Multiphase-field Model Research Articles

Numerical study of growth competition between twin grains during directional solidification by using multi-phase field method

A multi-phase field model is established to simulate the growth competition and evolution behavior between seaweed and columnar dendrites during directional solidification. According to the effects of surface tension and interfacial energy, we quantitatively analyze the influences of factors such as inclination angles, pulling velocity, and anisotropic strength on twin growth. The results demonstrate that the pulling velocity and anisotropic strength have an important influence on the morphology and evolution of the seaweed and dendritic growth. The low pulling velocity and anisotropic strength are both key parameters for maintaining the stable morphology of seaweed during competitive growth in a bicrystal, showing that the lateral branching behavior is the root of the dendrites that can ultimately dominate the growth. And it is clarified that the lateral branching behavior and lateral blocking are the root causes of the final dominant growth of dendrites. With the increase of anisotropy strength, the seaweed is eliminated fastest in case 1, the seaweed is transformed into degenerate dendritic morphology, and eliminates the seaweed by promoting the generation and lateral growth of the lateral branches of the dendrites. The increase of pulling velocity is to increase the undercooling of favorable oriented grain and accelerate the growth rate of dendrites, thus producing more new primary dendrites for lateral expansion and accelerating the elimination rate of unfavorable oriented grain.

Multiphase-field model for surface diffusion and attachment kinetics in the grand-potential framework.

Grand-potential based multiphase-field model is extended to include surface diffusion. Diffusion is elevated in the interface through a scalar degenerate term. In contrast to the classical Cahn-Hilliard-based formulations, the present model circumvents the related difficulties in restricting diffusion solely to the interface by combining two second-order equations, an Allen-Cahn-type equation for the phase field supplemented with an obstacle-type potential and a conservative diffusion equation for the chemical potential or composition evolution. The sharp interface limiting behavior of the model is deduced by means of asymptotic analysis. A combination of surface diffusion and finite attachment kinetics is retrieved as the governing law. Infinite attachment kinetics can be achieved through a minor modification of the model, and with a slight change in the interpretation, the same model handles the cases of pure substances and alloys. Relations between model parameters and physical properties are obtained which allow one to quantitatively interpret simulation results. An extensive study of thermal grooving is conducted to validate the model based on existing theories. The results show good agreement with the theoretical sharp-interface solutions. The obviation of fourth-order derivatives and the usage of the obstacle potential make the model computationally cost-effective.

Multiphase-field modeling of spinodal decomposition during intercalation in an Allen-Cahn framework

Owing to its unique thermodynamical description, phase separation has largely been modeled in the Cahn-Hilliard framework. In the present work, as a computationally efficient alternative, a multicomponent, multiphase-field model operating in Allen-Cahn framework is presented and subsequently employed to simulate spinodal decomposition. Stability analysis shows that the formulation can cover the effect of phase separation while simplifying the extension to multiple phases and incorporation of additional driving forces. Computational efficiency of the proposed approach is compared with the conventional technique by modeling intercalation in a representative one-dimensional domain. Moreover, intercalation within a multigrain system involving multiparticle interaction is studied. Our results suggest initiation of phase transformation at higher order junctions as well as a grain-by-grain intercalation behavior in a two-phase cathode material such as the well-studied ${\text{LiFePO}}_{4}$.

Effects of applied load on formation and reorientation of zirconium hydrides: A multiphase field modeling study

The effects of applied load on the formation and reorientation of δ hydrides in α-zirconium matrix are studied by a multiphase field model. Simulations with only one hydride seed indicate that the hydride reorientation occurs only when a tensile or compressive strain of ~0.02 is applied along 112¯0 (hoop or circumferential direction) or 0001 (radial direction aligned with the basal pole direction), respectively. However, two-seed and multi-seed hydride simulations show the significant influence of neighboring hydrides on the required load for reorientation, reducing it by more than one order of magnitude and making it comparable to the experimental results. Through phase-field simulations, this work suggests that the hydride reorientation happens more easily when clusters of hydrides are present, and the required external load for reorientation is within the elastic limit of the cladding material.

Defects in Active Nematics – Algorithms for Identification and Tracking

Abstract The growing interest in active nematics and the emerging evidence of the relevance of topological defects in biology asks for reliable data analysis tools to identify, classify and track such defects in simulation and microscopy data. We here provide such tools and demonstrate on two examples, on an active turbulent state in an active nematodynamic model and on emerging nematic order in a multi-phase field model, the possibility to compare statistical data on defect velocities with experimental results. The considered tools, which are physics based and data driven, are compared with each other.

