Search

The tool of phase-field modeling for the prediction of chemical as well as microstructural evolution during crystallization from a melt in a mineralogical system has been developed in this work. We provide a compact theoretical background and introduce new aspects such as the treatment of anisotropic surface energies that are essential for modeling mineralogical systems. These are then applied to two simple model systems - the binary olivine-melt and plagioclase-melt systems - to illustrate the application of the developed tools. In one case crystallization is modeled at a constant temperature and undercooling while in the other the process of crystallization is tracked for a constant cooling rate. These two examples serve to illustrate the capabilities of the modeling tool. The results are analyzed in terms of crystal size distributions (CSD) and with a view toward applications in diffusion chronometry; future possibilities are discussed. The modeling results demonstrate that growth at constant rates may be expected only for limited extents of crystallization, that breaks in slopes of CSD-plots should be common, and that the lifetime of a given crystal of a phase is different from the lifetime of this phase in a magmatic system. The last aspect imposes an inherent limit to timescales that may be accessed by diffusion chronometry. Most significantly, this tool provides a bridge between CSD analysis and diffusion chronometry - two common tools that are used to study timescales of magmatic processes.

Journal of materials research and technology : jmr&t 9(3), 3805-3816 (2020). doi:10.1016/j.jmrt.2020.02.007, Published by Elsevier, Rio de Janeiro

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.

Recent advances in the field of diffusion in multiprincipal element systems are critically reviewed, with an emphasis on experimental as well as theoretical approaches to determining atomic mobilities (tracer diffusion coefficients) in chemically complex multicomponent systems. The newly elaborated and augmented pseudobinary and pseudoternary methods provide a rigorous framework to access tracer, intrinsic, and interdiffusion coefficients in alloys with an arbitrary number of components. Utilization of the novel tracer-interdiffusion couple method allows for a high-throughput determination of composition-dependent tracer diffusion coefficients. A combination of these approaches provides a unique experimental toolbox to access diffusivities of elements that do not have suitable tracers. The pair-exchange diffusion model, which gives a consistent definition of diffusion matrices without specifying a reference element, is highlighted. Density-functional theory–informed calculations of basic diffusion properties—asrequired for the generation of extensive mobility databases for technological applications—are also discussed.

In additive manufacturing processes, solidification velocities are extremely high in comparison to ordinary directional solidification. Therefore, the dependencies of the primary dendrite arm spacing (PDAS) on the process parameters deviate from the dependencies predicted by standard analytical methods. In this work, we investigate the microstructure evolution and element distribution in Fe-18.9Mn and Fe-18.5Mn-Al-C alloys solidified during the selective laser melting process. A quantitative multicomponent phase-field model verified by Green-function calculations (Karma, Rappel: Phys. Rev. E, 1998, 57, 4323) and the convergence analysis is used. The resulting non-standard dependencies of the PDAS on the process parameters in a wide range of solidification velocities are compared with analytical calculations. It is shown that the numerical values of the PDAS are similar to the values predicted by the Kurz–Fisher method for the low and intermediate solidification velocities and are smaller for the solidification velocities higher than 0.03 m/s. The PDAS and the Mn distribution in a Fe-18.5Mn-Al-C alloy are compared to the experimental results and a very good agreement is found.

Shape memory alloys (SMAs), such as Nitinol (i.e., NiTi), are of great importance in biomedical and engineering applications due to their unique superelasticity and shape memory properties. In recent years, additive manufacturing (AM) processes have been used to produce complex NiTi components, which provide the ability to tailor microstructure and thus the critical properties of the alloys, such as the superelastic behavior and transformation temperatures (TTs), by selection of processing parameters. In biomedical applications, superelasticity in implants play a critical role since it gives the implants bone-like behavior. In this study, a methodology of improving superelasticity in Ni-rich NiTi components without the need for any kind of post-process heat treatments will be revealed. It will be shown that superelasticity with 5.62% strain recovery and 98% recovery ratio can be observed in Ni-rich NiTi after the sample is processed with 250 W laser power, 1250 mm/s scanning speed, and 80 µm hatch spacing without, any post-process heat treatments. This superelasticity in as-fabricated Ni-rich SLM NiTi was not previously possible in the absence of post-process heat treatments. The findings of this study promise the fast, reliable and inexpensive fabrication of complex shaped superelastic NiTi components for many envisioned applications such as patient-specific biomedical implants.

