Allen = Khan equation

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Allen = Khan equation (it presents Allen = Khan and comes British: Allen-Cahn equation) is a reaction diffusion equation in connection with the name of Sam Allen and John Warner Khan (English version) and expresses the phase separation process of the iron alloy including the order disorder metastasis.

The Allen = Khan equation,

${\displaystyle {{\partial \eta} \over {\partial t}}=M_{\eta }[\epsilon _{\eta }^{2}\nabla ^{2}\eta -f'(\eta)]}$

I am given で. Here ${\displaystyle M_{\eta}}$ は mobility (mobility),${\displaystyle f}$ Energy density free to do は,${\displaystyle \eta}$ I express は 非保存秩序 parameter (nonconserved order parameter).

This equation is an L2 incline style of the ギンツブルグ = orchid Dow = Wilson free energy Pan-function. When it is this expression, what mass does not save becomes the neck. Therefore it comes to be paid the attention Khan = Hilliard equations.

References

• http://www.ctcms.nist.gov/~wcraig/variational/node10.html

• Samuel M. Allen and John W. Cahn, "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions," Acta Met. 20, 423 (1972).

• J. W. Cahn and S. M. Allen, "A Microscopic Theory of Domain Wall Motion and Its Experimental Verification in Fe-Al Alloy Domain Growth Kinetics," J. de Physique 38, C7-51 (1977).

• S. M. Allen and J. W. Cahn, "A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening," Acta Met.27, 1085–1095 (1979).

• L. Bronsard & F. Reitich, On three-phase boundary motion and the singular limit of a vector valued Ginzburg–Landau equation, Arch. Rat. Mech. Anal. 124 (1993), 355–379.

• Xinfu Chen, Generation, propagation, and annihilation of metastable patterns, J. Diff. Eqns. 206 (2004), 399–437.

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