The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems.Here,we develop a new analytical approach based on the bifur...The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems.Here,we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models.Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions.To elucidate the idea,we apply this analytical approach to three representative phase field equations:the Allen-Cahn equation,the Cahn-Hilliard equation,and the Allen-Cahn-Ohta-Kawasaki system.The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.展开更多
Hydraulic stimulation technology is widely employed to enhance the permeability of geothermal reservoirs.Nevertheless,accurately predicting hydraulic fracture propagation in complex geological conditions remains chall...Hydraulic stimulation technology is widely employed to enhance the permeability of geothermal reservoirs.Nevertheless,accurately predicting hydraulic fracture propagation in complex geological conditions remains challenging,thereby hindering the effective utilization of existing natural fractures.In this study,a phase field model was developed utilizing the finite element method to examine the influence of fluid presence,stress conditions,and natural fractures on the initiation and propagation of hydraulic fractures.The model employs Biot's poroelasticity theory to establish the coupling between the displacement field and the fluid field,while the phase field theory is applied to simulate fracture behavior.The results show that whenσ_(x0)/σ_(y0)<3 or qf<20 kg/(m^(3)·s),the presence of natural fractures can alter the original propagation direction of hydraulic fractures.Conversely,in the absence of these conditions,the propagation path of natural fractures is predominantly influenced by the initial stress field.Furthermore,based on the analysis of breakdown pressure and damage area,the optimal intersection angle between natural fractures and hydraulic fractures is determined to range from 45°to 60°.Finally,once a dominant channel forms,initiating and propagating hydraulic fractures in other directions becomes increasingly difficult,even in highly fractured areas.This method tackles the challenges of initiating and propagating hydraulic fractures in complex geological conditions,providing a theoretical basis for optimizing Enhanced Geothermal System(EGS)projects.展开更多
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 ...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.展开更多
It is well known that coarse-grained super-elastic NiTi shape memory alloys(SMAs)exhibit localized rather than homogeneous martensite transformation(MT),which,however,can be strongly influenced by either internal size...It is well known that coarse-grained super-elastic NiTi shape memory alloys(SMAs)exhibit localized rather than homogeneous martensite transformation(MT),which,however,can be strongly influenced by either internal size(grain size,GS)or the external size(geometric size).The coupled effect of GS and geometric size on the functional properties has not been clearly understood yet.In this work,the super-elasticity,one-way,and stress-assisted two-way shape memory effects of the polycrystalline NiTi SMAs with different aspect ratios(length/width for the gauge section)and different GSs are investigated based on the phase field method.The coupled effect of the aspect ratio and GS on the functional properties is adequately revealed.The simulated results indicate that when the aspect ratio is lower than about 4:1,the stress biaxiality and stress heterogeneity in the gauge section of the sample become more and more obvious with decreasing the aspect ratio,which can significantly influence the microstructure evolution in the process involving external stress.Therefore,the corresponding functional property is strongly dependent on the aspect ratio.With decreasing the GS and the aspect ratio(to be lower than 4:1),both the aspect ratio and GS can affect the MT or martensite reorientation in each grain and the interaction among grains.Thus,due to the strong internal constraint(i.e.,the constraint of grain boundary)and the external constraint(i.e.,the constraint of geometric boundary),the capabilities of the functional properties of NiTi SMAs are gradually weakened and highly dependent on these two factors.展开更多
As the demand for advanced material design and performance prediction continues to grow,traditional phase-field models are increasingly challenged by limitations in computational efficiency and predictive accuracy,par...As the demand for advanced material design and performance prediction continues to grow,traditional phase-field models are increasingly challenged by limitations in computational efficiency and predictive accuracy,particularly when addressing high-dimensional and complex data in multicomponent systems.To overcome these challenges,this study proposes an innovative model,LSGWO-BP,which integrates an improved Grey Wolf Optimizer(GWO)with a backpropagation neural network(BP)to enhance the accuracy and efficiency of quasi-phase equilibrium predictions within the KKS phase-field framework.Three mapping enhancement strategies were investigated–Circle-Root,Tent-Cosine,and Logistic-Sine mappings-with the Logistic mapping further improved via Sine perturbation to boost global search