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Effects of the geometrical alignment of an idealized deformation microstructure on the migration of recrystallization boundaries are studied using a phase-field model. A deformation field consisting of two sets of regularly arranged bands is used as an example to mimic typical deformed microstructures, while the migration of both initially flat and circular recrystallization boundaries are considered. The results show that for both cases the local variations in the deformation field affect not only the local shape of the migrating boundary but also the overall distances traveled. It is revealed that boundary migration into heterogeneously distributed energy fields are not uniform neither in space nor time, and can therefore not be expressed by the classic migration model with a uniform migration assumption, even when averaged over the whole boundary and long time. [ABSTRACT FROM AUTHOR]

Tailoring the microstructural design of a metastable two-phase Ti-Cr dental implant alloy with the optimal quantity of Cr-rich LAVES structure requires an integration of computation with informatics. This research introduces a stepwise process for generating alloy microstructure data using the phase field method, followed by sparsification and tensor computation techniques to reconstruct a microstructural informatics database for Ti-Cr alloy. This results in a significant 28-fold reduction in data size. Characterized by inpainting accuracies of 94.38% (Tucker decomposition) and 90.38% (Canonical Polyadic), the Tucker method is declared as the superior one in preserving microstructural elastic deformation information. The findings suggest that tensor computation methods present a robust strategy for accurately reconstructing complex microstructures, offering valuable insights for the digital manufacturing of Ti-Cr dental implant alloy. • Cr-rich LAVES structure has preferred elasto-chemical properties. • Phase field method can quantify the area proportion of metastable phases in Ti-Cr alloy. • CP and Tucker decompositions for tensor reconstruction. • High degree of sparsity favors microstructure reconstruction. • Tensor ensured data portabilization by 28 folds. [ABSTRACT FROM AUTHOR]

The phase-field method has already proven its usefulness to simulate microstructural evolution for several applications, e.g., during solidification, solid-state phase transformations, fracture, etc. This wide variety of applications follows from its diffuse-interface approach. Moreover, it is straightforward to take different driving forces into account. The purpose of this paper is to give an introduction to the phase-field modelling technique with particular attention for models describing phenomena important in extractive metallurgy. The concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the phase-field equations are discussed. Some of the possibilities to solve the equations describing microstructural evolution are also described, followed by possibilities to make the phase-field models quantitative and the phase-field modelling of the microstructural phenomena important in extractive metallurgy, i.e., multiphase field models. Finally, this paper illustrates how the phase-field method can be applied to simulate several processes taking place in extractive metallurgy and how the models can contribute to the further development or improvement of these processes. [ABSTRACT FROM AUTHOR]

Microstructure evolution in non-conserved and volume-conserved isotropic two-phase grain structures, with equal interfacial energy for all interfaces and equal volume fractions of the two-phases, is compared based on large-scale phase-field simulations. Two important observations are as follows: (1) the growth mechanism has a larger effect on the grain size distributions of the two alloys than on their grain topology distributions; (2) the growth rate and normalized grain size relation of the grains per topological class are much more scattered for the conserved system than the non-conserved system. [ABSTRACT FROM AUTHOR]

The coarsening behavior of three-phase materials, such as eutectic alloys, is of high technological interest. In this study, 3D ternary three-phase polycrystalline materials were modeled to study the effect of bulk diffusion and phase arrangement on the coarsening kinetics. The diffusion mobilities were defined to be different in the three phases. By varying the phase boundary and grain boundary energies, microstructures with different phase arrangements were obtained, in which the different types of grains had a tendency to alternate or cluster. In all cases, a regime was reached where the average grain size follows a power growth law with growth exponent $$n=3$$ , indicating bulk diffusion-controlled coarsening. The overall growth rate and that of the individual phases were clearly affected by the phase arrangement, the magnitude of the phase boundary energy and the diffusion mobilities of the different phases. In all cases, the phase with the lowest diffusion mobility showed the highest growth rate and on average a larger number of grain faces. While the average number of grain faces became constant in time in systems with constant grain boundary energy, the average number of grain faces continued to increase during the whole simulation time when the grain boundary energy was misorientation dependent. [ABSTRACT FROM AUTHOR]

