Quantitative Phase-field Model Research Articles

A quantitative phase-field model combining with front-tracking method for polycrystalline solidification of alloys

As the orientation of an individual crystal is constant in solid and diffusive interface layer, a sharp interface model can be coupled to calculate the orientation in phase-field simulation of polycrystalline growth. Here, a two-dimensional phase-field model in combination with the front-tracking method is provided for simulating polycrystalline growth during solidification. In this model, the quantitative phase-field formulations for slow solidification of dilute binary alloys are employed to describe the dynamical evolution of diffusive solid–liquid interface while the front-tracking method is utilized to track the spatial dependent orientation of each grain. Because of the high computing efficiency that is resulted from only one order parameter used to depict the phase transformation during solidification, the model overcomes the disadvantage of reported phase-field approaches for polycrystalline growth. The built model was firstly solved to simulate a free equiaxed dendrite with different orientations growing from undercooled melt for benchmarking. The comparison results indicate that the model is able to compute the orientation exactly. Secondly, the model was extended to simulate the solidification with many equiaxed dendrites. The growth behaviors of the simulated crystals were characterized and analyzed, which demonstrate that the model is feasible to quantitatively and efficiently predict growth dynamics of crystals in a large number and scale during solidification of alloys.

Phase-field elasticity model based on mechanical jump conditions

Computational models based on the phase-field method typically operate on a mesoscopic length scale and resolve structural changes of the material and furthermore provide valuable information about microstructure and mechanical property relations. An accurate calculation of the stresses and mechanical energy at the transition region is therefore indispensable. We derive a quantitative phase-field elasticity model based on force balance and Hadamard jump conditions at the interface. Comparing the simulated stress profiles calculated with Voigt/Taylor (Annalen der Physik 274(12):573, 1889), Reuss/Sachs (Z Angew Math Mech 9:49, 1929) and the proposed model with the theoretically predicted stress fields in a plate with a round inclusion under hydrostatic tension, we show the quantitative characteristics of the model. In order to validate the elastic contribution to the driving force for phase transition, we demonstrate the absence of excess energy, calculated by Durga et al. (Model Simul Mater Sci Eng 21(5):055018, 2013), in a one-dimensional equilibrium condition of serial and parallel material chains. To validate the driving force for systems with curved transition regions, we relate simulations to the Gibbs-Thompson equilibrium condition (Johnson and Alexander, J Appl Phys 59(8):2735, 1986).

Effect of initial particle size distribution on the dynamics of transient Ostwald ripening: A phase field study

The coarsening of polydisperse particles with different initial particle size distributions (PSD) is studied using a quantitative phase field model in two dimensions with emphasis on the transient behavior before reaching steady state. The coarsening rate constant, scaled PSD, scaled evolution law and radial distribution function were systematically examined and compared with available theoretical and experimental results. It is found that the length of transient regime is directly correlated with the extension of the initial scaled PSD toward the large particle size region, i.e. the so called tail, rather than the width of scaled PSD which can be described in terms of standard deviation. Initial distributions with short tails evolve rapidly to the steady state even though the initial width of PSD is large. Whereas, after long time coarsening with a factor of 4–6 change in average radius, initial distributions with long tails still deviate from the steady-state form, yet may appear stable due to the slowly changing rate. Some inconsistencies about the transient coarsening behaviors in previous studies are clarified based on the present results. Moreover, the validity of circular shape assumption is certified when the particle volume fraction is not larger than 0.4.

Phase-field modeling of epitaxial growth with the Ehrlich-Schwoebel barrier: Model validation and application

In this paper, we introduce different forms of mobility into a quantitative phase-field model to produce arbitrary Ehrlich-Schwoebel (ES) effects. Convergence studies were carried out in the one-side step-flow model, which showed that the original mobility not only induces the ES effect, but also leads to larger numerical instability with increase of the step width. Thus, another modified form of the ES barrier is proposed, and is found to be more suitable for large-scale simulations. Model applications were performed on the wedding-cake structure, coarsening and coalescence of islands and spiral growth. The results show that the ES barrier exhibits more significant kinetic effects at the larger deposition rates by limiting motions of atoms on upper steps, leading to aggregation on the top layers, as well as the roughening of growing surfaces.