capability and convergence speed in large-scale,complex data scenarios.Evaluation results demonstrate that the LSGWO-BP model significantly outperforms conventional machine learning approaches in predicting quasi-phase equilibrium,achieving a 14%–28%reduction in mean absolute error(MAE).Substantial improvements were also observed in mean squared error,root mean squared error,and mean absolute percentage error,alongside a 7%–33%increase in the coefficient of determination(R2).Furthermore,the model exhibits strong potential for microstructural simulation applications.Overall,the study confirms the effectiveness of the LSGWO-BP model in materials science,especially in enhancing phase-field modeling efficiency and enabling accurate,intelligent prediction for multicomponent alloy systems,thereby offering robust support for microstructure prediction and control.展开更多
The commonly used incompressible phase field models for non-reactive,binary fluids,in which the Cahn-Hilliard equation is used for the transport of phase variables(volume fractions),conserve the total volume of each p...The commonly used incompressible phase field models for non-reactive,binary fluids,in which the Cahn-Hilliard equation is used for the transport of phase variables(volume fractions),conserve the total volume of each phase as well as the material volume,but do not conserve the mass of the fluid mixture when densities of two components are different.In this paper,we formulate the phase field theory for mixtures of two incompressible fluids,consistent with the quasi-compressible theory[28],to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase.In this formulation,the mass-average velocity is no longer divergence-free(solenoidal)when densities of two components in the mixture are not equal,making it a compressible model subject to an internal constraint.In one formulation of the compressible models with internal constraints(model 2),energy dissipation can be clearly established.An efficient numerical method is then devised to enforce this compressible internal constraint.Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions.Numerical comparisons show that(a)predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin;and(b)predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone.The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.展开更多
A thermodynamic consistent phase field model is developed to describe the sintering process with multiphase powders. In this model, the interface region is assumed to be a mixture of different phases with the same che...A thermodynamic consistent phase field model is developed to describe the sintering process with multiphase powders. In this model, the interface region is assumed to be a mixture of different phases with the same chemical potential, but with different compositions. The interface diffusion and boundary diffusion are also considered in the model. As an example, the model is applied to the sintering process with Fe-Cu powders. The free energy of each phase is described by the well-developed thermodynamic models, together with the published optimized parameters. The microstructure and solute distribution during the sintering process can both be obtained quantitively.展开更多
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 h...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.展开更多
The pre-precipitation processes of γ＇ phase in Ni-Al-Ti alloy with different chemical compositions are investigated by microscopic phase-field dynamics model. The simulation results show that the early precip...The pre-precipitation processes of γ＇ phase in Ni-Al-Ti alloy with different chemical compositions are investigated by microscopic phase-field dynamics model. The simulation results show that the early precipitated phase with L10 structure is formed before the formation of γ＇ phase. With the increase of Al content, the composition of precipitated phase is transformed from the co-existence of γ＇ phase and material phase into single y phase. Furthermore, excessive Al content can stimulate the production of γ＇ phase, and the conversion time from L10 phase to L12 phase will lag behind other samples with lower Al con-tent. By analyzing the occupation probability of Ti, it can be found that the solid solution strengthening effect of Ti is reduced with the increase of Al component.展开更多
In the present study, the influence of natural convection on the lamellar eutectic growth is determined by a phase-field-lattice Boltzmann study for Al-Cu eutectic alloy. The mass difference resulting from concentrati...In the present study, the influence of natural convection on the lamellar eutectic growth is determined by a phase-field-lattice Boltzmann study for Al-Cu eutectic alloy. The mass difference resulting from concentration difference led to the fluid flow, and a robust parallel and adaptive mesh refinement algorithm was employed to improve the computational efficiency without any compromising accuracy. Results show that the existence of natural convection would affect the growth undercooling and thus control the interface shape by adjusting the lamellar width. In particular, by alternating the magnitude of the solute expansion coefficient, the strength of the natural convection is changed. Corresponding microstructure patterns are discussed and compared with those under no-convection conditions.展开更多
