Multi-phase Field Research Articles

金属粉床積層造形プロセスに対する格子ボルツマン法とマルチフェーズフィールド法

A Multi-phase field lattice Boltzmann method for melting, gas/liquid flow and solidification in powder bed fusion process of Ni alloy additive manufacturing is developed. Improved lattice kinetic scheme is adapted for gas/liquid free surface flow simulation. High speed calculation is achieved by MPI&OpenMP hybrid parallelization. It is confirmed that this method has high capability to simulate the powder bed fusion process in detail.

Multi phase-field simulation study on microstructural grain development during inductive-based friction surfacing of aluminium

The computational solution is considered an effective tool for the analysis and a good understanding of the complex microstructural development that occurs during friction surfacing. In this article, ABACUS software was used to create a 3D-FE model of friction surfaced layering of AA 6063 aluminium on EN8 carbon steel, and the heat and strain rate collected throughout the operation were utilised for a computational investigation of microstructural recrystallization and grain development. The Multi-phase Field model (MFM) and constructive material model (CMM) were used for the prediction of the grain development during the process. A decrease in the incubation period from 0.839 sec to 0.578 sec was seen before recrystallization, after a temperature rises from 100°C to 300°C for substrate preheating. Validation of the reliability model obtained from the computational study was done using the image received from electron backscattered diffraction (EBSD) for grain size development and distribution. An appropriate assessment has been made between computational and experimental images which shows the maximum error of less than 10%. The development of grain structure during recrystallization was impacted extensively for increasing the coating strength, which was seen as inversely proportional to the average coating's grain diameter.

Hot Cracking Susceptibility Prediction from Quantitative Multi-Phase Field Simulations with Grain Boundary Effects

Hot Cracking Susceptibility Prediction from Quantitative Multi-Phase Field Simulations with Grain Boundary Effects

Multi-phase field simulation of competitive grain growth for directional solidification

The multi-phase field model of grain competitive growth during directional solidification of alloy is established. Solving multi-phase field models for thin interface layer thickness conditions, the grain boundary evolution and grain elimination during the competitive growth of SCN-0.24-wt% camphor model alloy bi-crystals are investigated. The effects of different crystal orientations and pulling velocities on grain boundary microstructure evolution are quantitatively analyzed. The obtained results are shown below. In the competitive growth of convergent bi-crystals, when favorably oriented dendrites are in the same direction as the heat flow and the pulling speed is too large, the orientation angle of the bi-crystal from small to large size is the normal elimination phenomenon of the favorably oriented dendrite, blocking the unfavorably oriented dendrite, and the grain boundary is along the growth direction of the favorably oriented dendrite. When the pulling speed becomes small, the grain boundary shows the anomalous elimination phenomenon of the unfavorably oriented dendrite, eliminating the favorably oriented dendrite. In the process of competitive growth of divergent bi-crystal, when the growth direction of favorably oriented dendrites is the same as the heat flow direction and the orientation angle of unfavorably oriented grains is small, the frequency of new spindles of favorably oriented grains is significantly higher than that of unfavorably oriented grains, and as the orientation angle of unfavorably oriented dendrites becomes larger, the unfavorably oriented grains are more likely to have stable secondary dendritic arms, which in turn develop new primary dendritic arms to occupy the liquid phase grain boundary space, but the grain boundary direction is still parallel to favorably oriented dendrites. In addition, the tertiary dendritic arms on the developed secondary dendritic arms may also be blocked by the surrounding lateral branches from further developing into nascent main axes, this blocking of the tertiary dendritic arms has a random nature, which can have aninfluence on the generation of nascent primary main axes in the grain boundaries.

A multi-phase-field model of topological pattern formation during electrochemical dealloying of binary alloys

Porous metal structures produced by electrochemical dealloying have been extensively studied for use in various applications that require high surface areas. A new comprehensive multi-phase-field (MPF) model is proposed to study topological porous patterns formed by spontaneously etching a bulk binary alloy that involves electrochemical reactions, bulk and surface diffusion, ion transport, applied electrode potential, and charge conservation. The governing equations for the alloy-porous cluster-electrolyte system account for a generalized Butler-Volmer electrochemical reaction and are in accordance with the classical nucleation theory. Based on a quantitative examination of the effects of electrode potential and precursor composition, the simulation results reproduce typical phenomena including passive surface dealloying, active porosity evolution, critical potential, and characteristic length scale in two-dimension (2-D) and three-dimension (3-D). To consider more complicated dealloying systems, the evolutions of a bimodal porous metal structure, nanocomposite, and nested porous network with a structured hierarchy are investigated. The proposed model can be a useful tool for understanding and predicting the morphology evolution of diverse porous structures during electrochemical dealloying.

