Three-dimensional Phase Field Research Articles

Role of modulus mismatch in vertically aligned nanocomposite formation during spinodal decomposition in constrained films

Self-assembly of vertically aligned nanocomposite films via spinodal decomposition is of potential interest in a range of technique applications. How the elastic interaction mediated by modulus mismatch, cubic anisotropy factor, and anisotropic lattice mismatch between the product phases facilitates the experimentally observed columnar nanocomposite formation was elucidated using three-dimensional phase field simulations of spinodal decomposition in an elastically inhomogeneous cubic binary epilayer. A phase diagram from vertically aligned columnar nanocomposites to vertically aligned laminar nanocomposites was constructed with respect to modulus mismatch, cubic anisotropy factor, and anisotropic lattice mismatch. The results provide a guideline for such nanocomposite structure formation.

Variant selection during α precipitation in Ti–6Al–4V under the influence of local stress – A simulation study

Variant selection of α (hexagonal close-packed, hcp) phase during its precipitation from β (body-centered cubic, bcc) matrix plays a key role in determining the microstructural state and mechanical properties of α/β titanium alloys. In this work, we develop a three-dimensional quantitative phase field model to predict variant selection and microstructural evolution during β→α transformation in Ti–6Al–4V (wt.%) under the influence of both external and internal stresses. The model links its inputs directly to thermodynamic and mobility databases, and incorporates the crystallography of bcc to hcp transformation, elastic anisotropy and defects within semi-coherent α/β interfaces in its total free energy formulation. It is found that, for a given undercooling, the development of a transformation texture (also called microtexture) of the α phase due to variant selection during precipitation is determined by the interplay between externally applied stress or strain and internal stress generated by the precipitation reaction itself. For example, the growth of pre-existing α precipitates is accompanied by selective nucleation and growth of secondary α plates of certain variants that may not be the ones preferred by the initially applied stress. Possible measures to reduce transformation texture are discussed.

The influence of crack face electrical boundary conditions on the nonlinear behavior of ferroelectric single crystal

The nonlinear electromechanical behavior of a cracked ferroelectric single crystal subjected to pure electrical loading is investigated by a three-dimensional phase field model for different crack face electrical boundary conditions. Phase field simulations show that crack face electrical boundary conditions have significant influence on the electrical and mechanical responses of the ferroelectric single crystal to an external electric field. The coercive field for the polarization switching of a single crystal with an electrically permeable crack is about 50% larger than that for a single crystal with an electrically impermeable crack. The remanent strain and the strain variation induced by polarization switching in a single crystal with a permeable crack are larger than those with an impermeable crack. The different macroscopic nonlinear behaviors are attributed to different polarization switching processes. It is found that domain switching takes place from the surface of a single crystal with a permeable crack, while it begins from the vicinity of the crack tip when the crack is impermeable. A ferroelectric single crystal with an impermeable crack exhibits strip 90° domain switching under a negative electric field, which is consistent with experimental observation.

Investigating thermal effects on morphological evolution during crystallisation of hcp metals: three-dimensional phase field study

Computational models simulating the crystallisation of hexagonal close packed (hcp) metals have traditionally used simple solid/fluid interface anisotropy relations. However, this method does not provide a proper representation of the crystallisation anisotropy, particularly for non-isothermal crystallisation. In this work, we used a phase field (PF) model to investigate the effects of thermal properties (thermal diffusivity and latent heat) on the morphological evolution of hcp metals during crystallisation. The governing equation of the PF model was coupled to the temperature evolution equation, and a three-dimensional model was developed in a finite element framework using COMSOL software. We applied the model to study the crystal formation and evolution of magnesium and yttrium. The unique aspect of this work is that we used Qin and Bhadeshia’s three-dimensional interface anisotropic model to accurately simulate solid/fluid interface anisotropy. The results showed that the thermal effects significantly influenced the shape evolution of the crystals and can control the formation of potential sites for void nucleation inside the crystal structures.