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 ...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.展开更多
Fracture is a very common failure mode of the composite materials,which seriously affects the reliability and service-life of composite materials.Therefore,the study of the fracture behavior of the composite materials...Fracture is a very common failure mode of the composite materials,which seriously affects the reliability and service-life of composite materials.Therefore,the study of the fracture behavior of the composite materials is of great significance and necessity,which demands an accurate and efficient numerical tool in general cases because of the complexity of the arising boundary-value or initial-boundary value problems.In this paper,a phase field model is adopted and applied for the numerical simulation of the crack nucleation and propagation in brittle linear elastic two-phase perforated/particulate composites under a quasi-static tensile loading.The phase field model can well describe the initiation,propagation and coalescence of the cracks without assuming the existence and the geometry of the initial cracks in advance.Its numerical implementation is realized within the framework of the finite element method(FEM).The accuracy and the efficiency of the present phase field model are verified by the available reference results in literature.In the numerical examples,we first study and discuss the influences of the hole/particle size on the crack propagation trajectory and the force-displacement curve.Then,the effects of the hole/particle shape on the crack initiation and propagation are investigated.Furthermore,numerical examples are presented and discussed to show the influences of the hole/particle location on the crack initiation and propagation characteristics.It will be demonstrated that the present phase field model is an efficient tool for the numerical simulation of the crack initiation and propagation problems in brittle two-phase composite materials,and the corresponding results may play an important role in predicting and preventing possible hazardous crack initiation and propagation in engineering applications.展开更多
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 p...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.展开更多
A phase field model is developed to simulate the grain evolution of 17-4PH steel during cyclic heat treatment （CHT）. Our simulations successfully reproduce the grain morphologies of every CHT. In the process of ever...A phase field model is developed to simulate the grain evolution of 17-4PH steel during cyclic heat treatment （CHT）. Our simulations successfully reproduce the grain morphologies of every CHT. In the process of every CHT, phase transformation recrystallization happens. The recrystallized grains appear mainly on the original grain boundaries. The average grain size of 13.2 μm obtained by 1040 ℃×1 h solution treatment for this experimental steel can be refined to 2.2 μm after five CHT＇s. Furthermore, the effects of phenomenological parameters in our model are discussed.展开更多
A model has been established to simulate the realistic spatio-temporal microstructure evolution in recrystallization of a magnesium alloy using the phase field approach. A set of rules have been proposed to decide the...A model has been established to simulate the realistic spatio-temporal microstructure evolution in recrystallization of a magnesium alloy using the phase field approach. A set of rules have been proposed to decide the real physical value of all parameters in the model. The thermodynamic software THERMOCALC is applied to determine the local chemical free energy and strain energy, which is added to the free energy density of grains before recrystallization. The Arrhenius formula is used to describe boundary mobility and the activity energy is suggested with a value of zinc segregation energy at the boundary. However, the mobility constant in the formula was found out by fitting to a group of grain size measurements during recrystallization of the alloy. The boundary range is suggested to decide the gradient parameters in addition of fitting to the experimental boundary energy value. These parameter values can be regarded as a database for other similar simulations and the fitting rules can also be applied to build up databases for any other alloy systems. The simulated results show a good agreement with reported experimental measurement of the alloy at the temperatures from 300 to 400℃ for up to 100 min but not at 250℃. This implies a mechanism variation in activity energy of the boundary mobility in the alloy at low temperature.展开更多
Domain switching around electrically permeable and impermeable cracks in ferro-electric single crystals subjected to a mechanical load is investigated by using a phase field model.It is found that the electrical bound...Domain switching around electrically permeable and impermeable cracks in ferro-electric single crystals subjected to a mechanical load is investigated by using a phase field model.It is found that the electrical boundary conditions have little effect on the polarization distribution without any external load when the initial polarization is parallel to the crack,which is totally different from previous studies where the initial polarization is perpendicular to the crack.How-ever,the simulation results show that the electrical boundary conditions have great influence on the polarization distribution as well as the domain switching behavior when a mechanical load is applied.The critical mechanical load of domain switching with a permeable crack is much smaller than that in the case of an impermeable crack.展开更多
