Sharp interface limits of a thermodynamically consistent solutal phase field model

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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Quantitative comparison of the phase-transformation kinetics in a sharp-interface and a phase-field model

Computational Materials Science ◽

10.1016/j.commatsci.2011.01.028 ◽

2011 ◽

Vol 50 (6) ◽

pp. 1846-1853 ◽

Cited By ~ 17

Author(s):

M.G. Mecozzi ◽

J. Eiken ◽

M. Apel ◽

J. Sietsma

Keyword(s):

Phase Transformation ◽

Phase Field ◽

Field Model ◽

Phase Field Model ◽

Quantitative Comparison ◽

Sharp Interface ◽

Transformation Kinetics ◽

Phase Transformation Kinetics

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Sharp-interface formation during lithium intercalation into silicon

European Journal of Applied Mathematics ◽

10.1017/s0956792517000067 ◽

2017 ◽

Vol 29 (1) ◽

pp. 118-145 ◽

Cited By ~ 2

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.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210509000699 ◽

2010 ◽

Vol 140 (6) ◽

pp. 1161-1186 ◽

Cited By ~ 6

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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Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.428 ◽

2018 ◽

Vol 849 ◽

pp. 805-833 ◽

Cited By ~ 16

Author(s):

Xianmin Xu ◽

Yana Di ◽

Haijun Yu

Keyword(s):

Boundary Condition ◽

Phase Field ◽

Field Model ◽

Phase Field Model ◽

Slip Boundary Condition ◽

Sharp Interface ◽

Navier Slip ◽

Slip Boundary ◽

Moving Contact ◽

Moving Contact Lines

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations. The effects of the mobility number as well as a phenomenological relaxation parameter on the boundary condition are considered. In asymptotic analysis, we consider both the cases that the mobility number is proportional to the Cahn number and the square of the Cahn number, and derive the sharp-interface limits for several set-ups of the boundary relaxation parameter. It is shown that the sharp-interface limit of the phase-field model is the standard two-phase incompressible Navier–Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

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Sharp-interface projection of a fluctuating phase-field model

Physical Review E ◽

10.1103/physreve.71.061603 ◽

2005 ◽

Vol 71 (6) ◽

Cited By ~ 5

Author(s):

R. Benítez ◽

L. Ramírez-Piscina

Keyword(s):

Phase Field ◽

Field Model ◽

Phase Field Model ◽

Sharp Interface

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A new phase-field model for strongly anisotropic systems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2008.0385 ◽

2009 ◽

Vol 465 (2105) ◽

pp. 1337-1359 ◽

Cited By ~ 112

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.

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