Laplacian Echo-State Networks for production analysis and forecasting in unconventional reservoirs

Production data analysis for low permeability shale reservoirs is crucial in characterizing flow regimes and reservoir properties, and the forecasting of production is essential for portfolio and reservoir management. However, traditional methods have failed due to incorrect physics or complicated convolution from the well control history. In this research, we provide a physics-assisted analytics workflow using Laplacian Eigenmaps Coupled Echo-State Network (LEESN) to facilitate and accelerate the analysis of noisy historical production data. Pressure-rate deconvolution is an ill-posed, complex time-series problem. When using the traditional Echo-State Network (ESN), the number of training sets is less than the number of neurons. To solve this problem, we apply LEESN to first deconvolve noisy variable-pressure variable-rate histories into smooth constant-pressure rate responses. The physics-based training features and training algorithm provide additional benefits in addition to the analytic approach by honoring transient flow physics. After training, the constant-pressure rate response can be predicted and used for reservoir characterization, and the trained model could be further used for production and EUR forecasting through long-term rate predictions to the economic limit. The proposed workflow was first validated by synthetic cases where the production data were obtained through simulation. The short-term flow rate history was obtained by specifying highly variable controlling pressures. We also added artificial white Gaussian noise to mimic measured signals collected in the field and input this information into LEESN for deconvolution. The constant-pressure rate response was generated after training to determine flow regimes and properties such as permeability using a traditional transient testing specialized plot. All outcomes from the analytics approach were validated by comparison against the input data of the synthetic simulation model. The advantages of the analytics approach were maintained with a moderate variation of noisy pressure-rate signals. For production forecasting, both the trained analytics model and simulator were used to predict for an extended period, and the results indicated good agreement between the response predictions. We performed a further sensitivity analysis on important parameters such as the training scale as well as the capability of Laplacian Eigenmaps handling moderate noise in training data. The comparison between the model predictions and simulation data showed significantly increased accuracy in production estimates. The efficacy was further demonstrated from additional single-phase and multiphase synthetic and field cases. This study shows that the LEESN approach is a powerful alternative to interpret pressure-rate-time information from production data. The discussion and comparison of LEESN with other traditional production analysis and forecast methods are included as well. Deconvolved pressure-rate data greatly enhances traditional rate-transient analysis (RTA) used to characterize reservoir parameters, and the trained model enables engineers to predict future production even with noisy, highly-variable production histories. The robustness of the proposed analytics methodology is strengthened by coupling the training features with transient flow physics and provides a unique approach for production analysis and forecasting for unconventional reservoirs.

Multiphase field models for collective cell migration.

Confluent cell monolayers and epithelia tissues show remarkable patterns and correlations in structural arrangements and actively driven collective flows. We simulate these properties using multiphase field models. The models are based on cell deformations and cell-cell interactions and we investigate the influence of microscopic details to incorporate active forces on emerging phenomena. We compare four different approaches, one in which the activity is determined by a random orientation, one where the activity is related to the deformation of the cells, and two models with subcellular details to resolve the mechanochemical interactions underlying cell migration. The models are compared with respect to generic features, such as coordination number distribution, cell shape variability, emerging nematic properties, as well as vorticity correlations and flow patterns in large confluent monolayers and confinements. All results are compared with experimental data for a large variety of cell cultures. The appearing qualitative differences of the models show the importance of microscopic details and provide a route towards predictive simulations of patterns and correlations in cell colonies.

Geometry of Triple Junctions during Grain Boundary Premelting.