A quantitative phase field method of multi-component diffusion-controlled phase transformations coupled with the Kim-Kim-Suzuki model was applied to study the effect of initial particle size distribution （PSD） in 3D...A quantitative phase field method of multi-component diffusion-controlled phase transformations coupled with the Kim-Kim-Suzuki model was applied to study the effect of initial particle size distribution （PSD） in 3D and space distribution in 2D on dissolution of α particles in Ti-6Al-4V alloy below β transus temperature in real time and length scale. The thermodynamic and mobility data were obtained from Thermo-Calc and DICTRA softwares, respectively. The results show that the volume fractions of α particles decay with time as： f =feq ＋ （f0 -feq） exp（-Ktn） for four cases of PSD. The sequence of dissolution kinetics from fast to slow is： uniform PSD, normal PSD, lognormal PSD and bimodal PSD. The space distribution is found to be a major factor affecting the dissolution kinetics and the microstructures. When the distance of the particles is less than critical value, the dissolution rates reduce with the decrease in distance. The Al and V concentration fields around the particles appear more obvious soft impingement.展开更多
Ab st ra ct Nanocrystalline materials exhibit unique properties due to their extremely high grain boundary(GB) density.However,this high-density characteristic induces grain coarsening at elevated temperatures,thereby...Ab st ra ct Nanocrystalline materials exhibit unique properties due to their extremely high grain boundary(GB) density.However,this high-density characteristic induces grain coarsening at elevated temperatures,thereby limiting the widespread application of nanocrystalline materials.Recent experimental observations revealed that GB segregation and second-phase pinning effectively hinder GB migration,thereby improving the stability of nanocry stalline materials.In this study,a mouified phase-field model that integrates mismatch strain,solute segregation and precipitation was developed to evaluate the influence of lattice misfit on the thermal stability of nanocrystalline alloys.The simulation results indicated that introducing a suitable mismatch strain can effectively enhance the microstructural stability of nanocrystalline alloys.By synergizing precipitation with an appropriate lattice misfit,the formation of second-phase particles in the bulk grains can be suppressed,thereby facilitating solute segregation/precipitation at the GBs.This concentrated solute segregation and precipitation at the GBs effectively hinders grain migration,thereby preventing grain coarsening.These findings provide a new perspective on the design and regulation of nanocrystalline alloys with enhanced thermal stability.展开更多
The morphology of Ni_(4)Ti_(3) precipitates is important in tuning the martensitic transformation(MT)behavior and mechanical properties of nitinol.Constrained ageing is effective in engineering the morphology of Ni_(4...The morphology of Ni_(4)Ti_(3) precipitates is important in tuning the martensitic transformation(MT)behavior and mechanical properties of nitinol.Constrained ageing is effective in engineering the morphology of Ni_(4)Ti_(3) precipitates due to the variant selection effect of external load which is still lacking.In this work,maps of variant selection effect of external load applied along all crystallographic directions are obtained by using a combination of theoretical analyses and phase field simulations.It is found that maps produced by uniaxial tension and uniaxial compression are quite different.The number and types of Ni_(4)Ti_(3) variants preferred by external load vary as the loading direction changes.Moreover,factors influencing the strength of variant selection effect are discovered.This work provides insights on understanding the Ni_(4)Ti_(3) precipitation process and sheds light on the engineering of morphology of Ni_(4)Ti_(3) precipitates for desired mechanical and functional properties.展开更多
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...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.展开更多
基金supported as part of the Computational Materials Sciences Program funded by the U.S.Department of Energy,Office of Science,Basic Energy Sciences,under Award No.DE-SC0020145Y.Z.would like to acknowledge support for his effort by the Simons Foundation through Grant No.357963 and NSF grant DMS-2142500.
文摘The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems.Here,we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models.Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions.To elucidate the idea,we apply this analytical approach to three representative phase field equations:the Allen-Cahn equation,the Cahn-Hilliard equation,and the Allen-Cahn-Ohta-Kawasaki system.The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.