A comprehensive analysis of three-dimensional normal grain growth of pure iron via multi-phase field simulation

A three-dimensional (3D) multi-phase field model was established to simulate normal grain growth in pure iron. The advanced visualization technology was used to extract the related data for individual grains, which can clearly display the grain morphology with distinct grain boundary surface as well as the space distribution of neighboring grains?Based on the simulation results, the grain growth kinetics model was described, which is in conformity with Burke and Turnbull?s parabolic law. The phenomenon of a ?Hillert regime? in 3D grain growth and the topological transformation mechanism were investigated. The grain size distributions under different time evolution showed a good agreement with the Hillert distribution. The details of grain growth, especially grain size distribution and volume growth rate, were analyzed. The models of von Neumann-Mullins and Hilgensfeldt for predicting the volumetric growth rate were compared. The volumetric growth rate was approximately zero when the number of grain sides was close to 13.7. The multi-phase field simulation can be used to analyze the dynamic evolution of the topological relationship of grains and reveal the general law of normal grain growth quantitatively.

Accuracy Evaluation of Some Multi–phase–field Models for Multiphase Flow Based on Conserved Allen–Cahn Equation

Accuracy Evaluation of Some Multi–phase–field Models for Multiphase Flow Based on Conserved Allen–Cahn Equation

Microstructural phase coexistence kinetics near the polymorphic phase boundary

By implementing a novel multiphase field model for ferroelectric systems, the phase coexistence of the tetragonal (T) and rhombohedral (R) phases in Pb-free BZT-40BCT was analyzed. Metastable coexistence of the T and R phases is predicted between a thermodynamic upper limit at TC,R=49.90∘C and a kinetic lower limit determined by the time-temperature-transformation behaviour. Predicted domain microstructures exhibit faceted domain walls and curved T-R phase interfaces that are consistent with recent TEM studies in the vicinity of the polymorphic phase boundary (PPB). Further, miniaturization of the domain structure near the PPB is a result of the relatively low interfacial energies and a pinning effect caused by the large metastable phase fraction that originates from the vanishing macroscopic driving force for phase transformation. Particularly, the vanishing of rhombohedral domain wall energies as T→TC,R enables a phase transformation-induced polarization rotation mechanism and predicts a hierarchical domain morphology for the R phase. These results are in agreement with the higher piezoelectric response reported near the maximum temperature for R+T coexistence and the observations of a miniaturized nanodomains structure within micron-sized, wedge-shaped domains in the R phase for the BZT-xBCT system.

Formation of wide-blocky calcite veins by extreme growth competition

Liassic limestones on the coast of Somerset in the UK contain dense arrays of calcite microveins with a common, but poorly understood microstructure, characterized by laterally wide crystals that form bridges across the vein. We investigated the mechanisms of formation and evolution of these ‘wide-blocky’ vein microstructures using a combination of high-resolution analytical methods, including virtual petrography, optical cathodoluminescence and scanning electron microscopy techniques (e.g. energy-dispersive X-ray spectrometry, back-scattered electron imaging, cathodoluminescence and electron back-scattered diffraction), laboratory experiments and multiphase field modelling. Our results indicate that the studied veins formed in open, fluid-filled fractures, each in a single opening and sealing episode. As shown by the optical and electron back-scattered diffraction images, the vein crystals grew epitaxially on grains of the wall rock and we hypothesize that their growth rates differed depending on whether the crystals were on a wall rock grain substrate that fractured intergranularly (slow growth rates) or transgranularly (rapid growth rates). Our multiphase field models support this hypothesis, showing that wide, blocky crystals only form where there are significant differences in the growth rate and are dependent on the type of seed grain. These results provide strong evidence for extreme growth competition, a process that we propose controls vein-filling in many micritic carbonate reservoirs, as well as demonstrate that the characteristics of the fracture wall can affect the filling processes in syntaxial veins. Supplementary material: The description and images of the studied thin sections are available at https://doi.org/10.6084/m9.figshare.c.5172371 . High-resolution optical microscopy mosaics (under plane-polarized- and crossed polarized light) of the thin section collection in PetroScan file format are available on request from the authors.