Grain boundaries (GBs) whose energy is larger than twice the energy of the solid-liquid interface exhibit the premelting phenomenon, for which an atomically thin liquid layer develops at temperatures slightly below the bulk melting temperature. Premelting can have a severe impact on the structural integrity of a polycrystalline material and on the mechanical high-temperature properties, also in the context of crack formation during the very last stages of solidification. The triple junction between a dry GB and the two solid-liquid interfaces of a liquid layer propagating along the GB cannot be defined from macroscopic continuum properties and surface tension equilibria in terms of Young's law. We show how incorporating atomistic scale physics using a disjoining potential regularizes the state of the triple junction and yields an equilibrium with a well-defined microscopic contact angle. We support this finding by dynamical simulations using a multiphase field model with obstacle potential for both purely kinetic and diffusive conditions. Generally, our results should provide insights on the dynamics of GB phase transitions, of which the complex phenomena associated with liquid metal embrittlement are an example.

Multi-phase field modeling in TRIP steels: Distributed vs. average stability and strain-induced transformation of retained austenite

Austenite stability and dispersion are key parameters controlling the strain-induced transformation of Retained Austenite (RA) to martensite in TRIP steels. Stabilization is achieved through a two-stage heat-treatment process, consisting of Intercritical Annealing (IA) followed by Bainitic Isothermal Treatment (BIT). In the present study, an integrated approach is developed, for the description of microstructural evolution of a TRIP700 steel during processing, using multi-phase field modeling. Two models are employed to assess RA stability, through MSσ, MS calculations, and strain-induced transformation kinetics of dispersed RA upon plastic deformation. The first model considers the grain size and composition distributions, while the second the respective average property values of RA obtained at the end of BIT. In-depth understanding of the influence of microstructural evolution on the transformation kinetics of RA can be achieved by taking into account the local properties of RA particles, presenting significant differences from that using average values. Results indicate that refined RA is more stable, transforming into martensite at late stages of deformation. Simulations of the RA particle size, stability and martensite transformation fraction resulting from uniaxial tension are validated against experiments conducted on TRIP700 steel under different BIT conditions, presenting good agreement. The proposed integrated approach can assist the design of TRIP steels by identifying optimal microstructural characteristics.

Ferroelectric switching behavior of nanoscale Hf0.5Zr0.5O2 grains

Grain size has shown significant effect on the ferroelectricity of HfO2-based thin films. However, its microscopic mechanism remains unclear. For this purpose, Hf0.5Zr0.5O2 film with different sized grains was prepared by radio frequency magnetron sputtering. The piezoresponse force microscope characterization of the domain switching properties of nanograins shows that domain switching ability of the grain becomes better as its size decreases within the size range we studied. Then, density functional theory (DFT) calculation of grains’ energy proves that the lower surface energy of ferroelectric phase is the key factor to make the ferroelectric phase more stable and larger proportion in smaller sized grain. Finally, a multi-phase coexistence phase field model is developed based on the DFT calculation. It is found out that if the contribution of surface energy to the total energy of nano-grain increases, the volume fraction of ferroelectric phase will increase. And (001) ferroelectric phase with the lowest surface energy density will be the most stable and account for the largest volume fraction. Our finding provides a microscopic insight into the “grain size effect” and also some guidelines for high-performance ferroelectric thin films preparation through “grain size engineering”.

Multiphase-field modeling of disproportionation performance of ZrCo alloy during hydrogen absorption

This work attempts to develop a multiphase field model to study ZrCoHy, ZrH2, and ZrCo2 phase formation paths in the ZrCoH0.5 matrix. Interestingly, one conserved phase-field variable (c) and four non-conserved phase-field variables (ϕi) are utilized to construct free energy function, where c represents the hydrogen concentration and ϕi stands for the ZrCoHx, ZrCoHy, ZrH2, and ZrCo2 phase. Some valuable material parameters, such as the Gibbs formation enthalpy, are also taken into account. Importantly, the system energy fluctuation is taken as the driving force of phase evolution, and then, governing equations are solved by self-programming complied with the Fortran language. The results demonstrate that disproportionation products are most likely to be derived from the initial ZrCoHy. The presence of ZrH2 and ZrCo2 promotes the nucleation incubation of disproportionation products. Besides, ZrCoHy is more prone to generate ZrH2 and ZrCo2 in comparison with ZrCoHx, which is attributed to the fact that the crystal structure of ZrCoHy is closer to that of ZrCo2. Because ZrCoHy could be regarded as the transition state when ZrCoHx is transformed into ZrCo2, ZrCoHy is easy to be produced in advance during the ZrCoHx disproportionation process, instead of ZrH2 and ZrCo2 directly generated by the ZrCoHx disproportionation.