Towards a Three-Dimensional Phase-Field Model of Dendritic Solidification with Physically Realistic Interface Width

We review the application of advanced numerical techniques such as adaptive mesh refinement, implicit time stepping, multigrid solvers and massively parallel implementations as a route to obtaining solutions to the three-dimensional phase-field problem with a domain size and interface resolution previously possible only in two dimensions. Using such techniques it is shown that such models are tractable even as the interface width approaches the solute capillary length.

The role of phase compatibility in martensite

Shape memory alloys inherit their macroscopic properties from their mesoscale microstructure originated from the martensitic phase transformation. In a cubic to orthorhombic transition, a single variant of martensite can have a compatible (exact) interface with the austenite for some special lattice parameters in contrast to conventional austenite/twinned martensite interface with a transition layer. Experimentally, the phase compatibility results in a dramatic drop in thermal hysteresis and gives rise to very stable functional properties over cycling. Here, we investigate the microstructures observed in Ti50Ni50−xPdx alloys that undergo a cubic to orthorhombic martensitic transformation using a three-dimensional phase field approach. We will show that the simulation results are in very good agreement with transmission electron microscopy observations. However, the understanding of the drop in thermal hysteresis requires the coupling of phase transformation with plastic activity. We will discuss this point within the framework of thermoelasticity, which is a generic feature of the martensitic transformation.

A phase-field model of anisotropic polycrystalline system and computer simulation – Part I: Theory

A three-dimensional phase field model of anisotropic polycrystalline solids has been developed. In this model the system’s free energy is derived in a generalized form based on Ginzburg–Landau’s (GL) theory [1], which is a function of lattice’s parameters and electron density. A grain boundary in this model is viewed as a discontinuity of periodic atoms array, represented by the Read–Shockley dislocation representation [3]. The Peierls–Nabbaro’s dislocation potential [84] has been built into the model to characterize grain boundary’s adhesion. Therefore, the model establishes the links between lattice’s structure, sub-atomic quantum physics, and the evolutions of grains and grain-boundaries, leading to a meso-scopic constitutive law that governs grains’ nucleation, growth, and distortion in a polycrystalline system. This also provides a means to pursue analytical solutions of an alloy’s micro-structure’s evolution and establishes a framework for corresponding numerical analysis.

The Twinning Modes of B2-B19′ Martensitic Transformation in NiTi Alloys - A Phase Field Study

A three-dimensional phase field model applicable for the B2-B19′ martensitic phase transformation of NiTi alloys was developed to predict the twinning modes in the B2-B19′ transition. The phase field simulation results showed that by taking into account the transformation induced elastic strain, the martensite variants were self-accommodated and sheared along the specific interface in the entire transition stage for reducing the elastic energy, whether the formed interface between the variants belongs to twinning plane could be determined by calculating the minimum value of long range elastic interaction energy, Bpq. Through comparison with the existing analytical solutions, it is demonstrated that the phase field model can be used to predict the type I twinning modes in the B2-B19′ phase transition with good precision.

Influence of the stacking fault energy surface on partial dislocations in fcc metals with a three-dimensional phase field dislocations dynamics model

We present simulations of the dissociation of perfect dislocations into extended partial dislocations in aluminum, palladium, and nickel using a phase field dislocation dynamics (PFDD) theory that incorporates the $\ensuremath{\gamma}$ surface. As expected from dislocation theory, the simulations show that increasing the intrinsic stacking fault energy, normalized by the product of the shear modulus and Burgers vector, decreases the equilibrium stacking fault width. Significantly, it is also found that increasing the unstable stacking fault energy has the same effect when the intrinsic stacking fault energy is held constant. Furthermore, our results show that the equilibrium configurations cannot be described only by the ratio between the intrinsic and unstable stacking fault energies as previously suggested but rather by their product.

Formation conditions of coarse columnar austenite grain structure in peritectic carbon steels by the discontinuous grain growth mechanism

The discontinuous grain growth leading to a coarse columnar austenite grain (CCG) structure in as-cast peritectic carbon steels is analyzed by means of two- and three-dimensional phase field simulations. On the basis of the theory of grain growth, the conditions for discontinuous grain growth to occur are elucidated in terms of cooling conditions and material properties. The theoretical analysis explains the results of the phase field simulations well and should be useful in predicting the formation of CCG structure in continuously cast slabs.