An important cause for the mechanical entrainment of copper droplets in slags during primary and secondary copper production is their interaction with solid spinel particles, hindering the sedimentation of the copper droplets. In the present study, the interactions between the three phases involved (slag-Cu droplets-spinel solids) were investigated using an adapted sessile drop experiment, combined with detailed microstructural investigation of the interaction zone. An industrially relevant synthetic PbO-CaO-SiO-CuO-AlO-FeO-ZnO slag system, a MgAlO spinel particle, and pure copper were examined with electron microscopy after their brief interaction at 1523 K (1250 °C). Based on the experimental results, a mechanism depending on the interlinked dissolved Cu and oxygen contents within the slag is proposed to describe the origin of the phenomenon of sticking Cu alloy droplets. In addition, the oxygen potential gradient across the phases ( i.e., liquid Cu, slag, and spinel) appears to affect the Cu entrainment, as deduced from a microstructural analysis. [ABSTRACT FROM AUTHOR]

The coarsening behavior of three-phase materials, such as eutectic material systems, is of high technological interest. Microstructure evolution simulations can help to understand the effect of different magnitudes of the diffusivities in the different phases. In this study, the evolution of a 3D three-phase morphology was modeled with equal interfacial energy and volume fraction and similar thermodynamic properties for the three phases, but the diffusion mobilities were taken different. It was observed that the phase with the lowest mobility has the highest growth rate and, on average, a larger number of grain faces, while the other two phases have a nearly equal growth rate and average number of grain faces. The simulation results are compared with results from experiments and simulation studies for single-phase and two-phase materials. [ABSTRACT FROM AUTHOR]

Abstract: Two-dimensional phase-field simulations were performed of grain growth in highly textured materials with equal fractions of two texture components, denoted as and grains, and assuming two values of the grain boundary energies, namely for the boundaries between grains of a different texture component and for boundaries between grains of a similar orientation, resulting in microstructures with alternating and grains and stable quadruple junctions. For different magnitudes of the anisotropy in grain boundary energy , the occurrence of the different types of triple and quadruple junctions and the distributions of the normalized grain size, the number of faces per grain, the normalized grain boundary length per grain and the dihedral angles at grain boundary junctions were investigated. [Copyright &y& Elsevier]

Abstract: A sparse bounding box algorithm is extended to perform efficient phase field simulations of grain growth in anisotropic systems. The extended bounding box framework allows to attribute different properties to different grain boundary types of a polycrystalline microstructure and can be combined with explicit, implicit or semi-implicit time stepping strategies. To illustrate the applicability of the software, the simulation results of a case study are analysed. They indicate the impact of a misorientation dependent boundary energy formulation on the evolution of the misorientation distribution of the grain boundary types and on the individual growth rates of the grains as a function of the number of grain faces. [Copyright &y& Elsevier]

Abstract: For the first time, the pinning effect of small spheroid particles with aspect ratios 1, 2 and 3 on grain growth in polycrystalline structures was studied by three-dimensional phase-field simulations. This was possible by using an efficient parallel sparse bounding box algorithm. Simulation results for different volume fractions of the second-phase particles show that distributions of particles with higher aspect ratios have a stronger pinning effect, since grain boundaries have the tendency to align with the largest cross-section of the particles. The number of particles at triple, quadruple or higher order junctions increases with volume fraction, and with aspect ratio. Nevertheless, the final grain size has a volume fraction dependence of the form , following theoretical predictions assuming random intersections between grain boundaries and particles. A generalised Zener relation with a prefactor depending on the aspect ratio is proposed. [Copyright &y& Elsevier]

Abstract: There exist different phase-field models for the simulation of grain growth in polycrystalline structures. In this paper, the model formulation, application and simulation results are compared for two of these approaches. First, we derive relations between the parameters in both models that represent the same set of grain boundary energies and mobilities. Then, simulation results obtained with both models, using equivalent model parameters, are compared for grain structures in 2D and 3D. The evolution of the individual grains, grain boundaries and triple junction angles is followed in detail. Moreover, the simulation results obtained with both approaches are compared using analytical theories and previous simulation results as benchmarks. We find that both models give essentially the same results, except for differences in the structure near small shrinking grains which are most often locally and temporary for large grain structures. [Copyright &y& Elsevier]

Abstract: The phase-field method has become an important and extremely versatile technique for simulating microstructure evolution at the mesoscale. Thanks to the diffuse-interface approach, it allows us to study the evolution of arbitrary complex grain morphologies without any presumption on their shape or mutual distribution. It is also straightforward to account for different thermodynamic driving forces for microstructure evolution, such as bulk and interfacial energy, elastic energy and electric or magnetic energy, and the effect of different transport processes, such as mass diffusion, heat conduction and convection. The purpose of the paper is to give an introduction to the phase-field modeling technique. The concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the kinetic phase-field equations are introduced. Furthermore, common techniques for parameter determination and numerical solution of the equations are discussed. To show the variety in phase-field models, different model formulations are exploited, depending on which is most common or most illustrative. [Copyright &y& Elsevier]