Multiphase Field Modeling of Dendritic Solidification of Low-Carbon Steel with Peritectic Phase Transition

In the present study, an improved multiphase field model with taking into consideration the S/L interface anisotropy is proposed to simulate the dendritic growth with peritectic transition of low-carbon steel, and a new random heterogeneous nucleation method is proposed to treat the γ phase nucleation during the peritectic solidification process. The γ phase growth around the dendrite (the single dendrite and polycrystalline ferrite) and the subsequent peritectic transformation during the peritectic solidification are simulated. The results show that when the temperature reaches the γ phase nucleation temperature, γ nuclei suddenly nucleate at the δ/L interface, and laterally grow with the movement of L/γ/δ triple point around the δ/L interface of δ dendrite. After the δ dendrite is fully wrapped by the continuous intervening γ phase, the γ phase continues to grow with direct phase transformation from both δ phase and liquid phase. With the increase of melt undercooling, more δ phase and liquid phase are consumed to form γ phase, and the intervening γ phase between the δ dendrite and liquid phase becomes more and more thick, but the δ dendrite arm becomes more and more thin. During the dendritic solidification process, the solute would be rejected from dendrite trunk with the dendritic growth and enriched at the dendrite boundary, especially at the dendrite root. The solute enrichment at the dendrite root would inhibit the γ phase growth at the dendrite root and thus the thickness of intervening γ phase at the dendrite root is thinnest around the dendrite. Moreover, the solute concentration in γ phase is among the δ phase and liquid phase, because the carbon solubility in γ phase is higher than that in δ phase, but lower than that in liquid phase.

A parabolic approximation scheme for multi-phase-filed simulation of non-isothermal solidification

A parabolic approximation scheme is proposed to implement the multi-phase-filed simulation of non-isothermal solidification. Based on the second-order fitting of free energy density function, this scheme constructs an explicit expression for the calculation of phase compositions, which is more efficient than the traditional Newton’s iteration scheme used in the Kim-Kim-Suzuki model. As an application example of this numeral scheme, the peritectic solidification of Ti-Al alloy is simulated to present the coupling of multi-phase field and concentration field, which reveals the non-isothermal evolution principle of microstructure and micro-segregation. The present study not only improves the calculation efficiency through a new numeral scheme, but also contributes to the understanding of the thermodynamics and kinetics of Ti-Al peritectic solidification.

A multi phase-field-cohesive zone model for laminated composites: Application to delamination migration

Failure processes in Laminated Fiber-Reinforced Composites (LFRCs) entail the development and progression of different physical mechanisms and, in particular, the interaction between inter-laminar and intra-laminar cracking. Reliable modeling of such complex scenarios can be achieved by developing robust numerical predictive tools that allow for the interaction of both failure modes. In this study, a novel Multi Phase-Field (MPF) model relying on the Puck theory of failure for intra-laminar failure at ply level is coupled with a Cohesive Zone Model (CZM) for inter-laminar cracking. The current computational method is numerically implemented as a system of non-linear coupled equations using the finite element method via user-defined UMAT and UEL subroutines in ABAQUS. The computational tool is applied to qualitatively predict delamination migration in long laminated fiber-reinforced polymers composites comprising 44 cross-ply laminates. The reliability of the current approach is examined via the correlation with experimental results. Finally, the present study is complemented with additional representative examples with the aim of providing further insight into the potential role of different aspects of the system in the delamination migration, including (i) the variation of the ply angle in the migration zone, (ii) the load application point, and (iii) initial crack length.