Many regions of the Earth’s mantle deform in grain size-sensitive creep regimes. The grain size right below the transition zone is believed to be very small, and the grain size should in subsequent depths be mainly controlled by normal grain growth. The grain size evolution is commonly predicted using either existing grain growth laws in combination with grain boundary diffusion coefficients or by extrapolating empirically determined grain growth laws. Effects of Zenner pinning and different ratios of second phases have been studied, while the role of anisotropic grain boundary properties is currently neglected1. The grain boundary energy varies with the orientation of the grain boundary plane, as expressed through the typical crystal habitae (Wulff-shapes). Individual crystals in a polycrystalline material maintain a grain boundary energy anisotropy during grain growth. Here we study how grain boundary anisotropy impacts grain boundary migration and normal grain growth rates by three-dimensional phase-field simulations2. We imply grain boundary energy minimization by faceting/varying the grain boundary plane to minimize the grain boundary energy. The ideal grain boundary energy anisotropy for the solid-solid interface is taken from experimentally investigated grain boundary plane distributions and grain boundary energy distributions on periclase (MgO). We compare the grain size evolution in simulations with isotropic and anisotropic grain boundary energy of cubic crystal symmetry. We found that the grain boundary energy anisotropy has a significant influence on grain boundary migration and grain growth kinetics2.2- 0>2= ktThe change in grain size is given as variation between the initial and final average grain radius, r. The time is t, and a material-specific parameter that accounts for the grain boundary energy anisotropy is extracted from the simulations ask = A·µgb·σgbWhere, the grain boundary energy, σgb varies with orientation, and the grain boundary mobility, µgb assumed to be isotropic. We found that the rate of grain growth for periclase, A is a factor of 3 smaller compared to an isotropic material.These results are of three-fold importance: The theory of grain growth is developed for purely isotropic materials1and clearly fail to predict grain growth for materials with anisotropic grain boundary energy correctly. Predicting grain growth using existing rate-laws and grain boundary diffusion studies can lead to an incomplete conclusion and a distorted picture of the real phenomenon. Because energy reduction associated with grain growth is small, already small energy variations, such as those associated with grain boundary energy anisotropy, can have a significant effect. A better prediction of grain size evolution will need to develop an anisotropic theory for grain growth, that address our lake in knowledge regarding pressure effects on both grain boundary energy anisotropy and diffusion.1. Rohrer, G. S. Annu. Rev. Mater. Res. 20052. Salama et al. Acta Materialia 2020

The recently proposed pair-exchange concept for multicomponent diffusion is analyzed in detail. The concept defines the differences of chemical potential gradients of two elements as general driving forces for interdiffusion and defines the corresponding proportionality coefficients as pair-mobilities for atomic exchange fluxes of a pair of elements at the mesoscopic scale. The total fluxes of alloying elements are given as the sum over corresponding pair-contributions, which rely on an interdependent set of forces to maintain a meaningful symmetric form. Onsager's reciprocal relations are satisfied as well as the consistency between description of chemical- and tracer-diffusion. It is confirmed that a consistent definition of interdiffusion coefficients requires the fulfillment of the Gibbs-Duhem relation by the applied thermodynamic Gibbs energy description.

The mathematical model for intercalation dynamics in phase-separating materials (Singh et al., 2008) is a powerful tool for the investigation of the spinodal decomposition in nanoparticles. By means of this model, we conduct a careful mathematical analysis of the intercalation dynamics in nanoparticles to study the dependence of spinodal gap on the boundary reaction rate and the particle size, which can be used for LiFePO 4 battery material application. Consistent with previous investigations, we found that for some range of the boundary reaction rate and the particle size the concentration spinodal gap is not continuous, but it has stable “islands” where no spinodal decomposition is expected. The new important observation is that the presence of an infinitesimally small boundary reaction rate will destabilize nanoparticles even for infinitesimal length. In particular for nanoparticles having the size of order or less than interphase width λ , the spontaneous charge or discharge will occur at the reaction rate of order 0.1 D / λ . The further raise of the intercalation rate will stabilize the system until some size limit of order two diffusion length. The intercalation effects are proven by means of numerical simulations. We also show that the increasing enthalpy of the spinodal mixture as well as increasing elastic energy due to the lattice misfit can destabilize the particles and increase the spinodal gap.