基金supported by the National Key Research and Development Program(2021YFB150740401)National Natural Science Foundation of China(42202336)the CAS Pioneer Hundred Talents Program in China(Y826031C01)。
文摘Hydraulic stimulation technology is widely employed to enhance the permeability of geothermal reservoirs.Nevertheless,accurately predicting hydraulic fracture propagation in complex geological conditions remains challenging,thereby hindering the effective utilization of existing natural fractures.In this study,a phase field model was developed utilizing the finite element method to examine the influence of fluid presence,stress conditions,and natural fractures on the initiation and propagation of hydraulic fractures.The model employs Biot's poroelasticity theory to establish the coupling between the displacement field and the fluid field,while the phase field theory is applied to simulate fracture behavior.The results show that whenσ_(x0)/σ_(y0)<3 or qf<20 kg/(m^(3)·s),the presence of natural fractures can alter the original propagation direction of hydraulic fractures.Conversely,in the absence of these conditions,the propagation path of natural fractures is predominantly influenced by the initial stress field.Furthermore,based on the analysis of breakdown pressure and damage area,the optimal intersection angle between natural fractures and hydraulic fractures is determined to range from 45°to 60°.Finally,once a dominant channel forms,initiating and propagating hydraulic fractures in other directions becomes increasingly difficult,even in highly fractured areas.This method tackles the challenges of initiating and propagating hydraulic fractures in complex geological conditions,providing a theoretical basis for optimizing Enhanced Geothermal System(EGS)projects.
文摘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.
基金supported by the National Natural Science Foundation of China (Grant Nos.12202294 and 12022208)the Project funded by China Postdoctoral Science Foundation (Grant No.2022M712243)the Fundamental Research Funds for the Central Universities (Grant No.2023SCU12098).
文摘It is well known that coarse-grained super-elastic NiTi shape memory alloys(SMAs)exhibit localized rather than homogeneous martensite transformation(MT),which,however,can be strongly influenced by either internal size(grain size,GS)or the external size(geometric size).The coupled effect of GS and geometric size on the functional properties has not been clearly understood yet.In this work,the super-elasticity,one-way,and stress-assisted two-way shape memory effects of the polycrystalline NiTi SMAs with different aspect ratios(length/width for the gauge section)and different GSs are investigated based on the phase field method.The coupled effect of the aspect ratio and GS on the functional properties is adequately revealed.The simulated results indicate that when the aspect ratio is lower than about 4:1,the stress biaxiality and stress heterogeneity in the gauge section of the sample become more and more obvious with decreasing the aspect ratio,which can significantly influence the microstructure evolution in the process involving external stress.Therefore,the corresponding functional property is strongly dependent on the aspect ratio.With decreasing the GS and the aspect ratio(to be lower than 4:1),both the aspect ratio and GS can affect the MT or martensite reorientation in each grain and the interaction among grains.Thus,due to the strong internal constraint(i.e.,the constraint of grain boundary)and the external constraint(i.e.,the constraint of geometric boundary),the capabilities of the functional properties of NiTi SMAs are gradually weakened and highly dependent on these two factors.
基金supported by the National Natural Science Foundation of China(Grant Nos.52161002,51661020 and 11364024)。
文摘As the demand for advanced material design and performance prediction continues to grow,traditional phase-field models are increasingly challenged by limitations in computational efficiency and predictive accuracy,particularly when addressing high-dimensional and complex data in multicomponent systems.To overcome these challenges,this study proposes an innovative model,LSGWO-BP,which integrates an improved Grey Wolf Optimizer(GWO)with a backpropagation neural network(BP)to enhance the accuracy and efficiency of quasi-phase equilibrium predictions within the KKS phase-field framework.Three mapping enhancement strategies were investigated–Circle-Root,Tent-Cosine,and Logistic-Sine mappings-with the Logistic mapping further improved via Sine perturbation to boost global search capability and convergence speed in large-scale,complex data scenarios.Evaluation results demonstrate that the LSGWO-BP model significantly outperforms conventional machine learning approaches in predicting quasi-phase equilibrium,achieving a 14%–28%reduction in mean absolute error(MAE).Substantial improvements were also observed in mean squared error,root mean squared error,and mean absolute percentage error,alongside a 7%–33%increase in the coefficient of determination(R2).Furthermore,the model exhibits strong potential for microstructural simulation applications.Overall,the study confirms the effectiveness of the LSGWO-BP model in materials science,especially in enhancing phase-field modeling efficiency and enabling accurate,intelligent prediction for multicomponent alloy systems,thereby offering robust support for microstructure prediction and control.