Study on the phase transition dynamics of HfO2-based ferroelectric films under ultrafast electric pulse

Wake-up effect and fatigue in HfO2-based ferroelectric films are closely related to the phase transition dynamics of the film subjected to ultrafast electric pulses. Here, we establish a multiphase coexistence phase field dynamics model for HfO2-based ferroelectric films in the ultrafast time scale and study the effects of the amplitude, width and frequency of the electric pulse on the phase transition dynamics. Based on the simulation results, we obtain the analytical equation of the volume fraction of switched c-domains under low fields as a function of pulse duration. And we found that monoclinic phase can transform into ferroelectric c-domains under high amplitude electric field (E ⩾ 2.8 MV cm−1). The electric pulse duration affects the film’s retention properties. When the duration of the electric pulse is less than 1.2 ns or longer than 1.8 ns, the ferroelectric c-domains will respectively invert into other phases or increase cumulatively after removing the electric field. The frequency of cyclic pulse is related to the degree of wake up effect. The lower the pulse frequency is, the more obvious the ‘wake up’ effect is.

Multiphase field modeling of grain boundary migration mediated by emergent disconnections

Knowledge about grain boundary migration is a prerequisite for understanding and ultimately modulating the properties of polycrystalline materials. Evidence from experiments and molecular dynamics (MD) simulations suggests that the formation and motion of disconnections is a mechanism for grain boundary migration. Here, grain boundary migration is modeled using a multiphase field model based on the principle of minimum dissipation potential with nonconvex boundary energy, along with a stochastic model for thermal nucleation of disconnection pairs. In this model, disconnections arise spontaneously in the presence of an elastic driving force, and that their motion mediates boundary migration. The effect is due to the fact that the formation of the disconnections pairs results in a stress concentration, causing the elastic driving force to exceed the threshold value and driving the propagation of the disconnection along the interface. The model is applied to study the propagation/annihilation of single disconnection pairs, the relaxation of a perturbed interface, and shear coupling at various temperatures. The results are consistent with the current understanding of disconnections, and capture the effect of thermal softening.

フェーズフィールド法による Ni 基超合金付加製造における凝固偏析予測

フェーズフィールド法による Ni 基超合金付加製造における凝固偏析予測

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.

Multi-phase-field lattice Boltzmann model for polycrystalline equiaxed solidification with motion

The formation process of equiaxed structures during alloy solidification is a complicated multiphysics problem as it includes solid–liquid phase transformations, transport of solute and heat, liquid flow, solid motion, solid–solid interactions, and grain growth after solidification completion. In this study, a multi-phase-field (MPF) model with the double-obstacle (DO) potential is introduced to our previous model [T. Takaki et al., Comput. Mater. Sci., 147 (2018) 124–131] to express the formation process of equiaxed structures accurately and efficiently, from the growth of multiple mobile dendrites to the grain growth with grain boundary anisotropy after solidification. The accuracy of the resulting MPF-lattice Boltzmann (MPF-LB) model with the DO potential is evaluated by comparison with the results obtained using a model with the double-well potential through simulations of purely diffusive dendrite growth, the growth of an immobile dendrite with forced convection, and the growth of a mobile dendrite under shear flow. Finally, the model is applied to both large-scale polycrystalline solidification and semi-solid deformation processes.

Quantitative phase-field simulation of the entire solidification process in one hypereutectic Al–Si alloy considering the effect of latent heat

The multi-phase-field (MPF) model coupled with reliable CALPHAD thermodynamic/atomic mobility descriptions, appropriate faceted anisotropy and key experimental validation is employed to perform the quantitative simulation of the microstructure evolution of the hypereutectic Al-16 ​wt%Si alloy during solidification. By considering the effect of latent heat released during the eutectic phase transformation in the simulation, the appropriate entire solidification sequence is obtained. All the characteristics in the as-cast microstructure can be nicely reproduced, including the primary (Si) encircled by the second primary (Al) halo, and the coupled eutectic grains appearing at the Liquid/(Al) interface rather than Liquid/(Si) interface. Moreover, three types of growth patterns of eutectic (Si) with different lengths are distinguished according to the distance between primary (Si) and the nucleation position of eutectic (Si). Furthermore, the evaluated ratios of width to length for 50 eutectic (Si) grains due to the MPF simulation are in good agreement with the experimental data. The quantitative phase-field simulation of the entire solidification process of the hypereutectic Al-16 ​wt%Si alloy in the present work is anticipated to pave the way for revealing the grain refinement mechanism in hypereutectic Al–Si alloys in the future studies.