) tip growth on a quantitative level, (ii) well resolved side-branching. The dilute binary alloy Al-4 at%Cu is used as a model alloy. The effects of the numerical resolution (the ratio of the capillary length to the grid spacing) on the growth velocity are studied by means of convergence tests for isothermal and directional solidification in comparison to the theoretical values calculated by the Green-function method (A. Karma, W.J. Rappel, Phys. Rev. E 57 (1998) 4323). An interface stability parameter is introduced as a measure for the estimation of the maximum value of the grid spacing for effective simulations. We show that predominantly the side-branching occurs at numerical resolution lower than the limit value needed to produce correct results in accordance to the convergence analysis. The best results for dendrite growth at a relevant numerical resolution are obtained for the double well potential. © 2019 Elsevier B.V.view abstract 10.1016/j.commatsci.2019.109197 **Concentration-dependent atomic mobilities in FCC CoCrFeMnNi high-entropy alloys**

Gaertner, D. and Abrahams, K. and Kottke, J. and Esin, V.A. and Steinbach, I. and Wilde, G. and Divinski, S.V.*Acta Materialia*166 (2019)The diffusion kinetics in a CoCrFeMnNi high entropy alloy is investigated by a combined radiotracer–interdiffusion experiment applied to a pseudo-binary Co15Cr20Fe20Mn20Ni25/Co25Cr20Fe20Mn20Ni15 couple. As a result, the composition-dependent tracer diffusion coefficients of Co, Cr, Fe and Mn are determined. The elements are characterized by significantly different diffusion rates, with Mn being the fastest element and Co being the slowest one. The elements having originally equiatomic concentration through the diffusion couple are found to reveal up-hill diffusion, especially Cr and Mn. The atomic mobility of Co seems to follow a S-shaped concentration dependence along the diffusion path. The experimentally measured kinetic data are checked against the existing CALPHAD-type databases. In order to ensure a consistent treatment of tracer and chemical diffusion a generalized symmetrized continuum approach for multi-component interdiffusion is proposed. Both, tracer and chemical diffusion concentration profiles are simulated and compared to the measurements. By using the measured tracer diffusion coefficients the chemical profiles can be described, almost perfectly, including up-hill diffusion. © 2018 Acta Materialia Inc.view abstract 10.1016/j.actamat.2018.12.033 **First Evidence for Mechanism of Inverse Ripening from In-situ TEM and Phase-Field Study of δ′ Precipitation in an Al-Li Alloy**

Park, J. and Kamachali, R.D. and Kim, S.-D. and Kim, S.-H. and Oh, C.-S. and Schwarze, C. and Steinbach, I.*Scientific Reports*9 (2019)In-situ TEM investigation of aging response in an Al–7.8 at.% Li was performed at 200 °C up to 13 hours. Semi-spherical δ′ precipitates growing up to an average radius of 7.5 nm were observed. The size and number of individual precipitates were recorded over time and compared to large-scale phase-field simulations without and with a chemo-mechanical coupling effect, that is, concentration dependence of the elastic constants of the matrix solid solution phase. This type of coupling was recently reported in theoretical studies leading to an inverse ripening process where smaller precipitates grew at the expense of larger ones. Considering this chemo-mechanical coupling effect, the temporal evolution of number density, average radius, and size distribution of the precipitates observed in the in-situ experiment were explained. The results indicate that the mechanism of inverse ripening can be active in this case. Formation of dislocations and precipitate-free zones are discussed as possible disturbances to the chemo-mechanical coupling effect and consequent inverse ripening process. © 2019, The Author(s).view abstract 10.1038/s41598-019-40685-5 **On crystal mosaicity in single crystal Ni-based superalloys**

Hallensleben, P. and Scholz, F. and Thome, P. and Schaar, H. and Steinbach, I. and Eggeler, G. and Frenzel, J.*Crystals*9 (2019)In the present work, we investigate the evolution of mosaicity during seeded Bridgman processing of technical Ni-based single crystal superalloys (SXs). For this purpose, we combine solidification experiments performed at different withdrawal rates between 45 and 720 mm/h with advanced optical microscopy and quantitative image analysis. The results obtained in the present work suggest that crystal mosaicity represents an inherent feature of SXs, which is related to elementary stochastic processes which govern dendritic solidification. In SXs, mosaicity is related to two factors: inherited mosaicity of the seed crystal and dendrite deformation. Individual SXs have unique mosaicity fingerprints. Most crystals differ in this respect, even when they were produced using identical processing conditions. Small differences in the orientation spread of the seed crystals and small stochastic orientation deviations continuously accumulate during dendritic solidification. Direct evidence for dendrite bending in a seeded Bridgman growth process is provided. It was observed that continuous or sudden bending affects the growth directions of dendrites. We provide evidence which shows that some dendrites continuously bend by 1.7° over a solidification distance of 25 mm. © 2019 by the authors. Licensee MDPI, Basel, Switzerland.view abstract 10.3390/cryst9030149 **Phase-field simulation of martensite microstructure in low-carbon steel**

Shchyglo, O. and Du, G. and Engels, J.K. and Steinbach, I.*Acta Materialia*175 (2019)We present three-dimensional phase-field simulations of martensite microstructure formation in low-carbon steel. In this study, a full set of 24 Kurdjumov-Sachs symmetry variants of martensite is considered. Three different carbon compositions are investigated in order to reveal the effect of carbon content on the martensite microstructure formation. The simulations are performed using the finite strain framework which allows considering real martensite transformation strains. Using Neuber elasto-plastic approximation to the mechanical equilibrium solution, realistic stresses and strains can be obtained during martensite formation resulting in realistic mechanical driving forces for the transformation. The simulated microstructures are compared to experimental results for three carbon compositions. Good agreement between simulated and experimental results is achieved. © 2019 Acta Materialia Inc.view abstract 10.1016/j.actamat.2019.06.036 **Solute trapping in non-equilibrium solidification: A comparative model study**

Reuther, K. and Hubig, S. and Steinbach, I. and Rettenmayr, M.*Materialia*6 (2019)A sharp interface model and a diffuse interface model describing rapid solidification are compared. Both models are based on the assumption of two independent processes at the solid/liquid interface that together consume the available driving force. These two processes are (1) the interface motion and (2) the redistribution of atoms between the two phases. The sharp interface model is based on an interface thermodynamics model (Hillert and Rettenmayr, [3,4]) and derives its driving force for the phase transformation directly from Gibbs free energy formulations of the liquid and solid phases. In contrast, the Finite Interface Disspiation model (Steinbach et al., [10]), a diffuse interface model, is based on a grand potential formulation. The models are benchmarked against experimental data on solute trapping in the Al–Sn system (Smith and Aziz, [23]). The sharp interface model reproduces the experimental data with a single set of kinetic parameters, the diffuse interface model requires a velocity dependent interface permeability. It is concluded that due to the investigated high Péclet number regime the permeability in the diffuse interface model does not only describe the transport processes through the interface, but also the diffusion processes in the layer directly in front of it. © 2019 Acta Materialia Inc.view abstract 10.1016/j.mtla.2019.100256 **Computationally Efficient Phase-field Simulation Studies Using RVE Sampling and Statistical Analysis**

Schwarze, C. and Darvishi Kamachali, R. and Kühbach, M. and Mießen, C. and Tegeler, M. and Barrales-Mora, L. and Steinbach, I. and Gottstein, G.*Computational Materials Science*147 (2018)For large-scale phase-field simulations, the trade-off between accuracy and computational cost as a function of the size and number of simulations was studied. For this purpose, a large reference representative volume element (RVE) was incrementally subdivided into smaller solitary samples. We have considered diffusion-controlled growth and early ripening of δ′ (Al3Li) precipitate in a model Al-Li system. The results of the simulations show that decomposition of reference RVE can be a valuable computational technique to accelerate simulations without a substantial loss of accuracy. In the current case study, the precipitate number density was found to be the key controlling parameter. For a pre-set accuracy, it turned out that large-scale simulations of the reference RVE can be replaced by simulating a combination of smaller solitary samples. This shortens the required simulation time and improves the memory usage of the simulation considerably, and thus substantially increases the efficiency of massive parallel computation for phase-field applications. © 2018 Elsevier B.V.view abstract 10.1016/j.commatsci.2018.02.005 **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 **Multi-phase-field method for surface tension induced elasticity**

Schiedung, R. and Steinbach, I. and Varnik, F.*Physical Review B*97 (2018)A method, based on the multi-phase-field framework, is proposed that adequately accounts for the effects of a coupling between surface free energy and elastic deformation in solids. The method is validated via a number of analytically solvable problems. In addition to stress states at mechanical equilibrium in complex geometries, the underlying multi-phase-field framework naturally allows us to account for the influence of surface energy induced stresses on phase transformation kinetics. This issue, which is of fundamental importance on the nanoscale, is demonstrated in the limit of fast diffusion for a solid sphere, which melts due to the well-known Gibbs-Thompson effect. This melting process is slowed down when coupled to surface energy induced elastic deformation. © 2018 American Physical Society.view abstract 10.1103/PhysRevB.97.035410 **Numerical Benchmark of Phase-Field Simulations with Elastic Strains: Precipitation in the Presence of Chemo-Mechanical Coupling**

