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.

Phase Field Simulation of Intragranular Microvoids Evolution Due to Surface Diffusion in Stress Field

Based on the bulk free energy density and the degenerate mobility constructed by the quartic double-well potential function,a phase field model is established to simulate the evolution of intragranular microvoids due to surface diffusion in a stress field.The corresponding phase field governing equations are derived.The evolution of elliptical microvoids with different stressesΛ,aspect ratiosβand linewidths hˉis calculated using the mesh adaptation finite element method and the reliability of the procedure is verified.The results show that there exist critical values of the stressΛc,the aspect ratioβc and the linewidth hˉc of intragranular microvoids under equivalent biaxial tensile stress.When Λ≥Λ_(c),β≥β_(c) or h≤h_(c),the elliptical microvoids are instable with an extending crack tip.WhenΛ<Λ_(c),β<β_(c) or hˉ>h_(c),the elliptical microvoids gradually cylindricalize and remain a stable shape.The instability time decreases with increasing the stress or the aspect ratio,while increases with increasing the linewidth.In addition,for the interconnects containing two elliptical voids not far apart,the stress will promote the merging of the voids.

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.

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 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 dendritic coarsening during isothermal solidification

Dendritic coarsening in AI-2mol%Si alloy during isothermal solidification at 880K was investigated by phase field modeling. Three coarsening mechanisms operate in the alloy： （a） melting of small dendrite arms; （b） coalescence of dendrites near the tips leading to the entrapment of liquid droplets; （c） smoothing of dendrites. Dendrite melting is found to be dominant in the stage of dendritic growth, whereas coalescence of dendrites and smoothing of dendrites are dominant during isothermal holding. The simulated results provide a better understanding of dendrite coarsening during isothermal solidification.

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 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.

Driving lithium to deposit inside structured lithium metal anodes:A phase field model

Lithium metal anode is one of the most important anode materials for next-generation high-specificenergy secondary batteries.Structured lithium metal anodes have received extensive attention in the development of practical lithium metal batteries.Methods of driving lithium metal to deposit inside the pores of structured lithium metal anodes have always been one of the most concerned issues,especially for highly conductive frameworks.An electrochemical phase field theory with galvanostatic lithium plating process is employed in this work,the mechanism that illustrates the preference of lithium metal to deposit at the top of the framework structure has been revealed,and through the simulation analysis of various regulating strategies,the strategies that can efficiently drive lithium to deposit inside structured pores are summarized.This work presents the theoretical calculation and analysis methods that can be used for the rational design of lithium metal batteries.

Electrocaloric effect in ferroelectric materials:From phase field to first-principles based effective Hamiltonian modeling

Electrocaloric effect(ECE)of ferroelectrics has attracted considerable interest due to its potential application in environmentally friendly solid-state refrigeration.The discovery of giant ECE in ferroelectric thin films has greatly renewed the research activities and significantly stimulated experimental and theoretical investigations.In this review,the recent progress on the theoretical modeling of ECE in ferroelectric and antiferroelectric materials are introduced,which mainly focuses on the phase field modeling and first-principles based effective Hamiltonian method.We firstly provide the theoretical foundation and technique details for each method.Then a comprehensive review on the progress in the application of two methods and the strategies to tune the ECE are presented.Finally,we outline the practical procedure on the development of multi-scale computational method without experiemtal parameters for the screening of optimized electrocaloric materials.

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.

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.

Experimental studies and phase field modeling of microstructure evolution during solidification with electromagnetic stirring

Thixocasting requires manufacturing of billets with non-dendritic microstructure.Aluminum alloy A356 billets were produced by rheocasting in a mould placed inside a linear electromagnetic stirrer.Subsequent heat treatment was used to produce a transition from rosette to globular microstructure.The current and the duration of stirring were explored as control parameters.Simultaneous induction heating of the billet during stirring was quantified using experimentally determined thermal profiles.The effect of processing parameters on the dendrite fragmentation was discussed.Corresponding computational modeling of the process was performed using phase-field modeling of alloy solidification in order to gain insight into the process of morphological changes of a solid during this process.A non-isothermal alloy solidification model was used for simulations.The morphological evolution under such imposed thermal cycles was simulated and compared with experimentally determined one.Suitable scaling using the thermosolutal diffusion distances was used to overcome computational difficulties in quantitative comparison at system scale.The results were interpreted in the light of existing theories of microstructure refinement and globularisation.

Numerical simulation for GMAW with a new model based on phase field model

In this paper,a numerical investigation about the metal transfer of GMAW is investigated based on the phase field model.Be different of most published work,we take the thermocapillary effect and mixture energy into the process of phase transfer and interface change which is different from volume of fluid( VOF) method.We discretize the whole model with a continuous finite element method and we also apply a penalty formulation to the continuity condition enhancing the stability of the pressure.Metal transfer of GMAW with constant and pulse current is computed as numerical examples which agrees well with the data of high-speed photography.The result shows that the computing process of the phase field model is stability and it has a higher precision in predicting the diameter of droplet.

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

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

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

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

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

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

Phase field crystal study of the crystallization modes within the two-phase region

Using the phase field crystal approach, the crystallization process within the liquid-solid coexistence region is inves- tigated for a square lattice on an atomic scale. Two competing growth modes, i.e., the diffusion-controlled growth through long-range atomic migration in liquid and the diffusionless growth through local atom rearrangement, which give rise to two completely different crystallization behaviors, are compared. In the diffusion-controlled regime, the interface migrates in a layerwise manner, leading to a gradual change of crystal morphology from truncated square to four-fold symmetric dendrite with the increase of driving force. For the diffusionless growth mode, a single crystal with no significant density change occupies the whole system at a faster rate while exhibiting a small growth anisotropy. The competition between these two modes is also discussed from the key input of the phase field crystal model： the correlation function.

Three-dimensional fracture of UO_(2)ceramic pellets by phase field modeling

Fracture is the primary failure type for the UO_(2)ceramic pellets,which are operated under irradiation and thermal processes,and physically confined by the cladding.In this work,by combining closely coupled physical processes of heat conduction,heat transfer,and mechanical deformation with the cohesive phase field framework,we proposed a high-precision modeling of quasistatic cracking of three-dimensional(3D)UO_(2)ceramic fuel pellets under high temperature and irradiation.The simulation results agree well with the experimental results,indicating that the proposed method has the potential to capture 3D crack modes in the interested time range.Further,we studied the relation of the 3D fracture patterns of fuel pellets with the peak power densities.It was found that the power level plays a critical role in determining the competition between radial and circumferential cracks,as well as transverse penetrating cracks.