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Corson (Cu-Ni-Si) alloys were unidirectionally solidified via the Mizuta method under air or water cooling. To replicate the microstructure of a Corson alloy during solidification, phase-field (PF) simulations were performed, and the numerical results were compared with the experimental findings. The comparison between the air- and water-cooling conditions revealed that the primary dendrite trunks were thinner, and the secondary dendrites were longer and more developed under air-cooling conditions. These results were consistent with the experimental observations. Furthermore, the secondary dendrite arm spacings were evaluated via PF simulations, and the calculated results were consistent with the experimental observations. The interface energy and its anisotropy adopted in the present PF simulations were reasonable for the Corson alloy when compared with the values reported in other studies. [ABSTRACT FROM AUTHOR]

Understanding the relation between the microstructure and the material's effective behavior is an important aspect in inverse computational materials engineering. Especially in the context of localized, inelastic phenomena like plasticity and crack growth, the microstructure morphology plays a crucial role. Due to the stochastic nature of heterogeneous media, a statistical analysis over multiple simulations is necessary, since even with the same material, the simulated crack paths and effective crack lengths are highly dependent on the specific locations of microstructural features. A relevant factor that limits this type of investigation is the high cost of real microstructure data. This work presents a digital workflow for exploring the fracture properties of materials. Therein, the required statistical analyses are facilitated by an algorithm that reconstructs multiple realization of a material structure given a single example. The reconstructed structures are discretized with a regular non-conforming mesh with a diffuse interface and crack representation. Crack phase-field simulations are conducted in order to analyze the effective response. An in-depth introduction to the required methods is given together with a statistical evaluation of the conducted numerical experiments. It is concluded that the statistical variation of the effective material behavior overshadows morphological trends in the presented case. This confirms the relevance and utility of complementing simulation-based workflows with microstructure reconstruction and statistical analyses. [ABSTRACT FROM AUTHOR]

In this paper, a mechanics-based phase-field model at the microscale is introduced for microstructure evolution during solidification. The couple phase-field model consists of Allen–Cahn equation for phase order parameter, Cahn–Hilliard equation for composition, heat conduction and elasticity equations. The introduced elastic energy allows for volumetric inelastic strains due to melting/solidification as well as a thermodynamically consistent solid-melt interface stress and consequently, residual stresses during solidification at the microscale and deformation can be captured. The computational microcell is considered at the melt-solid interface and the temperature as a time dependent function is used for its boundary conditions to solve the coupled phase-field model. Using COMSOL FE code, examples of columnar growth are studied. As result, the suppressive effect of elastic driving forces and the reduction in solidification rate, due to the volumetric inelastic strains, on solidification are revealed. The inelastic surface stress, concentrated inside the interface, can change the morphology of solidified structure but does not show a remarkable effect on the solidification rate. The thermal strain was included which reduced the effect of volumetric transformation strain and consequently, the internal stresses near constrained regions were decreased. The effect of undercooling was studied which showed that increasing the undercooling increased the temperature gradient in the vertical direction and near the interface and solidification rate and significantly changed the morphology of solidified structure, as a homogeneous growth was resolved for larger undercooling while a columnar growth was obtained for smaller undercooling. Solidification was studied under mechanical loading which showed external loading changes the stress distribution and magnitude and the morphology of solidified structure. Effect of an inclusion on solidification was also investigated. The inclusion represented a more homogeneous distribution of stress and temperature with different magnitudes compared to the rest of the sample, creating a directional solidification toward the inclusion. [ABSTRACT FROM AUTHOR]

Physical Vapor Deposited (PVD) TiAlN coatings are extensively utilized as protective layers for cutting tools, renowned for their excellent comprehensive performance. To optimize quality control of TiAlN coatings for cutting tools, a multi-scale simulation approach is proposed that encompasses the microstructure evolution of coatings considering the entire preparation and service lifecycle of PVD TiAlN coatings. This scheme employs phase-field simulation to capture the essential microstructure of the PVD-prepared TiAlN coatings. Moreover, cutting simulation is used to determine the service temperature experienced during cutting processes at varying rates. Cahn-Hilliard modeling is finally utilized to consume the microstructure and service condition data to acquaint the microstructure evolution of TiAlN coatings throughout the cutting processes. This methodology effectively establishes a correlation between service temperature and its impact on the microstructure evolution of TiAlN coatings. It is expected that the present multi-scale numerical simulation approach will provide innovative strategies for assisting property design and lifespan prediction of TiAlN coatings. [ABSTRACT FROM AUTHOR]

The utilization of the laser powder bed fusion (L-PBF) method in additive manufacturing experiments has revealed the intriguing potential to induce morphological evolutions within the solidification microstructure by manipulating the thermal conditions. Transitions of this nature can have a substantial impact on the ultimate texture and material properties of the product. This study utilizes numerical investigations to examine the microstructure evolution of the AlSi10Mg alloy, which is fabricated using the L-PBF process, under different thermal conditions. In order to investigate the changes in the microstructure, we employ a parameter phase-field (PF) model. This model effectively replicates the natural development of nuclei from inoculant particles and accurately simulates the transitions in the morphology. The PF model's validity is determined through the numerical investigation of morphological transitions during directional solidification of the AlSi10Mg alloy. The model's predictions are then compared to the analytical Hunt model. The formation of a columnar microstructure with a significantly reduced secondary dendrite arm spacing is observed when the ratio of the temperature gradient (G) to the solidification rate (R) is increased. [ABSTRACT FROM AUTHOR]