Investigation of the heteroepitaxial film growth for anisotropic systems has been carried out by means of the anisotropic phase-field crystal model. Numerical simulations have been performed by varying the anisotropic parameters, orientations, and lattice misfit between isotropic substrate and anisotropic film layers. The simulation results demonstrate the formation of dislocations during the growth of layers with non-cubic anisotropic lattice in the presence of the elastic strain caused by the lattice misfits. It is shown that for any particular choice of misfit, both the lattice anisotropy of the film and the orientation of the substrate influence the parameters of dislocation formation such as the number of dislocations, the characteristic distance at which dislocations begin to nucleate, and the excess elastic energy.

In selective laser melting (SLM), temperature gradients and cooling rates are extremely large in comparison to the ordinary directional solidification process. Therefore, the standard analytical methods are not able to predict the dependency of the dendrite arm spacing on the process parameters correctly. In the current research, we use a quantitative multicomponent phase-field model to investigate the arm spacing during the SLM process taking into account the dependency of the tip undercooling on the solidification velocity. It is found that the precision of the phase-field method can be estimated by a stability parameter which is defined as a ratio of the numerical resolution to the solidification velocity and should be chosen larger than a critical value. We show that our developed results are in good agreement with the theoretically obtained ones based on Kurz–Fisher method. We investigate the microstructure evolution and component distribution in Fe–Mn–Al–C solidified alloy during SLM process. The arm spacing and the Mn distribution are in a very good agreement with the experimental results. Additionally, the resulting non-standard dependencies of the arm spacing on the process parameters are compared with analytical calculations, which show excellent agreement between predictions and experimental measurements.

We study different forms of linear and non-linear field equations, so-called `phase-field' equations, in relation to the de~Broglie-Bohm double solution program. This defines a framework in which elementary particles are described by localized non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schr\"odinger 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 which 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., Comment: Submitted to Zeitschrift f\"ur Naturforschung A

The development of accurate kinetic databases by parametrizing the composition and temperature dependence of elemental atomic mobilities, is essential for correct multicomponent calculations and simulations. In this work the automated assessment procedure for the establishment of CALPHAD-type kinetic databases is proposed, including the storage of raw data and assessment results, automatic weighting of data, parameter selection and automated reassessments. This allows the establishment of reproducible up-to-date databases. The proposed software, written in python, is applied to the assessment of a kinetic database for the fcc Co–Cr–Fe–Mn–Ni high entropy alloy using only tracer diffusion data for a sharp separation of thermodynamic and kinetic data. The established database is valid for the whole composition range of the five-component high entropy alloy.

The recently proposed pair-exchange diffusion model for multicomponent diffusion in a random alloy is analyzed in detail. The model defines the differences of chemical potential gradients of two elements as general driving forces for interdiffusion and the corresponding proportionality coefficients as pair-mobilities for atomic exchange fluxes of a pair of elements at the mesoscopic scale. The total fluxes of alloying elements are given as the sum over corresponding pair-contributions, which rely on a set of independent forces and maintain a meaningful symmetric form to satisfy Onsagers reciprocal relations. It is demonstrated that a consistent definition of interdiffusion coefficients requires the fulfillment of the Gibbs-Duhem relation by the applied thermodynamic Gibbs energy description. The expression for the marker velocity for multicomponent random alloy is derived in the pair-wise form as a sum of the differences of chemical potential gradients of two elements with the linear coefficients which are differences between corresponding mobilities. The application of the model to High Entropy Alloys is demonstrated.

