A phase field model for snow crystal growth in three dimensions

At equilibrium, the shape of a strongly anisotropic crystal exhibits corners when for some orientations the surface stiffness is negative. In the sharp-interface problem, the surface free energy is traditionally augmented with a curvature-dependent term in order to round the corners and regularize the dynamic equations that describe the motion of such interfaces. In this paper, we adopt a diffuse interface description and present a phase-field model for strongly anisotropic crystals that is regularized using an approximation of the Willmore energy. The Allen–Cahn equation is employed to model kinetically controlled crystal growth. Using the method of matched asymptotic expansions, it is shown that the model converges to the sharp-interface theory proposed by Herring. Then, the stress tensor is used to derive the force acting on the diffuse interface and to examine the properties of a corner at equilibrium. Finally, the coarsening dynamics of the faceting instability during growth is investigated. Phase-field simulations reveal the existence of a parabolic regime, with the mean facet length evolving in t , with t the time, as predicted by the sharp-interface theory. A specific coarsening mechanism is observed: a hill disappears as the two neighbouring valleys merge.

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Study on Microstructural Evolution of Binary Alloy Crystal Growth in Directional Solidification Process

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.470.100 ◽

2013 ◽

Vol 470 ◽

pp. 100-103

Author(s):

Dong Sheng Chen ◽

Ming Chen ◽

Rui Chang Wang

Keyword(s):

Crystal Growth ◽

Directional Solidification ◽

Binary Alloy ◽

Phase Field ◽

Field Model ◽

Phase Field Model ◽

Field Method ◽

Phase Field Method ◽

Pure Substance ◽

Columnar Dendrite

PFM (phase field method) was employed to study microstructure evolution, and considering the effect of solute concentration to the undercooling, we developed a phase field model for binary alloy on the basis of pure substance model. In the paper, the temperature field and solute field were coupled together in the phase field model to calculate the crystal growth of magnesium alloy in directional solidification. The simulation results showed a non-planar crystal growth of planar to cellular to columnar dendrite, the comparison of different dendrite patterns were carried out in the numerical simulation, and with the increasing of the anisotropy, the second dendrite arms became more developed.

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Computer Simulation of Grain Growth in Three Dimensions by the Phase Field Model with Anisotropic Grain-Boundary Mobilities

Materials Science Forum ◽

10.4028/www.scientific.net/msf.539-543.2437 ◽

2007 ◽

Vol 539-543 ◽

pp. 2437-2442

Author(s):

Yoshihiro Suwa ◽

Yoshiyuki Saito ◽

Hidehiro Onodera

Keyword(s):

Grain Boundary ◽

Grain Growth ◽

Phase Field ◽

Large Scale ◽

Field Model ◽

Phase Field Model ◽

High Mobility ◽

Minor Component ◽

Three Dimensions ◽

A Minor

The kinetics and topology of grain growth in three dimensions were simulated using a phase-field model with anisotropic grain-boundary mobilities. In order to perform large scale calculations we applied both modifications of algorithms and parallel coding techniques to the Fan and Chen's phase-field algorithm. Kinetics of abnormal grain growth is presented. It is observed that the grains of a minor component which are at the beginning surrounded preferentially by boundaries of high mobility grow faster than the grains of a major component until the texture reverses completely. Additionally, topological results of grain structures, such as grain size distributions and grain face distributions, are discussed

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Phase Field Model Simulation of Grain Growth in Three Dimensions under Isotropic and Anisotropic Grain Boundary Energy Conditions

Materials Science Forum - Recrystallization and Grain Growth III ◽

10.4028/0-87849-443-x.1101 ◽

2007 ◽

pp. 1101-1106

Author(s):

Kyung Jun Ko ◽

Pil Ryung Cha ◽

Jong Tae Park ◽

Jae Kwan Kim ◽

Nong Moon Hwang

Keyword(s):

Grain Boundary ◽

Grain Growth ◽

Phase Field ◽

Field Model ◽

Model Simulation ◽

Boundary Energy ◽

Phase Field Model ◽

Three Dimensions ◽

Energy Conditions ◽

Grain Boundary Energy

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Phase field model for electrically induced damage using microforce theory

Mathematics and Mechanics of Solids ◽

10.1177/10812865211052078 ◽

2021 ◽

pp. 108128652110520

Author(s):

Elizaveta Zipunova ◽

Evgeny Savenkov

Keyword(s):

Phase Field ◽

Field Model ◽

Order Term ◽

Phase Field Model ◽

Special Problem ◽

Three Dimensions ◽

Electric Permittivity ◽

One Dimensional ◽

Codimension Two ◽

Problem Setting

In this paper, we present a consistent derivation of the phase field model for electrically induced damage. The derivation is based on Gurtin’s microstress and microforce theory and the Coleman–Noll procedure. The resulting model accounts for Ohmic currents, includes charge conservation law and allows for finite electric permittivity and conductivity distribution in the medium. Special attention is devoted to the case when the damaged region is a codimension-two object, i.e., a curve in three dimensions. It is shown that in this case the free energy of the model necessarily includes a high-order term, which ensures the well-posedness of the problem. A special problem setting is proposed to account for the prescribed charge distribution. Local features of the phase field distribution are illustrated with one-dimensional axisymmetric numerical experiments.

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