[Display omitted] • Microstructure evolution and characteristics of lamella structure is simulated by three-dimensional phase-field simulation. • The impact mechanism of Al content on lamellar spacing is revealed. • Models of Nucleation and growth of multi-variants γ lamella are investigated. • Evolution kinetics of γ lamella shows distinct larger rate in longitudinal direction. The D0 19 - α 2 / L1 0 - γ alternate lamellae microstructure in TiAl alloys brings the superior mechanical property, the lamellae with multi-variant γ phase are key characteristic of morphology-property causality. The nucleation, growth and coarsening of multi-variants γ are investigated in TiAl alloys by using the three-dimensional phase-field simulation, the γ morphology, lamellar spacing (LS) and the predicted yield strength of alloy are in agreement with the experiment. It is found that the γ variants can nucleate and grow up separately to form the single-variant lamella, or through the symbiotic nucleation and growth to form multi-variant lamella, the growth of γ lamella follows the terrace-ledge-kink mechanism. The growth rate along the longitudinal direction is distinctly larger than that of the thickness direction, resulting in the lamellar structure. A three-stage growth kinetics, nucleation, growth and coarsening of γ lamella in thickness is revealed. As Al content changes, the LS is enlarged with the reduced lamella number. The influence of the LS on the mechanical properties is evaluated by using the Hall-Petch relationship. The results give an insight for the formation and evolution mechanisms of multi-variant γ lamella, and discover the internal relations of lamellar morphology and mechanical properties of TiAl alloys. [ABSTRACT FROM AUTHOR]

During the process of laser cladding IN718/WC composite coatings, the dissolution and metallurgical reaction of WC particles significantly influence the microstructures of the coatings. However, the limited experimental methods restrict direct observation and prevent a deep understanding of this complicated process. Therefore, a novel numerical model for the solidification process of IN718/WC composite coatings was proposed. The model was established with the coupled multi-phase-field model and lattice Boltzmann method. Different kinds of microstructure around WC particles were simulated by the model and verified by experiments. Subsequently, microhardness and wear tests were carried out to investigate the improved mechanical properties of IN718 coatings reinforced by WC particles. The results show that the cellular alloy reaction layer, IN718 grains, and eutectic structure are formed, in turn, around WC particles. The convection in the laser molten pool can induce double-tail-like or spindle-like WC convection diffusion bands. The hardness of these bands is higher than that of the IN718 matrix. More importantly, WC convection diffusion bands can inhibit the growth of columnar crystals, because the dissolved WC can decrease the freezing temperature of the melt. Finally, mechanical property tests show that WC particles increase the hardness of the coating and significantly improve its wear resistance. [ABSTRACT FROM AUTHOR]

Representing snow equi‐temperature metamorphism (ETM) is key to model the evolution and properties of the snow cover. Recently, a phase‐field model describing mean curvature flow evolution on 3D microstructures was proposed (Bretin et al., 2019, https://doi.org/10.1051/m2an/2018075). In the present work, this model is used to simulate snow ETM at the pore scale, considering the only process of moving interfaces by sublimation‐deposition driven by curvatures. We take 3D micro‐tomographic images of snow as input in the model and obtain a time series of simulated microstructures as output. Relating the numerical time, as defined in the model, to the real physical time involves the condensation coefficient α, a poorly constrained parameter in literature. A calibration was performed by fitting simulations to experimental data through the evolution of specific surface area (SSA) of snow under ETM at −2°C. A value of the condensation coefficient was obtained: (9.8 ± 0.7) × 10−4 and was used in all the following simulations. We then show that the calibrated model enables to well reproduce an independent time series of ETM at −2°C in terms of SSA, covariance length, and mean curvature distribution. Finally, the calibrated model was used to investigate the effect of ETM on microstructure and effective transport properties (thermal conductivity, vapor diffusion, and permeability) for four different samples. As an interesting preliminary result, simulations show an enhancement of the structural anisotropy of snow in the case of initially anisotropic microstructures, such as depth hoar. Results highlight the potential of such microscale models for the development of snow property predictions for large‐scale snowpack models. Plain Language Summary: Snow on the ground is a skeleton of ice in air evolving continuously under different environmental constraints. When the temperature gradient within the snowpack is weak, the snow grains tend to gradually become smoother and rounder. This so‐called equi‐temperature metamorphism (ETM) is one of the main mechanisms of snow evolution and its correct representation is crucial for snow modeling. Here, we use a mean curvature flow model, describing the smoothing of 3D microstructures, to simulate snow ETM. 3D micro‐tomographic images of snow samples are used as input; the output is a time series of 3D images showing ETM evolution. Describing ETM classically relies on the condensation coefficient α, a poorly constrained parameter, which drives the intensity of the evolution. We estimate this parameter for ETM at −2°C by fitting simulations to experimental data. Based on comparisons with an independent data set, we show that the model enables to well reproduce ETM at −2°C when no significant densification occurs. Finally, we use the model to investigate the effect of ETM on microstructure and effective transport properties of snow for four different samples. Overall, this work presents promising tools for snow metamorphism study and the development of predictive means for large‐scale snow models. Key Points: A phase‐field model of mean curvature flow is applied on the process of sublimation‐deposition, describing equi‐temperature metamorphismThe model is calibrated through the condensation coefficient parameter by fitting experimental and simulated data at −2°CEqui‐temperature metamorphism is predicted on different snow microstructures; morphological and transport property changes are analyzed [ABSTRACT FROM AUTHOR]