基金partially supported by NSF grants DMS-0915066 and AFOSR FA9550-11-1-0328partially supported by NSF grants DMS-0819051,DMS-0908330,SC EPSCOR award+1 种基金the USC startup fundpartially supported by the ARO grant W911NF-09-1-0389 and the USC startup fund.
文摘The commonly used incompressible phase field models for non-reactive,binary fluids,in which the Cahn-Hilliard equation is used for the transport of phase variables(volume fractions),conserve the total volume of each phase as well as the material volume,but do not conserve the mass of the fluid mixture when densities of two components are different.In this paper,we formulate the phase field theory for mixtures of two incompressible fluids,consistent with the quasi-compressible theory[28],to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase.In this formulation,the mass-average velocity is no longer divergence-free(solenoidal)when densities of two components in the mixture are not equal,making it a compressible model subject to an internal constraint.In one formulation of the compressible models with internal constraints(model 2),energy dissipation can be clearly established.An efficient numerical method is then devised to enforce this compressible internal constraint.Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions.Numerical comparisons show that(a)predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin;and(b)predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone.The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.
基金Project(2011CB606306)supported by the National Basic Research Program of ChinaProject(51101014)supported by the National Natural Science Foundation of ChinaProject(SKLSP201214)supported by the State Key Laboratory of Solidification Processing in Northwestern Polytechnical University,China
文摘A thermodynamic consistent phase field model is developed to describe the sintering process with multiphase powders. In this model, the interface region is assumed to be a mixture of different phases with the same chemical potential, but with different compositions. The interface diffusion and boundary diffusion are also considered in the model. As an example, the model is applied to the sintering process with Fe-Cu powders. The free energy of each phase is described by the well-developed thermodynamic models, together with the published optimized parameters. The microstructure and solute distribution during the sintering process can both be obtained quantitively.
基金Project (51101059) supported by the National Natural Science Foundation of ChinaProject (20110490874) supported by the China Postdoctoral Science Foundation
文摘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.
基金National Natural Science Foundation of China(Nos.U1610123,51674226,51574207,51574206,51274175)International Cooperation Project of the Ministry of Science and Technology of China(No.2014DFA50320)Science and Technology Major Project of Shanxi Province(No.MC2016-06)
文摘The pre-precipitation processes of γ＇ phase in Ni-Al-Ti alloy with different chemical compositions are investigated by microscopic phase-field dynamics model. The simulation results show that the early precipitated phase with L10 structure is formed before the formation of γ＇ phase. With the increase of Al content, the composition of precipitated phase is transformed from the co-existence of γ＇ phase and material phase into single y phase. Furthermore, excessive Al content can stimulate the production of γ＇ phase, and the conversion time from L10 phase to L12 phase will lag behind other samples with lower Al con-tent. By analyzing the occupation probability of Ti, it can be found that the solid solution strengthening effect of Ti is reduced with the increase of Al component.
基金supported by the National Natural Science Foundation of China[grant number U1537202]the TsinghuaGeneral Motor International collaboration project[grant number 20153000354]+1 种基金the UK Royal Society through the Newton International Fellowship Schemethe National Laboratory for Information Science and Technology in Tsinghua University for access to supercomputing facilities
文摘In the present study, the influence of natural convection on the lamellar eutectic growth is determined by a phase-field-lattice Boltzmann study for Al-Cu eutectic alloy. The mass difference resulting from concentration difference led to the fluid flow, and a robust parallel and adaptive mesh refinement algorithm was employed to improve the computational efficiency without any compromising accuracy. Results show that the existence of natural convection would affect the growth undercooling and thus control the interface shape by adjusting the lamellar width. In particular, by alternating the magnitude of the solute expansion coefficient, the strength of the natural convection is changed. Corresponding microstructure patterns are discussed and compared with those under no-convection conditions.
基金supported by the Doctor Foundational Research Project in Shenyang Ligong University(Serial Number:0010).
文摘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.
