Phase-field models for anisotropic interfaces

McFadden, Geoffrey B.; Wheeler, A. A.; Braun, R. J.; Coriell, S. R.; Sekerka, Robert F.

doi:10.1103/physreve.48.2016

Cited by 328 publications

(218 citation statements)

References 21 publications

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“…Since the phase-field model is phenomenological, it needs to be validated by mapping onto a free-boundary problem of interest, i.e., a sharp-interface model. In early works, the mapping was achieved in the sharp-interface limit, where the interface thickness W was made extremely small [5,6]. However, a huge computational cost is required when a small value is assigned to W. Hence, it is common practice to employ a value of W several orders of magnitude larger than the realistic interface thickness.…”

Section: Introductionmentioning

confidence: 99%

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

2016

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We present the variational formulation of a quantitative phase-field model for isothermal low-speed solidification in a binary dilute alloy with diffusion in the solid. In the present formulation, cross-coupling terms between the phase field and composition field, including the so-called antitrapping current, naturally arise in the time evolution equations. One of the essential ingredients in the present formulation is the utilization of tensor diffusivity instead of scalar diffusivity. In an asymptotic analysis, it is shown that the correct mapping between the present variational model and a free-boundary problem for alloy solidification with an arbitrary value of solid diffusivity is successfully achieved in the thin-interface limit due to the cross-coupling terms and tensor diffusivity. Furthermore, we investigate the numerical performance of the variational model and also its nonvariational versions by carrying out two-dimensional simulations of free dendritic growth. The nonvariational model with tensor diffusivity shows excellent convergence of results with respect to the interface thickness.

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Section: Introductionmentioning

confidence: 99%

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

2016

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“…The model was originally proposed for simulating the dendrite growth from an undercooled pure melt and has been extended to solidification of alloys. [8][9][10][11][12][13][14] By adopting the thin interface limit to derive the phase-field mobility, the applicability of the phase-field model is considerably widened. 15,16) Since the thermal diffusivity is much larger than the solutal one in metal system, for example three hundred times difference in Fe-C alloy system, the effect of heat transfer has been regards as negligibly small comparing to that of solute in the existing phase-field calculations for alloys.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Initial Microstructure Evolution of Fe-C Alloys Using a Phase-field Model.

Ode

Suzuki

2002

ISIJ International

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Microstructure evolution during the rapid solidification of Fe-C and Fe-C-P alloys is simulated using the phase-field model for alloys with thin interface limit parameters. Heat transfer equation is solved simultaneously to study the heat flow and the effect of latent heat generation on the microstructure. The calculations have been carried out using a double grid method and parallel computing technique. The competitive growth of growing cells is reproduced, and the cellular/dendritic transition is also observed. Since there is a negative thermal gradient in front of a leading tip, the growth can be regarded as unidirectional free dendrite growth. The microstructure changes depending on the preferred growth orientation and impurity are also studied. The secondary arms grow preferably towards inside of the melt and develop well with increase of the tilted angle. The secondary and primary arm spacing decrease by the small amount of phosphorus addition. The time change of averaged surface temperature depending on the initial undercooling shows that the surface undercooling is always observed even when the initial value is zero.KEY WORDS: Fe-C alloy; rapid solidification; micro-structure; dendrite; phase-field model. Governing Equations Governing Equations using Dilute Solution ApproximationThe phase-field equation with k-hold interface energy anisotropy and the solute diffusion equation for a dilute alloy are given by, 14,16) . ...(6)where h(f)ϭf 3 (15Ϫ10fϪ6f 2 ), g(f)ϭf 2 (1Ϫf) 2 , q is the angle between normal direction of interface and the x-axis, n is the magnitude of anisotropy, the superscript of e represent equilibrium state and subscripts of xx, yy and xy represent partial derivative by x or y twice, and S and L represents solid and liquid phases, respectively, R is gas constant, V m is molar volume, D(f) is solute diffusion coefficient and W, e and M are phase-field parameters which are defined later. The prime of e represents the differential by q. When the preferred growth orientation is tilted from the x-axis with q 0 , Eq. (2) is modified as follows. Note that the terms of sin(2q) and cos(2q) in Eq. (1) are substituted for the function of the partial derivative of the phase-field and they are independent of tilted angle of preferred growth direction. For a dilute ternary alloy, the governing equations and parameters are modified as in the literature. 17) In this study, temperature field is calculated simultaneously using a double grid method, in which the grid size for temperature field is set to be larger than that for phase-field and concentration field, in order to save the calculation time. The latent heat generation is estimated by summing up the change of phase-field within the corresponding temperature grid and the thermal diffusion equation is obtained as follows.where a is thermal diffusivity, L is latent heat, c p is specific heat and A represents the area ratio of phase-field grid to thermal one.The phase-field parameters of e and W are related to interface energy, s, and interface width, 2...