The growth and interconnection of fission gas bubbles in the hotter central regions of U-(Pu)-Zr nuclear fuel has been simulated with a phase-field model. The Cahn-Hilliard equation was used to represent the two-phase microstructure, with a single defect species. The volume fraction of the bubble phase and surface area of the bubble-matrix interface were determined during growth and interconnection. Surface area increased rapidly during the initial stages of growth, then slowed and finally decreased as bubble interconnection began and coarsening acted to reduce surface area. The fraction of the bubbles vented to a simulation domain boundary, fV, was quantified as a measure of the microstructure's interconnectivity and plotted as a function of porosity p. The defect species diffusivity was varied; although changes in diffusivity significantly affected the microstructure, the plots of fV vs. p did not change significantly. The percolation threshold pc was calculated to be approximately 0.26, depending on the assumed diffusivity and using an initial bubble number density based on experimental observations. This is slightly smaller than the percolation threshold for continuum percolation of overlapping 3D spheres. The simulation results were used to parameterize two different engineering-scale swelling models for U-(Pu)-Zr in the nuclear fuel performance code BISON. [ABSTRACT FROM AUTHOR]

• A phase-field-informed surrogate model to predict microstructure morphologies from physical vapor deposition. • Two-point statistics, dimensionality reduction and polynomial chaos expansion are utilized to construct the surrogate model. • Demonstrated efficiency to rapidly predict microstructure as a function of deposition conditions. • The process-structure protocol compares well with high-fidelity, phase-field simulations of physical vapor deposition. Here, we present a surrogate model that rapidly predicts the microstructures of a binary-alloy thin film during physical vapor deposition. This surrogate model is constructed and trained from a data set produced by phase-field simulations of physical vapor deposition. It relies on a statistical representation of the microstructure, principal component analysis, polynomial chaos expansion, and a microstructure-reconstruction algorithm to estimate the microstructure as a function of the deposition parameters and properties of the materials being deposited. This protocol, exercised on a simplified physical vapor deposition model, demonstrates the efficacy of the surrogate model to rapidly predict a broad class of microstructures as a function of deposition conditions with good accuracy relative to high-fidelity models. The considerable computational gain from the surrogate model compared to the detailed phase-field approach highlights the importance of pursuing such approaches, especially when used for producing parameter-microstructure maps for rapid and accurate predictions of the microstructure. As such, this surrogate model can be used to guide the choice of deposition conditions and materials being deposited to fabricate functional thin films with targeted microstructures. [ABSTRACT FROM AUTHOR]

Morphological transition of a rod-shaped phase into a string of spherical particles is commonly observed in the microstructures of alloys during solidification (Ratke and Mueller, 2006). This transition phenomenon can be explained by the classic Plateau-Rayleigh theory which was derived for fluid jets based on the surface area minimization principle. The quintessential work of Plateau-Rayleigh considers tiny perturbations (amplitude much less than the radius) to the continuous phase and for large amplitude perturbations, the breakup condition for the rod-shaped phase is still a knotty issue. Here, we present a concise thermodynamic model based on the surface area minimization principle as well as a non-linear stability analysis to generalize Plateau-Rayleigh's criterion for finite amplitude perturbations. Our results demonstrate a breakup transition from a continuous phase via dispersed particles towards a uniform-radius cylinder, which has not been found previously, but is observed in our phase-field simulations. This new observation is attributed to a geometric constraint, which was overlooked in former studies. We anticipate that our results can provide further insights on microstructures with spherical particles and cylinder-shaped phases. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Microstructural evolution studied for non-conserved and conserved systems. • Continuous transition between interface-controlled and diffusion-limited regimes observed. • Influence of volume fraction, interfacial energy and diffusivity on the coarsening kinetics is analyzed. • Topological events influence the local coarsening behavior. Abstract A thermodynamically consistent phase-field model is exploited to demonstrate the influence of relative volume fractions and bulk diffusivity on the grain growth phenomena in two-phase polycrystalline systems. For very small and high volume fractions, the simulated morphology consists of a dispersion of isolated minor phase grains embedded in the matrix of major phase grains. At intermediate fractions, the obtained microstructure resembles an interpenetrating network-like structure. The performed large-scale 2-D simulations elucidate the governing mechanisms for the concurrent two-phase growth at low and high volume fractions, and the continuous transition between interface-controlled and diffusion-limited regimes. While the slowest kinetics is observed for the 50 / 50 volume fraction case, irrespective of the diffusivity, the fastest kinetics is displayed by the pure systems, with a slight difference, which is due to relative interfacial energies. The relative growth rates of the individual phases and the maximum attainable grain size to mean size ratio are observed to follow the well-known trends for isotropic systems. The obtained results for the concurrent growth of a minor phase with various diffusivities, reveal that it is difficult to reconcile all observed behaviors with a universal Zener relation, in contradiction to the previously made claims in the literature. A need for the full-fledged growth law, which takes into consideration the role of the diffusivity, the volume fractions, and the relative interfacial energies in multiphase polycrystalline systems, is pointed out. [ABSTRACT FROM AUTHOR]

Abstract Past models of stress-strain response under cyclic loading mainly rely on macroscopic equations which consider microstructure evolution indirectly or simply discard microstructure information. Modern materials science, on the other hand, seeks quantitative descriptions for the relations between microstructure and loading response. In the present work, we show a promising mesoscale phase-field framework which can describe co-evolution of phase/grain and defect microstructures, reveal microstructure mechanisms and simultaneously predict deformation properties as a natural outcome of microstrucuture interactions. The energy functionals for phase/grain and defect microstructures are constructed, followed by functional variation which leads to governing equations. Applying the developed framework to high temperature cyclic loading of single crystal Nickel-based superalloys, the simulated results show that cyclic loading-microstructure-property relations can be principally revealed. In the short term perspective (in one cycle), dislocations move back and forth, leading to cyclic loops consistent with characteristics observed in experiments. The plastic strains are one order of magnitude smaller than total strains, which explains why the cyclic loops are very "thin". In the long term perspective, all γ / γ ′ microstructures exhibit directional coarsening similar to creep under zero cyclic loading ratio, with the extent of rafting slightly dependents on cyclic waveform, period, etc. The plastic strains are sensitive to cyclic loading conditions both in terms of curve shape and in terms of magnitude. [ABSTRACT FROM AUTHOR]