Darvishi Kamachali, R. and Schwarze, C. and Lin, M. and Diehl, M. and Shanthraj, P. and Prahl, U. and Steinbach, I. and Raabe, D.*Computational Materials Science*155 (2018)Phase-field studies of solid-state precipitation under strong chemo-mechanical coupling are performed and benchmarked against the existing analytical solutions. The open source software packages OpenPhase and DAMASK are used for the numerical studies. Solutions for chemical diffusion and static mechanical equilibrium are investigated individually followed by a chemo-mechanical coupling effect arising due to composition dependence of the elastic constants. The accuracy of the numerical solutions versus the analytical solutions is quantitatively discussed. For the chemical diffusion benchmark, an excellent match, with a deviation <0.1%, was obtained. For the static mechanical equilibrium benchmark Eshelby problem was considered where a deviation of 5% was observed in the normal component of the stress, while the results from the diffuse interface (OpenPhase) and sharp interface (DAMASK) models were slightly different. In the presence of the chemo-mechanical coupling, the concentration field around a static precipitate was benchmarked for different coupling coefficients. In this case, it is found that the deviation increases proportional to the coupling coefficient that represents the strength of coupling concentration and elastic constants. Finally, the interface kinetics in the presence of the considered chemo-mechanical coupling were studied using OpenPhase and a hybrid OpenPhase–DAMASK implementation, replacing the mechanical solver of OpenPhase with DAMASK's. The observed deviations in the benchmark studies are discussed to provide guidance for the use of these results in studying further phase transformation models and implementations involving diffusion, elasticity and chemo-mechanical coupling effect. © 2018 Elsevier B.V.view abstract 10.1016/j.commatsci.2018.09.011 **Phase-field modeling of pores and precipitates in polycrystalline systems**

Kundin, J. and Sohaib, H. and Schiedung, R. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*26 (2018)In this work, we develop an efficient phase-field approach to simulate the grain growth in polycrystalline ceramic materials in the presence of pores with various mobilities and diffusion coefficients. The multi-phase-field model is coupled to the Cahn-Hilliard equation for pore dynamics by interaction functions which describe the interaction of pores with grain boundaries. Two types of the model are suggested with one and two order parameters responsible for the pores. We also show that the model can be applied to the simulation of the interaction of the grain boundaries with coherent and non-coherent particles. The parameters of the model allow us to reproduce the equilibrium dihedral angle in the triple-junction of a pore or a particle and a grain boundary. A drag velocity of the grain boundary in the presence of pores or precipitates was also measured for various diffusion coefficients and grain boundary mobilities. The effects of the pore dynamics on the grain size evolution in ceramic materials was investigated and compared with reported theoretical predictions and experimental data. © 2018 IOP Publishing Ltd.view abstract 10.1088/1361-651X/aacb94 **Rejuvenation of Single-Crystal Ni-Base Superalloy Turbine Blades: Unlimited Service Life?**

Ruttert, B. and Horst, O. and Lopez-galilea, I. and Langenkämper, D. and Kostka, A. and Somsen, C. and Goerler, J.V. and Ali, M.A. and Shchyglo, O. and Steinbach, I. and Eggeler, G. and Theisen, W.*Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science*(2018)Rejuvenation of the initially hot isostatic pressing (HIP) heat-treated single-crystal Ni-base superalloy (SX) ERBO/1 was examined experimentally and via phase field simulation to establish rejuvenation treatments as a cost-effective alternative for another interval of service life. Creep was performed at 950 °C and 350 MPa, and the specimens were crept to 0.6 pct (creep rate minimum) or 2 pct strain, respectively. A slight coarsening of the γ/γ′ microstructure was observed experimentally and via simulation at 0.6 pct and rafting at 2 pct strain. The damaged microstructures were rejuvenated in a novel hot isostatic press that provides fast quenching rates before the same specimens were recrept under the same initial creep conditions. High-resolution microscopy proves that the rejuvenation re-establishes the original γ/γ′ microstructure in the dendrite core of the precrept specimens (0.6 and 2 pct). However, the interdendritic areas of the 2 pct precrept and rejuvenated specimen still contain elongated γ′ particles enwrapped by interfacial dislocation networks that survived the applied rejuvenation. The subsequent experimental and simulated creep tests after rejuvenation demonstrated that the creep behavior is only reproducible by the proposed rejuvenation for specimens that had crept until the end of the primary creep regime. © 2018 The Minerals, Metals & Materials Society and ASM Internationalview abstract 10.1007/s11661-018-4745-6 **Multi-phase-field model for surface and phase-boundary diffusion**

Schiedung, R. and Kamachali, R.D. and Steinbach, I. and Varnik, F.*Physical Review E*96 (2017)The multi-phase-field approach is generalized to treat capillarity-driven diffusion parallel to the surfaces and phase boundaries, i.e., the boundaries between a condensed phase and its vapor and the boundaries between two or multiple condensed phases. The effect of capillarity is modeled via curvature dependence of the chemical potential whose gradient gives rise to diffusion. The model is used to study thermal grooving on the surface of a polycrystalline body. Decaying oscillations of the surface profile during thermal grooving, postulated by Hillert long ago but reported only in few studies so far, are observed and discussed. Furthermore, annealing of multi-nanoclusters on a deformable free surface is investigated using the proposed model. Results of these simulations suggest that the characteristic craterlike structure with an elevated perimeter, observed in recent experiments, is a transient nonequilibrium state during the annealing process. © 2017 American Physical Society.view abstract 10.1103/PhysRevE.96.012801 **On the evolution of cast microstructures during processing of single crystal Ni-base superalloys using a Bridgman seed technique**

Hallensleben, P. and Schaar, H. and Thome, P. and Jöns, N. and Jafarizadeh, A. and Steinbach, I. and Eggeler, G. and Frenzel, J.*Materials and Design*128 (2017)The present work takes a new look at a modified Bridgman process (Bridgman seed technique, BST) for the production of laboratory Ni-base single crystal (SX) superalloy cylinders of 12/120 mm diameter/length. This type of specimen is needed to perform inexpensive parametric studies for the development of new SX and for understanding the evolution of microstructures during SX casting. During melting, the seed partially melts back. The elementary segregation processes cause a so far unknown type of constitutional heating/cooling. Competitive growth eventually establishes a constant average dendrite spacing. In the present work it is documented how this dendrite spacing varies in one cylindrical ingot, and how it scatters when a series of SX ingots is produced. This type of information is scarce. The calculated temperature gradient across the solid/liquid interface (calculated by FEM) is in excellent agreement with predictions from the Kurz-Fisher equation which yields a dendrite spacing based on the experimental withdrawal rate and the microstructurally determined average dendrite spacing. The presence of small angle grain boundaries on cross sections which were taken perpendicular to the solidification direction can be rationalized on the basis of small deviations from the ideal growth directions of individual primary dendrites. © 2017 Elsevier Ltdview abstract 10.1016/j.matdes.2017.05.001 **On the numerical evaluation of local curvature for diffuse interface models of microstructure evolution**

Vakili, S. and Steinbach, I. and Varnik, F.*Procedia Computer Science*108 (2017)Within diffuse interface models for multiphase problems, the curvature of the phase boundary can be expressed as the difference of two terms, a Laplacian and a second, gradient, term of the diffuse interface variable, ø. In phase field simulations of microstructure evolution, the second term is often replaced by f'(ø) = df/dø, where f(ø) is the potential function in the free energy functional of the underlying physical model. We show here that this replacement systematically deteriorates the accuracy in local curvature evaluation as compared to a discretized evaluation of the second term. Analytic estimates reveal that the discretization errors in the Laplacian and in the second term have roughly the same spatial dependence across the interface, thus leading to a cancellation of errors in k. This is confirmed in a test case, where the discretization error can be determined via comparison to the exact solution. If, however, the second term is replaced by a quasi exact expression, the error in δø enters k without being compensated and can obscure the behavior of the local curvature. Due to the antisymmetric variations of this error term, approaches using the average curvature, as obtained from an integral along the interface normal, are immune to this problem. © 2017 The Authors. Published by Elsevier B.V.view abstract 10.1016/j.procs.2017.05.256 **Parallel multiphase field simulations with OpenPhase**

Tegeler, M. and Shchyglo, O. and Kamachali, R.D. and Monas, A. and Steinbach, I. and Sutmann, G.*Computer Physics Communications*215 (2017)The open-source software project OpenPhase allows the three-dimensional simulation of microstructural evolution using the multiphase field method. The core modules of OpenPhase and their implementation as well as their parallelization for a distributed-memory setting are presented. Especially communication and load-balancing strategies are discussed. Synchronization points are avoided by an increased halo-size, i.e. additional layers of ghost cells, which allow multiple stencil operations without data exchange. Load-balancing is considered via graph-partitioning and sub-domain decomposition. Results are presented for performance benchmarks as well as for a variety of applications, e.g. grain growth in polycrystalline materials, including a large number of phase fields as well as Mg–Al alloy solidification. Program summary Program Title: OpenPhase Program Files doi: http://dx.doi.org/10.17632/2mnv2fvkkk.1 Licensing provisions: GPLv3 Programming language: C++ Nature of problem: OpenPhase[1] allows the simulation of microstructure evolution during materials processing using the multiphase field method. In order to allow an arbitrary number of phase fields active parameter tracking is used, which can cause load imbalances in parallel computations. Solution method: OpenPhase solves the phase field equations using an explicit finite difference scheme. The parallel version of OpenPhase provides load-balancing using over-decomposition of the computational domain and graph-partitioning. Adaptive sub-domain sizes are used to minimize the computational overhead of the over-decomposition, while allowing appropriate load-balance. Additional comments including Restrictions and Unusual features: The distributed-memory parallelism in OpenPhase uses MPI. Shared-memory parallelism is implemented using OpenMP. The library uses C++11 features and therefore requires GCC version 4.7 or higher. [1] www.openphase.de © 2017 Elsevier B.V.view abstract 10.1016/j.cpc.2017.01.023 **Quantum-Phase-Field Concept of Matter: Emergent Gravity in the Dynamic Universe**