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“…The anisotropic Allen-Cahn equation [19,20] is then derived from the linear response of the interface to the local drive given by the Gibbs-Thompson condition, [σ(ψ) + σ ′′ (ψ)]κ, were σ is the surface tension, and κ the local interface curvature. The stiffness, σ + σ ′′ , reflects the local change of extent and orientation of the interface due to a deformation.…”

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Anisotropic Coarsening: Grain Shapes and Nonuniversal Persistence

Rutenberg

Vollmayr-Lee

1999

Phys. Rev. Lett.

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We solve a coarsening system with small but arbitrary anisotropic surface tension and interface mobility. The resulting size-dependent growth shapes are significantly different from equilibrium microcrystallites, and have a distribution of grain sizes different from isotropic theories. As an application of our results, we show that the persistence decay exponent depends on anisotropy and hence is nonuniversal.PACS numbers: 05.70. Ln, 64.60.Cn, 81.30.Hd The geometrical Wulff construction [1] gives an explicit relation between the anisotropic surface tension and the resulting equilibrium crystal shape. This marks an early and dramatic success in quantitatively connecting morphology to the interfacial properties of a material. However, distinct Wulff microcrystallites must be in 'splendid isolation' -with negligible exchange between them in comparison to the internal dynamics required to equilibrate [2]. In contrast, dilute phase separating alloys and coarsening polycrystallites exhibit growing microcrystalline droplets or grains with non-negligible interactions. While it has been shown for these and other coarsening systems that anisotropy influences the morphology [3,4], such effects have not been quantitatively understood for even the simplest models of curvaturedriven growth.The understanding of interacting isotropic phases (see, e.g., [5]) was significantly advanced by the models of Lifshitz and Slyozov [6] and Wagner [7] for diffusive and curvature-driven coarsening, respectively. These meanfield theories correctly capture a remarkable amount of coarsening phenomenology, and are exact in the dilute limit. With this inspiration, we generalize Wagner's model -an interacting ensemble of coarsening droplets, evolving to continually lower their surface energy without changing their total volume -to include arbitrary anisotropy in the surface tension and the interface mobility. We solve the model perturbatively in anisotropy strength, and relate the interfacial properties to the resulting non-trivial grain shapes. These "growth shapes" are contrasted with those of equilibrium (Wulffconstructed) grains to highlight the connection between dynamics and microcrystallite morphology. We then compare our results on the ensemble of grains to Wagner's isotropic solution to demonstrate anisotropy effects on coarsening correlations, including the effect on persistence exponents.Our model is applicable to single-phase polycrystallite coarsening, where distinct grains are distinguished only by their crystallographic orientation (see [8][9][10]). Most theoretical studies of polycrystallites focus on their cellular structure, specifically on the static and dynamical description of the vertices where three or more grain boundaries meet. However, vertex-based models have significant shortcomings when anisotropy is included, since it modifies both the distribution of the number of vertices per grain [4] and the otherwise fixed angles formed where three grains adjoin [8]. Furthermore, von Neumann's law, a direct relationship in tw...

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Phase-field models for anisotropic interfaces

Cited by 328 publications

References 21 publications

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

Numerical Simulation of Initial Microstructure Evolution of Fe-C Alloys Using a Phase-field Model.

Anisotropic Coarsening: Grain Shapes and Nonuniversal Persistence

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