Graphical abstract Abstract A multi-order-parameter phase-field model was built by coupling a phase-field model with a physically-based statistical nucleation model to predict the microstructure evolution and flow stress responses of discontinuous dynamic recrystallization in 304L stainless steel. Individual growth kinetics simulations of recrystallized nucleus showed that the critical nuclei size was determined by balance between the local stored energy difference and grain boundary energy. This was different from the widely-used semi-analytical Roberts-Alhblom model of nucleation criterion and showed good agreement with Bailey-Hirsch model. The migration rate of the recrystallization interface did not follow the monotone change but strongly depended on the deformation conditions. The overall simulations of dynamic recrystallization agreed well with experimental observation. The characteristic features such as effect of deformation conditions on the peak stress, critical strains and grain size were quantitatively captured by the model. The sensitivity of grain boundary mobility to both temperature and strain rate was found from simulation. If the initial grain size decreased to a critically small value, the enhanced work hardening effect due to grain refinement maybe results in the dramatic increase of nucleation density, and hence finer steady-state grain size. The transition from single peak flow behaviors to multiple peak implies the change of dominating recrystallization behavior from nucleation to interface migration. [ABSTRACT FROM AUTHOR]

Mechanical properties of nickel-base superalloys can be greatly influenced by spatial variation of γ′ microstructure, calling for models at mesoscale to study additional complexity in the underlying microstructure/property relationship. We combine a FFT-based elasto-viscoplasticity (FFT-EVP) model with a phase-field (PF) model to study plastic deformation of γ/γ′ superalloys. The model is applied to Haynes 282 (H282), where the low volume fraction ( < 20 % ) of dispersed spherical γ′ particles results in a dislocation-particle interaction of either Orowan looping at tensile conditions or climb-bypass at creep conditions. The incorporation of these mechanisms is achieved through the framework of a dislocation density based constitutive model in FFT-EVP, together with the introduction of a location-dependent inter-particle spacing based on the k-nearest neighbor algorithm. Features of non-uniform γ′ microstructures, including γ′ volume fraction variation and non-uniform γ′ distribution due to element microsegregation observed in welded H282 samples, are modeled using the PF method and then passed to FFT-EVP to explore their influence on the tensile and creep properties in a parametric manner. It is found that if the particle distribution is uniform, γ′ volume fraction variation can exhibit a nonlinear effect on both the tensile strength and creep rate; for a given overall γ′ volume fraction, the degree of particle distribution non-uniformity can also exhibit nonlinear influence on the tensile strength and creep rate. It is also shown that the location-dependent inter-particle spacing serves as a better microstructure descriptor than the conventional analytical expression of the average inter-particle spacing in terms of establishing a homogenized but microstructure-sensitive constitutive microstructure-property relationship for fast-acting engineering applications. [ABSTRACT FROM AUTHOR]

This work presents simulation of microstructure evolution in the nugget zone (NZ) of a AZ31-Mg-alloy friction stir weld. The process parameters (tool geometrical characteristics, rotational speed, travel speed, applied load) have been correlated with the resulting microstructural features in the NZ of the weld (grain size and population) with the aid of the MICRESS software, which provides the ability to simulate both nucleation and grain growth during dynamic recrystallization phenomena evolving in the NZ during the weld thermal cycle. The input parameters of the developed model include the tool geometry, the welding conditions as well as the recrystallization energy, the grain boundary mobility and specific material properties. NZ microstructure obtained by simulation shows good agreement with experimental measurements for both grain population and size. [ABSTRACT FROM AUTHOR]

The phase-field method coupling with thermodynamic data has become a trend for predicting the microstructure formation in technical alloys. Nevertheless, the frequent access to thermodynamic database and calculation of local equilibrium conditions can be time intensive. The extrapolation methods, which are derived based on Taylor expansion, can provide approximation results with a high computational efficiency, and have been proven successful in applications. This paper presents a high precision second order extrapolation method for calculating the driving force in phase transformation. To obtain the phase compositions, different methods in solving the quasi-equilibrium condition are tested, and the M-slope approach is chosen for its best accuracy. The developed second order extrapolation method along with the M-slope approach and the first order extrapolation method are applied to simulate dendrite growth in a Ni-Al-Cr ternary alloy. The results of the extrapolation methods are compared with the exact solution with respect to the composition profile and dendrite tip position, which demonstrate the high precision and efficiency of the newly developed algorithm. To accelerate the phase-field and extrapolation computation, the graphic processing unit (GPU) based parallel computing scheme is developed. The application to large-scale simulation of multi-dendrite growth in an isothermal cross-section has demonstrated the ability of the developed GPU-accelerated second order extrapolation approach for multiphase-field model. [ABSTRACT FROM AUTHOR]

Ameliorating the computing efficiency is always of importance in phase-field simulations of material microstructure formation and evolution. Borrowing from the nonlinear preconditioning treatment of diffuse interface models, the usual quantitative phase-field model for a binary alloy has been transformed to make it easier to compute accurately. The transformation yields a new variable whose value changes linearly across the interface. The dependences of simulated results of the nonlinearly preconditioned phase-field formula on the interface grid size and the discretization time step have been examined in detail through numerical experiments, including the growth velocity, the radius and the solute concentration of a steady tip. The results show that the new evolution equations are able to be solved on a computational mesh with interface grids 2–4 times coarser than those used in the conventional method. In combination with the front-tracking method to capture the crystallographic orientation of each crystal, the orientation gradient energy is incorporated into the nonlinearly preconditioned phase-field model, which enables simulations of grain boundary behaviors. The algorithm of the distributed parallel finite element method on an adaptive mesh is applied to further raise the computing efficiency. Simulations of multi-dendrites growth of Al-4 wt.%Cu alloy in undercooled melt cooling down continuously are performed. The results demonstrate that the proposed fast simulation approaches allow quantitative simulations of a large number of dendrites growth on the scale of centimeters or millimeters, respectively in two or three dimensions, just using an ordinary workstation instead of clusters or supercomputers. [ABSTRACT FROM AUTHOR]