Dynamics of grain boundary premelting

The mechanical strength of a polycrystalline material can be drastically weakened by a phenomenon known as grain boundary (GB) premelting that takes place, owing to the so-called disjoining potential, when the dry GB free energy sigma _{gb} exceeds twice the free energy of the solid–liquid interface sigma _{sl}. While previous studies of GB premelting are all limited to equilibrium conditions, we use a multi-phase field model to analyze premelting dynamics by simulating the steady-state growth of a liquid layer along a dry GB in an insulated channel and the evolution of a pre-melted polycrystalline microstructure. In both cases, our results reveal the crucial influence of the disjoining potential. A dry GB transforms into a pre-melted state for a grain-size-dependent temperature interval around T_m, such that a critical overheating of the dry GBs over T_m should be exceeded for the classical melting process to take place, the liquid layer to achieve a macroscopic width, and the disjoining potential to vanish. Our simulations suggest a steady-state velocity for this transformation proportional to sigma _{gb} -2 sigma _{sl}. Concerning the poly-crystalline evolution, we find unusual grain morphologies and dynamics, deriving from the existence of a pre-melted polycrystalline equilibrium that we evidence. We are then able to identify the regime in which, due to the separation of the involved length scales, the dynamics corresponds to the same curvature-driven dynamics as for dry GBs, but with enhanced mobility.

Phase-field simulations of grain boundary grooving under diffusive-convective conditions

Grain boundaries are potential sites for the development of morphological instabilities along a solidifying interface, and play an important role in the microstructural properties of cast alloys. In the present study, we employ a multiphase-field model to address the grain boundary grooving phenomenon under the cooperative effect of convection and volume diffusion in the liquid phase. Under pure diffusive conditions, we first benchmark our phase-field model by illustrating and comparing the formation of symmetric groove profiles across the grain boundary with Mullins’s groove. In the presence of an additional convective transport mechanism, we systematically depict that the asymmetricity of groove profile increases with the increase in the convection velocity. Particularly, it is identified that the groove kinetics as well as the grain boundary grooving mechanism is significantly modified in the diffusive-convective regime. The present two-dimensional simulations, while providing considerable insights into the mechanism of grain boundary grooving, also closes the gap with the experimental and theoretical observations reported earlier. In addition, the role of relative surface energies on the groove deepening along with the ridge morphology is discussed in detail. The simulated grain boundary features are of practical significance for understanding and controlling the nanocrystalline thin film experiments.

Morphological stability of three-dimensional cementite rods in polycrystalline system: A phase-field analysis

Transformations accompanying shape-instability govern the morphological configuration and distribution of the phases in a microstructure. Owing to the influence of the microstructure on the properties of a material, in the present work, the stability of three-dimensional rods in a ‘representative’ polycrystalline system is extensively analysed. A multiphase-field model, which recovers the physical laws and sharp-interface relations, and includes grain boundary diffusion, is adopted to investigate the morphological evolution of the precipitate. Moreover, the efficiency of the numerical approach is ensured by establishing the volume-preserving chemical equilibrium through the incorporation TCFe8 (CALPHAD) data and solving phase-field evolution in the Allen-Cahn framework. The morphological evolution of the rod in the representative multiphase system exhibits a unique transformation mechanism which is significantly different from the evolution of an isolated finite-structure. It is realised that, in a polycrystalline arrangement, irrespective of the initial size of the rod, the shape-change begins with the energy-minimising events at the triple junctions. This early transformation renders a characteristic morphology at the longitudinal ends of the structure, which introduces sufficient driving-force through the curvature-difference for the subsequent morphological changes. The continued mass transfer to the terminations, ultimately, breaks-off the rod into separate entities that are entangled in the grain boundary. With increase in the aspect ratio of the rod, it is identified that the source of mass transfer, which turns into the ovulation site, shifts from the centre. This increases the number of fragmentation events and introduces satellite particle. The size of the satellite particle is dictated by a definite ovulation criterion, which is ascertained by examining the transformation of different-sized rods. A comprehensive understanding of the transformation kinetics and mechanism governing the morphological evolution of the rods in a polycrystalline system is rendered in this work.

Simulation of turbulent bubbly pipe flow with high density ratio and high reynolds number by using the lattice boltzmann method and a multi-phase field model

Direct numerical simulation (DNS) has been widely employed to study the dynamics of turbulent bubbly flows; however, it is used in a limited setting, namely, turbulent bubbly channel flows with low density ratio and low Reynolds number. The present study aims to overcome these limitations and simulate turbulent bubbly pipe flow of a water-air system, a common experimental setting, with high density ratio and Reynolds number. Recently, we developed a cumulant lattice Boltzmann method for two-phase flows and simulated violent two-phase flows with high density ratio and high Reynolds number (Sitompul and Aoki, 2019). In this study, that method is extended by incorporating a multi-phase field model for simulating dispersed bubbles. The proposed method was evaluated by using several cases, namely, 3D bubble rising, turbulent single-phase channel and pipe flows with friction Reynolds number (Reτ≈180,550), turbulent bubbly channel flows with (Reτ≈180) and void fraction (α=1.5%,19.4%), and turbulent bubbly pipe flow with (Reτ=550,α=9.5%). The proposed method can stably simulate the cases, and the results obtained by the proposed method agree well with numerical and experimental results given in the references for given computational domain and grid sizes.

