ISSN:
1572-9613
Keywords:
Grain boundary
;
internal layers
;
motion by curvature
;
Cahn-Allen model
;
Allen-Cahn model
;
Ginzburg-Landau functional
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract The Cahn-Allen model for the motion of phase-antiphase boundaries is generalized to account for nonlinearities in the kinetic coefficient (relaxation velocity) and the coefficient of the gradient free energy. The resulting equation is $$\varepsilon ^2 u_l = \alpha (u)(\varepsilon [\kappa (u)]^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \nabla \cdot \{ [\kappa (u)]^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \nabla u\} - f(u))$$ wheref is bistable. Hereu is an order parameter and κ and α are physical quantities associated with the system's free energy and relaxation speed, respectively. Grain boundaries, away from triple junctions, are modeled by solutions with internal layers when ε≪1. The classical motion-by-curvature law for solution layers, well known when κ and α are constant, is shown by formal asymptotic analysis to be unchanged in form under this generalization, the only difference being in the value of the coefficient entering into the relation. The analysis is extended to the case when the relaxation time for the process vanishes for a set of values ofu. Then α is infinite for those values.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02186837
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