Steinbach, I.*Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences*72 (2017)A monistic framework is set up where energy is the only fundamental substance. Different states of energy are ordered by a set of scalar fields. The dual elements of matter, mass and space, are described as volume-and gradient-energy contributions of the set of fields, respectively. Time and space are formulated as background-independent dynamic variables. The evolution equations of the body of the universe are derived from the first principles of thermodynamics. Gravitational interaction emerges from quantum fluctuations in finite space. Application to a large number of fields predicts scale separation in space and repulsive action of masses distant beyond a marginal distance. The predicted marginal distance is compared to the size of the voids in the observable universe. © 2017 Walter de Gruyter GmbH, Berlin/Boston.view abstract 10.1515/zna-2016-0270 **Topological phase inversion after long-term thermal exposure of nickel-base superalloys: Experiment and phase-field simulation**

Goerler, J.V. and Lopez-Galilea, I. and Mujica Roncery, L. and Shchyglo, O. and Theisen, W. and Steinbach, I.*Acta Materialia*124 (2017)Ni-base superalloys are materials which are designed to resist extreme thermal and mechanical conditions. In this regard, an essential factor is their microstructure consisting of γ′ precipitates embedded in a γ matrix. The application of superalloys at high temperatures can however induce the topological phase inversion, where the γ′-phase topologically becomes the matrix phase, resulting in subpar material properties. In this work, the topological inversion is analyzed via experiment and phase-field simulation. The evolution of the microstructure has been quantified in the second generation single crystal Ni-base superalloy ERBO/1, which belongs to the family of CMSX-4, submitted to long-term aging at 1100° C for up to 250 h. Phase-field simulations carried out using a multi phase-field approach deliver insight into the microstructure evolution driven by the loss of coherency of the γ′ precipitates, which is induced by the accumulation of dislocations at the γ/γ′ interfaces. The obtained simulation results are in good agreement with the experimental results, and indicate that the mechanisms causing the topological inversion are linked to the accommodation of the lattice misfit, which enables coalescence and ripening of γ′ precipitates. © 2016 Acta Materialia Inc.view abstract 10.1016/j.actamat.2016.10.059 **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 **Microstructure design of tempered martensite by atomistically informed full-field simulation: From quenching to fracture**

Borukhovich, E. and Du, G. and Stratmann, M. and Boeff, M. and Shchyglo, O. and Hartmaier, A. and Steinbach, I.*Materials*9 (2016)Martensitic steels form a material class with a versatile range of properties that can be selected by varying the processing chain. In order to study and design the desired processing with the minimal experimental effort, modeling tools are required. In this work, a full processing cycle from quenching over tempering to mechanical testing is simulated with a single modeling framework that combines the features of the phase-field method and a coupled chemo-mechanical approach. In order to perform the mechanical testing, the mechanical part is extended to the large deformations case and coupled to crystal plasticity and a linear damage model. The quenching process is governed by the austenite-martensite transformation. In the tempering step, carbon segregation to the grain boundaries and the resulting cementite formation occur. During mechanical testing, the obtained material sample undergoes a large deformation that leads to local failure. The initial formation of the damage zones is observed to happen next to the carbides, while the final damage morphology follows the martensite microstructure. This multi-scale approach can be applied to design optimal microstructures dependent on processing and materials composition. © 2016 by the authors.view abstract 10.3390/ma9080673 **Modeling of Gibbs energies of pure elements down to 0 K using segmented regression**

Roslyakova, I. and Sundman, B. and Dette, H. and Zhang, L. and Steinbach, I.*Calphad: Computer Coupling of Phase Diagrams and Thermochemistry*55 (2016)A novel thermodynamic modeling strategy of stable solid alloy phases is proposed based on segmented regression approach. The model considers several physical effects (e.g. electronic, vibrational, etc.) and is valid from 0 K up to the melting temperature. The preceding approach has been applied for several pure elements. Results show good agreement with experimental data at low and high temperatures. Since it is not a first attempt to propose a “universal” physical-based model down to 0 K for the pure elements as an alternative to current SGTE description, we also compare the results to existing models. Analysis of the obtained results shows that the newly proposed model delivers more accurate description down to 0 K for all studied pure elements according to several statistical tests. © 2016 Elsevier Ltdview abstract 10.1016/j.calphad.2016.09.001 **Phase field modeling of intercalation kinetics: A finite interface dissipation approach**

Zerihun, N.A. and Steinbach, I.*MRS Communications*6 (2016)When two materials interact, the processes between the phases determine the functional properties of the compound. Pivotal interface phenomena are diffusion and redistribution of atoms (molecules). This is especially of interest in Lithium ion batteries where the interfacial kinetics determines the battery performance and impact cycling stability. A new phase field model, which links the atomistic processes at the interface to the mesoscale transport by a redistribution flux controlled by the so called 'interface permeability' was developed. The model was validated with experimental data from diffusion couples. Calculations of the concentration profiles of the species at the electrode-electrolyte interface are reported. Active particle size, morphology and spatial arrangement were put in correlation with diffusion behavior for use in reverse engineering. © Materials Research Society 2016.view abstract 10.1557/mrc.2016.31 **Phase-field simulation of liquid phase migration in the WC-Co system during liquid phase sintering**

Cheng, K. and Zhang, L. and Schwarze, C. and Steinbach, I. and Du, Y.*International Journal of Materials Research*107 (2016)Liquid phase sintering is a process for forming high performance, multiple-phase components from powders. The process includes very complex interactions between various mass transportation phenomena, among which the liquid phase migration represents an important one in the aspect of forming a gradient structure in cemented carbide. In the present work, phase-field simulation of the liquid phase migration phenomenon during liquid phase sintering is performed in the WC - Co based cemented carbide. The simulation results are analyzed and compared with the experimentally determined key factors of microstructural evolution, such as contiguity and liquid phase migration rate. The diffusion-controlled solution-precipitation mechanism of the liquid phase migration process in the cemented carbide system is confirmed from the current simulation result, which provides deeper understanding of the microstructural evolution during the liquid phase migration process. These simulations can offer guidance in preventing the liquid phase migration process during liquid phase sintering of cellular cemented carbide. © Carl Hanser Verlag GmbH & Co. KG.view abstract 10.3139/146.111353 **Phase-field study of zener drag and pinning of cylindrical particles in polycrystalline materials**

Schwarze, C. and Darvishi Kamachali, R. and Steinbach, I.*Acta Materialia*106 (2016)Zener drag and pinning in composites reinforced with cylindrical particles is investigated using three-dimensional phase-field simulations. Detailed systematic studies clarify the effect of relative orientation of the particle and length/diameter ratio on the kinetics of drag. It is shown that a combination of local equilibrium at junctions in contact with the particles, initial driving force of the migrating grain boundaries, and configuration of the particles within the polycrystalline matrix determine the intensity and persistence of drag and pinning effects. © 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2015.10.045 **Response to "Comment on 'Viscous coalescence of droplets: A lattice Boltzmann study'" [Phys. Fluids 28, 079101 (2016)]**

Gross, M. and Steinbach, I. and Raabe, D. and Varnik, F.*Physics of Fluids*28 (2016)view abstract 10.1063/1.4958828 **Divorced Eutectic Solidification of Mg-Al Alloys**

Monas, A. and Shchyglo, O. and Kim, S.-J. and Yim, C.D. and Höche, D. and Steinbach, I.*JOM*67 (2015)We present simulations of the nucleation and equiaxed dendritic growth of the primary hexagonal close-packed α-Mg phase followed by the nucleation of the β-phase in interdendritic regions. A zoomed-in region of a melt channel under eutectic conditions is investigated and compared with experiments. The presented simulations allow prediction of the final properties of an alloy based on process parameters. The obtained results give insight into the solidification processes governing the microstructure formation of Mg-Al alloys, allowing their targeted design for different applications. © 2015, The Minerals, Metals & Materials Society.view abstract 10.1007/s11837-015-1418-4 **Dual-scale phase-field simulation of Mg-Al alloy solidification**