Ni-rich γ cells and Nb-rich eutectic droplets that form during laser power bed fusion solidification of Ni-Nb alloys are studied using experiments and simulations. Finite element simulations estimate the cooling rates in the melt pool and phase-field simulations predict the resulting cellular microstructures. The cell and droplet spacings are determined as a function of cooling rate and fit to a power law. The formation of Laves phase is predicted for a critical composition of Nb in the liquid droplets. Finally, our simulations demonstrate that anisotropy in the γ orientation influences the Laves fraction significantly. [ABSTRACT FROM AUTHOR]

The growth of Si precipitates in Mg 2 Si matrix under an external stress has been simulated by the phase-field method. The temporal evolution of a microstructure was determined by solving the time-dependent Cahn-Allen equations for the concentration and Ginzburg-Landau equations for the order parameters for precipitates. We observed that the external stress accelerates the Si precipitate growth, which is mainly due to reduction of the free energy of the Si precipitates under an external stress. Under a tensile stress, the Si precipitates preferentially align along the stress direction. Increasing the tensile stress accelerates the growth and decreases the aspect ratio of the Si precipitates. Finally, the Si precipitates aggregate to form a lath structure for all cases. Under a compressive stress, the growth rate is larger and the aspect ratio of the Si precipitates is smaller than those in the case of tensile stress at the same absolute value of the applied stress. We found the obtained microstructure contains uniformly dispersed Si lenticular precipitates in the Mg 2 Si matrix, which is suitable for thermoelectric materials because the phonons are frequently scattered at the interface between the Si precipitates and the Mg 2 Si matrix. Our present simulation results suggest that thermoelectric properties of this type of material will be significantly improved compared to the same material without stress. [ABSTRACT FROM AUTHOR]

Dynamic recrystallization (DRX) can in principle serve as an alternative way of controlling grain structure via a single route of hot working instead of the traditional cold working followed by annealing at elevated temperatures; in reality, its widespread application is hindered by the lack of quantitative understanding and prediction of the process. Using a recently developed model (Zhao et al., 2016) that integrates a fast Fourier transform-based elasto-viscoplastic model and a phase-field recrystallization model, we investigate the evolution of both microstructural and micromechanical fields in polycrystal copper during uniaxial compression at various elevated temperatures. Quantitative analysis based on the simulation results confirms that stress redistribution upon the formation of a new grain can significantly lower the dislocation density of neighboring grains, leading the so-called “DRX-enhanced recovery”, while the new grain itself undergoes accelerated work hardening as compared to the matrix. Numerical analysis using current simulation data reveals a macroscopic kinetic equation describing the average dislocation density evolution during DRX softening. The critical strain for the onset of DRX and the Zener–Hollomon parameter are found to obey a power law, with the model predicted exponent being consistent with that found in experiments. Temperature-dependence of the Avrami exponents have also been predicted using the simulation data, which agrees with the experimental finding. The population of grain boundaries and triple and quadruple junctions are shown to evolve with deformation and be temperature-dependent. [ABSTRACT FROM AUTHOR]

The continuum decomposition of the Fe-Cr alloys from initial phase separation to steady-state coarsening with concentrations varying from 25 at % Cr and 30 at % Cr to 33 at % Cr aged at 750 K was studied by utilizing three-dimensional phase-field simulations. The dynamic stages of separation of nanoscale Cr-enriched α phase were distinguished by the evolution of the volume fraction, particle number density and the average particle radius of the α phase. The stage of steady-state coarsening was characterized with an equilibrium volume fraction and decreasing particle number density. The coarsening rate constant by linear fitting of the cube of average radius and aging time shows an increase with the increasing Cr concentration. The time exponents decrease from the growth and coarsening stage to the steady-state coarsening stage and show a dependence on the particles number density at different concentrations. The quantitative evolutions of α phase via nucleation growth and spinodal decomposition are theoretically helpful for understanding the microstructure evolution with aging time in Fe-Cr alloys. [ABSTRACT FROM AUTHOR]

The solidification of eutectic alloys generally produces two-phase microstructures. Their morphology is determined by the dynamics of the solid-liquid interfaces at the crystallization front. At the triple lines, where the liquid and the two solid phases meet, solid-liquid and solid-solid surface free energies are in local equilibrium. We perform three-dimensional phase-field simulations with a multi-phase-field model in which the surface free energies can be independently controlled. We find that an anisotropy of the interphase boundary (solid-solid) energy has a strong effect on the microstructural patterns. The lamellae tend to align with directions of minimal interphase boundary energy. For a two-fold anisotropy, as generally expected for interphase boundaries between two cubic crystals, regular lamellar arrays are formed, in strong contrast to the labyrinth patterns observed in isotropic systems. If two different grains compete, the one with the lowest interphase boundary energy always overgrows the other. These results are consistent with observations in bulk metallic eutectic samples, namely, the frequent occurrence of large regular lamellar arrays and the prevalence of grains with special orientation relationships. [ABSTRACT FROM AUTHOR]

