Phase field modeling of multiple dendrite growth of Al-Si binary alloy under isothermal solidification

Phase field method offers the prospect of being able to perform realistic numerical experiments on dendrite growth in metallic systems. In this study, the growth process of multiple dendrites in AI-2-mole-%-Si binary alloy under isothermal solidification was simulated using phase field model. The simulation results showed the impingement of arbitrarily oriented crystals and the competitive growth among the grains during solidification. With the increase of growing time, the grains begin to coalesce and impinge the adjacent grains. When the dendrites start to impinge, the dendrite growth is obviously inhibited.

Prediction of mushy zone permeability of Al-4.5wt%Cu alloy during solidification by phase field model and CFD simulation

Liquid permeability of the mushy zone is important for porosity formation during the solidification process. In order to investigate the permeability of the mushy zone, an integrated model was developed by combining the phase field model and computational fluid dynamics (CFD) model. The three-dimensional multigrain dendrite morphology was obtained by using the phase field model. Subsequently, the computer-aided design (CAD) geometry and mesh were generated based on calculated dendrite morphologies. Finally, the permeability of the dendritic mushy zone was obtained by solving the Navier-Stokes and continuity equations in ANSYS Fluent software. As an example, the dendritic mushy zone permeability of Al-4.5wt%Cu alloy and its relationship with the solid fractions were studied in detail. The predicted permeability data can be input to the solidification model on a greater length scale for macro segregation and porosity simulations.

Phase field modeling of dendrite growth

Single dendrite and multi-dendrite growth for Al-2 mol pct Si alloy during isothermal solidification are simulated by phase field method. In the case of single equiaxed dendrite growth, the secondary and the necking phenomenon can be observed. For multi-dendrite growth, there exists the competitive growth among the dendrites during solidification. As solidification proceeds, growing and coarsening of the primary arms occurs, together with the branching and coarsening of the secondary arms. When the diffusion fields of dendrite tips come into contact with those of the branches growing from the neighboring dendrites, the dendrites stop growing and being to ripen and thicken.

Discussions on the non-equilibrium effects in the quantitative phase field model of binary alloys

All the quantitative phase field models try to get rid of the artificial factors of solutal drag, interface diffusion and interface stretch in the diffuse interface. These artificial non-equilibrium effects due to the introducing of diffuse interface are analysed based on the thermodynamic status across the diffuse interface in the quantitative phase field model of binary alloys. Results indicate that the non-equilibrium effects are related to the negative driving force in the local region of solid side across the diffuse interface. The negative driving force results from the fact that the phase field model is derived from equilibrium condition but used to simulate the non-equilibrium solidification process. The interface thickness dependence of the non-equilibrium effects and its restriction on the large scale simulation are also discussed.

Phase field modeling of the ring-banded spherulites of crystalline polymers: The role of thermal diffusion

The ring-banded spherulite is a special morphology of polymer crystals and has attracted considerable attention over recent decades. In this study, a new phase field model with polymer characteristics is established to investigate the emergence and formation mechanism of the ring-banded spherulites of crystalline polymers. The model consists of a nonconserved phase field representing the phase transition and a temperature field describing the diffusion of the released latent heat. The corresponding model parameters can be obtained from experimentally accessible material parameters.Two-dimensional calculations are carried out for the ring-banded spherulitic growth of polyethylene film under a series of crystallization temperatures. The results of these calculations demonstrate that the formation of ring-banded spherulites can be triggered by the self-generated thermal field. Moreover, some temperature-dependent characteristics of the ring-banded spherulites observed in experiments are reproduced by simulations, which may help to study the effects of crystallization temperature on the ring-banded structures.

Transport diffuse interface model for simulation of solid-fluid interaction

For solid-fluid interaction, one of the phase-density equations in diffuse interface models is degenerated to a "0 = 0" equation when the volume fraction of a certain phase takes the value of zero or unity. This is because the conservative variables in phasedensity equations include volume fractions. The degeneracy can be avoided by adding an artificial quantity of another material into the pure phase. However, nonphysical waves,such as shear waves in fluids, are introduced by the artificial treatment. In this paper,a transport diffuse interface model, which is able to treat zero/unity volume fractions, is presented for solid-fluid interaction. In the proposed model, a new formulation for phase densities is derived, which is unrelated to volume fractions. Consequently, the new model is able to handle zero/unity volume fractions, and nonphysical waves caused by artificial volume fractions are prevented. One-dimensional and two-dimensional numerical tests demonstrate that more accurate results can be obtained by the proposed model.

