THE CAHN HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC

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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


Published : Monday, June 18, 2012
Reading/s : 14
Origin : univ-rouen.fr
Number of pages: 40
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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 :
tu = 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 ) = 14 ( 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 11 + ss 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
Derivation of the Cahn-Hilliard system :
Mass balance : ut = div h
h : mass flux
Constitutive equation : h = κ r w
Ginzburg-Landau free energy : Ψ GL ( u , r u ) = R Ω ( α 2 |r u | 2 + F ( u )) dx
Ω R N , N 3 : domain occupied by the material
Usual definition of w : derivative of Ψ GL w.r.t. u
No longer valid
New definition : variational derivative of Ψ GL w.r.t. u
w = α Δ u + F ( u )
Usual boundary conditions :
w ν = 0 on Γ u ∂ν = 0 on Γ
Γ = Ω ν : unit outer normal vector Mass conservation : ddt R Ω udx = 0
Equivalently :
u Δ u 0 on Γ = = ∂ν ∂ν
Regular potentials :
Well-posedness, regularity : C.M. Elliott-S. Zheng, B. Nicolaenko-B. Scheurer, D. Li-C. Zhong, ...
Existence of finite-dimensional attractors : B. Nicolaenko-B. Scheurer-R. Temam, D. Li-C. Zhong, ...
Convergence of solutions to steady states : S. Zheng, P. Rybka-K.-H. Hoffmann
Logarithmic (singular) potentials :
Main difficulty : prove that u remains in ( 1 , 1 )
Remark : Not true for regular potentials
Well-posedness, regularity : C.M. Elliott-S. Luckhaus, C.M. Elliott-H. Garcke, A. Debussche-L. Dettori, A. Miranville-S. Zelik
Existence of finite-dimensional attractors : A. Debussche-L. Dettori, A. Miranville-S. Zelik
Convergence of solutions to steady states : H. Abels-M. Wilke
Dynamic boundary conditions :
Influence of the walls for confined systems
Mainly studied for polymer mixtures
Technological applications
Problem : define the boundary conditions (we need 2 boundary conditions)
First boundary condition : no mass flux at the boundary :
w = 0 on Γ ∂ν
Bulk mass conservation : ddt R Ω udx = 0
Second boundary condition : we consider, in addition to the Ginzburg-Landau free energy
Ψ GL ( u , r u ) = Z Ω ( 2 α |r u | 2 + F ( u )) dx
the surface free energy
Ψ Γ ( u , r u ) = Z Γ ( α 2 Γ |r Γ u | 2 + G ( u )) dx
α Γ > 0 r Γ : surface gradient
Original surface potential : G ( s ) = 21 a Γ s 2 b Γ s
a Γ > 0 : accounts for a modification of the effective interaction between the components b Γ : characterizes the preferential attraction of one of the components by the walls
Total energy : Ψ = Ψ GL + Ψ Γ
The system tends to minimize the excess surface energy :
1 d u α Γ Δ Γ u + g ( u ) + αu ν = 0 on Γ t d > 0 : relaxation parameter Δ Γ : Laplace-Beltrami operator g = G 0
Dynamic boundary condition
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