In single crystal Nickel-based superalloys subject to creep loading, the γ / γ ′ phase microstructure co-evolves with the system of dislocations under load. Computational modeling thus requires multiphysics approaches capable of describing and simulating both phase and defect microstructures within a common conceptual framework. To do so, we formulate a coupled continuum model of the evolution of phase and dislocation microstructures. The simulated γ / γ ′ phase microstructure accounts for concentration as well as crystallographic order parameters. Dislocation microstructure evolution is described in terms of dislocation densities and associated stress-driven dislocation fluxes. The creep strain curve is obtained as a natural by-product of the microstructure evolution equations. We perform simulations of γ / γ ′ evolution for different dislocation densities and establish the driving forces for microstructure evolution by analyzing in detail the changes in different contributions to the elastic energy and chemical free energy density, as well as the evolution of stress concentrations that may trigger the transition from dislocation flow in the γ channels towards shearing of the γ ′ precipitates. Our investigation reveals the mechanisms controlling the process of directional coarsening (rafting) and demonstrates that the kinetics of rafting significantly depends on characteristics of the dislocation microstructure. In addition to rafting under constant load, we investigate the effect of changes in loading conditions and explore the possibility of improving creep properties by pre-rafting along a different loading path. [ABSTRACT FROM AUTHOR]

Numerical simulations based on phase-field methods are indispensable in order to investigate interesting and important phenomena in the evolution of microstructures. Microscopic phase transitions are highly affected by mechanical driving forces and therefore the accurate calculation of the stresses in the transition region is essential. We present a method for stress calculations within the phase-field framework, which satisfies the mechanical jump conditions corresponding to sharp interfaces, although the sharp interface is represented as a volumetric region using the phase-field approach. This model is formulated for finite deformations, is independent of constitutive laws, and allows using any type of phase inherent inelastic strains. [ABSTRACT FROM AUTHOR]

This paper presents a study of two lightweight steels, Fe-15Mn-10Al-0.8C-5Ni and Fe-15Mn-10Al-0.8C (in wt.%) where strength is dependent upon the microstructure of 2 nd phase precipitates. We investigate the effects of annealing temperature from 500 °C to 1050 °C on the precipitation of ordered phases size and morphology through phase-field modelling and experimental studies based on laboratory scale annealing and characterization. The chemical composition of carbides and B2 compounds as a function of isothermal annealing temperature and the matrix within which they formed are elucidated in this study. It is found that nano-sized disk-shaped B2 particles form at higher annealing temperatures (e.g. 900 °C and 1050 °C). The simulation results on carbides demonstrated the effects of energetic competition between interfacial energy and elastic strain energy on the morphological evolution of carbides. In addition to that, different ordering behaviours observed depending on the Ni content into the steel. The results demonstrate processing route designed through the phase-field simulations led to a better combination of strength and ductility. The tensile testing results indicate an increase in the strength and elongation when B2 precipitate morphology changes from micro-size faceted shape to nano-size disk-like particles. [ABSTRACT FROM AUTHOR]

Solidification is an important process in many alloy processing routes. The solidified microstructure of alloys is usually made up of dendrites, eutectics or a combination of both. The evolving morphologies are largely determined by the solidification process and thus many materials properties are dependent on the processing conditions. While the growth of either type of microstructure is well-investigated, there is little information on the coupled growth of both microstructures. This work aims to close this gap by formulating a phase-field model capable of reproducing dendritic, eutectic as well as dendritic–eutectic growth. Following this, two-dimensional simulations are conducted which show all three types of microstructures depending on the composition and processing conditions. The effect of the dendritic–eutectic growth on the microstructural lengths, which determine materials properties, is investigated and the morphological hysteresis between eutectic growth and dendritic–eutectic growth is studied by employing solidification velocity jumps. Finally, qualitative three-dimensional simulations are conducted to test for morphological changes in the eutectic. [Display omitted] [ABSTRACT FROM AUTHOR]

Phase-field (PF) models are one of the most powerful tools to simulate microstructural evolution in metallic materials, polymers, and ceramics. However, existing PF approaches rely on rigorous mathematical model development, sophisticated numerical schemes, and high-performance computing for accuracy. Although recently developed surrogate microstructure models employ deep-learning techniques and reconstruction of microstructures from lower-dimensional data, their accuracy is fairly limited as spatiotemporal information is lost in the pursuit of dimensional reduction. Given these limitations, we present a novel data-driven emulator (DDE) for predicting microstructural evolution, which combines an image-based convolutional and recurrent neural network (CRNN) with tensor decomposition, while leveraging previously obtained PF datasets for training. To assess the robustness of DDE, we also compare the emulation sequence and the scaling behavior with phase-field simulations for several noisy initial states. Finally, we discuss the effectiveness of our microstructure emulation technique in the context of runtime speed-up while also highlighting its trade-off with accuracy. [Display omitted] • Novel data-driven emulator for predicting microstructural evolution. • The approach combines Convolutional Recurring Neural Networks with Tensor Decomposition. • Accurate emulation of phase-separating microstructures including the diffuse interface profiles. [ABSTRACT FROM AUTHOR]

