scholarly journals Phase Field Model for Solidification with Boundary Interface Interaction

2016 ◽  
Vol 9 ◽  
pp. 1-8
Author(s):  
Jie Liao
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Contact Region ◽  
Contact Point ◽  
Boundary Surface ◽  
Phase Field Model ◽  
Heat Diffusion ◽  
Diffuse Interface ◽  
Interface Problem

By incorporation the surface free energy in the free energy functional, a phase field model for solidification with boundary interface intersection is developed. In this model, the bulk equation is appropriately modified to account for the presence of heat diffusion inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

scholarly journals A Phase Field Model for Binary Alloy Solidification with Boundary Interface Intersection

2016 ◽  
Vol 8 ◽  
pp. 9-18
Author(s):  
Jie Liao
Keyword(s):  
Binary Alloy ◽  
Phase Field ◽  
Field Model ◽  
Contact Region ◽  
Contact Point ◽  
Boundary Surface ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Interface Problem ◽  
Alloy Solidification

A phase field model for binary alloy solidification with boundary interface intersection is developed. In the phase field model, the heat and solute conservation equations are appropriately modified to account for the presence of heat and solute rejection inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The thin interface asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

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.

scholarly journals On the Diffuse Interface Models for High Codimension Dispersed Inclusions

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2206
Author(s):  
Elizaveta Zipunova ◽  
Evgeny Savenkov
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Three Dimensional ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
State Variables ◽  
Interface Models ◽  
Dispersed Inclusions ◽  
Derivatives Of

Diffuse interface models are widely used to describe the evolution of multi-phase systems of various natures. Dispersed inclusions described by these models are usually three-dimensional (3D) objects characterized by phase field distribution. When employed to describe elastic fracture evolution, the dispersed phase elements are effectively two-dimensional (2D) objects. An example of the model with effectively one-dimensional (1D) dispersed inclusions is a phase field model for electric breakdown in solids. Any diffuse interface field model is defined by an appropriate free energy functional, which depends on a phase field and its derivatives. In this work we show that codimension of the dispersed inclusions significantly restricts the functional dependency of the free energy on the derivatives of the problem state variables. It is shown that to describe codimension 2 diffuse objects, the free energy of the model necessarily depends on higher order derivatives of the phase field or needs an additional smoothness of the solution, i.e., its first derivatives should be integrable with a power greater than two. Numerical experiments are presented to support our theoretical discussion.

Explicit Dynamics of Diffuse Interface in Phase‐Field Model

2020 ◽  
pp. 2000162
Author(s):  
Chao Yang ◽  
Houbing Huang ◽  
Wenbo Liu ◽  
Junsheng Wang ◽  
Jing Wang ◽  
...  
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Explicit Dynamics

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.

CONVECTIVE PATTERNS IN LIQUID–VAPOR SYSTEMS WITH DIFFUSE INTERFACE

2006 ◽  
Vol 16 (09) ◽  
pp. 2705-2711 ◽  
Author(s):  
RODICA BORCIA ◽  
DOMNIC MERKT ◽  
MICHAEL BESTEHORN
Keyword(s):  
Phase Field ◽  
Compressible Fluid ◽  
Field Model ◽  
Marangoni Convection ◽  
Van Der Waals ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Fully Nonlinear ◽  
Nonlinear Simulations ◽  
Liquid Vapor

Recently, we have developed a phase field model to describe Marangoni convection with evaporation in a compressible fluid of van der Waals type away from criticality [Eur. Phys. J. B44 (2005)]. Using this model, we report now 2D fully nonlinear simulations where we emphasize the influence of evaporation on convective patterns.

scholarly journals Phase Transitions: Explicit Dynamics of Diffuse Interface in Phase‐Field Model (Adv. Theory Simul. 1/2021)

2021 ◽  
Vol 4 (1) ◽  
pp. 2170001
Author(s):  
Chao Yang ◽  
Houbing Huang ◽  
Wenbo Liu ◽  
Junsheng Wang ◽  
Jing Wang ◽  
...  
Keyword(s):  
Phase Transitions ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Explicit Dynamics

Thermodynamic Model Formulations for Inhomogeneous Solids with Application to Non-isothermal Phase Field Modelling

2016 ◽  
Vol 41 (2) ◽  
Author(s):  
Svyatoslav Gladkov ◽  
Julian Kochmann ◽  
Stefanie Reese ◽  
Markus Hütter ◽  
Bob Svendsen
Keyword(s):  
Phase Field ◽  
Thermodynamic Model ◽  
Field Model ◽  
Phase Field Model ◽  
Antiphase Boundary ◽  
Diffuse Interface ◽  
Equilibrium Thermodynamics ◽  
Field Methods ◽  
Macroscopic Theory ◽  
Non Equilibrium

AbstractThe purpose of the current work is the comparison of thermodynamic model formulations for chemically and structurally inhomogeneous solids at finite deformation based on “standard” non-equilibrium thermodynamics [SNET: e. g. S. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland, 1962] and the general equation for non-equilibrium reversible–irreversible coupling (GENERIC) [H. C. Öttinger, Beyond Equilibrium Thermodynamics, Wiley Interscience, 2005]. In the process, non-isothermal generalizations of standard isothermal conservative [e. g. J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial energy. J. Chem. Phys. 28 (1958), 258–267] and non-conservative [e. g. S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979), 1085–1095; A. G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983] diffuse interface or “phase-field” models [e. g. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Modern Phys. 49 (1977), 435–479; N. Provatas and K. Elder, Phase Field Methods in Material Science and Engineering, Wiley-VCH, 2010.] for solids are obtained. The current treatment is consistent with, and includes, previous works [e. g. O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D 43 (1990), 44–62; O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Phys. D 69 (1993), 107–113] on non-isothermal systems as a special case. In the context of no-flux boundary conditions, the SNET- and GENERIC-based approaches are shown to be completely consistent with each other and result in equivalent temperature evolution relations.

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