Allen–Cahn equation

The Allen–Cahn equation (after John W. Cahn and Sam Allen) is a reaction-diffusion equation of mathematical physics which describes the process of phase separation in multi-component alloy systems, including order-disorder transitions.

The equation describes the time evolution of a scalar-valued state variable $\eta$ on a domain $\Omega$ during a time interval ${\mathcal {T}}$ , and is given by:^[1]^[2]

${{\partial \eta } \over {\partial t}}=M_{\eta }[{\rm {{div}(\epsilon _{\eta }^{2}\nabla \,\eta )-f'(\eta )]\quad {\rm {{on}\,\Omega \times {\mathcal {T}},\quad \eta ={\bar {\eta }}\quad {\rm {{on}\,\partial _{\eta }\Omega \times {\mathcal {T}},\quad -(\epsilon _{\eta }^{2}\nabla \,\eta )\cdot m=q\quad {\rm {{on}\,\partial _{q}\Omega \times {\mathcal {T}},\quad \eta =\eta _{o}\quad {\rm {{on}\,\Omega \times \{0\}}}}}}}}}}}$

where $M_{{\eta }}$ is the mobility, $f$ is the free energy density, ${\bar \eta }$ is the control on the state variable at the portion of the boundary $\partial _{\eta }\Omega$ , $q$ is the source control at $\partial _{q}\Omega$ , $\eta _{o}$ is the initial condition, and $m$ is the outward normal to $\partial\Omega$ .

It is the L2 gradient flow of the Ginzburg–Landau–Wilson Free Energy Functional. It is closely related to the Cahn–Hilliard equation. In one space-dimension, a very detailed account is given by a paper by Xinfu Chen.^[3]

References

↑ S. M. Allen and J. W. Cahn, "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions," Acta Met. 20, 423 (1972).
↑ S. M. Allen and J. W. Cahn, "A Correction to the Ground State of FCC Binary Ordered Alloys with First and Second Neighbor Pairwise Interactions," Scripta Met. 7, 1261 (1973).
↑ X. Chen, "Generation, propagation, and annihilation of metastable patterns", J. Differential Equations 206, 399–437 (2004).

http://www.ctcms.nist.gov/~wcraig/variational/node10.html
S. M. Allen and J. W. Cahn, "Coherent and Incoherent Equilibria in Iron-Rich Iron-Aluminum Alloys," Acta Met. 23, 1017 (1975).
S. M. Allen and J. W. Cahn, "On Tricritical Points Resulting from the Intersection of Lines of Higher-Order Transitions with Spinodals," Scripta Met. 10, 451–454 (1976).
S. M. Allen and J. W. Cahn, "Mechanisms of Phase Transformation Within the Miscibility Gap of Fe-Rich Fe-Al Alloys," Acta Met. 24, 425–437 (1976).
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, 355–379 (1993).

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