In order to simulate multi-component diffusion controlled precipitation of ordered phases in low density steels using the phase-field method, the Gibbs free energy of the γ , α and κ phases in the quaternary Fe-Mn-Al-C system was linked to the CALPHAD method using a three-sublattice model which is based on the accumulation of considerable thermodynamic data in multi-component systems and the assurance of continuous variation of the interface area. This model includes the coherent precipitation of κ phase from a disordered FCC γ phase and semi-coherent precipitation of the same κ phase from a disordered BCC α structure. The microstructure evolution of κ -carbide was simulated with three-dimensional phase-field model. The simulation was first performed for a single particle in both γ and α phases to investigate the evolution of interfacial and elastic strain energy during the precipitation process. The simulation results show that κ has a cuboidal morphology in γ and elongated plate-like morphology in α which is in agreement with the morphologies reported in the literature. The multi-particle simulations were also performed for the precipitation of κ phase from both disordered γ and α . The results also demonstrate that the size of κ precipitates in γ is remarkably smaller than that in α phase. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• A mesoscale model for structural alloy corrosion in molten salts is developed. • Model is implemented using the MOOSE framework. • Sensitivity analysis, verification, and validation are performed on the model. • Quantitatively predicts mass-loss and captures selective Cr depletion. [Display omitted] Understanding the impact of microstructure on corrosion rates can aid the development of corrosion-resistant alloys for molten salt reactors. In this work, we develop an electrochemical phase-field model for capturing the microstructure-dependent corrosion of structural alloys by molten salts. As a demonstration problem, we apply this model to capture the selective depletion of Cr from Ni-Cr grain boundaries during corrosion in molten FLiBe salt. We perform sensitivity analysis and model verification on 1D simulations to confirm that the model predicts diffusion-limited kinetics. The model is validated using 1D, 2D, and 3D simulations against experimental data for Ni-5Cr and Ni-20Cr corrosion in molten FLiBe. The 1D simulations predict the corrosion behavior with reasonable accuracy when using an effective diffusion coefficient that accurately represents the grain boundary diffusion. 2D simulations that represent the grain structure underpredict the corrosion. 3D simulations that represent the grain structure predict the corrosion with reasonable accuracy. The corrosion rate predicted by the 3D simulations is proportional to the average grain size at the alloy/salt interface. [ABSTRACT FROM AUTHOR]

Samarium-cobalt (Sm-Co) sintered magnets have high magnetic energy densities, great resistance to demagnetization and corrosion, and excellent thermal stability in a wide temperature range (–50–550 °C). However, the utilization of these magnets is restricted by their brittleness. Based on micromechanical and the Zener pinning model, Sm-Co sintered magnets with improved mechanical properties have been designed and fabricated via microstructure engineering. A small amount of fine Sm 2 O 3 particulates (0–3 wt%) has been incorporated into Sm 2 (CoFeCuZr) 17 sintered magnets to refine the grain size by up to approximately 50% (from 45 to 22 µm) and narrow the grain size distribution. Doping with 3 wt% Sm 2 O 3 increased the flexural strength by 62% while maintaining magnetic performance. Both grain-refined unimodal microstructure and heterogeneous laminated coarse/fine grain microstructure were formed by strategically designed assemblies of Sm 2 O 3 -added Sm-Co powder feedstock mixtures. The technology is compatible with existing magnet manufacturing processes. Numerical micromechanics simulation indicates that the fracture is dominated by intragranular mode. The mechanical strength is mainly enhanced by the additive-induced grain refinement, while the small amount of Sm 2 O 3 addition has a small direct positive contribution. These magnets will be more cost-effective, efficient, and robust for various functional applications. • This work addresses the mechanical resilience of a critical functional material (i.e., Sm-Co magnets). • We developed mechanically robust, high magnetic-performance Sm-Co magnets by Sm 2 O 3 doped microstructure engineering. • The correlation between microstructure and mechanical properties is revealed by a micromechanical simulation. • These magnets will be more cost-effective, efficient, and robust for various functional applications. [ABSTRACT FROM AUTHOR]

The phase separation and microstructure evolution of undercooled Fe 62 Sn 34 Ge 4 immiscible alloy droplets have been investigated by drop tube method and numerical simulations. A conspicuous phase separation has taken place within alloy droplets during free fall. The solidification microstructures are mainly characterized by core/shell and dispersed structures. The cooling rates and microstructure morphologies of undercooled alloy droplets are strongly dependent on droplet size. With the decrease of droplet size, the cooling rate increases and the microstructure morphology transfers from core/shell structure to dispersed structure. A modified Model H, which considers the spinodal decomposition, subsequent coarsening and migration of second-phase globules under the effects of Marangoni migrations, is developed to explore the dynamic mechanisms of liquid/liquid phase separation. Numerical results reveal that the dispersed structures, in which Sn-rich globules are homogeneously dispersed within Fe-rich matrix, are the intermediate and metastable phase separation morphologies before the formation of stable core/shell morphology. The Marangoni migrations are the major dynamic mechanisms responsible for the microstructure evolution from dispersed structure into core/shell structure. The chemical gradient drives the phase separation and microstructure evolution, which lowers the overall energy of alloy melt. Theoretical calculations show that the solutal Marangoni migration contributes to the aggregation and coalescence of neighboring globules, whereas thermal Marangoni migration induces the coalescence between periphery globules and inner globules, and subsequently forces these globules to aggregate at droplet center to form a Sn-rich core. [ABSTRACT FROM AUTHOR]

Computational models based on the phase-field method typically operate on a mesoscopic length scale and resolve structural changes of the material and furthermore provide valuable information about microstructure and mechanical property relations. An accurate calculation of the stresses and mechanical energy at the transition region is therefore indispensable. We derive a quantitative phase-field elasticity model based on force balance and Hadamard jump conditions at the interface. Comparing the simulated stress profiles calculated with Voigt/Taylor (Annalen der Physik 274(12):573, ), Reuss/Sachs (Z Angew Math Mech 9:49, ) and the proposed model with the theoretically predicted stress fields in a plate with a round inclusion under hydrostatic tension, we show the quantitative characteristics of the model. In order to validate the elastic contribution to the driving force for phase transition, we demonstrate the absence of excess energy, calculated by Durga et al. (Model Simul Mater Sci Eng 21(5):055018, ), in a one-dimensional equilibrium condition of serial and parallel material chains. To validate the driving force for systems with curved transition regions, we relate simulations to the Gibbs-Thompson equilibrium condition (Johnson and Alexander, J Appl Phys 59(8):2735, ). [ABSTRACT FROM AUTHOR]

