Three-dimensional Phase-field Simulations Research Articles

Three-Dimensional Phase-Field Simulation of Stress-Assisted Two-Way Shape Memory Effect and Its Cyclic Degradation of Single-Crystal NiTi Shape Memory Alloy

Three-Dimensional Phase-Field Simulation of Stress-Assisted Two-Way Shape Memory Effect and Its Cyclic Degradation of Single-Crystal NiTi Shape Memory Alloy

Accelerating three-dimensional phase-field simulations via deep learning approaches

Accelerating three-dimensional phase-field simulations via deep learning approaches

Numerical modeling of shrinkage induced cracking in cementitious materials with three-dimensional simulation and phase-field method

This study investigates the damage and cracking of concrete induced by drying shrinkage. While previous studies have primarily focused on two-dimensional configurations, this research employs three-dimensional phase-field simulations of concrete samples undergoing the drying process. The distribution of inclusions is derived from tomographic images of real samples. The analysis includes the evolution of the damage field and the propagation of cracked zones. The study introduces the formulation and implementation of the numerical method and applies it to elucidate the cracking process resulting from drying shrinkage in concrete composites with various types of inclusions. Numerous numerical simulations were conducted to accurately depict the cracking process induced by drying shrinkage in the samples, evaluating the effects of different inclusions with varying stiffness on the concrete’s cracking phenomenon and comparing the impact of various energy degradation methods on crack simulation. Moreover, the study analyzes the influence of the spatial distribution of the inclusions in the three-dimensional simulation and compares the numerical results with experimental observations.

Evolution dynamics of thin liquid structures investigated using a phase-field model.

Liquid structures of thin-films and torus droplets are omnipresent in daily lives. The morphological evolution of liquid structures suspending in another immiscible fluid and sitting on a solid substrate is investigated by using three-dimensional (3D) phase-field (PF) simulations. Here, we address the evolution dynamics by scrutinizing the interplay of surface energy, kinetic energy, and viscous dissipation, which is characterized by Reynolds number Re and Weber number We. We observe special droplet breakup phenomena by varying Re and We. In addition, we gain the essential physical insights into controlling the droplet formation resulting from the morphological evolution of the liquid structures by characterizing the top and side profiles under different circumstances. We find that the shape evolution of the liquid structures is intimately related to the initial shape, Re, We as well as the intrinsic wettability of the substrate. Furthermore, it is revealed that the evolution dynamics are determined by the competition between the coalescence phenomenology and the hydrodynamic instability of the liquid structures. For the coalescence phenomenology, the liquid structure merges onto itself, while the hydrodynamic instability leads to the breakup of the liquid structure. Last but not least, we investigate the influence of wall relaxation on the breakup outcome of torus droplets on substrates with different contact angles. We shed light on how the key parameters including the initial shape, Re, We, wettability, and wall relaxation influence the droplet dynamics and droplet formation. These findings are anticipated to contribute insights into droplet-based systems, potentially impacting areas like ink-jet printing, drug delivery systems, and microfluidic devices, where the interplay of surface energy, kinetic energy, and viscous dissipation plays a crucial role.

Three-dimensional phase-field simulation of intergranular bubble evolution in UO2 during irradiation

In this study, a three-dimensional phase-field model was developed to investigate the evolution of intergranular bubbles during irradiation. The model considered several grain structures and multiple fission gas bubbles, with the microstructure evolution being governed by improved dynamic equations. Verification of the phase-field model showed good agreement with the theory in describing the properties of grain boundary (GB). The study examined the dependency of bubble percolation on bubble shape, fission rate, and average grain size. The simulation results revealed that the changes in these three factors have significant effects on the shape of GB bubbles, the percolation rate, and the connectivity threshold of GB bubbles. The bubble connection threshold was obtained from the curve of triple junction coverage versus grain boundary coverage. The results indicate that the connectivity threshold increases with the higher fission rate and lager average grain size. This investigation into bubble percolation during irradiation provides valuable insights for further study on fission gas release in UO2 fuel within nuclear reactors.