Phase-field modeling of free dendritic growth of magnesium based alloy

In this paper, the process of the free dendritic growth of Mg-0.5 wt.%Al alloy in the basal plane (0001) is simulated in two-dimensional system by using a quantitative phase-field model. A convergence study is carried out to choose the optimal coupling parameter λ and grid width Δx/W0 in simulation. Then we systematically discuss the effects of the anisotropic strength ε and the supersaturation Ω on dendritical tip growth velocity, radius, Péclet number, and stability parameter σ *. Results show that the stability parameter σ * defined by the theory of microscopic solvability is a function of the anisotropy strength ε, i.e., σ* ≅ ε1.81905, which is obviously closest to σ * (ε) ≅ ε 1.75 obtained from the analytical solution. Moreover, for Ω σ * is approximately a constant while it sharply and monotonically decreases with the augment of the value of ε for Ω > 0.6. This indicates that there is a transition from solute-controlled dendrite to kinetic dendrite as Ω increases. Furthermore, the transition of the growth pattern from the snow-like to the circle-like patterns occurs as Ω increases.

A quantitative phase-field model for two-phase elastically inhomogeneous systems

Solid-state phase transformations are influenced by strains that are generated internally or applied externally. The stress state, composition, and microstructure evolution, which together determine the properties of solid materials can be studied using phase-field models coupled with micro-elasticity theory in the small strain limit. This coupling has been implemented using various schemes in literature. In a previous article (Durga et al., 2013), the authors evaluated three main existing schemes for a two-phase system and concluded that these schemes are not quantitative for inhomogeneous anisotropic elastic properties of the two phases. The stress states predicted by these models deviate from the expected values due to the generation of extra interfacial energy, which is an artefact of the models resulting from interfacial conditions different from local mechanical equilibrium conditions. In this work, we propose a new scheme with interfacial conditions consistent with those of the analytical results applicable to a general system where shear strains may be present. Using analytical solutions for composition and stress evolution, we validate this model for 2D and 3D systems with planar interface in the presence of misfit between phases and applied strains, and a 2D system with an elliptical second-phase particle. This extended scheme can now be applied to simulate quantitatively the microstructural evolution with coupled chemical and mechanical behaviour in any 2D or 3D two-phase system subject to internal or external strains irrespective of interface curvature.

Thin interface analysis of a phase-field model for epitaxial growth with nucleation and Ehrlich–Schwoebel effects

In this paper, we perform thin interface analysis of a quantitative phase field model for epitaxial growth where nucleation and the Ehrlich–Schwoebel barrier have been considered. Results show that once the nucleation term is introduced into the phase-field model, modification must be carried out to get rid of the extrinsic “kinetic nucleation effect”. While in the ES effect, the asymmetric diffusivity accounts for an irrational step motion that leads the model to deviate from the sharp-interface approximation, hence another modification for the attachment time should be carried. Attributed to these modifications, the phase-field model is more quantitative in describing step flow dynamics in the sharp-interface limit, as well as exhibiting the more convergence of the steady-state velocity with respect to the step width for larger scale simulations. Our analysis and modifications explore the quantitative linking between atom motions and step dynamics.

Peculiarities of Phase Transitions and Structure Formation in a Ternary Al-Cu-Ni Alloy with Four-Phase Peritectic Reaction