The grain boundary diffusion and segregation can influence the grain growth kinetics, the grain size distribution, and therefore the mechanical properties of the ceramic matrix composites. The present paper proposes a phase-field modeling approach to simulate the grain growth in polycrystalline alumina fibers embedded in alumina matrix at temperatures above 1000 °C in presence of the grain boundary diffusion of dopants from the matrix to the fiber and vice versa. The multi-phase-field model for grain growth [I. Steinbach and F. Pezzolla, Phys. D, 134 (1999) 385] is extended by the incorporation of the grain boundary diffusion, grain boundary segregation model, and the dependence of the interface mobility on the segregation concentration. The kinetic parameters of the model which allow describing the real microstructure evolution were estimated by the comparison to the experimental measurements. The simulation and experimental results of the grain growth with the diffusion of dopants in Nextel 610 fibers show the significant effect of the grain boundary diffusion on the grain size distribution. The results of numerical tests were used to adjust the values of the grain boundary diffusion coefficients by the experimental data at different temperatures by means of an inverse method. From the simulations, the diffusion coefficient of Mg was estimated to be 6–7 times higher than that of Si.

Metal additive manufacturing (AM) improves the design flexibility of titanium aluminides (TiAl-based alloys) as a new class of high-temperature alloys towards widespread applications. In this work, the underlying mechanisms responsible for the site-specific thermal history and grain evolution during laser powder bed fusion (LPBF) of TiAl-based alloys are investigated through an integrated computational and experimental effort. In specific, a multiphysics modeling framework integrating a finite element thermal model with a highly efficient phase-field method is developed to simulate the solidification microstructure at different locations within the melt pool during LPBF processing. The investigation of process-microstructure relationship is accomplished using a Ti-45Al (at.%) alloy for a binary approximation, with a focus on site-specific primary dendrite arm spacing (PDAS) and non-equilibrium microsegregation. The microstructural sensitivity to spatial variations, individual processing parameters, and misorientation angle between the preferred crystalline orientation and the temperature gradient direction are studied to thoroughly understand the rapid solidification during LPBF. LPBF experiments are carried out to validate the modeling results in terms of melt pool dimensions and site-specific PDAS across the melt pool. The knowledge gained in this work will benefit the development of AM processing routine for fabrication of high-performance TiAl-based alloys towards extensive applications.

An extended non-steady-state model for the interaction between a solid particle and an advancing solid/liquid interface based on the dynamic model of Catalina et al. (Metall Mater Trans A 31:2559â2568, 2000) is used to calculate the critical velocities for the pushing/engulfment transition in Si-SiC system under microgravity and under normal gravity conditions. The aim of this study was to explain the abnormal behavior of the critical velocity in experiments. The simulations were carried out for two cases of the drag force formulation. The effects of the non-spherical form of the particles as well as the cluster formation were also taken into account. It is found that in the presence of the gravity force, the particles will be engulfed when the particle size exceeds a certain limit which does not depend on the choice of the drag force formulation.

We present a phase-field model for particle capture during directional solidification. Its predictions for critical growth velocities for particles of different sizes are compared with experimental results for capture of silicon carbide (SiC) particles during directional solidification of silicon. The phase-field model allows us to systematically test the influence of different assumptions about attractive and repulsive forces and the capture mechanisms, including the effects of particle shape and of partial engulfment of the particle by the interface. We identify common properties of models that show agreement with experiments, trying to determine the underlying physical effects by abductive inference. We find that predictions vary only slightly between models with different repulsive forces and that the shape of the particle can have a larger effect on the critical growth velocity than the exact nature of the repulsive force or the capture process. We assess to what extent a good description of experimental critical growth velocities implies that the model accurately describes the actual physical processes and propose additional ways to test the validity of models.

The present work proposes a phase-field approach to realistically simulate abnormal grain growth in polycrystalline ceramic fibers at temperatures above 1000 °C. Under these conditions, grain growth is characterized by the formation of large elongated rectangular grains in a matrix of normal grains. The multi-phase-field model is extended by mechanisms which are responsible for the abnormal anisotropic growth: anisotropic interface energy, anisotropic interface mobility and a recrystallization driving force. Two types of the mobility anisotropy are considered: anisotropy due to the misorientation between grains and anisotropy due to the dependency on inclination angles. The experimental data from heat treatments of the ceramic fiber Nextel 610 at 1200–1300 °C with different dwell times are examined by the proposed model. The comparison of the experiment and the simulation results at various times shows that the extended multi-phase-field model is able to simulate the microstructure realistically. In most model cases, abnormal grains have a rectangular shape, similar to the experiment. The best fit of the experimental grain size distribution and the grain shape is achieved by the simulation with the misotientation dependency of the interface mobility. Furthermore, it was shown that a strong decrease on the grain growth rate with time, observed in the experiment, can be reproduced by the simulations taking into account the segregation of impurities on grain boundaries that results in the decreasing grain boundary mobility.