Multiphase-field modelling of crack propagation in geological materials and porous media with Drucker-Prager plasticity

A multiphase-field approach for elasto-plastic and anisotropic brittle crack propagation in geological systems consisting of different regions of brittle and ductile materials is presented and employed to computationally study crack propagation. Plastic deformation in elasto-plastic materials such as frictional, granular or porous materials is modelled with the pressure-sensitive Drucker-Prager plasticity model. This plasticity model is combined with a multiphase-field model fulfilling the mechanical jump conditions in diffuse solid-solid interfaces. The validity of the plasticity model with phase-inherent stress and strain fields is shown, in comparison with sharp interface finite element solutions. The proposed model is capable of simulating crack formation in heterogeneous multiphase systems comprising both purely elastic and inelastic phases. We investigate the influence of different material parameters on the crack propagation with tensile tests in single- and two-phase materials. To show the applicability of the model, crack propagation in a multiphase domain with brittle and elasto-plastic components is performed.

Multiphase-field modeling of temperature-driven intermetallic compound evolution in an Al–Mg system for application to solid-state joining processes

Solid-state joining of dissimilar materials results typically in the formation of intermetallic compounds at the weld interface, which strongly determines the resulting mechanical properties. To tailor the joint strength, understanding of the formation of the intermetallic compound and their driving mechanisms is crucial. In this study, the evolution of temperature-driven Al3Mg2 and Al12Mg17 intermetallic compounds in an Al–Mg system for application to solid-state joining processes via a multiphase-field approach is numerically investigated. To this end, the CALPHAD approach to obtain the thermodynamic parameters of the relevant phases is used in conjunction with the multiphase-field model. The simulation results are qualitatively compared with experimental results in the literature in terms of thickness and morphology of intermetallic grains, exhibiting a reasonable agreement. The influence of grain boundary diffusion and interface energy on the morphology and kinetics of the intermetallic compound grains is investigated in detail with the established multiphase-field model.

Multi-phase field modeling for various fracture mechanisms in composites

Composite possesses wide spectrum of mechanical responses along with remarkable fracture strength and toughness. Such properties evolve due to combination of different failure mechanisms across multiple phases. Understanding the contribution of these mechanisms on overall behavior is necessary to fabricate the composite with desired fracture properties. Although numerous studies have been performed to correlate the fracture response with crack propagation characteristics, they fail to account several underlying micro-mechanical events. Therefore, in the present study, a multi-phase field fracture model is proposed. We consider that constituents are homogeneously distributed within the matrix, while macro-scale phase field descriptors are utilized to represent the failure of its constituents. Further, the reinforced elements are assumed to be anisotropic in nature with preferential fracture direction. Therefore, by employing the rule of mixture in free energies and applying thermodynamic principles, the governing equations together with evolution laws of phase field parameters are derived. Using multi-field approach and extracting the weak form for governing equations, we develop finite element based numerical framework. Coupled sets of linear equations are solved using staggered scheme in parallel computing architecture. Proposed scheme is verified with analytical solution of a composite where separate evolutions of damage within phases are correlated with emergent stress–strain response. Numerical simulations are performed to predict the mechanical response by interpreting the competitive involvement of fracture mechanisms within matrix and inclusion phases. The present findings and future scopes highlight the predictive capability of the proposed scheme for analyzing fracture behavior of composites.

Precipitation and Growth Simulation of γ′ Phase in Single Crystal Superalloy DD6 with Multiphase-Field Method and Explicit Nucleation Algorithm