The structure formation in peritectic Al-4.5at.%Cu-11at.%Ni ternary alloy with four-phase peritectic reaction was investigated using the quantitative phase-field model of eutectic growth. This model, extended to an arbitrary number of phases, guarantees the stability requirements on individual interfaces. The thermal noise terms disturb the stability and produce the heterogeneous nucleation of secondary phases in accordance to the energetic and concentration conditions. In our recent work it was shown that in differential thermal analysis (DTA) experiments specific microstructure parts in Al-4.5at.%Cu-11at.%Ni alloy with a four-phase peritectic reaction were observed, which cannot be explained by Scheil calculation or simple phase-field modeling. In this work, it was found by numerical experiments that, due to the formation of anisotropic quasi-primary Al3Ni2 crystals and the suppression of the nucleation of (Al) phase, the eutectic-like coupled growth of Al3Ni2 and Al3Ni phases can be observed. In addition, at further cooling the anisotropic shape of quasi-secondary Al3Ni crystals promotes the nucleation of the (Al) phase. The simulated final structure is comparable to the experimental one which is characterized by large Al3Ni2 crystals enveloped by the Al3Ni phase.

Onset of the initial instability during the solidification of welding pool of aluminum alloy under transient conditions

Onset of initial morphological instability is predicted by using a new analytic model and quantitative phase field model during the solidification of the welding pool of Al–Cu alloy under transient conditions. In the linear growth stage of the welding pool, the dynamic evolution of the interface instability is analyzed, and the interface behaviors under infinitesimal fluctuations are also investigated. The results show that the mean wavelength at the crossover time evaluated from this analytic model is in good agreement with those obtained by the quantitative phase field simulations and the experiments. The linear growth stage takes up quite a long time of the whole solidification of welding pool and thus it should be primarily considered in investigating the transient growth of welding pool. This study establishes a valid numerical framework for studying the dendrite growth under transient solidification conditions and provides a new approach for studying the transient solidification of welding pool.

Dynamic evolution of initial instability during non-steady-state growth.

Dynamic evolution of initial instability is investigated by an analytic model obtained by modifying the theory of Warren and Langer [Phys. Rev. E 47, 2702 (1993)] and the quantitative phase-field model in directional solidification under transient conditions for realistic parameters of a dilute alloy. The evolutions of tip velocity and concentration in the liquid side of the interface predicted by the analytic model agree very well with that from the phase-field simulation in the linear growth stage of the non-steady-state growth, indicating that the model could be used as a convenient method to study the initial instability during non-steady-state growth. The influences of non-steady-state conditions which include the increasing rate of pulling speed and temperature gradient at the onset of initial instability are investigated, and we find that, the initial instability seems to depend strongly on the non-steady-state conditions and the non-steady-state history, and thus, it should be primarily considered in the study of the transient growth.

Quantitative phase field model for dislocation sink strength calculations

An original phase-field model dedicated to the calculation of dislocation sink strength is presented. Through a three-step validation procedure, it is shown to predict with an excellent accuracy the value of the sink strength with or without elasticity in numerous test cases for which an analytical solution is known. Further analysis shows that existing analytical values of dislocation sink strength are significantly underestimated, an effect that results from simplifying assumptions made to take into account irradiation in the diffusion equation.

Dendrite to symmetry-broken dendrite transition in directional solidification of non-axially oriented crystals

In this paper, the morphological transition from dendrite to symmetry-broken dendrite is investigated in the directional solidification of non-axially-oriented crystals using a quantitative phase-field model. The effects of pulling velocity and crystal orientation on the morphological transition are investigated. The results indicate the orientation dependence of the symmetry-broken double dendrites. A dendrite to symmetry-broken dendrite transition is found by varying the pulling velocity at different crystal orientations and the symmetry-broken multiple dendrites emerge as a transition state for the symmetry-broken double dendrites. The state region during the transition can be well characterized through the variations of the characteristic angle and the average primary dendritic spacing.

Phase‐Field Modeling of Diffusion Coupled Crack Propagation Processes

Computational models based on the phase-field method operate on a mesoscopic length scale providing microstructure and mechanical property relations. We derive a quantitative phase-field model capable of describing the coupled diffusion of the surrounding water and the stress evolution during dynamic fracture formation under applied load in glass materials.