[Display omitted] The local migration of recrystallization boundaries into spatially varying deformation fields which are typical for metals deformed to low and high strains is investigated using quantitative phase-field simulations. It is found that the average migration velocities as well as the local velocities of the recrystallization boundaries critically depend on the spacing between planar geometrically necessary boundaries (GNBs) which are present in the deformation microstructure. Also the morphology of the deformation field reflecting low and high strain deformation cases strongly affect the migrating boundary during recrystallization. Additional simulations were performed as a function of the two critical parameters- stored energy and GNB spacing - independently. Two regimes are found suggesting that these two parameters are closely linked. Previous experimental results are interpreted based on these novel simulation results. [ABSTRACT FROM AUTHOR]

This paper reports on 3D phase field simulations of IMC growth in Co/Sn and Cu/Sn solder systems. In agreement with experimental micrographs, we obtain uniform growth of the CoSn3 phase in Co/Sn solder joints and a non-uniform wavy morphology for the Cu6Sn5 phase in Cu/Sn solder joints. Furthermore, simulations were performed to obtain an insight in the impact of Sn grain size, grain boundary versus bulk diffusion, IMC/Sn interface mobility and Sn grain boundary mobility on IMC morphology and growth kinetics. It is found that grain boundary diffusion in the IMC or Sn phase have a limited impact on the IMC evolution. A wavy IMC morphology is obtained in the simulations when the grain boundary mobility in the Sn phase is relatively large compared to the interface mobility for the IMC/Sn interface, while a uniform IMC morphology is obtained when the Sn grain boundary and IMC/Sn interface mobilities are comparable. For the wavy IMC morphology, a clear effect of the Sn grain size is observed, while for uniform IMC growth, the effect of the Sn grain size is negligible. [ABSTRACT FROM AUTHOR]

Three different phase fields are predicted and experimentally detected in the Mg rich corner of the Mg-Ca-Si ternary diagram. The present phases are Mg + MgCaSi + Mg 2 Si in phase field 1, Mg + MgCaSi in phase field 2 and Mg + Mg 2 Ca + MgCaSi in phase field 3. The focus of this study is on the formation and evolution of the intermetallic phases. The final microstructures have been related with their solidification process and with the alloys mechanical properties. A clear influence of the observed intermetallic phases on the mechanical performance was found. A bigger size and higher amounts of the MgCaSi intermetallic phase increase the alloys strength and make them brittle, while in its fine morphology MgCaSi reduces the strengthening effect and slightly decreases the ductility compared to pure Mg. Mg 2 Si phase in its needle-like small size morphology contributes to an increase of the hardness and compressive strength. Its presence reduces the alloys ductility making them brittle. Finally, the highest values for compressive strength and hardness are related to the Mg 2 Ca presence. [ABSTRACT FROM AUTHOR]

In this work the microstructure and degradation behavior of several as -cast alloy compositions belonging to the Mg rich corner of the Mg-Si-Sr system are presented and related. The intermetallic phases are identified and analyzed describing the microstructure evolution during solidification. It is intended in this work to obtain insight in the behavior of the ternary alloys in in vitro tests and to analyze the degradation behavior of the alloys under physiologically relevant conditions. The as-cast specimens have been exposed to immersion tests, both mass loss (ML) and potentiodynamic polarization (PDP). The degradation rate (DR) have been assessed and correlated to microstructure features, impurity levels and alloy composition. The initial reactions resulted to be more severe while the degradation stabilizes with time. A higher DR is related with a high content of the Mg 17 Sr 2 phase and with the presence of coarse particles of the intermetallics Mg 2 Si, MgSiSr and MgSi 2 Sr. Specimens with a higher DR typically have higher levels of impurities and alloy contents. [ABSTRACT FROM AUTHOR]

A detailed investigation was carried out to study the effect of volume fraction on microstructure evolution in segregation-free isotropic volume-conserved two-phase alloys. Large scale two-dimensional (2D) and three-dimensional (3D) simulations were performed for 10/90, 30/70 and 50/50 alloys with a modified quantitative phase-field model. The grain growth in the two-phase alloys was consistent with volume-diffusion controlled growth with growth exponent m ∼ 3 . The kinetic coefficient, mean grain size, mean number of faces, grain size distribution (GSD), grain topology distribution (GTD) and grain size/topological class relation were all influenced by the volume fraction of the minor phase ( α ) in the two-phase alloys. In the 10/90 alloy, a bimodal distribution in the GSD and GTD for the 2D total grain structure was found. The timelines of evolution of major-phase grains in 3D simulations for all alloys were examined. ‘Particle-like’ grain migration of isolated α -phase grains was observed in the 2D simulations. The formation and evolution of non-constant curvatures on interphase boundaries in the isotropic volume-conserved two-phase alloys is discussed. The microstructure evolution in the volume-conserved two-phase alloys was compared with that in non-conserved two-phase alloys ( m ∼ 2 ). Simulation results are found to be in good agreement with a wide range of previous experimental and computer simulation results. [ABSTRACT FROM AUTHOR]

