 
Summary: Continuum approach to selfsimilarity and scaling in morphological relaxation
of a crystal with a facet
Dionisios Margetis,1
Michael J. Aziz,2
and Howard A. Stone2
1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
Received 1 April 2004; revised manuscript received 5 October 2004; published 26 April 2005
The morphological relaxation of axisymmetric crystal surfaces with a single facet below the roughening
transition temperature is studied analytically for diffusionlimited DL and attachmentdetachmentlimited
ADL kinetics with inclusion of the EhrlichSchwoebel barrier. The slope profile F r,t , where r is the polar
distance and t is time, is described via a nonlinear, fourthorder partial differential equation PDE that
accounts for step linetension energy g1 and stepstep repulsive interaction energy g3; for ADL kinetics, an
effective surface diffusivity that depends on the step density is included. The PDE is derived directly from the
stepflow equations and, alternatively, via a continuum surface free energy. The facet evolution is treated as a
freeboundary problem where the interplay between g1 and g3 gives rise to a region of rapid variations of F, a
boundary layer, near the expanding facet. For long times and g3/g1 O 1 singular perturbation theory is
applied for selfsimilar shapes close to the facet. For DL kinetics and a class of axisymmetric shapes, a the
boundarylayer width varies as g3/g1