Current development in quantitative phase-field modeling of solidification

The quantitative phase-field simulations were reviewed on the processes of solidification of pure metals and alloys.The quantitative phase-field equations were treated in a diffuse thin-interface limit,which enabled the quantitative links between interface dynamics and model parameters in the quasi-equilibrium simulations.As a result,the quantitative modeling is more effective in dealing with microstructural pattern formation in the large scale simulations without any spurious kinetic effects.The development of the quantitative phase-field models in modeling the formation of microstructures such as dendritic structures,eutectic lamellas,seaweed morphologies,and grain boundaries in different solidified conditions was also reviewed with the purpose of guiding to find the new prospect of applications in the quantitative phase-field simulations.

Phase field calculation of interface mobility in a ternary alloy

A novel approach based on the quantitative phase field model was proposed to calculate the interface mobility and applied to the α/β interface of a ternary Ti-6Al-4V alloy.Phase field simulations indicate that the higher interface mobility leads to the faster transformation rate,but only a unique value of interface mobility matches the diffusion equation under the diffusion-controlled condition.By comparing the transformation kinetics from phase field simulations with that from classical diffusion equation,the interface mobility at different temperatures can be obtained.The results show that the calculated interface mobility increases with increasing temperature and accords with Arrhenius equation very well.

Phase-field simulation for the evolution of solid/liquid interface front in directional solidification process

In this study, the phase field method was used to study the multi-controlling factors of dendrite growth in directional solidification. The effects of temperature gradient, propelling velocity, thermal disturbance and growth orientation angle on the growth morphology of the dendritic growth in the solid/liquid interface were discussed. It is found that the redistribution of solute leads to multilevel cavity and multilevel fusion to form multistage solute segregation, and the increase of temperature gradient and propelling velocity can accelerate the dendrite growth of directional solidification, and also make the second dendrites more developed, which reduces the primary distance and the solute segregation. When the temperature gradient is large, the solid-liquid interface will move forward in a flat interface mode,and the thermal disturbance does not affect the steady state behavior of the directionally solidified dendrite tip. It only promotes the generation and growth of the second dendrites and forms the asymmetric dendrite. Meanwhile, it is found that the inclined dendrite is at a disadvantage in the competitive growth compared to the normal dendrite, and generally it will disappear. When the inclination angle is large, the initial primary dendrite may be eliminated by its secondary or third dendrite.

MICRO-DESCRIPTION OF THE SOLUTE-FIELD AND THE PHASE-FIELD MODEL FOR ISOTHERMAL PHASE TRANSITION IN BINARY ALLOYS

A new phase field method for two-dimensional simulations of binary alloy solidification was studied. A model basing on solute conservative in every unit was developed for solving the solute diffusion equation during solidification. Two-dimensional computations were performed for ideal solutions and Ni-Cu dendritic growth into an isothermal and highly supersaturated liquid phase.

EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS

In this paper,we study linearly first and second order in time,uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel"scalar auxiliary variable"(SAV)approach,a new developed efficient and accurate method for a large class of gradient flows.The schemes are based on the first order Euler method and the second order backward differential formulas(BDF2)for time discretization,and finite element methods for space discretization.The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps.Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.

Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with applications to crystal growth.New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.

Effects of temperature gradient on the interface microstructure and diffusion of diffusion couples:Phase-field simulation

The temporal interface microstructures and diffusions in the diffusion couples with the mutual interactions of the temperature gradient, concentration difference and initial aging time of the alloys are studied by phase-field simulation, and the diffusion couples are produced by the initial aged spinodal alloys with different compositions. Temporal composition evolution and volume fraction of the separated phase indicate the element diffusion direction through the interface under the temperature gradient. The increased temperature gradient induces a wide single-phase region on two sides of the interface.The uphill diffusion proceeds through the interface, no matter whether the diffusion direction is up or down with respect to the temperature gradient. For an alloy with short initial aging time, phase transformation accompanying the interdiffusion results in the straight interface with the single-phase regions on both sides. Compared with the temperature gradient,composition difference of diffusion couple and initial aging time of the alloy show greater effects on diffusion and interface microstructure.

Phase field simulation of the columnar dendritic growth and microsegregation in a binary alloy

This paper applies a phase field model for polycrystalline solidification in binary alloys to simulate the formation and growth of the columnar dendritic array under the isothermal and constant cooling conditions. The solidification process and microsegregation in the mushy zone are analysed in detail. It is shown that under the isothermal condition solidification will stop after the formation of the mushy zone, but dendritic coarsening will progress continuously, which results in the decrease of the total interface area. Under the constant cooling condition the mushy zone will solidify and coarsen simultaneously. For the constant cooling solidification, microsegregation predicted by a modified Brody- Flemings model is compared with the simulation results. It is found that the Fourier number which characterizes microsegregation is different for regions with different microstructures. Dendritic coarsening and the larger area of interface should account for the enhanced Fourier number in the region with well developed second dendritic arms.

