Non-axisymmetric growth of dendrite with arbitrary symmetry in two and three dimensions: sharp interface model vs phase-field model

2020 ◽  
Vol 229 (19-20) ◽  
pp. 2899-2909
Author(s):  
L. V. Toropova ◽  
P. K. Galenko ◽  
D. V. Alexandrov ◽  
M. Rettenmayr ◽  
A. Kao ◽  
...  
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Sharp Interface ◽  
Three Dimensions ◽  
Interface Model ◽  
Sharp Interface Model

scholarly journals Sharp-interface formation during lithium intercalation into silicon

2017 ◽  
Vol 29 (1) ◽  
pp. 118-145 ◽  
Author(s):  
E. MECA ◽  
A. MÜNCH ◽  
B. WAGNER
Keyword(s):  
Free Boundary ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Lithium Ion ◽  
Sharp Interface ◽  
Ion Concentration ◽  
Interface Model ◽  
Sharp Interface Model ◽  
Time Dynamics

In this study, we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as amorphous silicon (a-Si). The governing equations couple a viscous Cahn–Hilliard-Reaction model with elasticity in the framework of the Cahn–Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in lithium ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface intersects the free boundary of the Si layer. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

A new phase-field model for strongly anisotropic systems

2009 ◽  
Vol 465 (2105) ◽  
pp. 1337-1359 ◽  
Author(s):  
Solmaz Torabi ◽  
John Lowengrub ◽  
Axel Voigt ◽  
Steven Wise
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Three Dimensional ◽  
Phase Field Model ◽  
Sharp Interface ◽  
Interface Model ◽  
Sharp Interface Model ◽  
Anisotropic Surface ◽  
Sharp Corners ◽  
New Phase

We present a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn–Hilliard-type equations. Such problems arise during the growth and coarsening of thin films. When the anisotropic surface energy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn–Hilliard equations become ill-posed. Our models contain a high-order Willmore regularization, where the square of the mean curvature is added to the energy, to remove the ill-posedness. The regularized equations are sixth order in space. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. Using the method of matched asymptotic expansions, we show the convergence of our phase-field model to the general sharp-interface model. We present two- and three-dimensional numerical results using an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the dynamics of the new phase-field model and the sharp-interface model. The computed equilibrium shapes using the new model also match a recently developed analytical sharp-interface theory that describes the rounding of the sharp corners by the Willmore regularization.

PHASE-FIELD MODELING OF ELASTIC EFFECTS IN EUTECTIC GROWTH WITH MISFIT STRESSES

2012 ◽  
Vol 04 (01) ◽  
pp. 1250024
Author(s):  
ZOHREH EBRAHIMI ◽  
JOAO REZENDEH
Keyword(s):  
Phase Field ◽  
Elastic Energy ◽  
Field Model ◽  
Phase Field Model ◽  
Solidification Process ◽  
Eutectic Growth ◽  
Point Of View ◽  
Eutectic Solidification ◽  
Elastic Model ◽  
Sharp Interface Model

Elastic interactions, arising from a difference of lattice spacing between two coherent phases in eutectic alloys with misfit stresses, can have an influence on microstructural pattern formation of eutectic colonies during solidification process. From a thermodynamic point of view the elastic energy contributes to the free energy of the phases and modifies their mutual stability. Therefore, the elastic stresses will have an effect on stability of lamellae, lamellae spacing and growth modes. In this paper, a phase-field model is employed to investigate the influence of elastic misfits in eutectic growth. The model reduces to the traditional sharp-interface model in a thin-interface limit, where the microscopic interface width is small but finite. An elastic model is designed, based on linear microelasticity theory, to incorporate the elastic energy in the phase-field model. Theoretical and numerical approaches, required to model elastic effects, are formulated and the stress distributions in eutectic solidification structures are evaluated. The two-dimensional simulations are performed for directed eutectic growth and the simulation results for different values of the misfit stresses are illustrated.

scholarly journals A ternary Cahn–Hilliard–Navier–Stokes model for two-phase flow with precipitation and dissolution

2020 ◽  
pp. 1-35
Author(s):  
Christian Rohde ◽  
Lars von Wolff
Keyword(s):  
Phase Field ◽  
Stokes Equations ◽  
Solid Phase ◽  
Immiscible Fluids ◽  
Sharp Interface ◽  
Ion Concentration ◽  
Navier Stokes ◽  
Interface Model ◽  
Sharp Interface Model ◽  
Two Phase

We consider the incompressible flow of two immiscible fluids in the presence of a solid phase that undergoes changes in time due to precipitation and dissolution effects. Based on a seminal sharp interface model a phase-field approach is suggested that couples the Navier–Stokes equations and the solid’s ion concentration transport equation with the Cahn–Hilliard evolution for the phase fields. The model is shown to preserve the fundamental conservation constraints and to obey the second law of thermodynamics for a novel free energy formulation. An extended analysis for vanishing interfacial width reveals that in this limit the sharp interface model is recovered, including all relevant transmission conditions. Notably, the new phase-field model is able to realize Navier-slip conditions for solid–fluid interfaces in the limit.

scholarly journals A regularized phase-field model for faceting in a kinetically controlled crystal growth

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200227
Author(s):  
T. Philippe ◽  
H. Henry ◽  
M. Plapp
Keyword(s):  
Crystal Growth ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Interface Problem ◽  
Kinetically Controlled ◽  
Interface Theory ◽  
Coarsening Dynamics

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.

On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit

2010 ◽  
Vol 140 (6) ◽  
pp. 1161-1186 ◽  
Author(s):  
Wolfgang Dreyer ◽  
Christiane Kraus
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Van Der Waals ◽  
Phase Field Model ◽  
Entropy Inequality ◽  
Sharp Interface ◽  
Energy Estimates ◽  
Sharp Interface Limit ◽  
Entropy Flux

We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible.We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.

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