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 ${\displaystyle \eta }$ on a domain ${\displaystyle \Omega }$ during a time interval ${\displaystyle {\mathcal {T}}}$, and is given by:[1][2]

${\displaystyle {{\partial \eta } \over {\partial t}}=M_{\eta }[\operatorname {div} (\varepsilon _{\eta }^{2}\nabla \,\eta )-f'(\eta )]\quad {\text{on }}\Omega \times {\mathcal {T}},\quad \eta ={\bar {\eta }}\quad {\text{on }}\partial _{\eta }\Omega \times {\mathcal {T}},}$

${\displaystyle \quad -(\varepsilon _{\eta }^{2}\nabla \,\eta )\cdot m=q\quad {\text{on }}\partial _{q}\Omega \times {\mathcal {T}},\quad \eta =\eta _{o}\quad {\text{on }}\Omega \times \{0\},}$

where ${\displaystyle M_{\eta }}$ is the mobility, ${\displaystyle f}$ is a double-well potential, ${\displaystyle {\bar {\eta }}}$ is the control on the state variable at the portion of the boundary ${\displaystyle \partial _{\eta }\Omega }$, ${\displaystyle q}$ is the source control at ${\displaystyle \partial _{q}\Omega }$, ${\displaystyle \eta _{o}}$ is the initial condition, and ${\displaystyle m}$ is the outward normal to ${\displaystyle \partial \Omega }$.

It is the L² gradient flow of the Ginzburg–Landau free energy functional.[3] It is closely related to the Cahn–Hilliard equation.

References

1. Allen, S. M.; Cahn, J. W. (1972). "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions". Acta Metall. 20 (3): 423–433. doi:10.1016/0001-6160(72)90037-5.
2. Allen, S. M.; Cahn, J. W. (1973). "A Correction to the Ground State of FCC Binary Ordered Alloys with First and Second Neighbor Pairwise Interactions". Scripta Metallurgica. 7 (12): 1261–1264. doi:10.1016/0036-9748(73)90073-2.
3. Veerman, Frits (March 8, 2016). "What is the L² gradient flow?". MathOverflow.

Further reading

• 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". Journal 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. S2CID 123291032.