Study of the twinned dendrite tip shape I: Phase-field modeling

The growth kinetics advantage of twinned aluminum dendrites over regular ones is still an unsolved problem of solidification. Although it is linked to the tip geometry, the influence of a coherent (1 1 1) twin plane on a 〈1 1 0〉 twinned dendrite tip is unclear, despite several past experimental observations. In the present contribution, a three-dimensional phase field model implemented on a cluster of parallel computers has been used to simulate the growth of a twinned dendrite under various directional solidification conditions. Only half a dendrite was modeled by replacing the coherent twin plane by a boundary with an appropriate condition on the phase parameter that is equivalent to the Young–Laplace equilibrium condition along the triple line between twinned solid, untwinned solid and liquid. It is found that the small liquid cusp present at the tip rapidly evolves into a doublon-type morphology, i.e. a 〈1 1 0〉 dendrite split in its center by a deep and thin liquid pool with the triple line at the root. At high growth rates, the two sides of the doublon tend to coalescence and form small isolated liquid droplets. The positive concentration gradient near the doublon root appears to be rapidly smeared out by back-diffusion in the solid, thus making difficult its quantification through experimental methods. These simulation results are correlated with new experimental evidence presented in a companion paper.

PHASE FIELD SIMULATION OF EFFECTS OF PORES ON B2-R PHASE TRANSFORMATION IN NiTi SHAPE MEMORY ALLOY

Recently, porous NiTi shape memory alloys (SMAs) have drawn great interest in vari- ous engineering fields, in particular for biomedical applications as one of the promising biomaterials for hard-tissue replacements and orthopedic implants. It is well known that the porous NiTi SMAs exhibit three transformations, B2-B19 � , B2-R and R-B19 � . Among these phase transformations B2-B19 � and R-B19 � involve high lattice distortion and large transformation hysteresis. Consequently, these distortion and transformations usually introduce structural defects which may result in degradation of mechanical stability for the functional application. On the contrary, B2-R transformation is governed by small lattice distortion which indicates less damage to the microstructure and lower sensitivity to the defects, and thus generates higher reversibility and mechanical stability. Due to these unique virtues, it is of great importance to study the R-phase transformation behavior in porous NiTi SMAs, since the pore has a significant influence on B19 � martensitic transformation as well as the R-phase transforma- tion. In this paper, a three-dimensional phase field model aiming at accounting for the pore effect on phase transformation in NiTi SMAs was developed to study the B2-R phase transformation behavior in porous NiTi SMAs. The model was applied to characterize the microstructure evolution of B2-R phase

Large-Scale 3D Phase Field Dislocation Dynamics Simulations On High-Performance Architectures

In this paper we present the development and performance of a three-dimensional phase field dislocation dynamics (3D PFDD) model for large-scale dislocation-mediated plastic deformation on high-performance architectures. Through the parallelization of this algorithm, efficient run times can be achieved for large-scale simulations. The algorithm’s performance is analyzed over several computing platforms including Infiniband, GigE, and proprietary (SiCortex) interconnects. Scalability is considered on data sets up to 2,0483, along with the efficiency on up to 2,048 processors. Results show that scalability improves as the size of the data set increases and that the overall performance is best on the Infiniband interconnect. In addition, a performance model has been developed to predict run times and efficiency on large sets of data running on multiple processors. This performance analysis shows that this parallel code is capable of harnessing the greater computer power available from petascale systems.

Phase field simulation of the 180° domain-switching process in PbTiO3 single crystal under an antiparallel electric field

The process of 180° domain switching in PbTiO3 single crystal under an antiparallel electric field was investigated by the three-dimensional phase field simulation, especially the effect of electric field on the type and duration of domain switching. It is found that the polarization reversal of domains takes place under an antiparallel electric field in PbTiO3 single crystal. The results of the phase field simulation indicate that there is only 90° domain switching under a weak electric field. With the rise of the electric field, 180° domain switching appears. If the electric field is strengthened further, 90° domain switching disappears and the duration of domain switching is shortened.