Composition pathway in Fe–Cu–Ni alloy during coarsening

In this work the microstructure evolution for a two phase Fe–Cu–Ni ternary alloy is studied in order to understand the kinetic composition paths during coarsening of precipitates. We have employed a quantitative phase-field model utilizing the CALPHAD database to simulate the temporal evolution of a multi-particle system in a two-dimensional domain. The paths for the far-field matrix and for precipitate average compositions obtained from simulation are found to be rectilinear. The trends are compared with the corresponding sharp interface theory, in the context of an additional degree of freedom for determining the interface compositions due to the Gibbs–Thomson effect in a ternary alloy.

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.

Spatiotemporal Dynamics of Oscillatory Cellular Patterns in Three-Dimensional Directional Solidification

We report results of directional solidification experiments conducted on board the International Space Station and quantitative phase-field modeling of those experiments. The experiments image for the first time in situ the spatially extended dynamics of three-dimensional cellular array patterns formed under microgravity conditions where fluid flow is suppressed. Experiments and phase-field simulations reveal the existence of oscillatory breathing modes with time periods of several 10's of minutes. Oscillating cells are usually noncoherent due to array disorder, with the exception of small areas where the array structure is regular and stable.

Phase-field simulations and geometrical characterization of cellular solidification fronts

The structure and dynamics of cellular solidification fronts produced during the directional solidification of dilute binary alloys are studied by phase-field simulations. A quantitative phase-field model in conjunction with a multi-scale simulation algorithm allows us to simulate arrays with 10–40 cells in three dimensions on time scales that are long enough to allow for a significant reorganization of the array. We analyze the geometry of the complex two-phase structure (mushy zone) and extract the fraction of solid and the connectivity of the two phases as a function of depth. We find a transition from stable arrays at high values of the crystalline anisotropy to unsteady arrays at low anisotropy that continuously exhibit tip splitting and cell elimination events.

Phase-field modeling of vacancy cluster evolution in Fe

A quantitative phase-field model is developed to study the evolution of vacancy cluster in Fe. Total energy of the system is constructed based on the assumption of ideal gas state equation, and an approach to linking the computational parameters in the phase-field model to the experimental properties of Fe is provided. Such a phase filed model is employed to quantitatively investigate the nucleations, growths, and coalescences of voids in single and polycrystalline Fe. The effects of grain boundary on voids evolution are also investigated. These results provide a way of further studying the evolution behaviors of both H/He gas atoms and voids in Fe.

Influence of solid-solid interface anisotropy on three-phase eutectic growth during directional solidification

We utilize a quantitative phase-field model to simulate three-phase eutectic growth and the oscillatory instabilities at large spacings. We analyze the effect of symmetry in the selection of the oscillatory mode for the case of directional solidification. Going further from our previous article (Choudhury A.et al., Phys. Rev. E, 83 (2011) 051608), we manipulate the symmetry elements in a given configuration through the use of solid-solid anisotropy. Characteristic modes are compared between the case of isotropic surface energies and those retrieved in the presence of solid-solid anisotropy. We end with certain general arguments with regards to the growth rate of oscillatory modes and symmetry elements of the obtained microstructure depending on the symmetry elements of the starting configuration.

Guidance to design grain boundary mobility experiments with molecular dynamics and phase-field modeling

Quantitative phase-field modeling can play an important role in designing experiments to measure the grain boundary (GB) mobility. In this work, molecular dynamics (MD) simulation is employed to determine the GB mobility using Cu bicrystals. Two grain configurations are considered: a shrinking circular grain and a half-loop grain. The results obtained from the half-loop configuration approach asymptotically to that obtained from the circular configuration with increasing half-loop width. We then verify the phase-field model by direct comparison to the MD simulation results, obtaining excellent agreement. Next, this phase-field model is used to predict the behavior in a common experimental setup that utilizes a half-loop grain configuration in a bicrystal to measure the GB mobility. With a 3-D simulation, we identify the two critical times within the experiments to reach an accurate value of the GB mobility. We use a series of 2-D simulations to investigate the impact of the notch angle on these two critical times. We also show that if the notch does not have a sharp tip, it may immobilize the GB migration indefinitely. Finally, we demonstrate that our approach for the quarter-loop configuration eliminates some disadvantages of the half-loop.