The prediction of the equilibrium and metastable phase/grain morphologies during selective laser melting (SLM) of Ni-base superalloys can be accomplished using phase-field modeling (PF). In this paper, a model for binary and multi-component systems is presented and validated via the steady-state growth problem by comparison to theoretical predictions of the Green-function calculations and the Kurz–Fisher model. By means of the PF simulations, a phenomenological dependency of the growth velocity on undercooling during the rapid solidification is evaluated. It is found that the growth velocity does not achieve steady-state values for real process and material parameters. The final microstructure predicted by the PF is in good agreement with the experimental observations. Additionally, the simulations provide concentration profiles, which show a pronounced segregation of solutes to the dendrite core. The simulated microstructures and concentration fields can be used as inputs for the simulation of the precipitation of secondary phases.

The coarsening kinetics of γ′-precipitates in single crystalline Ni-based superalloys is studied using phase-field simulations. At first, we discuss interdiffusion-limited γ′-coarsening in technologically relevant superalloys with the explicit inclusion of up to nine chemical components. The simulations show that an additional influence from the coherency strain leads to a substantially faster coarsening-evolution compared to the predictions from the LSW-theory. Second, we perform a virtual experiment to determine the influence of varying rhenium (Re) and ruthenium (Ru) additions on the temporal evolution of the γ-γ′ microstructure under thermo-mechanical loads. We observe that a change in the Re content strongly alters the coarsening kinetics. On the contrary, it is found that Ru variations have no significant effect on the coarsening kinetics.

We study the contributions from solute segregation to rafting in CMSX4 and CMSX6 superalloys. For this purpose, we use a phase-field model that takes into account the cross interaction between all solutes in Ni-base superalloys. The thermodynamic formulation in the phase-field model is validated by comparing phase-field simulations to ThermoCalc equilibrium calculations and DICTRA sharp interface simulations. Additionally, an elastic term is coupled to the model to account for the misfit, elastic inhomogeneity and applied loading. A technique to quantify the kinetics of microstructures with anisotropic γ ′ -precipitates is integrated in the model. Our simulations reveal that the rafting process is considerably slower in CMSX4 than in CMSX6. The presence of slow diffusing elements in CMSX4, particularly the refractory element Re, plays a major role in reducing the rate of rafting. The results suggest that Re slows interface mobility by accumulating along the growth path, a behavior we attribute to its low diffusivity and low solubility in the γ ′ -precipitate.

We have investigated the microstructure evolution during the isothermal and non-isothermal solidification of ternary Al–Cu–Ni alloys by means of a general multi-phase-field model for an arbitrary number of phases. The stability requirements for the model functions on every dual interface guarantee the absence of “ghost” phases. The aim was to generate a realistic microstructure by coupling the thermodynamic parameters of the phases and the thermodynamically consistent phase-field evolution equations. It is shown that the specially constructed thermal noise terms disturb the stability on the dual interfaces and can produce heterogeneous nucleation of product phases at energetically favorable points. Similar behavior can be observed in triple junctions where the heterogeneous nucleation of a fourth phase is more favorable. Finally, the model predicts the growth of a combined eutectic-like and peritectic-like structure that is comparable to the observed experimental microstructure in various alloys.

Continuous models for the prediction of the grain growth and recrystallization kinetics can be roughly classified into two methods. One method is multi-phase-field approaches. The second method is the phase-field model of Kobayashi, Warren and Carter (KWC) with two or three order parameters that make the model more efficient and more promising for the description of polycrystals. The present work proposes an extension to the KWC model to incorporate the recrystallization process by means of an additional order parameter for recrystallized grains. The results of the simulation of the grain growth and recrystallization process obtained by this extended KWC model are compared to existing results in the literature obtained by the original KWC phase-field model. The procedure of the parameter estimation for a simple isotropic case (no orientation dependence of the surface energy and kinetic parameters) is suggested. The ability of the model to describe the grain boundaries with low misorientation and to incorporate the grain boundary diffusion with the impurity segregation is discussed.

