Bifurcation Analysis Reveals Solution Structures of Phase Field Models

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.

Phase field model for electric-thermal coupled discharge breakdown of polyimide nanocomposites under high frequency electrical stress

In contrast to conventional transformers, power electronic transformers, as an integral component of new energy power system, are often subjected to high-frequency and transient electrical stresses, leading to heightened concerns regarding insulation failures. Meanwhile, the underlying mechanism behind discharge breakdown failure and nanofiller enhancement under high-frequency electrical stress remains unclear. An electric-thermal coupled discharge breakdown phase field model was constructed to study the evolution of the breakdown path in polyimide nanocomposite insulation subjected to high-frequency stress. The investigation focused on analyzing the effect of various factors, including frequency, temperature, and nanofiller shape, on the breakdown path of Polyimide(PI) composites. Additionally, it elucidated the enhancement mechanism of nano-modified composite insulation at the mesoscopic scale. The results indicated that with increasing frequency and temperature, the discharge breakdown path demonstrates accelerated development, accompanied by a gradual dominance of Joule heat energy. This enhancement is attributed to the dispersed electric field distribution and the hindering effect of the nanosheets. The research findings offer a theoretical foundation and methodological framework to inform the optimal design and performance management of new insulating materials utilized in high-frequency power equipment.

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.

Mass and Volume Conservation in Phase Field Models for Binary Fluids

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.

Thermodynamic consistent phase field model for sintering process with multiphase powders

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.

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 study for effect of chemical composition on pre-precipitation process of γ′ phase in Ni-Al-Ti alloy

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.

Phase-field-lattice Boltzmann study for lamellar eutectic growth in a natural convection melt

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.

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.

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.

A Phase-field Model to Simulate Recrystallization in an AZ31 Mg Alloy in Comparison of Experimental Data

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.

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.

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.

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.

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.

Simulation of size effects by a phase field model for fracture

In phase field fracture models the value of the order parameter distin- guishes between broken and undamaged material. At crack faces the order param- eter interpolates smoothly between these two states of the material, which can be regarded as phases. The crack evolution follows implicitly from the time inte- gration of an evolution equation of the order parameter, which is coupled to the mechanical field equations. Among other phenomena phase field fracture mod- els are able to reproduce crack nucleation in initially sound materials. For a 1D setting it has been shown that crack nucleation is triggered by the loss of stability of the unfractured, spatially homogeneous solution, and that the stability point depends on the size of the considered structure. This work numerically investi- gates to which extend size effects are reproduced by the 2D phase field model. Exemplarily, a finite element study of the hole size effect is performed and the simulation results are compared to exnerimental data.

Deconfinement Phase Transition with External Magnetic Field in the Friedberg-Lee Model

The deconfinement phase transition with external magnetic field is investigated in the Friedberg-Lee model. We expand the potentiM around the two locM minima of the first-order deconfinement phase transition and extract the ground state of the system in the frame of functional renormalization group. By solving the flow equations we find that the magnetic field displays a catalysis effect and it becomes more difficult to break through the confinement.

Phase field modeling of grain stability of nanocrystalline alloys by explicitly incorporating mismatch strain

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.

Variant selection map of external load during Ni_(4)Ti_(3) precipitation in nitinol:a theoretical and phase field study

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.

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.