Equiaxed growth of interacting Al–Cu dendrites in thin samples: a phase-field study at copper concentrations relevant for practical applications

We perform three-dimensional phase-field simulations of equiaxed solidification in Al–Cu thin samples. Purely diffusive conditions are considered in order to describe systems where convection and gravity effects can be neglected. The use of a parallel adaptive finite element algorithm introduced recently [Gong et al., Comput. Mater. Sci. 147 (2018) p. 338-352] allows us to reach the domain of copper concentrations used in practical applications (c≥3 wt% Cu). We compare the present results with those of a previous study which was restricted to lower copper concentrations (c≤2 wt% Cu) [Boukellal et al., Materialia 1 (2018) p. 62-69] due to the use of a finite difference code. In the fast dendritic growth regime, our results confirm that the dimensionless growth length Λ is independent of the copper concentration and the average separation distance between the dendrite nuclei. The new data obtained at higher copper concentrations lead to a more accurate estimate of Λ. Physical arguments are developed to specify the meaning of Λ and the grounds of the scaling law Λ=cst. Comparisons with available experimental results of the literature give additional support to this scaling law.

Complex hexagonal close-packed dendritic growth during alloy solidification by graphics processing unit-accelerated three-dimensional phase-field simulations: demo for Mg–Gd alloy

Complex hexagonal close-packed dendritic growth during alloy solidification by graphics processing unit-accelerated three-dimensional phase-field simulations: demo for Mg–Gd alloy

Pressure Control of Nonferroelastic Ferroelectric Domains in ErMnO3.

Mechanical pressure controls the structural, electric, and magnetic order in solid-state systems, allowing tailoring of their physical properties. A well-established example is ferroelastic ferroelectrics, where the coupling between pressure and the primary symmetry-breaking order parameter enables hysteretic switching of the strain state and ferroelectric domain engineering. Here, we study the pressure-driven response in a nonferroelastic ferroelectric, ErMnO3, where the classical stress-strain coupling is absent and the domain formation is governed by creation-annihilation processes of topological defects. By annealing ErMnO3 polycrystals under variable pressures in the MPa regime, we transform nonferroelastic vortex-like domains into stripe-like domains. The width of the stripe-like domains is determined by the applied pressure as we confirm by three-dimensional phase field simulations, showing that pressure leads to oriented layer-like periodic domains. Our work demonstrates the possibility to utilize mechanical pressure for domain engineering in nonferroelastic ferroelectrics, providing a lever to control their dielectric and piezoelectric responses.

Scaling law for growth of misoriented equiaxed Al-Cu dendrites: A phase-field study with in situ experiment validation

We consider the effect of a mutual misorientation between two interacting equiaxed dendrites in polycrystalline materials on the scaling law of growth from undercooled melt. This effect is investigated by three-dimensional quantitative phase-field simulations of Al-Cu alloy solidification in thin samples. It has been found that the equiaxed dendritic growth kinetics changes due to the reduced solute interaction when considering mutual grain misorientation, and both the characteristic growth rate and primary dendritic arm length can be approximately correlated linearly to the misorientation of two dendrites. The scaling law, which originally uses the solute composition and the distance between two nuclei as physical parameters to describe the dynamics of primary dendritic arms, is then adapted with consideration of the misorientation. Compared with the previous scaling law that only concerns face-to-face growth of two dendrites, the predictions of the newly proposed scaling law taking the grain orientation in account are in better agreement with the experimental data.