基金the National Natural Science Foundation of China(Grants U1333201 and U1833116)。
文摘Fracture is a very common failure mode of the composite materials,which seriously affects the reliability and service-life of composite materials.Therefore,the study of the fracture behavior of the composite materials is of great significance and necessity,which demands an accurate and efficient numerical tool in general cases because of the complexity of the arising boundary-value or initial-boundary value problems.In this paper,a phase field model is adopted and applied for the numerical simulation of the crack nucleation and propagation in brittle linear elastic two-phase perforated/particulate composites under a quasi-static tensile loading.The phase field model can well describe the initiation,propagation and coalescence of the cracks without assuming the existence and the geometry of the initial cracks in advance.Its numerical implementation is realized within the framework of the finite element method(FEM).The accuracy and the efficiency of the present phase field model are verified by the available reference results in literature.In the numerical examples,we first study and discuss the influences of the hole/particle size on the crack propagation trajectory and the force-displacement curve.Then,the effects of the hole/particle shape on the crack initiation and propagation are investigated.Furthermore,numerical examples are presented and discussed to show the influences of the hole/particle location on the crack initiation and propagation characteristics.It will be demonstrated that the present phase field model is an efficient tool for the numerical simulation of the crack initiation and propagation problems in brittle two-phase composite materials,and the corresponding results may play an important role in predicting and preventing possible hazardous crack initiation and propagation in engineering applications.
基金financially supported by the Educational Department of Liaoning Province (No.20060744)the Shenyang Nurturing Young Scientific Technological Talents Items (No.1081230-1-00)
文摘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.
基金supported by National Natural Science Foundation of China(No.51071061)NSAF(No.11176011)
文摘A phase field model is developed to simulate the grain evolution of 17-4PH steel during cyclic heat treatment （CHT）. Our simulations successfully reproduce the grain morphologies of every CHT. In the process of every CHT, phase transformation recrystallization happens. The recrystallized grains appear mainly on the original grain boundaries. The average grain size of 13.2 μm obtained by 1040 ℃×1 h solution treatment for this experimental steel can be refined to 2.2 μm after five CHT＇s. Furthermore, the effects of phenomenological parameters in our model are discussed.
基金the National Natural Science Foundation of China for the financial support under the grant Nos.50771028 and 50471024Education Ministry of China for an outstanding teacher research fund to this study.
文摘A model has been established to simulate the realistic spatio-temporal microstructure evolution in recrystallization of a magnesium alloy using the phase field approach. A set of rules have been proposed to decide the real physical value of all parameters in the model. The thermodynamic software THERMOCALC is applied to determine the local chemical free energy and strain energy, which is added to the free energy density of grains before recrystallization. The Arrhenius formula is used to describe boundary mobility and the activity energy is suggested with a value of zinc segregation energy at the boundary. However, the mobility constant in the formula was found out by fitting to a group of grain size measurements during recrystallization of the alloy. The boundary range is suggested to decide the gradient parameters in addition of fitting to the experimental boundary energy value. These parameter values can be regarded as a database for other similar simulations and the fitting rules can also be applied to build up databases for any other alloy systems. The simulated results show a good agreement with reported experimental measurement of the alloy at the temperatures from 300 to 400℃ for up to 100 min but not at 250℃. This implies a mechanism variation in activity energy of the boundary mobility in the alloy at low temperature.
基金supported by the National Natural Science Foundation of China (Nos. 10832009,11002123 and 11090333)the Zhejiang Provincial Natural Science Foundation under grant R6110115+1 种基金the Chinese Universities Scientific Fund under grant 2009QNA4035the program of Key Team of Technological Innovation of Zhejiang Province under grant2011R09025-07
文摘Domain switching around electrically permeable and impermeable cracks in ferro-electric single crystals subjected to a mechanical load is investigated by using a phase field model.It is found that the electrical boundary conditions have little effect on the polarization distribution without any external load when the initial polarization is parallel to the crack,which is totally different from previous studies where the initial polarization is perpendicular to the crack.How-ever,the simulation results show that the electrical boundary conditions have great influence on the polarization distribution as well as the domain switching behavior when a mechanical load is applied.The critical mechanical load of domain switching with a permeable crack is much smaller than that in the case of an impermeable crack.