The structure formation in peritectic Al-4.5at.%Cu-11at.%Ni ternary alloy with four-phase peritectic reaction was investigated using the quantitative phase-field model of eutectic growth. This model, extended to an arbitrary number of phases, guarantees the stability requirements on individual interfaces. The thermal noise terms disturb the stability and produce the heterogeneous nucleation of secondary phases in accordance to the energetic and concentration conditions. In our recent work it was shown that in differential thermal analysis (DTA) experiments specific microstructure parts in Al-4.5at.%Cu-11at.%Ni alloy with a four-phase peritectic reaction were observed, which cannot be explained by Scheil calculation or simple phase-field modeling. In this work, it was found by numerical experiments that, due to the formation of anisotropic quasi-primary Al3Ni2 crystals and the suppression of the nucleation of (Al) phase, the eutectic-like coupled growth of Al3Ni2 and Al3Ni phases can be observed. In addition, at further cooling the anisotropic shape of quasi-secondary Al3Ni crystals promotes the nucleation of the (Al) phase. The simulated final structure is comparable to the experimental one which is characterized by large Al3Ni2 crystals enveloped by the Al3Ni phase.

The investigation of solidification in ternary Al-Cu-Ni alloys is carried out by means of experiments and phase-field modeling. For three alloys in the Al-rich corner of the phase diagram differential thermal analysis (DTA) is performed. Then the alloys were analyzed using scanning electron microscopy with energy dispersive X-ray microanalysis. For the understanding of the general features of the alloy solidification quantitative phase-field simulations are carried out additionally to the theoretical Scheil calculation. It is found that many experimental DTA signals and the microstructure parts cannot be explained by simple Scheil calculation. We apply the multi-phase-field model previously developed for the simulation of peritectic reaction and extended it to three components and four-phases reactions. The main advantage of this model is the application of the equilibrium parameters evaluated from the refined free energies of the phases. It is shown that the simulated microstructure is comparable to the experimental one for two investigated alloys. The final phase fractions in the modeling correspond to the theoretical predictions of Scheil calculation but the time evolution of fractions is more complicated. In particular the kinetics (relation between tangential and normal growth velocities) of the peritectic-like reaction in ternary Al-Cu-Ni alloys shows differences compared to the peritectic reaction in binary Al-Ni alloys.

In this paper we present the application of an anisotropic phase-field crystal model to the process of the heteroepitaxial growth in anisotropic systems. During the growth of the thin films with sheared non-cubic lattice the elastic strain gives rise to dislocations and grain boundaries due to the difference between the film and the substrate lattice in terms of lattice constant (misfit) and lattice shearing (anisotropy difference). The numerical simulations for various misfits and anisotropy difference demonstrate that the anisotropic evolution equation for the phase-field variables allows to simulate the growth of the layers in anisotropic systems. We investigate this phenomenon and derive the relationships for the phenomenological dependencies such as the dependence of the characteristic layer thickness and the number of defects on the misfit for various anisotropy difference based on our recently developed anisotropic phase-field crystal model [28] .

A kinetic Monte Carlo (KMC) model coupled with the vapor diffusion above the Al-polar (0001) surface of AlN is constructed for the physical vapor transport (PVT) growth of AlN crystals. Most of the important surface events and the vapor diffusion of Al atoms are taken into account. Based on the numerical simulations, an analytical model of the step-flow growth on the (0001) surface is attained and the time evolution of random terrace widths under homogeneous and linearly inhomogeneous vapor fluxes of Al atoms is explored. Using the KMC model and the analytical model it is found that under the growing conditions in this work the rate limiting step for the PVT growth of AlN is the supply of Al atoms due to the tiny flow of Al atoms in the vapor phase (Alg) at the steady state. The energy barriers for adsorbed AlN (AlNad) incorporated at different configurations of neighboring AlN dimers can influence the growth morphology significantly. If the adsorption rate of Alg is much slower than the rates of the surface events, the step-bunching caused by the randomness of the terrace widths can be avoided under either the homogeneous or linearly inhomogeneous flux of Alg.

