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Continuum approach to self-similarity and scaling in morphological relaxation of a crystal with a facet

Summary: Continuum approach to self-similarity 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
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 diffusion-limited DL and attachment-detachment-limited
ADL kinetics with inclusion of the Ehrlich-Schwoebel barrier. The slope profile F r,t , where r is the polar
distance and t is time, is described via a nonlinear, fourth-order partial differential equation PDE that
accounts for step line-tension energy g1 and step-step 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
step-flow equations and, alternatively, via a continuum surface free energy. The facet evolution is treated as a
free-boundary 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 self-similar shapes close to the facet. For DL kinetics and a class of axisymmetric shapes, a the
boundary-layer width varies as g3/g1


Source: Aziz, Michael J.- School of Engineering and Applied Sciences, Harvard University
Margetis, Dionisios - Institute for Physical Science and Technology & Department of Mathematics, University of Maryland at College Park


Collections: Materials Science; Mathematics; Physics