Monas, A. and Shchyglo, O. and Höche, D. and Tegeler, M. and Steinbach, I.*IOP Conference Series: Materials Science and Engineering*84 (2015)Phase-field simulations of the nucleation and growth of primary α-Mg phase as well as secondary, β-phase of a Mg-Al alloy are presented. The nucleation model for α- and β-Mg phases is based on the "free growth model" by Greer et al.. After the α-Mg phase solidification we study a divorced eutectic growth of α- and β-Mg phases in a zoomed in melt channel between α-phase dendrites. The simulated cooling curves and final microstructures of α-grains are compared with experiments. In order to further enhance the resolution of the interdendritic region a high-performance computing approach has been used allowing significant simulation speed gain when using supercomputing facilities. © Published under licence by IOP Publishing Ltd.view abstract 10.1088/1757-899X/84/1/012069 **From wetting to melting along grain boundaries using phase field and sharp interface methods**

Sai Pavan Kumar Bhogireddy, V. and Hüter, C. and Neugebauer, J. and Shchyglo, O. and Steinbach, I. and Spatschek, R.*Computational Materials Science*108 (2015)Abstract We investigate the ability of a multi-order parameter phase field model with obstacle potentials to describe grain boundary premelting in equilibrium situations. In agreement with an energetic picture we find that the transition between dry and wet grain boundaries at the bulk melting point is given by the threshold 2σsl=σgb, with σsl being the solid-melt interfacial energy and σgb the energy of a dry grain boundary. The predictions for premelting are confirmed by simulations using the phase field package OpenPhase. For the prediction of the kinetics of melting along grain boundaries in pure materials, taking into account the short ranged interactions which are responsible for the grain boundary premelting, a sharp interface theory is developed. It confirms that for overheated grain boundaries the melting velocity is reduced (increased) for non-wetting (wetting) grain boundaries. Numerical steady state predictions are in agreement with a fully analytical solution in a subset of the parameter space. Phase field simulations confirm the predictions of the sharp interface theory. © 2015 Elsevier B.V.view abstract 10.1016/j.commatsci.2015.02.040 **Gamma-channel stabilization mechanism in Ni-base superalloys**

Goerler, J.V. and Brinckmann, S. and Shchyglo, O. and Steinbach, I.*Philosophical Magazine Letters*95 (2015)A mechanism is presented which opposes coalescence of γ′-precipitates in Ni-base superalloys. The mechanism is based on the non-linear behaviour of the elastic energy in γ-channels, caused by the misfit strain between matrix and precipitate, as a function of the channel width. Variation of the channel width causes a disjoining pressure dependent on the density of misfit dislocations. © 2015 Taylor & Francis.view abstract 10.1080/09500839.2015.1109716 **Geometrical grounds of mean field solutions for normal grain growth**

Darvishi Kamachali, R. and Abbondandolo, A. and Siburg, K.F. and Steinbach, I.*Acta Materialia*90 (2015)The classical mean field approach for normal grain growth in polycrystalline materials is revisited. We re-drive and study possible self-similar solutions and show that the grain size distribution can be determined only by the geometry of neighbouring grains for any given configuration. In three dimensions, it is shown that a single index 〈R〉2/〈R2〉 can represent the geometrical characteristic of grains and has a one-to-one relationship with the mean field parameter γ. We reinvestigate the results of our recent phase-field study [Darvishi Kamachali R, Steinbach I. Acta Mater 2012;60:2719] in the light of new analytical results and found a value γ≈3.5-3.2 for the stable regime. © 2015 Acta Materialia Inc. All rights reserved.view abstract 10.1016/j.actamat.2015.02.025 **Incorporating the CALPHAD sublattice approach of ordering into the phase-field model with finite interface dissipation**

Zhang, L. and Stratmann, M. and Du, Y. and Sundman, B. and Steinbach, I.*Acta Materialia*88 (2015)A new approach to incorporate the sublattice models in the CALPHAD (CALculation of PHAse Diagram) formalism directly into the phase-field formalism is developed. In binary alloys, the sublattice models can be classified into two types (i.e., "Type I" and "Type II"), depending on whether a direct one-to-one relation between the element site fraction in the CALPHAD database and the phase concentration in the phase-field model exists (Type I), or not (Type II). For "Type II" sublattice models, the specific site fractions, corresponding to a given mole fraction, have to be established via internal relaxation between different sublattices. Internal minimization of sublattice occupancy and solute evolution during microstructure transformation leads, in general, to a solution superior to the separate solution of the individual problems. The present coupling technique is validated for Fe-C and Ni-Al alloys. Finally, the model is extended into multicomponent alloys and applied to simulate the nucleation process of VC monocarbide from austenite matrix in a steel containing vanadium. © 2014 Acta Materialia Inc.view abstract 10.1016/j.actamat.2014.11.037 **Large deformation framework for phase-field simulations at the mesoscale**

Borukhovich, E. and Engels, P.S. and Mosler, J. and Shchyglo, O. and Steinbach, I.*Computational Materials Science*108 (2015)Abstract A large-strain plasticity framework is set up for phase-field simulations at the mesoscopic scale. The approach is based on an Eulerian setting with remeshing after each time step to keep a fixed structured mesh. Rotations, as evaluated from the antisymmetric part of the deformation gradient tensor, are integrated to capture the process history. Special emphasis is also given to the homogenization of the diffuse interface region to ensure the Hadamard jump condition and 2-dimensional scaling of the interface. The approach is applied to deformation of a polycrystal. © 2015 Elsevier B.V.view abstract 10.1016/j.commatsci.2015.06.021 **Modeling the flow in diffuse interface methods of solidification**

Subhedar, A. and Steinbach, I. and Varnik, F.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*92 (2015)Fluid dynamical equations in the presence of a diffuse solid-liquid interface are investigated via a volume averaging approach. The resulting equations exhibit the same structure as the standard Navier-Stokes equation for a Newtonian fluid with a constant viscosity, the effect of the solid phase fraction appearing in the drag force only. This considerably simplifies the use of the lattice Boltzmann method as a fluid dynamics solver in solidification simulations. Galilean invariance is also satisfied within this approach. Further, we investigate deviations between the diffuse and sharp interface flow profiles via both quasiexact numerical integration and lattice Boltzmann simulations. It emerges from these studies that the freedom in choosing the solid-liquid coupling parameter h provides a flexible way of optimizing the diffuse interface-flow simulations. Once h is adapted for a given spatial resolution, the simulated flow profiles reach an accuracy comparable to quasiexact numerical simulations. © 2015 American Physical Society.view abstract 10.1103/PhysRevE.92.023303 **Primary combination of phase-field and discrete dislocation dynamics methods for investigating athermal plastic deformation in various realistic Ni-base single crystal superalloy microstructures**

Gao, S. and Kumar Rajendran, M. and Fivel, M. and Ma, A. and Shchyglo, O. and Hartmaier, A. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*23 (2015)Three-dimensional discrete dislocation dynamics (DDD) simulations in combination with the phase-field method are performed to investigate the influence of different realistic Ni-base single crystal superalloy microstructures with the same volume fraction of γ;precipitates on plastic deformation at room temperature. The phase-field method is used to generate realistic microstructures as the boundary conditions for DDD simulations in which a constant high uniaxial tensile load is applied along different crystallographic directions. In addition, the lattice mismatch between the γand γ;phases is taken into account as a source of internal stresses. Due to the high antiphase boundary energy and the rare formation of superdislocations, precipitate cutting is not observed in the present simulations. Therefore, the plastic deformation is mainly caused by dislocation motion in γ; matrix channels. From a comparison of the macroscopic mechanical response and the dislocation evolution for different microstructures in each loading direction, we found that, for a given γ;phase volume fraction, the optimal microstructure should possess narrow and homogeneous γ; matrix channels. © 2015 IOP Publishing Ltd Printed in the UK.view abstract 10.1088/0965-0393/23/7/075003 **Simulations of the Eutectic Transformations in the Platinum–Carbon System**

Monas, A. and Bloembergen, P. and Dong, W. and Shchyglo, O. and Steinbach, I.*International Journal of Thermophysics*36 (2015)In this paper, we present the simulation of the eutectic phase transitions in the Pt–C system, in terms of both freezing and melting, using the multi-phase-field model. The experimentally obtained heat-extraction and -injection rates associated with the induction of freezing and melting are converted into the corresponding rates for microstructure-scale simulations. In spite of the extreme differences in the volume fractions of the FCC–Pt-rich phase on the one hand and graphite (C) on the other, satisfactory results for the kinetics of solidification and melting have been obtained, involving reasonable offsets in temperature, inducing freezing and melting, with respect to the equilibrium eutectic temperature. For freezing in the simulations, the needle/rod-like morphology, as experimentally observed, was reproduced for different heat extraction rates. The seemingly anomalous peak characterizing the simulated freezing curves is ascribed to the speed up of the solidification process due to the curvature effect. Similarly, a peak is observed in the experimental freezing curves, also showing up more clearly with increasing freezing rates. Melting was simulated starting from a frozen structure produced by a freezing simulation. The simulations reproduce the experimental melting curves and, together with the simulated freezing curves, help to understand the phase transition of the Pt–C eutectic. Finally, the effect of metallic impurities was studied. As shown for Au, impurities affect the morphology of the eutectic structure, their impact increasing with the impurity content, i.e., they can act as modifiers of the structure, as earlier reported for irregular eutectics. © 2015, Springer Science+Business Media New York.view abstract 10.1007/s10765-015-1999-8 **Texture evolution in deformed AZ31 magnesium sheets: Experiments and phase-field study**

