Effect of forced lamina flow on microsegregation simulated by phase field method quantitatively

The influence of supercooled melt forced lamina flow on microsegregation was investigated. The concentration distribution at solid-liquid boundary of binary alloy Ni-Cu was simulated using phase field model coupled with flow field. The microsegregation, concentration maximum value, boundary thickness of concentration near upstream dendrite and normal to flow dendrite, and downstream dendrite were studied quantitatively in the case of forced lamia flow. The simulation results show that solute field and flow field interact complexly. Compared with melt without flow, in front of upstream dendrite tip, the concentration boundary thickness is the lowest and the concentration maximum value is the smallest for melt with flow. However, in front of downstream dendrite tip, the results are just the opposite. The zone of poor Cu in upstream dendrite where is the most severely microsegregation and shrinkage cavity is wider and the concentration is lower for melt with flow than that without flow.

Ageing process of pre-precipitation phase in Ni_(0.75)Al_(0.05)Fe_(0.2) alloy based on phase field method

By utilizing phase field method combined with analysis on free energy and interatomic potentials, pre-precipitation phase formation and transformation process of Ni0.75Al0.05Fe0.2 alloy in early precipitation stage during the ageing process under 1 000 K were studied. And free energy, microstructures, compositions and volume fractions of pre-precipitation phase and equilibrium phase were analyzed. The simulation results indicate that nonstoichiometric Llo pre-precipitation phase formed first, and then would gradually transform into L12 equilibrium phase. It is discovered that the phase transformation process was closely related to free energy and interatomic potentials. Additionally, it is revealed that free energy of Llo pre-precipitation phase was higher and interatomic potential was smaller than that of L12 equilibrium phase. Therefore, it is concluded that Llo phase was unstable, and phase transformation would occur to L12 which was more stable.

Phase field method simulation of faceted dendrite growth with arbitrary symmetries

A numerical simulation based on a regularized phase field model is developed to describe faceted dendrite growth morphology. The effects of mesh grid, anisotropy, supersaturation and fold symmetry on dendrite growth morphology were investigated, respectively. These results indicate that the nucleus grows into a hexagonal symmetry faceted dendrite. When the mesh grid is above 640×640, the size has no much effect on the shape. With the increase in the anisotropy value, the tip velocities of faceted dendrite increase and reach a balance value, and then decrease gradually. With the increase in the supersaturation value, crystal evolves from circle to the developed faceted dendrite morphology. Based on the Wulff theory and faceted symmetry morphology diagram, the proposed model was proved to be effective, and it can be generalized to arbitrary crystal symmetries.

Simulation of faceted dendrite growth of non-isothermal alloy in forced flow by phase field method

Numerical simulation based on a new regularized phase field model was presented to simulate the dendritic shape of a non-isothermal alloy with strong anisotropy in a forced flow. The simulation results show that a crystal nucleus grows into a symmetric dendrite in a free flow and into an asymmetry dendrite in a forced flow. As the forced flow velocity is increased, both of the promoting effect on the upstream arm and the inhibiting effects on the downstream and perpendicular arms are intensified, and the perpendicular arm tilts to the upstream direction. With increasing the anisotropy value to 0.14, all of the dendrite arms tip velocities are gradually stabilized and finally reach their relative saturation values. In addition, the effects of an undercooling parameter and a forced compound flow on the faceted dendrite growth were also investigated.

Anisotropic growth of multigrain in equiaxial solidification simulated with the phase field method

The phase field method has been mainly used to simulate the growth of a single crystal in the past. But polycrystalline materials predominate in engineering. In this work, a phase field model for multigrain solidification is developed, which takes into account the random crystallographic orientations of crystallites and preserves the rotational invariance of the free energy. The morphological evolution of equiaxial multigrain solidification is predicted and the effect of composition on transformation kinetics is studied. The numerical results indicate that due to the soft impingement of grains the Avrami exponent varies with the initial melt composition and the solidification fraction.

Analysis on high-temperature oxidation and growth stress of iron-based alloy using phase field method

High-temperature oxidation is an important property to evaluate thermal protection materials. However, since oxidation is a complex process involving microstructure evolution, its quantitative analysis has always been a challenge. In this work, a phase field method （PFM） based on the thermodynamics theory is developed to simulate the oxidation behavior and oxidation induced growth stress. It involves microstructure evolution and solves the problem of quantitatively computational analysis for the oxidation behavior and growth stress. Employing this method, the diffusion process, oxidation performance, and stress evolution axe predicted for Fe-Cr-A1-Y alloys. The numerical results agree well with the experimental data. The linear relationship between the maximum growth stress and the environment oxygen concentration is found. PFM provides a powerful tool to investigate high-temperature oxidation in complex environments.