High-performance GPU computing of phase-field lattice Boltzmann simulations for dendrite growth with natural convection

The effect of natural convection on dendrite morphology is investigated through three-dimensional large-scale phase-field lattice Boltzmann simulations using a block-structured adaptive mesh refinement scheme with the mother-leaf method in a parallel-GPU environment. The simulations confirmed that downward buoyancy enhances the growth of the primary and secondary arms, and upward buoyancy delays the growth of those arms. In addition, the effect of natural convection on the solidification morphologies gradually decreased as the primary arm tip reached the top of the computational domain and finally stopped. Furthermore, in the longer simulation under purely isothermal diffusive conditions, detachment of the secondary arms owing to curvature-driven fragmentation was observed. A large-scale non-isothermal dendrite growth simulation was also conducted, wherein it was observed that the tip growth rate of the primary arm was delayed, and the secondary arm spacing was larger than that in the isothermal condition.

Insight into elemental diffusion in rafting of Ni-based superalloys by three-dimensional multicomponent phase field simulation

Elemental diffusion is the underlying mechanism in rafting during creep process of Ni-based single crystal superalloys. A three-dimensional elastoplastic multicomponent multiphase-field model is established to investigate γ' rafting morphology. The distribution and diffusion of elements will be discussed according to the simulation results. Finally the rafting process has been illustrated in an elemental diffusion view.

Generation mechanism and motion behavior of sliver defect in single crystal Ni-based superalloy

Sliver is a common but easily neglected defect in single crystal Ni-based superalloy castings. To date, there is still no unified viewpoint on its formation mechanism and generation causes. In this work, the orientation discontinuity and motion behavior of sliver defects were studied through experiments and numerical simulations. The ultrathin wedge-shaped specimen containing the grain boundary of the sliver and the matrix was prepared at the initial position of the sliver defect for the observation of equal thickness fringes. The discontinuity of equal thickness fringes on both sides of the grain boundary was observed through a transmission electron microscope, which directly confirms the abrupt change in the orientation between the sliver and matrix from the nanoscale. The crystal lattices at the smooth area and the bulging area of the grain boundary were found to have unusually different arrangements. The irregular lattice arrangement at the bulging area shows that the grain boundary has experienced high-stress deformation. Statistical results of sliver orientation deviation with a further composition analysis show the micro protuberance of the mold shell has a noticeable inductive effect on the sliver generation. Furthermore, a self-developed three-dimensional phase-field simulation model coupled with the spatial topology algorithm is established to simulate the orientation deflection behavior and orientation deviation threshold of fractured dendrites. The simulation results indicated that there is an upper limit of the cross-section solid fraction at the fracture position for the motion of the fractured dendrites. When the cross-section solid fraction at the fracture position is higher than this upper limit, it will be difficult to produce large deviation slivers due to the structural limitation of surrounding dendrites. This upper limit does not change with the solidification temperature gradient.

Ni coarsening in Ni-yttria stabilized zirconia electrodes: Three-dimensional quantitative phase-field simulations supported by ex-situ ptychographic nano-tomography

In-depth understanding of nickel (Ni) coarsening is helpful for improving the service life of Ni-yttria stabilized zirconia (YSZ) electrodes in solid oxide cells. Unfortunately, very few quantitative experimental/theoretical descriptions of Ni coarsening in Ni-YSZ electrodes during long-term operation exist. In this paper, quantitative modeling of Ni coarsening in Ni-YSZ electrodes was achieved through three-dimensional (3D) phase-field simulation supported by ex-situ ptychographic nano-tomography and input of reliable thermophysical parameters. A pragmatic procedure was proposed to refine and verify the materials parameters for the simulations. Moreover, the microstructures of the Ni-YSZ electrode in the pristine and annealed states obtained via the ex-situ ptychographic nano-tomography were used as the initial input and experimental validation for the phase-field simulations. After that, comprehensive comparison between the simulated and the experimental 3D microstructures was conducted, indicating the successful quantitative phase-field simulation of Ni coarsening in Ni-YSZ electrodes presented here. The success of this work is expected to pave the way for accurate prediction of the service life and even design of high-performance Ni-YSZ electrodes.