Darvishi Kamachali, R. and Kim, S.-J. and Steinbach, I.*Computational Materials Science*104 (2015)Experimental and phase-field studies are performed to investigate mechanisms of preferential growth which lead to improved formability in AZ31 Mg sheets. A compression/annealing treatment is specialized to modify the initial texture in thin sheets. The texture and stress states of materials are studied via electron back scattered diffraction (EBSD) technique before and after annealing. Using the EBSD data on microstructure and residual stresses, a phase-field model is constructed to simulate the texture evolution after initial compression. The results suggest that the residual-stresses induced by in-plane compression are the main driving force for recrystallization and grain growth. The inhomogeneous stress distribution leads to preferential growth of {21¯1¯0} texture along the normal to the sheet, which are at lowest stress state, at the expense of initial basal texture. Limited mobility of twin boundaries changes the mixture of textures but the non-basal textures are still preferred. The formability tests confirm a significant enhancement of the final product compared to as-received sheets. © 2015 Elsevier B.V. All rights reserved.view abstract 10.1016/j.commatsci.2015.04.006 **Applications of scale-bridging to computational materials design**

Drautz, R. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*22 (2014)view abstract 10.1088/0965-0393/22/3/030201 **DFT-supported phase-field study on the effect of mechanically driven fluxes in Ni4Ti3 precipitation**

Kamachali, R.D. and Borukhovich, E. and Hatcher, N. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*22 (2014)Formation of the Ni4Ti3 precipitate has a strong effect on the shape memory properties of NiTi alloys. In this work, growth of this precipitate is studied using phase-field modelling and density functional theory (DFT) calculations. Using first-principles calculations, the composition-dependent stability and elastic properties of the B2 phase are obtained. Composition-dependent elastic constants are incorporated into our phase-field model to investigate the interplay between stress and concentration fields around the precipitate. The model introduces a source of diffusion due to mechanical relaxation which is accompanied by local softening/hardening of the B2 phase. The results are discussed in light of previous experimental and simulation studies. © 2014 IOP Publishing Ltd.view abstract 10.1088/0965-0393/22/3/034003 **Large scale 3-D phase-field simulation of coarsening in Ni-base superalloys**

Rajendran, M.K. and Shchyglo, O. and Steinbach, I.*MATEC Web of Conferences*14 (2014)In this study we present a large scale numerical simulation of γ-γ′ microstructure evolution in Ni-base superalloy using the multi-phase field method in three dimensions. We numerically simulated precipitation hardening heat treatment cycles. Large scale three dimensional simulations are necessary in order to get sufficient statistics for predicting the morphological evolution, average γ′ precipitate size, precipitates size distribution over time and ripening exponent for a given temperature and composition. A detailed analysis of obtained result is presented emphasising the effect of elastic interaction on the coarsening kinetics in Ni-base superalloy. The study is performed using the phase-field modelling library "OpenPhase" which is based on a multi-phase field multi-component model. © 2014 Owned by the authors, published by EDP Sciences.view abstract 10.1051/matecconf/20141411001 **Large strain elasto-plasticity for diffuse interface models**

Borukhovich, E. and Engels, P.S. and Böhlke, T. and Shchyglo, O. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*22 (2014)Most solid-state phase transformations are accompanied by large deformations, stemming either from external load, transformation strains or plasticity. The consideration of such large deformations will affect the numerical treatment of such transformations. In this paper, we present a new scheme to embed large deformations in an explicit phase-field scheme and its implementation in the open-source framework OpenPhase. The suggested scheme combines the advantages of a spectral solver to calculate the mechanical boundary value problem in a small strain limit and an advection procedure to transport field variables over the calculation grid. Since the developed approach should be used for various sets of problems, e.g. simulations of thermodynamically driven phase transformations, the mechanic formulation is kept general. However, to ensure compatibility with phase-field methods using the concept of diffuse interface, the latter is treated with special care in the present work. © 2014 IOP Publishing Ltd.view abstract 10.1088/0965-0393/22/3/034008 **Phase-field modeling for 3D grain growth based on a grain boundary energy database**

Kim, H.-K. and Kim, S.G. and Dong, W. and Steinbach, I. and Lee, B.-J.*Modelling and Simulation in Materials Science and Engineering*22 (2014)A 3D phase-field model for grain growth combined with a grain boundary (GB) energy database is proposed. The phase-field model is applied to a grain growth simulation of polycrystalline bcc Fe to investigate the effect of anisotropic GB energy on the microstructural evolution and its kinetics. It is found that the anisotropy in the GB energy results in different microstructures and slower kinetics, especially when the portion of low-angle, low-energy GBs is large. We discuss the applicability of the proposed phase-field simulation technique, based on the GB or interfacial energy database to simulations for microstructural evolution, including abnormal grain growth, phase transformations, etc., in a wider range of polycrystalline materials. © 2014 IOP Publishing Ltd.view abstract 10.1088/0965-0393/22/3/034004 **Phase-field modeling of grain-boundary premelting using obstacle potentials**

Bhogireddy, V.S.P.K. and Hüter, C. and Neugebauer, J. and Steinbach, I. and Karma, A. and Spatschek, R.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*90 (2014)We investigate the multiorder parameter phase field model of Steinbach and Pezzolla [Physica D 134, 385 (1999)PDNPDT0167-278910.1016/S0167-2789(99)00129-3] concerning its ability to describe grain boundary premelting. For a single order parameter situation solid-melt interfaces are always attractive, which allows us to have (unstable) equilibrium solid-melt-solid coexistence above the bulk melting point. The temperature-dependent melt layer thickness and the disjoining potential, which describe the interface interaction, are affected by the choice of the thermal coupling function and the measure to define the amount of the liquid phase. Due to the strictly finite interface thickness the interaction range also is finite. For a multiorder parameter model we find either purely attractive or purely repulsive finite-ranged interactions. The premelting transition is then directly linked to the ratio of the grain boundary and solid-melt interfacial energy. © 2014 American Physical Society.view abstract 10.1103/PhysRevE.90.012401 **Tertiary dendritic instability in late stage solidification of Ni-based superalloys**

Franke, M.M. and Singer, R.F. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*22 (2014)Derivatives of the commercial alloy CMSX-4 were directionally solidified and characterized with respect to their final dendrite microstructure. The results indicate that Ni-based superalloys with high segregation levels show significant instability in secondary dendrite arms and an increased tendency for tertiary arm formation, respectively. Phase-field simulations were used to explore the impact of chemical composition on morphological instability and tertiary arm formation during the directional solidification of Ni-based superalloys. It is found that an increase in specific alloying elements in the overall alloy composition leads to pronounced segregation at the end of solidification. This causes strong growth restriction of the secondary arms and triggers tertiary arm formation. The proposed mechanism explains experimental microstructures found in modifications of the base alloy CMSX-4. © 2014 IOP Publishing Ltd.view abstract 10.1088/0965-0393/22/2/025026 **A permeation model for the electrochemical interface**

Preiss, U. and Borukhovich, E. and Alemayehu, N. and Steinbach, I. and LaMantia, F.*Modelling and Simulation in Materials Science and Engineering*21 (2013)We show the transferability of the recently introduced concept of permeation from the context of finite dissipation in simple metallic interfaces to much more complicated electrochemical interfaces. The phenomenological bridge is formed by the exchange current, which can be measured by either impedance spectroscopy or by cyclic voltammetry. In a proof-of-concept phase field model, Nernst-Planck diffusion and transport of charged species in a potential gradient as the solution of the Poisson equation are considered. It is shown that charges build up on the outer electrode surface in a fashion resembling the electrochemical double layer. © 2013 IOP Publishing Ltd.view abstract 10.1088/0965-0393/21/7/074006 **Diffuse-interface modeling of solute trapping in rapid solidification: Predictions of the hyperbolic phase-field model and parabolic model with finite interface dissipation**

Zhang, L. and Danilova, E.V. and Steinbach, I. and Medvedev, D. and Galenko, P.K.*Acta Materialia*61 (2013)Two recently developed phase-field models, a hyperbolic model and a parabolic model with finite interface dissipation, are employed to study the solute trapping in a Si-0.25 at.% As alloy during rapid solidification. The hyperbolic model is applied at the nanometer scale of the interface width δ. The parabolic model is derived by a coarse-graining procedure and is intended to operate with mesoscopic resolution of the interface η. The coarse-graining numerical parameters, namely interface width η and the interface permeability P, are adjusted in the parabolic model to fit the segregation coefficient calculated by the microscopic model on the nanoscale. Based on the optimal sets of η and P selected at small interface velocity, a linear relation between their logarithm values is obtained. This logarithmic relation provides a theoretical basis for choosing the appropriate values of η and P in the numerical phase-field simulation in three spatial dimensions. © 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2013.03.042 **Phase-field model for microstructure evolution at the mesoscopic scale**