基金support from the National Natural Science Foundation of China (Nos. 51601078 and 51201147)the National Key R&D Program of China (Grant No. 2016YFB0701302)+1 种基金the Informalization Construction Program of Chinese Academy of Science (INFO-115-B01)Shenyang Supercomputer Centers for the computational resource of Chinese Academy of Science
文摘A quantitative phase field method of multi-component diffusion-controlled phase transformations coupled with the Kim-Kim-Suzuki model was applied to study the effect of initial particle size distribution （PSD） in 3D and space distribution in 2D on dissolution of α particles in Ti-6Al-4V alloy below β transus temperature in real time and length scale. The thermodynamic and mobility data were obtained from Thermo-Calc and DICTRA softwares, respectively. The results show that the volume fractions of α particles decay with time as： f =feq ＋ （f0 -feq） exp（-Ktn） for four cases of PSD. The sequence of dissolution kinetics from fast to slow is： uniform PSD, normal PSD, lognormal PSD and bimodal PSD. The space distribution is found to be a major factor affecting the dissolution kinetics and the microstructures. When the distance of the particles is less than critical value, the dissolution rates reduce with the decrease in distance. The Al and V concentration fields around the particles appear more obvious soft impingement.
基金financially supported by the National Natural Science Foundation of China (Nos.52122408, 51901013,51971018,52101188,52225103,52071023 and U20B2025)the Funds for Creative Research Groups of NSFC (No.51921001)the financial support from the Fundamental Research Funds for the Central Universities (University of Science and Technology Beijing,Nos.FRF-TP-2021-04C1 and 06500135)。
文摘Ab st ra ct Nanocrystalline materials exhibit unique properties due to their extremely high grain boundary(GB) density.However,this high-density characteristic induces grain coarsening at elevated temperatures,thereby limiting the widespread application of nanocrystalline materials.Recent experimental observations revealed that GB segregation and second-phase pinning effectively hinder GB migration,thereby improving the stability of nanocry stalline materials.In this study,a mouified phase-field model that integrates mismatch strain,solute segregation and precipitation was developed to evaluate the influence of lattice misfit on the thermal stability of nanocrystalline alloys.The simulation results indicated that introducing a suitable mismatch strain can effectively enhance the microstructural stability of nanocrystalline alloys.By synergizing precipitation with an appropriate lattice misfit,the formation of second-phase particles in the bulk grains can be suppressed,thereby facilitating solute segregation/precipitation at the GBs.This concentrated solute segregation and precipitation at the GBs effectively hinders grain migration,thereby preventing grain coarsening.These findings provide a new perspective on the design and regulation of nanocrystalline alloys with enhanced thermal stability.
基金supported by the National Natural Science Foundation of China(Grant No.12372152)the Qilu Young Talent Program of Shandong University,Zhejiang Lab Open Research Project(Grant No.K2022PE0AB05)+1 种基金Shandong Provincial Natural Science Foundation(Grant No.ZR2023MA058)Guangdong Basic and Applied Basic Research Foundation(Grant No.2023A1515011819).
文摘The morphology of Ni_(4)Ti_(3) precipitates is important in tuning the martensitic transformation(MT)behavior and mechanical properties of nitinol.Constrained ageing is effective in engineering the morphology of Ni_(4)Ti_(3) precipitates due to the variant selection effect of external load which is still lacking.In this work,maps of variant selection effect of external load applied along all crystallographic directions are obtained by using a combination of theoretical analyses and phase field simulations.It is found that maps produced by uniaxial tension and uniaxial compression are quite different.The number and types of Ni_(4)Ti_(3) variants preferred by external load vary as the loading direction changes.Moreover,factors influencing the strength of variant selection effect are discovered.This work provides insights on understanding the Ni_(4)Ti_(3) precipitation process and sheds light on the engineering of morphology of Ni_(4)Ti_(3) precipitates for desired mechanical and functional properties.
基金supported by the National Natural Science Foundation of China (Grant No 50401013)Doctorate Foundation of Northwestern Polytechnical University,China
文摘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.