Phase-field-based models have become common in material science, mechanics, physics, biology, chemistry, and engineering for the simulation of microstructure evolution. Yet, they suffer from the drawback of being computationally very costly when applied to large, complex systems. To reduce such computational costs, a Unet-based artificial neural network is developed as a surrogate model in the current work. Training input for this network is obtained from the results of the numerical solution of initial–boundary-value problems (IBVPs) based on the Fan–Chen model for grain microstructure evolution. In particular, about 250 different simulations with varying initial order parameters are carried out and 200 frames of the time evolution of the phase fields are stored for each simulation. The network is trained with 90% of this data, taking the i th frame of a simulation, i.e. order parameter field, as input, and producing the (i + 1) -th frame as the output. Evaluation of the network is carried out with a test dataset consisting of 2200 microstructures based on different configurations than originally used for training. The trained network is applied recursively on initial order parameters to calculate the time evolution of the phase fields. The results are compared to the ones obtained from the conventional numerical solution in terms of the errors in order parameters and the system's free energy. The resulting order parameter error averaged over all points and all simulation cases is 0.005 and the relative error in the total free energy in all simulation boxes does not exceed 1%. [Display omitted] [ABSTRACT FROM AUTHOR]

In modern technical applications various multiphase mixtures are used to meet demanding mechanical, chemical and electrical requirements. To understand their structural properties as continuous macroscopic materials, it is important to capture the microstructure of these mixtures. Due to their vast range of applications multicomponent systems are subjected to microstructural changes such as phase separation and coarsening. Therefore the ultimate microstructural arrangement depends on the system's configuration and on exterior driving forces. In addition to this, random physical imperfections within the material and random noise in the exterior thermodynamic fields influence in essence the microstructural evolution. Since all physical processes are subjected to a certain degree of random inhomogeneity under realistic conditions, the influence of random phenomena cannot be neglected in modern physical models. An advanced mathematical description and an implementation of these stochastic processes are required to adapt simulation results based on deterministic mathematical models to experimental observations. In our contribution we will present an operator-scaling anisotropic random field embedded in the Cahn-Hilliard phase-field model to describe the phase evolution in a binary mixture. The arising nonlinear diffusion equation will be solved numerically in the innovative framework of the isogeometric finite element method. To illustrate the flexibility and versatility of our approach, numerical and experimental results for a eutectic Sn-Pb alloy are contraposed. This is the first time that the microstructural evolution in a multicomponent system has been associated with operator-scaling anisotropic random fields. Due to its enormous potential as an essential ingredient in stochastic mathematical and physical modeling it is only a matter of time until these processes will become prevalent in engineering applications. [ABSTRACT FROM AUTHOR]

Abstract: A review surveying the developments in the application of the phase-field method in the study of phase transitions and microstructure evolution in multi-component systems is presented. Starting with a brief description of historical developments in the modelling method, we continue to take a tour into the applications of the method, both, in the understanding of the physical phenomena behind structure formation and the investigation of the problems relevant for the processing of technically important alloys. An outlook on the future applications of the method and the related challenges is put forth. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

A scheme for integrating grain growth model in phase transition and grain distortion calculation in plastic deformation is proposed, which enables the simulation of microstructure evolution in steel processing when phase transition takes place during rolling. The grain shape deformation is calculated by transforming grid coordinates according to homogeneous deformation matrix. Grain growth in phase transition is computed by phase-field model. Apart from the direct distortion effect of rolling on grain geometry, the change of interface geometry affects the interface propagation due to the anisotropic property of the interface energy. Finally, the method for integration of microstructure simulation with steel crystallography is discussed. [ABSTRACT FROM AUTHOR]

Abstract: A cubic to tetragonal transformation leads to simultaneous nucleation and growth of several orientation variants. The size and morphology of a critical nucleus, comprising of one or more variants, are determined by the competition among bulk thermodynamic driving force, elastic strain energy, and interfacial energy. In this work, a diffuse-interface model is developed to predict both the critical nucleus morphology of a single variant tetragonal precipitate described by a conserved field and the critical morphology of a two-variant nucleus and equilibrium multivariant twinning microstructure resulted from a structural transformation described by non-conserved fields. A constrained string method was employed to compute the probable minimum energy paths and the microstructures along each path. Numerical experiments indicate that our approach works effectively for predicting the critical nucleus morphology both the precipitation processes and structural transformations. [Copyright &y& Elsevier]

The article presents a study which investigates the microstructure formation and microsegregation of an austenitic steel based on the iron-manganese-carbon alloy system. In this study, a phase-field microstructure simulation model was employed to simulate the dendritic solidification of the alloy. Results reveal that the alloy is very susceptible to manganese segregation.

Under directional growth conditions, a strong crystallographic texture is observed in Mg-based alloys due to the anisotropy of the hcp-phase. The aim of the present work is to investigate the fundamental mechanisms of this growth texture evolution by microstructure simulations. For this purpose, a phase-field model has been extended to allow for complex 3D anisotropic interfacial energies and mobilities, calibrated by data from molecular dynamics studies. The model is first applied in 2D to AZ31 (Mg–3%Al–1%Zn). Bicrystal simulations are performed for a systematic study of grain interactions. Extended simulations are compared to experiments. Comprehensive texture evolution is investigated in 3D for Mg–6%Al. Texture is found to evolve mainly due to retarded growth in basal <0001> orientations. Grains with the fast growing <1120> orientation most closely aligned to the temperature gradient commonly prevail, but process dependent also other orientations of the basal plane (between <1120> and <1010>) may coexist. [ABSTRACT FROM AUTHOR]