Steinbach, I.*Annual Review of Materials Research*43 (2013)This review presents a phase-field model that is generally applicable to homogeneous and heterogeneous systems at the mesoscopic scale. Reviewed first are general aspects about first- and second-order phase transitions that need to be considered to understand the theoretical background of a phase field. The mesoscopic model equations are defined by a coarse-graining procedure from a microscopic model in the continuum limit on the atomic scale. Special emphasis is given to the question of how to separate the interface and bulk contributions to the generalized thermodynamic functional, which forms the basis of all phase-field models. Numerical aspects of the discretization are discussed at the lower scale of applicability. The model is applied to spinodal decomposition and ripening in Ag-Cu with realistic thermodynamic and kinetic data from a database. © Copyright © 2013 by Annual Reviews. All rights reserved.view abstract 10.1146/annurev-matsci-071312-121703 **Simulating mobile dendrites in a flow**

Medvedev, D. and Varnik, F. and Steinbach, I.*Procedia Computer Science*18 (2013)We propose a scheme for simulation of the solute-driven dendritic solidification which accounts for the flows of liquid and motion of growing dendrites. The scheme is based on the multiphase-field method for calculating the solidification and the lattice Boltzmann method to simulate the fluid flows. Motion and rotation of solid grains is possible. © 2013 The Authors. Published by Elsevier B.V.view abstract 10.1016/j.procs.2013.05.431 **Simulation of viscous sintering using the lattice Boltzmann method**

Varnik, F. and Rios, A. and Gross, M. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*21 (2013)The viscous flow is one of the important mass transport mechanisms in sintering powder materials such as porous glass. The main underlying driving force here is the tendency of the system to minimize the overall surface free energy. In this work, we first address the fundamental problem of the coalescence of two suspending drops and provide an alternative derivation of Frenkel's formula by explicitly taking account of driving forces which arise from the curvature of the contact zone. The result thus obtained is compared with non-ideal (two phase) fluid lattice Boltzmann simulations. Furthermore, we use this simulation tool to address the more complex case of the sintering of a compact of viscous spherical particles. We observe a significantly faster densification dynamics than in experiments but obtain qualitative agreement upon rescaling the unit of time. We attribute this faster dynamics in simulations to the absence of intergranular friction and discuss a possible improvement of the model. We also propose, for the present LB model, a simple way of imposing shear flow to determine the time dependence of the effective viscosity during the sintering process. The method is benchmarked via a simple analytic case and first simulation results are presented for situations for which no analytic theory exists. © 2013 IOP Publishing Ltd.view abstract 10.1088/0965-0393/21/2/025003 **Solutal gradients in strained equilibrium**

Darvishi Kamachali, R. and Borukhovich, E. and Shchyglo, O. and Steinbach, I.*Philosophical Magazine Letters*93 (2013)We demonstrate that the distortion of a crystal, caused by secondary phase precipitates, can stabilize a solutal gradient around the precipitate. The gradient persists in the quasi-static state stabilized by the gradient of the elastic energy around the precipitate. The peak concentration at the interface between precipitate and matrix hereby is independent of the radius of the precipitate and no mechanism of ripening is active in an arrangement of precipitates of different size. The model offers an explanation of experimental observations of the anomalous stability of nano-precipitates in Al-Cu. © 2013 Taylor & Francis.view abstract 10.1080/09500839.2013.847288 **Viscous coalescence of droplets: A lattice Boltzmann study**

Gross, M. and Steinbach, I. and Raabe, D. and Varnik, F.*Physics of Fluids*25 (2013)The coalescence of two resting liquid droplets in a saturated vapor phase is investigated by Lattice Boltzmann simulations in two and three dimensions. We find that, in the viscous regime, the bridge radius obeys a t1/2-scaling law in time with the characteristic time scale given by the viscous time. Our results differ significantly from the predictions of existing analytical theories of viscous coalescence as well as from experimental observations. While the underlying reason for these deviations is presently unknown, a simple scaling argument is given that describes our results well. © 2013 AIP Publishing LLC.view abstract 10.1063/1.4803178 **Why solidification? Why phase-field?**

Steinbach, I.*JOM*65 (2013)"Solidification" is a branch of pattern formation in theoretical physics. "Phase-field" is an applied tool in engineering. This strange combination of basic and applied research is reviewed against its historical background: a story of failure and success. The main achievements in both fields are highlighted, and future perspectives are briefly discussed. © 2013 The Author(s).view abstract 10.1007/s11837-013-0681-5 **3-D phase-field simulation of grain growth: Topological analysis versus mean-field approximations**

Darvishi Kamachali, R. and Steinbach, I.*Acta Materialia*60 (2012)The characteristics of 3-D grain growth are investigated by a topological analysis of phase-field simulation results compared with theoretical mean-field theories. We found that the size distribution of the grains starting from an arbitrary narrow distribution crosses the self-similar Hillert distribution, and ends in a distribution with relatively longer tails of large grains in which the central peak shifted towards smaller grain size. The distribution of topological classes, as characterized by the number of facets per grain, is found to be time-invariant for the process as a whole. The obtained shape function is in good agreement with the analytical distribution function derived based on the average N-hedron model [Rios PR, Glicksman ME. Act Mater 2008;56:1165]. The volumetric growth rate per topological class also correlates well with the analytical approach obtained by Mullins [Mullins WW. Acta Mater 1956;3:900]. The relationship between grain size and its shape, however, deviates from theoretical predictions. © 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2012.01.037 **Microsegregation and secondary phase formation during directional solidification of the single-crystal Ni-based superalloy LEK94**

Lopez-Galilea, I. and Huth, S. and Fries, S.G. and Warnken, N. and Steinbach, I. and Theisen, W.*Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science*43 (2012)A multicomponent phase-field method coupled to thermodynamic calculations according to the CALPHAD method was used to simulate microstructural evolution during directional solidification of the LEK94 commercial single-crystal Ni-based superalloy using a two-dimensional unit cell approximation. We demonstrate quantitative agreement of calculated microsegregation profiles and profiles determined from casting experiments as well as calculated fraction solid curves with those determined in differential thermal analysis (DTA) measurements. Finally, the role of solidification rate on dendrite morphology and precipitation of the secondary phases is investigated and a new measure of the dendrite morphology is presented to quantify the effect of back diffusion on the amount of secondary phases. © 2012 The Minerals, Metals & Materials Society and ASM International.view abstract 10.1007/s11661-012-1327-x **Pearlite revisited**

Steinbach, I. and Plapp, M.*Continuum Mechanics and Thermodynamics*24 (2012)Zener's model of pearlite transformation in steels can be viewed as the prototype of many microstructure evolution models in materials science. It links principles of thermodynamics and kinetics to the scale of the microstructure. In addition it solves a very practical problem: How the hardness of steel is correlated to the conditions of processing. Although the model is well established since the 1950s, quantitative explanation of growth kinetics was missing until very recently. The present paper will shortly review the classical model of pearlite transformation. Zener's conjecture of maximum entropy production will be annotated by modern theoretical and experimental considerations of a band of stable (sometimes oscillating) states around the state of maximum entropy production. Finally, an explanation of the growth kinetics observed in experiments is proposed based on diffusion fluxes driven by stress gradients due to large transformation strain. © Springer-Verlag 2011.view abstract 10.1007/s00161-011-0204-y **Phase-field model with finite interface dissipation**

Steinbach, I. and Zhang, L. and Plapp, M.*Acta Materialia*60 (2012)In rapid phase transformations, interfaces are often driven far from equilibrium, and the chemical potential may exhibit a jump across the interface. We develop a model for the description of such situations in the framework of the phase-field formalism, with separate concentration fields in each phase. The key novel feature of this model is that the two concentration fields are linked by a kinetic equation which describes the exchange of components between the phases, instead of an equilibrium partitioning condition. The associated rate constant influences the interface dissipation. For rapid exchange between the phases, the chemical potentials are equal in both coexisting phases at the interface as in previous models, whereas in the opposite limit strong non-equilibrium behavior can be modeled. This is illustrated by simulations of a diffusion couple and of solute trapping during rapid solidification. © 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2012.01.035 **Phase-field model with finite interface dissipation: Extension to multi-component multi-phase alloys**

Zhang, L. and Steinbach, I.*Acta Materialia*60 (2012)A previously developed phase-field model with finite interface dissipation for binary dual-phase alloys out of chemical equilibrium is generalized to a multi-component multi-phase model in the framework of the multi-phase-field formalism, allowing the description of multiple junctions with an arbitrary number of phases and components. In multiple junctions, each phase concentration is assigned to a dynamic equation to account for finite interface dissipation, and its formulation is proposed in two different models. The overall mass conservation between the phases of a multiple junction is used in model I, whereas the concentrations of each pair of phases have to be conserved during the transformations for model II. Both models demonstrate the decomposition of the nonlinear interactions between different phases into pairwise interaction of phases in multiple junctions. They converge to the same equilibrium state while in non-equilibrium states different predictions are given. © 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2012.02.032 **An analytical study of the static state of multi-junctions in a multi-phase field model**