Pyrometallurgical industries encounter production losses due to the attachment of metallic droplets to solid particles in liquid slags. Experimental work on this topic remains very challenging. Simulations based on a phase field model can circumvent this lack of experimental data and allow a more systematic insight into the role of the different parameters on the observed phenomenon. In the present work, a recently developed phase-field model to simulate the attachment of liquid metal droplets to solid particles in slags is extended to consider real microstructures of solid particles in liquid slags. Furthermore, it is investigated which initialization method for the liquid metal droplets corresponds best to the experimental conditions. One of the initialization methods used spinodal decomposition of a supersaturated slag to introduce the metallic droplets, whereas the other initialization consists of positioning metallic droplets in the slag in a random way. The simulations showed that both initialization methods result in microstructures that correspond with experimental observations, which points to the existence of several origins for the attachment of metal droplets to solid particles in slags. [ABSTRACT FROM AUTHOR]

Engineering alloys generally exhibit multi-phase microstructures. For simulating their microstructure evolution during solid-state phase transformation, CALPHAD-guided multi-phase-field models coupled with micro-mechanics have proven to be a reliable simulation tool. Nevertheless, their efficiency and accuracy still depend on the homogenization scheme used to interpolate the elastic properties in the interfacial regions. In this paper, we present a phase-field model for multi-phase and multi-component solids using a partial rank-one homogenization scheme that enforces static and kinematic compatibilities in the interfacial regions. To this end, we first extend the rank-one homogenization scheme to multi-phase systems. Moreover, for computational efficiency, we analytically solve the static compatibility equations for linear elastic three-phase solids. For quantitative accuracy, a coupling technique is used to extract the prerequisite thermodynamic and kinetic properties from CALPHAD databases. The model is solved numerically in an open source finite-element framework. As numerical applications, the microstructure of two elastically stressed intermetallic-containing three-phase alloys: Ni–Al and Al–Cr–Ni, are simulated. The accuracy of the model is verified against analytically obtained solutions for planar and concentric ring interfaces. We show that the simulation results remain unaltered with varying interface width. Except for one simulation, all cases show better or nearly equal convergence using the partial rank-one scheme compared to the Voigt–Taylor scheme. [Display omitted] • Multi-phase-field model ensuring interfacial static and kinematic compatibilities. • Local chemical equilibrium is also ensured in the interfacial regions. • Efficiently implemented in an open-source finite element framework. • Simulations show improve or equal convergence compared to Voigt–Taylor scheme. [ABSTRACT FROM AUTHOR]

Experimental results show that voids inside additive manufactured (i.e. 3D printed) materials are unavoidable. As such samples are often post-printing annealed, it is thus of interest to understand how voids affect boundary migration. In this work, phase field simulations of recrystallization are carried out for systems containing voids and typical inhomogeneous microstructures. The simulation results show that the voids significantly affect the shape of the boundary and its migration kinetics during recrystallization. The location of the voids in relation to the inhomogeneous deformation microstructure is furthermore found to affect local boundary migration. The results are not only of importance for optimizing annealing of printed samples, but also for understanding local boundary migration in particle containing alloys. [Display omitted] [ABSTRACT FROM AUTHOR]

Large-scale phase-field simulations were performed of the evolution of grain structures with nonuniform grain boundary energy. A novel approach is proposed to determine the average grain boundary dihedral angles between the grain faces along triple and quadruple line junctions from 3-D voxel-based microstructures. We examine the effect of grain boundary energy nonuniformity on the distributions of the grain size, number of faces per grain, and dihedral angles between grain faces. We study the effect of the initial grain size distribution on the evolution toward steady state for both nonuniform and uniform boundary energy systems. The steady-state grain size and number of faces distributions remain unimodal under all conditions investigated, whereas the dihedral angle distribution is found to become multi-modal when the ratio R = σ H / σ L between high and low grain boundary energies lies in the range of 1.39–1.81. In addition, when R ≈ 2 a topological transition is observed from a structure with grain faces meeting at triple lines (which themselves terminate in quadruple points) toward one in which the grain faces meet primarily at quadruple line junctions (ending at compact regions of triple junctions or very short triple lines). [ABSTRACT FROM AUTHOR]