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.

A thermodynamically consistent method for the investigation of the coarsening behavior and in particular for the prediction of the secondary dendrite arm spacing (SDAS) in multi-component alloys is proposed which is based on the numerical simulation by means of a phase-field model. Existing variants of the phase-field model equations for multi-component systems were considered and their advantages and disadvantages were discussed. For the investigation of the coarsening behavior the variant described by the mixture composition and the entropy change was chosen. The method is applied to a high-alloy tool steel where it was found that elements such as C, Si, Mn decrease the SDAS whereas Cr increases. The resulting dependencies of the SDAS on alloy composition were compared to the analytical prediction of the coarsening model. For this aim the analytical model of the coarsening behavior in multi-component alloys [Rappaz and Boettinger, Acta Mater. 47 (1990)] was extended by taking into account the cross dependencies between the components in multi-component diffusion and the case of slow diffusion in the solid phase. The equilibrium parameters used in the phase-field model and in the analytical model were obtained from Thermo-Calc through global equilibrium calculations using the database TCFE7. The difference between both methods was found to be smaller than 2 pct in the investigated composition region.

We modified a multi-phase-field model previously developed by the authors and applied it to investigate the peritectic microstructure formation and corresponding kinetics in an Al–Ni alloy during directional solidification. The model presented uses the parameters derived directly from the free energy functions of individual phases. The thermodynamic driving force is treated as a function of the difference between the current diffusion potential of the solute on an interface and the corresponding equilibrium diffusion potential. The time evolution of the concentration field is simplified and is given by one term. The main advantage of the modified model arises through the reduced number of parameters depending on phase-field variables. The proposal model is then extended to multi-component systems. In this context two methods for the evaluation of the thermodynamic parameters are proposed. The resulting model can be employed for large scale parameter studies even for demanding multi-component systems.

The phase-field crystal (PFC) technique is a widely used approach for modeling crystal growth phenomena with atomistic resolution on mesoscopic time scales. We use a two-dimensional PFC model for a binary system based on the work of Elder et al. [Phys. Rev. B 75, 064107 (2007)PRBMDO1098-012110.1103/PhysRevB.75.064107] to study the effect of the curved, diffuse solid-liquid interface on the interfacial energy as well as the nucleation barrier. The calculation of the interfacial energy and the nucleation barrier certainly depends on the proper definition of the solid-liquid dividing surface and the corresponding nucleus size. We define the position of the sharp interface at which the interfacial energy is to be evaluated by using the concept of equimolar dividing surface (r^{e}) and the minimization of the interfacial energy (r^{s}). The comparison of the results based on both radii shows that the difference r^{e}-r^{s} is always positive and has a limit for large cluster sizes which is comparable to the Tolman length. Furthermore, we found the real nucleation barrier for small cluster sizes, which is defined as a function of the radius r^{s}, and compared it with the classical nucleation theory. The simulation results also show that the extracted interfacial energy as function of both radii is independent of system size, and this dependence can be reasonably described by the nonclassical Tolman formula with a positive Tolman length.

The eutectic microstructure formed during the solidification of a binary titanium–iron alloy (Ti–29.5 at.% Fe) has been simulated using the phase-field method. The model uses the chemical free energy contributions of phases with different thermodynamic factors. It is demonstrated that the simulated microstructure exhibits phenomena which are also observed during eutectic solidification in experiments. The obtained microstructure consist of a combination of circular and lamellar phase structures. The growth behavior and growth morphology of the phases are found to be a strong function of the front undercooling and the temperature gradient in the system. The simulated dependency of the steady state lamellar width versus the front undercooling corresponds to the theoretical prediction following from the “Jackson–Hunt” model. It is also shown that the relation between the mobilities of the phases strongly affects the eutectic microstructure.

A general formulation is presented to derive the equation of motion and demonstrate thermodynamic consistency for several classes of phase-field (PF) and PF crystal (PFC) models. It can be applied to models with a conserved and non-conserved phase-field variable, describing either locally uniform or periodic stable states, and containing slow as well as fast thermodynamic variables. The approach is based on an entropy functional formalism previously developed in the context of PF models for locally uniform states [P. Galenko and D. Jou, Phys. Rev. E 71 (2005) p.046125] and thus allows to extend several properties of the latter to PF models for periodic states, i.e., PFC models.

