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Summary: Continuum Theory of Nanostructure Decay Via a Microscale Condition
Dionisios Margetis,1,* Pak-Wing Fok,1
Michael J. Aziz,2
and Howard A. Stone2
1
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
(Received 29 June 2006; published 1 September 2006)
The morphological relaxation of faceted crystal surfaces is studied via a continuum approach. Our
formulation includes (i) an evolution equation for the surface slope that describes step line tension, g1, and
step repulsion energy, g3; and (ii) a condition at the facet edge (a free boundary) that accounts for discrete
effects via the collapse times, tn, of top steps. For initial cones and tn ~tn4, we use ~tg from step
simulations and predict self-similar slopes in agreement with simulations for any g g3=g1 > 0. We
show that for g 1, (i) the theory simplifies to an equilibrium-thermodynamics model; (ii) the slope
profiles reduce to a universal curve; and (iii) the facet radius scales as gÿ3=4
.
DOI: 10.1103/PhysRevLett.97.096102 PACS numbers: 68.35.Md, 61.46.Hk, 61.50.Ah
Below the roughening temperature TR, various struc-
tures are created on crystal surfaces, including mounds,
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