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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