Numerical Simulations of Equiaxed Dendrite Growth Using Phase Field Method

Phase field method offers the prospect of being able to perform realistic numerical experiments on dendrite growth in a metallic system. In this paper, the equiaxed dendrite evolution during the solidification of a pure material was numerically simulated using the phase field model. The equiaxed dendrite growth in a two-dimensional square domain of undercooled melt (nickel) with four-fold anisotropy was simulated. The phase field model equations was solved using the explicit finite difference method on a uniform mesh. The formation of various equiaxed dendrite patterns was shown by a series of simulations, and the effect of anisotropy on equiaxed dendrite morphology was investigated.

Crack propagation simulation in brittle elastic materials by a phase field method

To overcome the difficulties of re-meshing and tracking the crack-tip in other computational methods for crack propagation simulations,the phase field method based on the minimum energy principle is introduced by defining a continuous phase field variable(x)∈[0,1]to characterize discontinuous cracks in brittle materials.This method can well describe the crack initiation and propagation without assuming the shape,size and orientation of the initial crack in advance.In this paper,a phase field method based on Miehe's approach[Miehe et al.,Comp.Meth.App.Mech.Eng.(2010)]is applied to simulate different crack propagation problems in twodimensional(2D),isotropic and linear elastic materials.The numerical implementation of the phase field method is realized within the framework of the finite element method(FEM).The validity,accuracy and efficiency of the present method are verified by comparing the numerical results with other reference results in literature.Several numerical examples are presented to show the effects of the loading type(tension and shear),boundary conditions,and initial crack location and orientation on the crack propagation path and force-displacement curve.Furthermore,for a single edge-cracked bi-material specimen,the influences of the loading type and the crack location on the crack propagation trajectory and force-displacement curve are also investigated and discussed.It is demonstrated that the phase field method is an efficient tool for the numerical simulation of the crack propagation problems in brittle elastic materials,and the corresponding results may have an important relevance for predicting and preventing possible crack propagations in engineering applications.

3D anisotropy simulation of dendrites growth with phase field method

The anisotropy problem of 3D phase-field model was studied,and various degrees of anisotropy were simulated by numerical calculation method.The results show that with the change of interface anisotropy coefficients,from smooth transition to the appearance of angle,equilibrium crystals shape morphology has a critical value,and 3D critical value is 0.3.The growth of dendrites is stable and the interface is smooth when it is less than critical value;the interface is unstable,rolling edge appears and the growth is discontinuous when it is more than critical value.With the increase of anisotropy coefficients,the dendrites grow faster under the same condition.

Simulation of fluctuation effect on dendrite growth by phase field method

The dendrite growth process was simulated with the phase field model coupling with the fluctuation.The effect of fluctuation intensity on the dendrite morphology and that of the thermal fluctuation together with the phase field fluctuation on the forming of side branches were investigated.The results indicate that with the decrease of thermal fluctuation amplitude,the furcation of dendrite tip also decreases,transverse dendrites become stronger,longitudinal dendrites become degenerated,Doublon structure disappears,and a quite symmetrical dendrite structure appears finally.Thermal fluctuation can result in the unsteadiness of dendrites side branches,and it is also the main reason for forming side branches.The phase field fluctuation has a little contribution to the side branches,and it is usually ignored in calculation.When the thermal fluctuation amplitude(F_u) is appropriate,the thermal noise can result in the side branches,but cannot change the steady behavior of the dendrites tip.

Paraelectric Doping Simultaneously Improves the Field Frequency Adaptability and Dielectric Properties of Ferroelectric Materials:A Phase-Field Study

Recent years,the polarization response of ferroelectrics has been entirely studied.However,it is found that the polarization may disappear gradually with the continually applied of electric field.In this paper,taking K0.48Na0.52NbO3(KNN)as an example,it was demonstrated that the residual polarization began to decrease when the electric field frequency increased to a certain extent using a phase-field methods.The results showed that the content of out-of-plane domains increased first and then decreased with the increase of applied electric field frequency,the maximum polarization disappeared at high frequencies,and the hysteresis loop became elliptical.In order to further study the abnormal changes of hysteresis loops of ferroelectrics under high electric field frequency,we analyzed the hysteresis loop and dielectric response of solid solution 0.1SrTiO_(3)-0.9K_(0.48)Na_(0.52)NbO_(3).It was found that the doped hysteresis loop maintained its shape at higher frequency and the dielectric constant increased.This kind of doping has a higher field frequency adaptability,which has a key guiding role in improving the dielectric properties of ferroelectric thin films and expanding the frequency application range of ferroelectric nano memory。

Volumetric lattice Boltzmann method for pore-scale mass diffusionadvection process in geopolymer porous structures

