Solutions to a model for interface motion by interface diffusion

2008 ◽  
Vol 138 (5) ◽  
pp. 923-955 ◽  
Author(s):  
Hans-Dieter Alber ◽  
Peicheng Zhu
Keyword(s):  
Field Model ◽  
Phase Field Model ◽  
Order Equation ◽  
Interface Model ◽  
Sharp Interface Model ◽  
Deformable Solid ◽  
Interface Motion ◽  
Interface Diffusion ◽  
Bulk Free Energy ◽  
Degenerate Parabolic

Existence of weak solutions is proved for a phase field model describing an interface in an elastically deformable solid, which moves by diffusion of atoms along the interface. The volume of the different regions separated by the interface is conserved, since no exchange of atoms across the interface occurs. The diffusion is driven only by reduction of the bulk free energy. The evolution of the order parameter in this model is governed by a degenerate parabolic fourth-order equation. If a regularizing parameter in this equation tends to zero, then solutions tend to solutions of a sharp interface model for interface diffusion. The existence proof is valid only for a 1½-dimensional situation.

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.

Stress- and diffusion-induced interface motion: Modelling and numerical simulations

2007 ◽  
Vol 18 (6) ◽  
pp. 631-657 ◽  
Author(s):  
HARALD GARCKE ◽  
ROBERT NÜRNBERG ◽  
VANESSA STYLES
Keyword(s):  
Grain Boundary ◽  
Numerical Simulations ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Boundary Motion ◽  
Interface Motion ◽  
Stress Effects ◽  
Grain Boundary Motion ◽  
And Diffusion

We propose a phase field model for stress and diffusion-induced interface motion. This model, in particular, can be used to describe diffusion-induced grain boundary motion and generalizes a model of Cahn, Fife and Penrose as it more accurately incorporates stress effects. In this paper we will demonstrate that the model can also be used to describe other stress-driven interface motion. As an example, interface motion resulting from interactions of interfaces with dislocations is studied.

Sign in / Sign up

Export Citation Format

Share Document