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 iron alloys, including order-disorder transitions.

The equation is given by:^[1][2]

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

where $M_{\eta}$ is the mobility, $f$ is the free energy density, and $\eta$ is the nonconserved order parameter.

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

1. S. M. Allen and J. W. Cahn, "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions," Acta Met. 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 Met. 7, 1261 (1973).
3. 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).