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Level Sets

  The functionals in equations 11 or 10 with a multiple well form for tex2html_wrap_inline1266 , can give spinodal decomposition and order-disorder kinetics as equations such as 12, 13, and 14. The solutions to these equations describe the formation of interfaces and their motion for small tex2html_wrap_inline1100 .

The motion of such interfaces during gradient flow in the tex2html_wrap_inline1064 norm reduces to the same motion as surface diffusion, if the mobility in equation 13 is independent of C within the interface where the gradient of C is large but decreases to a negligible amount in the bulk phases. In that case, equation 10 for gradient flow in the tex2html_wrap_inline1064 norm results in level sets of C within the interface that move with approximate normal velocity:

  equation740

The correspondence with surface diffusion is exact in the asymptotic limit of sharp interfaces, even when M depends interface orientation. This is the equation of volume conserving motion of an anisotropic surface by surface diffusion, and in the isotropic case reduces to Mullins' equation of motion by (minus) the Laplacian of the geometric mean curvature[10]. Equation 15 is obtained from tex2html_wrap_inline1064 gradient flow on the surface integral tex2html_wrap_inline1406 , where tex2html_wrap_inline1408 is the anisotropic surface energy. Sharp and diffuse interface motion equations in tex2html_wrap_inline1064 gradient flow give equivalent expressions for volume conserving interface motion.

The motion such interfaces during gradient flow in the tex2html_wrap_inline1016 norm is appropriate for grain boundaries, domain walls, or any other boundary for which long-range diffusion is not required for conservation. In the limit of sharp interfaces, motion of the level sets during tex2html_wrap_inline1016 gradient flow is the same as that which is derived from the surface integral approach in table III:

  equation748

which holds whether or not the interface is anisotropic.

The variational principle which leads to this motion is used in the next section to calculate the evolution of triple junctions.


next up previous
Next: Triple Junction Motion for Up: Motion of Surfaces Previous: Motion of Surfaces

W. Craig Carter
Tue Sep 30 16:07:27 EDT 1997