Guo, W. and Spatschek, R. and Steinbach, I.*Physica D: Nonlinear Phenomena*240 (2011)We investigate the properties of the multi-order parameter phase field model of Steinbach and Pezzolla [I. Steinbach, F. Pezzolla, A generalized field method for multi-phase transformations using interface fields, Physica D 134 (1999) 385393] with respect to the behavior in triple and higher order junctions. From the structure of this model, it was speculated that "dynamical" solutions may exist in the triple junction, which could lead to a violation of Young's law. Here we confirm analytically recent numerical simulations showing that such dynamical states do not exist, and that an equilibrium solution therefore does indeed correspond to a minimum of the free energy; this implies that Young's law must be satisfied in the framework of the model. We show that Young's law is a consequence of the interface kinetic equilibrium and not due to a mechanical force balance, in agreement with earlier predictions [C. Caroli, C. Misbah, On static and dynamical Young's condition at a trijunction, J. Phys. I France 7 (1997) 12591265]. © 2010 Elsevier B.V. All rights reserved.view abstract 10.1016/j.physd.2010.09.014 **Contact angle dependence of the velocity of sliding cylindrical drop on flat substrates**

Moradi, N. and Varnik, F. and Steinbach, I.*EPL*95 (2011)Via numerical simulations, we investigate the dynamics of cylindrical drops on flat substrates. In agreement with the common understanding, in the limit of Stokes flow and negligible drop deformation, the drop's centre-of-mass velocity scales linearly with the applied force and with the second power of drop radius. In this paper, we go a step further and perform a detailed study of dissipation loss inside the drop. An important result is that the dominant part of viscous dissipation arises from the region below the drop's centre of mass. Based on this observation, we propose a simple analytic model which allows capturing the dependence of the steady-state drop velocity on the equilibrium contact angle. Simulation results are in good agreement with the predictions of this model. © 2011 Europhysics Letters Association.view abstract 10.1209/0295-5075/95/44003 **Morphologies of small droplets on patterned hydrophobic substrates**

Moradi, N. and Gross, M. and Varnik, F. and Zikos, G. and Steinbach, I.*Modelling and Simulation in Materials Science and Engineering*19 (2011)For situations, in which the size of a droplet is comparable to the roughness scale of the solid substrate, we explore possible wetting morphologies on patterned hydrophobic substrates and investigate their dependence on the initial droplet position, droplet volume and the surface geometry. For a regular array of cubical pillars, small perturbations of a symmetric droplet state are restored by capillary forces. Larger deviations, on the other hand, may lead to completely new morphologies. Our studies also suggest that the previously reported 'reentrant transition' upon quasi-static evaporation (a transition from the suspended state to partial penetration and then back to the suspended state) (Gross, et al. 2009 Europhys. Lett. 88 26002) is not restricted to a symmetric initial state but may occur for quite non-symmetric morphologies as well. In contrast, a change in the substrate geometry may lead to a completely different behavior, fully precluding the reentrant transition. This underlines the importance of substrate design for the use of reentrant transition as a self-cleaning mechanism. © 2011 IOP Publishing Ltd.view abstract 10.1088/0965-0393/19/4/045005 **On the effect of superimposed external stresses on the nucleation and growth of Ni 4Ti 3 particles: A parametric phase field study**

Guo, W. and Steinbach, I. and Somsen, C. and Eggeler, G.*Acta Materialia*59 (2011)The effect of a superimposed stress on the coarsening of interacting Ni 4Ti 3 particles is studied using the multi-phase field method. It is found that the thickness/diameter ratio of a Ni 4Ti 3 particle in a (1 1 1) B2 plane increases with an increasing [1 1 1] B2 stress component. The particle shape can change from a disk to a sphere with increasing applied stress. It is also found that diffusional and mechanical interactions between two Ni 4Ti 3 particles can promote the nucleation of new particles. This provides an explanation for the autocatalytic nature of nucleation reported previously. Compressive stresses along [1 1 1] B2 increase the volume fraction and growth velocity of the Ni 4Ti 3 particles of the (1 1 1) B2 plane. Misoriented particles disappear during particle growth. The simulation results are discussed in the light of previous experimental results. © 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2011.02.002 **Phase-field modelling of microstructure evolution in solids: Perspectives and challenges**

Steinbach, I. and Shchyglo, O.*Current Opinion in Solid State and Materials Science*15 (2011)Phase-field modelling is maturing to become a universal tool for modelling microstructure evolution in materials science. In solidification applications it has been proven to give quantitative predictions. In solid state, however, the mechanisms of phase transformation and microstructure evolution in are much more involved due to the existence of mechanical interactions, high interface anisotropies, large densities of defects, and retarded kinetics of diffusion and growth. The paper gives an overview of actual developments in phase-field modelling of solid-state microstructure evolution and highlights necessary directions of future development in order to meet the challenge of quantitative predictions. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.cossms.2011.01.001 **Phase-field simulation of diffusion couples in the Ni-Al system**

Zhang, L. and Steinbach, I. and Du, Y.*International Journal of Materials Research*102 (2011)By linking thermodynamic and atomic mobility databases with two-dimensional phase-field simulation, the evolution of interdiffusion microstructures in a series of Ni-Al diffusion couples associated with the γ, γ', and b-phases was studied. The formation and subsequent growth of the γ'- phase layer in β/γ and γ' + β/γ diffusion couples reproduced the experimental observations well. Moreover, the effect of coherent strain on the γ - γ' microstructural evolution, as well as that of an external compressive force on the γ + γ'/γ + γ' diffusion couple, was investigated. The phase-field simulated concentration profiles of some of the Ni-Al diffusion couples were also compared with the corresponding experimental data and the results of one-dimensional DICTRA (DIffusion Controlled TRAnsformations) simulations. A discussion of the rafting direction was also made by comprehensively comparing the phase-field simulations with the predicted results from an elastic model. © Carl Hanser Verlag GmbH & Co. KG.view abstract 10.3139/146.110493 **Simulation of the external pressure influence on the micro-Structural evolution of a Single Crystal Ni-Based superalloy**

Lopez-Galilea, I. and Huth, S. and Fries, S.G. and Steinbach, I. and Theisen, W.*Advanced Materials Research*278 (2011)The phase field method has been applied to simulate the microstructural evolution of a commercial single crystal Ni-based superalloy during both, HIP and annealing treatments. The effects of applying high isostatic pressure on the microstructural evolution, which mainly retards the diffusion of the alloying elements causing the loss of the orientational coherency between the phases is demonstrated by the simulation and experimental results. © (2011) Trans Tech Publications, Switzerland.view abstract 10.4028/www.scientific.net/AMR.278.247 **Stability and dynamics of droplets on patterned substrates: Insights from experiments and lattice Boltzmann simulations**

Varnik, F. and Gross, M. and Moradi, N. and Zikos, G. and Uhlmann, P. and Müller-Buschbaum, P. and Magerl, D. and Raabe, D. and Steinbach, I. and Stamm, M.*Journal of Physics Condensed Matter*23 (2011)The stability and dynamics of droplets on solid substrates are studied both theoretically and via experiments. Focusing on our recent achievements within the DFG-priority program 1164 (Nano-and Microfluidics), we first consider the case of (large) droplets on the so-called gradient substrates. Here the term gradient refers to both a change of wettability (chemical gradient) or topography (roughness gradient). While the motion of a droplet on a perfectly flat substrate upon the action of a chemical gradient appears to be a natural consequence of the considered situation, we show that the behavior of a droplet on a gradient of topography is less obvious. Nevertheless, if care is taken in the choice of the topographic patterns (in order to reduce hysteresis effects), a motion may be observed. Interestingly, in this case, simple scaling arguments adequately account for the dependence of the droplet velocity on the roughness gradient (Moradi et al 2010 Europhys. Lett. 8926006). Another issue addressed in this paper is the behavior of droplets on hydrophobic substrates with a periodic arrangement of square shaped pillars. Here, it is possible to propose an analytically solvable model for the case where the droplet size becomes comparable to the roughness scale (Gross et al 2009 Europhys. Lett. 8826002). Two important predictions of the model are highlighted here. (i)There exists a state with a finite penetration depth, distinct from the full wetting (Wenzel) and suspended (Cassie-Baxter, CB) states. (ii)Upon quasi-static evaporation, a droplet initially on the top of the pillars (CB state) undergoes a transition to this new state with a finite penetration depth but then (upon further evaporation) climbs up the pillars and goes back to the CB state again. These predictions are confirmed via independent numerical simulations. Moreover, we also address the fundamental issue of the internal droplet dynamics and the terminal center of mass velocity on a flat substrate. © 2011 IOP Publishing Ltd.view abstract 10.1088/0953-8984/23/18/184112 **Atomic mobilities and diffusivities in the fcc, L12 and B2 phases of the Ni-Al system**

Zhang, L. and Du, Y. and Chen, Q. and Steinbach, I. and Huang, B.*International Journal of Materials Research*101 (2010)A phenomenological model was utilized to describe diffusivities in the γ (fcc)/γ′ (L12) and A2/B2 phases of the Ni-Al system. An effective strategy, which takes the homogeneity range and defect concentration into account, was developed in the present work to optimize the atomic mobilities of γ′ phase. Such a strategy results in a dramatic decrease in the number of atomic mobility parameters to be evaluated for the L12 phase. The measured composition-and temperature-dependent diffusivities in the Ni-Al system have been well replicated by the present mobility descriptions. For the L12 phase, comprehensive comparisons show that with fewer model parameters the presently obtained mobilities yield a better fit to experimental diffusivities, compared with previous assessments. The mobility descriptions are further validated by comparing calculated and measured concentration profiles for various diffusion couples. The time-dependent Al composition profile for the annealed vapor Al/γ couple is accurately described for the first time. © Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282.view abstract 10.3139/146.110428 **Diffusivities of an Al-Fe-Ni melt and their effects on the microstructure during solidification**

