Allen–Cahn equation explained

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

η on a domain \Omega during a time interval l{T} , and is given by:^{[1] [2]} {{\partial η }\over{\partial t}} = M ^{2} _{η} \nabla η ) - f'( η )] on \Omega x l{T}, η= \bar η on \partial _{η\Omega x l{T},} - (\varepsilon ^{2} _{η \nabla η ) ⋅} m = q on \partial _{q} \Omega x l{T}, η=η _{o} on \Omega x \{ 0 \}, where M _{η} is the mobility, f is a double-well potential, \bar η is the control on the state variable at the portion of the boundary \partial _{η\Omega} , q is the source control at \partial _{q\Omega} , η _{o} is the initial condition, and m is the outward normal to \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.

Mathematical description

Let

\Omega\subset \R ^{n} be an open set,v _{0(x)\in} L ^{2(\Omega)} an arbitrary initial function,\varepsilon> 0 and T> 0 two constants.

A function

v(x,t):\Omega x [ 0 ,T]\to \R is a solution to the Allen–Cahn equation if it solves^[4] \partial _{t} v - \Delta _{x} v =- (1)/(\varepsilon ^{2}) f(v), \Omega x [ 0 ,T] where\Delta _{x} is the Laplacian with respect to the space x ,f(v) = F'(v) is the derivative of a non-negative F\in C ^{1(\R)} with two minima F(\pm 1 ) =0 .

Usually, one has the following initial condition with the Neumann boundary condition

\begin{cases} v(x, 0 ) = v _{0(x),} & \Omega x \{ 0 \}\\ \partial _{n} v = 0 , & \partial \Omega x [ 0 ,T]\\ \end{cases} where \partial _{n} v is the outer normal derivative.

For

F(v) one popular candidate isF(v) = ((v ^{2-1)} ^{2})/(4) , f(v) = v ^{3-v.}

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 . 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 . 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 . 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 . 10.1016/0001-6160(79)90196-2.
• Bronsard . L. . Lia Bronsard . 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 . 10.1007/bf00375607 . 1993ArRMA.124..355B. 123291032 .

External links

• Simulation by Nils Berglund of a solution of the Allen–Cahn equation

Notes and References

1. S. M. . Allen . J. W. . Cahn . Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions . Acta Metall. . 20 . 3 . 423–433 . 1972 . 10.1016/0001-6160(72)90037-5 .
2. S. M. . Allen . J. W. . Cahn . A Correction to the Ground State of FCC Binary Ordered Alloys with First and Second Neighbor Pairwise Interactions . Scripta Metallurgica . 7 . 12 . 1261–1264 . 1973 . 10.1016/0036-9748(73)90073-2 .
3. Web site: Frits . Veerman . What is the L² gradient flow? . March 8, 2016 . .
4. Book: Bartels, Sören. Springer International Publishing . Numerical Methods for Nonlinear Partial Differential Equations . Deutschland . 2015 . 153.