Recently, we derived a generalized model for isotropic as well as anisotropic crystal lattice systems of arbitrary Poisson's ratio within the framework of the continuum phase-field crystal (PFC) approach [R. Prieler, J. Hubert, D. Li, B. Verleye, R. Haberkern, H. Emmerich, J. Phys.: Condens. Matter 21 (2009) p.464110] and showed how its parameters can be derived from classical density functional theory [M.A. Choudhary, D. Li, H. Emmerich and H. Lowen, J. Phys.: Condens. Matter 23 (2011) p.265005]. Here, we present a general procedure to model anisotropic material systems of arbitrary Poisson's ratios. In that way we can for the first time identify PFC solutions of arbitrary Poisson's ratios and thereby extend the applicability of the PFC method to a larger class of material systems.

We have developed a microelastical phase-field model to incorporate elastic energy and misfit stresses in eutectic growth. We apply the model to assess the formation of eutectic structures in Ti–Fe alloy, which exhibit high lattice mismatch owing to difference between lattice parameters of β - Ti and FeTi phases. Numerical simulations of both directional and free eutectic growth are performed by applying cubic anisotropy to the Ti–Fe system. The resulted microstructures are presented and the corresponding stress distributions are evaluated.

Microstructural changes during heat treatment of the Ni-based CMSX-4 and CMSX-6 superalloys have been investigated experimentally and simulated using a phase-field multi-component model incorporating elastic driving forces in the presence of a lattice misfit. Furthermore, a theoretical model of the coarsening of anisotropic particles is proposed for the prediction of the main kinetic parameters of the coarsening process. A comparison of the main characteristics of the microstructural evolution during non-directional γ′-coarsening, provided from both experiments and phase-field simulations, gives a good agreement of the coarsening kinetics of the CMSX-4 superalloy. However, for the CMSX-6 superalloy, phase-field simulation results and theoretical predictions are not entirely consistent with experimental results, and show that additional effects, for example, those caused by plastic deformation, might be a reason for the slow coarsening rate.

In the present work we describe a kinetic Monte-Carlo method (KMC) in the framework of the lattice gas model for the simulation of the adsorption in multicomponent systems from a liquid solution. The algorithm is subsequently used for an investigation of the influence of an adhesion promoter additive on the adsorption of amine-curing epoxide components on the aluminium oxide surface. The KMC method is coupled to the Metropolis algorithms to accelerate the simulations and take into account effects of various adsorption sites and neighboring adsorbates. An inclusion of the bulk diffusion in the simulation scheme allows to study the distribution of components in the solution near the interface. Parameters used in the simulations are reaction barriers and reaction energies for the surface complex formation. They were calculated in previous works by using density functional based tight binding (1–3). Results of the simulations show that the small additions of the adhesion promoter (˜2%), which adsorption is mo...

A modified phase-field model for quantitative simulations of low-speed phase transitions in multiphase systems is proposed, which takes into account the difference between thermodynamic factors in all the phases. The presented model is based on the quantitative phase-field concept developed by Steinbach et al. [I. Steinbach, F. Pezolla, B. Nestler, M. Seeelberg, R. Prieler, G.J. Schmitz, J.L.L. Rezende, A phase field concept for multiphase systems, Physica D 94 (1996) 135] for multiphase systems allowing to consider the multiphase transition as a superposition of pairwise interactions between two phases. We complete this approach and develop a model, which uses parameters derived from chemical free energy functions of individual phases evaluated from experimental data by the CALPHAD method Lukas et al. (2007) [17] . Because the thermodynamic factors are different in various phases we need to evaluate a special form of total chemical free energy function of a multiphase mixture and use it in the phase-field model. It is shown, that for the developed model the thin-interface asymptotic and the anti-trapping term developed previously for the solidification of pure substances can be applied. The model is verified by an example of the Al–Ni system whose peritectic structural morphology during the directional solidification is investigated. The suggested model can be also extended to multicomponent systems.

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.