Porous materials present significant advantages for absorbing radioactive isotopes in nuclear waste streams.To improve absorption efficiency in nuclear waste treatment,a thorough understanding of the diffusion-advection process within porous structures is essential for material design.In this study,we present advancements in the volumetric lattice Boltzmann method(VLBM)for modeling and simulating pore-scale diffusion-advection of radioactive isotopes within geopolymer porous structures.These structures are created using the phase field method(PFM)to precisely control pore architectures.In our VLBM approach,we introduce a concentration field of an isotope seamlessly coupled with the velocity field and solve it by the time evolution of its particle population function.To address the computational intensity inherent in the coupled lattice Boltzmann equations for velocity and concentration fields,we implement graphics processing unit(GPU)parallelization.Validation of the developed model involves examining the flow and diffusion fields in porous structures.Remarkably,good agreement is observed for both the velocity field from VLBM and multiphysics object-oriented simulation environment(MOOSE),and the concentration field from VLBM and the finite difference method(FDM).Furthermore,we investigate the effects of background flow,species diffusivity,and porosity on the diffusion-advection behavior by varying the background flow velocity,diffusion coefficient,and pore volume fraction,respectively.Notably,all three parameters exert an influence on the diffusion-advection process.Increased background flow and diffusivity markedly accelerate the process due to increased advection intensity and enhanced diffusion capability,respectively.Conversely,increasing the porosity has a less significant effect,causing a slight slowdown of the diffusion-advection process due to the expanded pore volume.This comprehensive parametric study provides valuable insights into the kinetics of isotope uptake in porous structures,facilitating the development of porous materials for nuclear waste treatment applications.

Integration of machine learning with phase field method to model the electromigration induced Cu_(6)Sn_(5) IMC growth at anode side Cu/Sn interface

Currently,in the era of big data and 5G communication technology,electromigration has become a serious reliability issue for the miniaturized solder joints used in microelectronic devices.Since the effective charge number(Z*)is considered as the driving force for electromigration,the lack of accurate experimental values for Z* poses severe challenges for the simulation-aided design of electronic materials.In this work,a data-driven framework is developed to predict the Z* values of Cu and Sn species at the anode based LIQUID,Cu_(6)Sn_(5) intermetallic compound(IMC)and FCC phases for the binary Cu-Sn system undergoing electromigration at 523.15 K.The growth rate constants(kem)of the anode IMC at several magnitudes of applied low current density(j=1×10^6 to 10×10^6A/m^2)are extracted from simulations based on a 1D multi-phase field model.A neural network employing Z* and j as input features,whereas utilizing these computed kemdata as the expected output is trained.The results of the neural network analysis are optimized with experimental growth rate constants to estimate the effective charge numbers.For a negligible increase in temperature at low j values,effective charge numbers of all phases are found to increase with current density and the increase is much more pronounced for the IMC phase.The predicted values of effective charge numbers Z* are then utilized in a 2D simulation to observe the anode IMC grain growth and electrical resistance changes in the multi-phase system.As the work consists of the aspects of experiments,theory,computation,and machine learning,it can be called the four paradigms approach for the study of electromigration in Pb-free solder.Such a combination of multiple paradigms of materials design can be problem-solving for any future research scenario that is marked by uncertainties regarding the determination of material properties.

A numerical implementation of the length-scale independent phase field method

The phase field method for fracture integrates the Griffith theory and damage mechanics approach to predict crack initiation and propagation within one framework.It replaced the discrete representation of crack by diffusive damage and solved it based on a minimization of the global energy storage functional.As a result,no crack tracking topology is needed,and complex crack shapes can be captures without user intervention.However,it is also reported to have an inconsistency between the predicted fracture toughness and the material strength.Recently,a novel energetic degradation function was proposed in literature to handle this issue.This research does some further modifications to the global energy storage functional so that Newton's method can be directly used to solve the energy minimization.With the new energy form,direct implementation of the length-scale independent phase field method into finite element packages like LS-DYNA becomes possible.This paper presents the framework and details of implementing the length-scale independent phase field method into LS-DYNA through a user-defined element and material subroutine.Several numerical examples are presented to compare with the experiment crack shape.Most importantly,this paper is one of the first ones to quantitatively predict accurate force response compared to experiments.These examples verify the accuracy of the new energy form and implementation.

A modified damage and fracture phase field model considering heterogeneity for rock‐like materials

Damage and fracture are the most extensive failure modes of rock materials,which may easily induce disaster and instability of engineering structures.This study developed a nonlocal damage fracture phase field model for rocks considering the heterogeneity of rocks.The modified phase field model introduced the heterogeneity of fracture parameters and modified the governing equations.Meanwhile,the free energy was constructed by the elastic strain energy sphere‐bias decomposition and the plastic strain energy.As for the numerical implementation,the three layers finite elements method structure was used in the frame of the finite element method.The ability of the modified phase field model has been illustrated by reproducing the experiment results of rock samples with pre‐existing cracks under compression.