A novel computational model for isotropic interfacial energies in multicomponent alloys and its coupling with phase-field model with finite interface dissipation

• A CALPHAD model for isotropic interfacial energies was developed and programmed. • Direct coupling between interfacial energy model and phase-field model was realized. • Isotropic interfacial energy databases in Ni-Al-Cr systems were established. • 3-D phase-field simulations of γ dendritic growth in two Ni alloys were conducted. • Effect of different interfacial energies on γ dendritic growth was analyzed. In this work, a novel computational model for the description of the temperature- and composition-dependent isotropic interfacial energy in multicomponent alloys was first developed in the framework of the CALculation of PHAse Diagram (CALPHAD) approach and implemented in a home-made code. By linking to the open-source code for interfacial energy calculation in alloys, OpenIEC, the databases for isotropic γ /liquid and γ / γ' interfacial energies in Ni-Al, Ni-Cr, Al-Cr, and Ni-Al-Cr systems were then efficiently established. After that, a direct coupling strategy between the current CALPHAD interfacial energy database and the phase-field model with finite interface dissipation was proposed and applied to three-dimensional (3-D) phase-field simulations of the primary γ dendritic growth in both Ni-Al and Ni-Al-Cr alloys during isothermal solidification. The effect of the interfacial energy on the morphology, tip growth rate, and partitioning coefficients in primary γ dendrites of binary Ni-Al and ternary Ni-Al-Cr alloys was investigated by comprehensively comparing the phase-filed simulation results using the composition-/temperature-dependent interfacial energies with those using the constant value. It is anticipated that the presently developed CALPHAD model for interfacial energy is of general validity for different multicomponent alloys and should be integrated with the phase-field model for quantitative simulation of their microstructure evolution.

Precipitation mechanism of nanoscale Cu-rich particles in restrained welding conditions and its effect on the mechanical properties of HAZs

The effect of welding constraint on nanoscale Cu-rich precipitation in welding heat affected zones (HAZs) of Cu-bearing steel was studied in this work. Due to the constraint tensile stress of 150 MPa, α-Fe transformation start temperature (Ts) of CGHAZ and FGHAZ increased by 55 °C and 42 °C, respectively. The volume fraction and average size of 9R-structural Cu-rich particles in FGHAZ increased from 0.3 % to 0.8 % and from 5 nm to 8 nm, respectively. Three-dimensional phase field simulation shows that the increment of Ts promoted the precipitation of Cu-rich particles in α-Fe matrix. However, the Ts of restrained CGHAZ was too low to favor the precipitation of Cu-rich particles. The tensile strength of FGHAZ increased from 970 MPa to 1032 MPa due to the nanoscale Cu-rich precipitation strengthening.

Origin of morphological variation of grain boundary precipitates in titanium alloys

Morphological variations of grain boundary (GB) allotriomorphs and widmanstatten sideplates (WS) of α precipitate in a β titanium alloy, have been systematically investigated by combining three-dimensional (3D) phase-field simulations, 3D experimental characterization and crystallographic orientation analysis. The inclination angle, θ, between the habit plane of the GBα and the hosting GB plane is found to dictate the morphologies of GB precipitates. For the first time, three distinct regimes of α morphology at a β GB, separated by two critical angles (θc1, θc2) are observed: θ≤θc1, GBα alone decorates the GB as a continuous layer; θc1<θ≤θc2, α precipitate appears as a mixture of GBα and a WS emanating from it; θ>θc2, WS alone grows directly from bare β GBs. The dramatic morphological variation owes its origin to the dynamic interplay between the spreading along GBP and the intragranular growth of the GB precipitate, and its dependence on θ.

