THE CAHN HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC

Alain Miranville

Description

THE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS Alain Miranville Université de Poitiers, France Collaborators : L. Cherfils, G. Gilardi, G.R. Goldstein, G. Schimperna, S. Zelik

• chemical potential


• cahn-hilliard system


• ginzburg-landau free


• mobility ?


• domain occupied


• surfaces tension

Sujets

• Université de Poitiers
• Chemical potential

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

THE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS

Alain Miranville

Université de Poitiers, France

Collaborators : L. Cherfils, G. Gilardi, G.R. Goldstein, G. Schimperna, S. Zelik

Cahn-Hilliard system :

Equivalently :

∂ ∂ t u = κ Δ w , κ > 0 w = − α Δ u + f ( u ) , α > 0

∂ u Δ 2 u − κ Δ f ( u ) = 0 + α ∂ t κ Describes the phase separation process in a binary alloy : spinodal decomposition, coarsening

u : order parameter w : chemical potential κ : mobility α : related to the surface tension at the interface

f : derivative of a double-well potential F

Typical choice :

2 F ( s ) = 1 4 ( s − 1 ) 2 f ( s ) = s 3 − s

Thermodynamically relevant potential :

F ( s ) = − θ 0 s 2 + θ 1 (( 1 + s ) ln ( 1 + s ) +( 1 − s ) ln ( 1 − s )) f ( s ) = − 2 θ 0 s + θ 1 ln 1 1 − + s s s ∈ ( − 1 , 1 ) , 0 < θ 1 < θ 0

Remark : κ should more generally depend on u and degenerate :

∂ u = div ( κ ( u ) r w ) ∂ t

κ ( s ) = 1 − s 2 Restricts the diffusion process to the interfacial region Is observed when the movements of atoms are confined to this region