The microstructure evolution of Ni-based superalloys during heat treatment is of great significance for obtaining better service performance. However, heat treatment experimentation is costly and time-consuming, and sometimes fails to reveal physical mechanisms well. In the present study, a multiphase-field model coupled with an explicit nucleation algorithm was established to simulate the precipitation and growth of γ′ phase in DD6 superalloy, which can be applied to a multicomponent elastic-inhomogeneous system. The PanNickel© database was used to calculate thermodynamic and kinetic data in multicomponent superalloys. First, the coupling method of multiphase-field model and explicit nucleation algorithm was introduced. The coupled model was used to simulate the precipitation of γ′ phase under isothermal and non-isothermal conditions. It was found that a unimodal microstructure was formed under isothermal conditions and there was a “soft impingement” phenomenon, while a bimodal distribution composed of cuboidal γ′ precipitates and fine secondary γ′ precipitates was formed during a cooling process of 25–125 °C/min. The effect of cooling rate was studied. Then, the chemical and elastic driving forces were analyzed. It was found that Al and Ta contributed most to the chemical driving force, while Re and W gathered at the γ/γ′ interface and inhibited the growth of γ′ phase. γ′ precipitates had a cuboidal shape under the influence of elastic driving force. Finally, the growth and coarsening process of γ′ phase was studied and compared with the well-known Lifshitz−Slyosov−Wagner (LSW) theory. The growth of γ′ phase can be divided into rapid growth, coarsening and quasi-static coarsening stages according to the simulation results, among which the coarsening stage is basically consistent with the LSW theory. The current model can be used to simulate the precipitation and growth of single crystal superalloys where multicomponent and elastic effects are considered.

S-PFM model for ideal grain growth

The phase-field method is increasingly used for studying grain growth in metallic alloys. Such an approach offers a thermodynamically consistent framework for studying microstructure evolution during grain growth without the need to explicitly track the interface positions. However, the numerical solution of the models requires a grid spacing much smaller than the interface width. This leads to computationally very intensive simulations, especially when a large number of grains is required to accurately measure statistical quantities. The recently proposed S-PFM approach provides a new inherently discrete formulation of phase field models where the interface width can be as small as the grid spacing, thus drastically improving the numerical performances of the method. Here, we show that this approach can be extended to a multi-phase field model for ideal grain growth. Then, we perform two dimensional simulations to analyse in detail the kinetics of both grain boundaries and triple junctions. We compare our model to a classical phase field formulation and we demonstrate that, for a prescribed accuracy, the memory requirement and simulation times are both reduced by a factor of 4D, where D is the space dimension. Finally, we perform a large-scale simulation of grain growth in two dimensions and show that the method is able to reveal the specific shape of the grain size distribution in the scale-invariant regime displaying two peaks around the mean grain size and quantitative measurements of the underlying topological classes are performed.

Multi-phase field simulation of multi-grain peritectic transition in multiple phase transformation

Taking Fe-C binary alloy as an example, based on the multi-phase field model, the nucleation and growth of δ phase, peritectic reaction, peritectic transformation, and the growth of subsequent austenite are simulated. Effects of the nucleation site of austenite on the peritectic reaction rate and the starting time of the peritectic transformation were studied. The simulation results show that the γ phase, as a shell, surrounds the δ phase and grows rapidly when the peritectic reaction occurs between the dendritic δ grains, and a layer of γ phase shell is formed around δ phase after the peritectic reaction. After the δ phase is surrounded by γ phase completely, the membrane shell separates the L phase from the δ phase, so that the phase transfers from peritectic reaction to peritectic transformation. During the peritectic transformation, since the solute diffusion coefficient of the liquid phase is much greater than that of the solid phase, the average growth rate of austenite in the liquid phase is visibly higher than that of the δ phase. The peritectic reaction rate is related to the curvature of the nucleation site of the γ phase on the δ phase grains. The peritectic reaction rate at the large curvatures is faster than that at small curvatures.

A two-dimensional phase-field study on dendritic growth competition under convective conditions

In the present work, the growth competition of columnar dendrites is investigated in the presence of melt convection. By using a multiphase-field model, the microstructural evolution of unidirectionally solidified dendrites with different crystal orientations is addressed in detail. Generally, for converging bicrystal grains, the dendritic growth competition follows a conventional overgrowth model, where a favorably oriented dendrite eliminates an unfavorably oriented dendrite at the grain boundary. However, in the presence of lateral melt flow in the liquid phase, we discover that the conventional overgrowth mechanism transforms into an unusual overgrowth mechanism as the melt velocity gradually increases. Recently observed in bi-crystal studies, the unfavorably oriented dendrites overgrow at the expense of the favorably oriented dendrites during an unusual overgrowth behavior. From our simulations we elucidate that the presence of an additional convective transport in the liquid phase modifies the solute distribution at the grain boundary, which in turn affects the overgrowth mechanism of converging dendrites. We show that there exists a critical melt velocity to determine the growth competition at the grain boundary. In addition, the role of interfacial anisotropy on growth competition is analyzed in detail, where we discern that the conventional overgrowth behavior is dominant at large anisotropic strengths. Lastly, a morphological selection map is depicted to predict the crossover region between the two overgrowth behaviors.