Microstructural coarsening in dense binary systems

The kinetics of microstructural evolution driven by phase coarsening (Ostwald ripening) in a binary, two-phase system were studied systematically at ultra high volume fractions of the dispersed phase. Three-dimensional phase-field simulations and theoretical analysis are compared. Late-stage microstructure evolution at ultra high volume fractions (VV>0.9) are reported here. One interesting finding is that the scaling exponents, m, for the kinetics of phase coarsening at ultra high volume fractions take values in the range 2<m<3, depending on the precise volume fraction of the dispersed phase, when varied over the narrow range 0.9<VV<1. This result is confirmed by a comparative study of simulated microstructures and measurements of interfacial energy decay rates with theoretical coarsening analysis. Scaled particle-size distributions derived from simulations at ultra high volume fractions agree neither with Wagner’s 3D particle-size distribution for interface-reaction controlled phase coarsening, nor with Lifshitz and Slyozov’s particle-size distribution for diffusion-controlled phase coarsening. The research reported substantiates that the kinetics of microstructure evolution at ultra high volume fractions, approaching true grain growth, exhibits a varying blend of both interface reaction-controlled and volume diffusion-controlled phase coarsening.

Strain phase diagram and physical properties of (110)-oriented PbTiO3 thin films by phase-field simulations

Extensive studies have shown that (110)-oriented ferroelectric films exhibit unique structures and excellent physical properties compared with (001)-oriented ferroelectric films. Showing how the domain structure evolve as the function of epitaxial strain, or establishing a strain phase diagram, should provide valuable information for the controlling of domain structures. In this work, three-dimensional phase-field simulations were performed to investigate the phase (domain) structures and physical properties in (110)-oriented PbTiO3 thin films. We constructed the temperature-misfit strain phase diagram of (110)-oriented PbTiO3 thin films. Typical phase (domain) structures were analyzed in detail and various types of topological flux-closure domains were found to exist in mixed phases. The experimental results on a special substrate KTaO3, in particular the orientation of domain walls, corroborate the simulation results. We found that the misfit strain could be used to engineer the phase (domain) structures and tune the dielectric, piezoelectric and thermal conductivity properties in (110)-oriented PbTiO3 thin films. These results deepen the understanding and provide a theoretical guidance for the strain engineering of the structures and properties of high-index (110)-oriented PbTiO3 thin films.

Parallel-GPU-accelerated adaptive mesh refinement for three-dimensional phase-field simulation of dendritic growth during solidification of binary alloy

In the phase-field simulation of dendrite growth during the solidification of an alloy, the computational cost becomes extremely high when the diffusion length is significantly larger than the curvature radius of a dendrite tip. In such cases, the adaptive mesh refinement (AMR) method is effective for improving the computational performance. In this study, we perform a three-dimensional dendrite growth phase-field simulation in which AMR is implemented via parallel computing using multiple graphics processing units (GPUs), which provide high parallel computation performance. In the parallel GPU computation, we apply dynamic load balancing to parallel computing to equalize the computational cost per GPU. The accuracy of an AMR refinement condition is confirmed through the single-GPU computations of columnar dendrite growth during the directional solidification of a binary alloy. Next, we evaluate the efficiency of dynamic load balancing by performing multiple-GPU parallel computations for three different directional solidification simulations using a moving frame algorithm. Finally, weak scaling tests are performed to confirm the parallel efficiency of the developed code.

Modeling of stoichiometric phases in off-eutectic compositions of directional solidifying NbSi-10Ti for phase-field simulations

To meet the requirements of modern technical and scientific applications, the development and investigation of high-performance materials with locally defined properties gets more and more into focus of actual research. These materials often contain phases with at least one stoichiometric component. To investigate such materials within grand potential-based phase-field simulations, a specialized method is needed, which allows to model the required Gibbs energy formulations for the stoichiometric phases. In this work, a modeling approach is introduced, based on the equilibrium conditions of the involved phases, to generate a Gibbs energy formulation for the stoichiometric (Nb,Ti)3Si phase in Nb-Si-Ti. With this formulation, the eutectic reaction in the ternary NbSi-10Ti system is investigated within two- and three-dimensional phase-field simulations, with eutectic and off-eutectic melt compositions. The solidified microstructures are qualitatively assessed by measuring the average front undercoolings, the adjusting phase fractions and the concentrations during the simulations. The validation of the results shows the ability of the generated material model to simulate the obtained system and proves the suitability of the presented approach to model stoichiometric phases.