Zhang, L. and Du, Y. and Steinbach, I. and Chen, Q. and Huang, B.*Acta Materialia*58 (2010)A systematical investigation of the diffusivities in an Al-Fe-Ni melt was presented. Based on the experimental and theoretical data about diffusivities, the temperature- and composition-dependent atomic mobilities were evaluated for the elements in Al-Ni, Al-Fe, Fe-Ni and Al-Fe-Ni melts via an effective approach. Most of the reported diffusivities can be reproduced well by the obtained atomic mobilities. In particular, for the first time the ternary diffusivity of the liquid in a ternary system is described in conjunction with the established atomic mobilities. The effect of the atomic mobilities in a liquid on microstructure and microsegregation during solidification was demonstrated with one Al-Ni binary alloy. The simulation results indicate that accurate databases of mobilities in the liquid phase are much needed for the quantitative simulation of microstructural evolution during solidification by using various approaches, including DICTRA and the phase-field method. © 2010 Acta Materialia Inc.view abstract 10.1016/j.actamat.2010.03.002 **Efficient and reliable finite element techniques for phase field models**

Stiemer, M. and Grosse-Wohrmann, A. and Gladkov, S. and Svendsen, B. and Spatschek, R. and Steinbach, I.*International Journal of Materials Research*101 (2010)In this work auto-adaptive finite-element techniques are presented that allow for quantitatively reliable numerical computation of phase field models. These techniques are based on goal oriented error estimation for finite elements with dual weighted residuals and algorithms for auto-adaptive mesh adaptation in space and in time. It is discussed for which kind of problems the computational overhead of such methods is expected to be justified by a sufficient reduction of the problem size.view abstract 10.3139/146.110301 **Modeling of hot ductility during solidification of steel grades in continuous casting - Part I**

Senk, D. and Stratemeier, S. and Böttger, B. and Göhler, K. and Steinbach, I.*Advanced Engineering Materials*12 (2010)The present paper gives an overview of the simultaneous research work carried out by RWTH Aachen University and ThyssenKrupp Steel Europe AG. With a combination of sophisticated simulation tools and experimental techniques it is possible to predict the relations between temperature distribution in the mould, solidification velocity, chemical steel composition and, furthermore, the mechanical properties of the steel shell. Simulation results as well as experimentally observed microstructure parameters are used as input data for hot tearing criteria. A critical choice of existing hot tearing criteria based on different approaches, like critical strain and critical strain rate, are applied and developed. The new "damage model" is going to replace a basic approach to determine hot cracking susceptibility in a mechanical FEM strand model for continuous slab casting of ThyssenKrupp Steel Europe AG. Critical strains for hot cracking in continuous casting were investigated by in situ tensile tests for four steel grades with carbon contents in the range of 0.036 and 0.76 wt%. Additionally to modeling, fractography of laboratory and industrial samples was carried out by SEM and EPMA and the results are discussed. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/adem.201000021 **Modelling of hot ductility during solidification of steel grades in continuous casting - Part II**

Böttger, B. and Stratemeier, S. and Subasic, E. and Göhler, K. and Steinbach, I. and Senk, D.*Advanced Engineering Materials*12 (2010)In continuous casting, the probability of hot cracks developing strongly depends on the local solidification process and the microstructure formation. In ref. [1], an integrative model for hot cracking of the initial solid shell is developed. This paper focuses on solidification modelling, which plays an important role in the integrated approach. Solidification is simulated using a multiphase-field model, coupled online to thermodynamic and diffusion databases and using an integrated 1D temperature solver to describe the local temperature field. Less-complex microsegregation models are discussed for comparison. The results are compared to EDX results from strand samples of different steel grades. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/adem.201000022 **Multi-phase field study of the equilibrium state of multi-junctions**

Guo, W. and Steinbach, I.*International Journal of Materials Research*101 (2010)Multiple junctions between different grains of a pure material are investigated using the multi-phase field model in 2D and 3D. According to theoretical considerations of the multi-phase field model equations, there may be quasi-static solutions that depend on the ratio of the interface mobilities. Numerical calculations in 2D and 3D indicate that the system always converges to the static solution as described by Young's law independent of the interface mobilities. No quasi-static solutions are found, which is attributed to the necessity of continuous solutions within the interface region of a phase-field model. The effect of interface mobility on the dynamics of the interface angle is discussed. © 2010 Carl Hanser Verlag.view abstract 10.3139/146.110298 **Multiscale simulations on the grain growth process in nanostructured materials**

Kamachali, R.D. and Hua, J. and Steinbach, I. and Hartmaier, A.*International Journal of Materials Research*101 (2010)In this work, multi-phase field and molecular dynamics simulations have been used to investigate nanoscale grain growth mechanisms. Based on experimental observations, the combination of grain boundary expansion and vacancy diffusion has been considered in the multi-phase field model. The atomistic mechanism of boundary movement and the free volume redistribution during the growth process have been investigated using molecular dynamics simulations. According to the multi-phase field results, linear grain growth in nanostructured materials at low temperature can be explained by vacancy diffusion in the stress field around the grain boundaries. Molecular dynamics simulations confirm the observation of linear grain growth for nanometresized grains. The activation energy of grain boundary motion in this regime has been determined to be of the order of onetenth of the self-diffusion activation energy, which is consistent with experimental data. Based on the simulation results, the transition from linear to normal grain growth is discussed in detail and a criterion for this transition is proposed. © Carl Hanser Verlag GmbH & Co. KG.view abstract 10.3139/146.110419 **Phase-field model with plastic flow for grain growth in nanocrystalline material**

Steinbach, I. and Song, X. and Hartmaier, A.*Philosophical Magazine*90 (2010)A phase-field model is presented which considers the accumulation of structural defects in grain boundaries by an isotropic eigenstrain associated with the grain boundaries. It is demonstrated that the elastic energy caused by dilatation of the grain boundary with respect to the bulk crystal contributes largely to the grain boundary energy. The sign of this contribution can be both positive and negative dependent on the local stress state in the grain boundary. Self-diffusion of atoms is taken into account to relax the stress caused by the dilatation of the grain boundary. Application of the model to discontinuous grain growth in pure nanocrystalline cobalt material is presented. Linear grain growth is found in the nanocrystalline state, which is explained by the interpretation of grain boundary motion as a diffusive process defining an upper limit of the grain boundary velocity independent of the grain boundary curvature but dependent on temperature. The transition to regular grain growth at a critical temperature, as observed experimentally, is explained by the drop of theoretical grain boundary velocity due to its mean curvature during coarsening of the nanograin structure below the maximum velocity.view abstract 10.1080/14786430903074763 **Roughness-gradient-induced spontaneous motion of droplets on hydrophobic surfaces: A lattice Boltzmann study**

Moradi, N. and Varnik, F. and Steinbach, I.*EPL*89 (2010)The effect of a step-wise change in the pillar density on the dynamics of droplets is investigated via three-dimensional lattice Boltzmann simulations. For the same pillar density gradient but different pillar arrangements, both motion over the gradient zone as well as complete arrest are observed. In the moving case, the droplet velocity scales approximately linearly with the texture gradient. A simple model is provided reproducing the observed linear behavior. The model also predicts a linear dependence of droplet velocity on surface tension. This prediction is clearly confirmed via our computer simulations for a wide range of surface tensions. Copyright © 2010 EPLA.view abstract 10.1209/0295-5075/89/26006 **Small droplets on superhydrophobic substrates**

Gross, M. and Varnik, F. and Raabe, D. and Steinbach, I.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*81 (2010)We investigate the wetting behavior of liquid droplets on rough hydrophobic substrates for the case of droplets that are of comparable size to the surface asperities. Using a simple three-dimensional analytical free-energy model, we have shown in a recent letter that, in addition to the well-known Cassie-Baxter and Wenzel states, there exists a further metastable wetting state where the droplet is immersed into the texture to a finite depth, yet not touching the bottom of the substrate. Due to this new state, a quasistatically evaporating droplet can be saved from going over to the Wenzel state and instead remains close to the top of the surface. In the present paper, we give an in-depth account of the droplet behavior based on the results of extensive computer simulations and an improved theoretical model. In particular, we show that releasing the assumption that the droplet is pinned at the outer edges of the pillars improves the analytical results for larger droplets. Interestingly, all qualitative aspects, such as the existence of an intermediate minimum and the "reentrant transition," remain unchanged. We also give a detailed description of the evaporation process for droplets of varying sizes. Our results point out the role of droplet size for superhydrophobicity and give hints for achieving the desired wetting properties of technically produced materials. © 2010 The American Physical Society.view abstract 10.1103/PhysRevE.81.051606

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

#### SPP 1713: Chemomechanics: Strong coupling of thermo-chemical and thermo-mechanical states in applied materials (concluded)

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

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

#### Stabilität von Produktionsprozessen am Beispiel metallischer Halbzeuge (concluded)

#### diffusion

#### kinetics

#### microstructure

#### modelling and simulation

#### multiscale modelling

#### phase transformations

#### phase-field

#### precipitation

#### solidification

#### thermodynamics