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 Metall. 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 Metallurgica 7, 1261 (1973).
^ Chen, Xinfu (2004). "Generation, propagation, and annihilation of metastable patterns". Journal of Differential Equations. 206 (2): 399–437. Bibcode:2004JDE...206..399C. doi:10.1016/j.jde.2004.05.017.

http://www.ctcms.nist.gov/~wcraig/variational/node10.html
Allen, S. M.; Cahn, J. W. (1975). "Coherent and Incoherent Equilibria in Iron-Rich Iron-Aluminum Alloys". Acta Metall. 23 (9): 1017. doi:10.1016/0001-6160(75)90106-6.
Allen, S. M.; Cahn, J. W. (1976). "On Tricritical Points Resulting from the Intersection of Lines of Higher-Order Transitions with Spinodals". Scripta Metallurgica. 10 (5): 451–454. doi:10.1016/0036-9748(76)90171-x.
Allen, S. M.; Cahn, J. W. (1976). "Mechanisms of Phase Transformation Within the Miscibility Gap of Fe-Rich Fe-Al Alloys". Acta Metall. 24 (5): 425–437. doi:10.1016/0001-6160(76)90063-8.
Cahn, J. W.; Allen, S. M. (1977). "A Microscopic Theory of Domain Wall Motion and Its Experimental Verification in Fe-Al Alloy Domain Growth Kinetics". J. De Physique. 38: C7–51.
Allen, S. M.; Cahn, J. W. (1979). "A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening". Acta Metall. 27 (6): 1085–1095. doi:10.1016/0001-6160(79)90196-2.
Bronsard, L.; Reitich, F. (1993). "On three-phase boundary motion and the singular limit of a vector valued Ginzburg–Landau equation". Arch. Rat. Mech. Anal. 124 (4): 355–379. Bibcode:1993ArRMA.124..355B. doi:10.1007/bf00375607.

[1] S. M. Allen and J. W. Cahn, "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions," Acta Metall. 20, 423 (1972).

[2] 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 Metallurgica 7, 1261 (1973).

[3] Chen, Xinfu (2004). "Generation, propagation, and annihilation of metastable patterns". Journal of Differential Equations. 206 (2): 399–437. Bibcode:2004JDE...206..399C. doi:10.1016/j.jde.2004.05.017.

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