Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

Warren, James A.; Boettinger, William J.

doi:10.1016/0956-7151(94)00285-p

Cited by 741 publications

(505 citation statements)

References 21 publications

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“…In a simulation study based on the alloy phase-field model of Wheeler et al [188], Warren and Boettinger [189] demonstrated the power of the phase-field method to reproduce qualitatively many observed microstructural features of dendritic equiaxed grains. These authors also emphasized the difficulty of obtaining simulation results that are independent of interface thickness.…”

Section: Binary Alloysmentioning

confidence: 99%

Atomistic and continuum modeling of dendritic solidification

Hoyt

2003

Materials Science and Engineering: R: Reports

398

220

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Section: Binary Alloysmentioning

confidence: 99%

Atomistic and continuum modeling of dendritic solidification

Hoyt

2003

Materials Science and Engineering: R: Reports

398

220

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“…The phase-field equation with k-hold interface energy anisotropy and the solute diffusion equation for a dilute alloy are given by, 14,16) . …”

Section: Governing Equations Using Dilute Solution Approximationmentioning

confidence: 99%

“…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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“…A phase field model is a powerful tool that can describe the complex interface pattern evolution. [9][10][11][12][13][14][15][16] It treats the interface as a finite area where the phase gradually changes from solid to liquid. Hence all the governing equations can be written in unified forms without distinguishing the interface and it has a great advantage in numerical calculations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of the Critical Velocity for Particle Pushing/Engulfment Transition in Fe-C alloys Using a Phase-field Model.

et al. 2000

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The particle/interface problem is numerically analyzed using a phase-field model in thin interface limit. A new double-well potential function in free energy density of the alloy is defined using a dilute solution approximation. With the function, a negative value of double-well potential height is usable and the mesh size restriction is largely relaxed. A pushing force for alloy systems is also introduced so as to relate to the interface energy change caused by the deformation of interface shape. With the pushing and drag forces calculated from the interface shape, the acceleration and velocity of the particle are estimated and the particle movement relative to the interface is analyzed. Using the model, the particle pushing and engulfment behaviors are successfully reproduced for the system of Fe-C alloys and an alumina particle. The critical velocities for the pushing/engulfment transition are determined for the particles with different diameters. The effect of initial carbon content on critical velocity is also examined and discussed in terms of the pushing force.KEY WORDS: phase-field model; thin interface limit; particle-interface problem; critical velocity for pushing/engulfment transition; Fe-C alloy.

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Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

Cited by 741 publications

References 21 publications

Atomistic and continuum modeling of dendritic solidification

Atomistic and continuum modeling of dendritic solidification

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

Numerical Simulation of the Critical Velocity for Particle Pushing/Engulfment Transition in Fe-C alloys Using a Phase-field Model.

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