Dendritic Morphology Simulation Using the Phase Field Method

Dendritic morphology was simulated using a macro- and micro-coupled method. Since the microstructure of a whole casting cannot be easily analyzed, a scheme was developed to calculate the temperature of the whole casting with the microstructure analyzed by selecting one cell in the central region of the casting. The heterogeneous nucleation was described using a Gaussian distribution with the dendritic growth controlled by the solution of the phase field equation. The initial temperature distribution in the micro- domain was obtained by interpolating the cell temperatures near the selected cell with the interface undercooling assumed to be the sum of thermal, solute, and curvature effects. The solute distribution was calculated from the mixed solute conservation equation with noise introduced to produce the side branches. The simulation results agree well with experimental results.

Evaluation of dielectric and piezoelectric behavior of unpoled and poled barium titanate polycrystals with oxygen vacancies using phase field method

This article studies the dielectric and piezoelectric behavior of unpoled and poled barium titanate(BaTiO_(3))polycrystals with oxygen vacancies.A phase field model is employed for BaTiO_(3) polycrystals,coupled with the time-dependent Ginzburg–Landau theory and the oxygen vacancies diffusion,to demonstrate the interaction between oxygen vacancies and domain evolutions.To generate grain structures,the phase field model for grain growth is also used.The hysteresis loop and butterfly curve are predicted at room and high temperatures.The permittivity,and longitudinal and transverse piezoelectric constants of the BaTiO_(3) polycrystals are then examined for various grain sizes and oxygen vacancy densities.

Phase-field Simulation of Microstructural Evolution during Preparation of Semi-solid Metal by Electromagnetic Stirring Method

In the process of preparation of semi-solid metal materials, a variety of factors would influence the preparing time and the morphology of non-dendritic microstructure. The aim of this work is using phase-field method to simulate non-dendritic growth during preparation of AI-4Cu-Mg semi-solid alloy by electromagnetic stirring method （EMS method）. Several factors such as the disturbance intensity, anisotropy, the thickness of the interface and the ratio of diffusivity in solid and liquid were considered. It is shown that decreasing the thickness of the interface results in more circular outline of particles, and increasing the diffusivity in solid can reduce degree of microsegregation. The disturbance intensity in the model can be connected with current intensity of stator or magnetic induction density impressed. Simulation results show that the larger the disturbance intensity or magnetic induction density, the more globular morphology the original phase in the matrix.

Simulation of interphase boundary of Ni_(75)Al_xV_(25-x )alloys using microscopic phase-field method

The evolution of ordered interphase boundary (IPB) of Ni75AlxV25-x alloys was simulated using the microscopic phase-field method. Based on the atomic occupation probability figure on 2D and order parameters, it was found that the IPB formed by different directions ofθ phase has great effect on the precipitation of γ ′ phase. The γ ′ phase precipitated at the IPB that is formed by [1 00]θ direction where the ( 001)θ plane is opposite, and then grows up and the shape is strap at final. The IPB structure between γ ′phase andθ phase is the same. There is no γ ′ phase precipitate at the IPB where the ( 002)θ and ( 001)θ planes are opposite, the ordered IPB is dissolved into disordered area. There is γ ′ phase precipitation at the IPB formed by the [ 001]θ and [1 00]θ directions, and the IPB structure is different between γ ′ phase and the different directions ofθ phase. The IPB where ( 001)γ′ and (1 00)θ plane opposite does not migrate during the γ ′ phase growth, and γ ′ phase grows along [1 00]θdirection.

Effect of fluid flow on solidified equiaxed dendrite morphology evolution based on phase field-lattice Boltzmann method

Fluid flow can significantly change the evolution of microstructural morphology. However, relatively little is known how the fluid flow, concentration and microstructure affect each other quantitatively, which is essential to optimize processing parameters. A quantitative simulation study of Al-Cu solidified equiaxed dendrite evolution under forced flow based on phase field-lattice Boltzmann method(PF-LBM) is carried out. Results obtained are validated by Gibbs-Thomson relation at the dendrite tip. Compared with the equiaxed dendrite evolution without flow, the upstream dendrite arm is enhanced while the downstream arm is inhibited. Besides, as the inlet flow rate increases, the secondary arms attached onto the upstream primary arm and the upstream side of the primary arm normal to the inflow velocity has been well developed. Results show that sidewise instabilities of the primary dendrite arm and onset of secondary arm is caused by the local concentration perturbation and will be enhanced or inhabited by the flow. It is believed that the coupled PF-LBM method is able to handle dendrite evolution under forced flow quantitatively, which helps in investigating the solidified dendrite morphology evolution.