THREE-DIMENSIONAL PHASE FIELD STUDY ON STRAIN SELF-ACCOMMODATION IN MARTENSTIC TRANSFORMATION

The number of martensitic variants depends on the grain size and orientation of parent phase. Although the growth of polytwin structure with the combination of different martensitic variants was studied by in situ dynamic observation, the energy change during the growth of martensite cannot be revealed, and in turn the strain self-accommodation of martensitic transformation (MT) cannot be confirmed. Based on the recent study of the phase field model, describing proper martensitic transformation for a single-crystal system under a fully constrained boundary, a three-dimensional (3D) phase field simulation is performed to demonstrate the microstructure evolution of plolytwin structure in this paper, meanwhile, the energy change during the growth of martensite is calculated. The results indicate that during the growth of two martensitic variants the total strain energy increases with prolonging microstructural evolution time, however, the opposite change occurs during the growth of three martensite variants. The 3D phase field simulation reveals that the growth of three martensitic variants possesses the best strain self-accommodation effect.

Extended phase diagram of the three-dimensional phase field crystal model

We determine the phase diagram of the phase field crystal model in three dimensions byusing numerical free energy minimization methods. Previously published results,based on single mode approximations, have indicated that in addition to theuniform (liquid) phase, there would be regions of stability of body-centered cubic,hexagonal and stripe phases. We find that in addition to these, there are also regionsof stability of face-centered cubic and hexagonal close packed structures in thismodel.

Three-dimensional adaptive phase field modeling of directional solidification of a binary alloy: 2D–3D transitions

Morphological instability is an important subject in alloy directional solidification, and the development of cell allay is important to alloy properties. In nature, as a result of cell arrangement, this problem is often three-dimensional (3D), even for thin-film, and its simulation is rather difficult. We have developed a 3D adaptive phase field model to simulate efficiently the directional solidification of a binary alloy. The morphological transition from 2D to 3D approximations are illustrated, and compared with available analytical and simulated results. The effect of anisotropy on the cell arrangement is discussed as well.

Large-scale three-dimensional phase field simulation of γ ′-rafting and creep deformation

Three-dimensional phase field simulations of coupled γ/γ ′ microstructural evolution and plastic deformation in single crystal Ni-Al are carried out at micrometer scales. Coherent γ/γ ′ microstructures and plastic deformation in γ-channels are described using a single, consistent methodology based on Khachaturyan's phase field microelasticity approach to coherent precipitates and dislocations. In particular, a new set of phase fields is introduced to characterize local density of dislocations from individual active slip systems. To increase the length scale of the phase field simulations, the Kim–Kim–Suzuki (KKS) treatment of γ/γ ′ interfaces was adopted. The rafting kinetics, precipitate-matrix inversion process and the corresponding creep deformation are characterized with respect to parameters such as applied stress and lattice misfit. The simulation results on γ ′-rafting kinetics and morphological evolution of the γ/γ ′ microstructures are compared with available experiment. The model can be used to carry out parametric studies of the effects of material and processing parameters such as alloy composition, external stress and working temperature on γ ′-rafting kinetics, morphological evolution and the corresponding creep deformation.

Three-dimensional Phase Field Modeling of the Faceted Cellular Growth

The new phase field model for strongly anisotropic systems proposed by Torabi et al. was employed to simulate the faceted cellular growth in three dimensions. Simulation reveals the whole formation process of the faceted cellular clearly. Simulation results also show that a linear relation of undercooling and growth velocity when the shape selection is completed at the late stage of the evolution, but a nonlinear relation holds during the shape selection stage. During the facet cellular formation, the crystal–melt interface is kept isothermal although the interface is sawtooth, and areas with negative temperature gradient appear in the melts, particularly at the ravine bottom of the sawtooth interface.

Efficient adaptive three-dimensional phase field simulation of free dendritic growth under natural convection

A great challenge in simulating dendritic crystal growth from various supercoolings is the large difference in time and length scales. Dendrite growth is also known to be affected by natural convection. By using adaptive mesh refinement and rescaling, the effects of gravity and its orientation on tip growth rate and dendrite morphology are investigated. Our simulation results are consistent with previous theories and experimental observations.