Microstructure evolution and phase transitions in metals simulated by the multiphase-field method

The phase-field method has emerged as the method of choice for the description of microstructure evolution and phase transitions in metallic materials.Following general thermodynamic laws a set of evolution equations for the structural variables of the system,the so called phase-fields,are derived.The paper reviews shortly the theoretical background of the multi-phase-field.Different examples demonstrating the applicability of the method to technical steels will be presented ranging from deformation of the dendritic strand shell during peritectic transformation,grain growth in Austenite to stress driven growth of Pearlite.

Effects of rotation and gravity on an electro-magneto-thermoelastic medium with diffusion and voids by using the Lord-Shulman and dual-phase-lag models

The effects of rotation and gravity on an electro-magneto-thermoelastic medium with diffusion and voids in a generalized thermoplastic half-space are studied by using the Lord-Shulman (L-S) model and the dual-phase-lag (DPL) model. The analytical solutions for the displacements, stresses, temperature, diffusion concentration, and volume fraction field with different values of the magnetic field, the rotation, the gravity, and the initial stress are obtained and portrayed graphically. The results indicate that the effects of gravity, rotation, voids, diffusion, initial stress, and electromagnetic field are very pronounced on the physical properties of the material.

Retrieving Topological Information of Implicitly Represented Diffuse Interfaces with Adaptive Finite Element Discretization

We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.

晶界效应下的焊接热裂倾向判据适用性研究

材料焊接热裂倾向评判在产品制造中具有重要作用,以往热裂倾向判据并未考虑晶界效应,故其适用性一直是讨论的热点问题。文中以Al-Cu合金为例,通过多晶凝固定量相场模型获取熔池凝固过程枝晶形貌,提取凝固路径。基于凝固路径开展了目前常用热裂倾向判据的适用性研究,包含RDG判据、Clyne-Davies判据和Sindo Kou判据。研究发现,RDG判据同实际焊接热裂倾向一致,且裂纹产生主要受孤立晶界段枝晶间压力降主导;Clyne-Davies判据及Sindo Kou判据评判由于计算过程忽略部分孤立晶界区间段,导致评判结果出现失真。通过对三个判据的研究,为材料热裂倾向评判提供了参考。

Towards data-efficient mechanical design of bicontinuous composites usinggenerative AI

The distribution of material phases is crucial to determine the composite’s mechanical property.While the full structure-mechanics relationship of highly ordered material distributions can be studied with finite number of cases,this relationship is difficult to be revealed for complex irregular distributions,preventing design of such material structures to meet certain mechanical requirements.The noticeable developments of artificial intelligence(AI)algorithms in material design enables to detect the hidden structure-mechanics correlations which is essential for designing composite of complex structures.It is intriguing how these tools can assist composite design.Here,we focus on the rapid generation of bicontinuous composite structures together with the stress distribution in loading.We find that generative AI,enabled through fine-tuned Low Rank Adaptation models,can be trained with a few inputs to generate both synthetic composite structures and the corresponding von Mises stress distribution.The results show that this technique is convenient in generating massive composites designs with useful mechanical information that dictate stiffness,fracture and robustness of the material with one model,and such has to be done by several different experimental or simulation tests.This research offers valuable insights for the improvement of composite design with the goal of expanding the design space and automatic screening of composite designs for improved mechanical functions.

钛铝合金包晶相变的多相场模拟研究

包晶相变使传统钛铝合金在铸造过程中极易出现成分偏析、显微组织粗大等现象,导致塑性较差且各向异性明显,显著影响了钛铝合金的使役性能。为深入理解包晶相变过程并进一步合理调控微观组织,本文建立了适用于可描述钛铝合金包晶相变的多相场模型,基于定量化多相自由能输入、多相界面处理、考虑反溶质截流及相界面处虚拟相成分的简化计算策略,针对Ti-45Al(原子分数)合金的包晶相变过程,主要研究了过冷度对包晶相变时微观组织演变的影响规律。研究发现,发生包晶反应(β+L→α)时,α相沿着β/L界面快速增长。考虑到α向L与β相的生长速度存在差异,使得α相片层生长时表现出非对称性。包晶转变过程由扩散控制,α/L、α/β相界面的迁移满足xij=Aijt1/2规律。随着过冷度增大,两种相界面迁移速率均增大,且α/L界面迁移速率的增幅较为显著;较低过冷度下,在α片层前端可见β相的重熔现象。增大过冷度后,其重熔现象逐渐消失,但L/β/α三相区的形状保持不变。三相交界区域形状主要由界面能(界面张力)之间的平衡决定,而与界面迁移速率的差异无关。