Underlying Equations

Based on method developed by Ref. 4, Ref. 5, and Ref. 6, we used the model for surface diffusion and evaporation/condensation. The weak statement can be written as

(1)

where J is the mass flux, I is the mass displacement, M is the mobility of atom on the surface, j is the volume of matter added per unit area of the solid surface per unit time, i is the volume of matter added per unit area and m is the evaporation-condensation rate, and is the energy change due to matter relocation and exchange on the surface. The first term in the integral associates with surface diffusion an the second term associates with the condensation/evaporation process.

The quantity can be obtained from the free energy consideration. Here we consider a) the surface energy, , b) the elastic energy, w , due to a misfit between the film and the substrate, and c) free energy per unit volume of atoms in a bulk solid phase, g , to model deposition from the vapor phase. The expression is

, (2)

where l is the length of the surface, is the normal displacement of the surface.

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[4] B. Sun, Z. Suo, and W. Yang, Acta Materialia 45 , 1907 (1997).

[5] B. Sun and Z. Suo, Acta Materialia 45 , 4953 (1997).

[6] J. H. Prevost, T. J. Baker, J. Liang, and Z. Suo, Int. J. Solids Structures 38 , 5185 (2001).