1. 4/26/2002, Friday Evolution of Surface Roughness

2. Model

3. Chemical Potential for Roughness Evolution

4. Interface Energy Change
Thickness = t
g: solid-etchant interface energy density
Curvature

5. Strain Energy Change Strain energy difference
w0 is the strain energy of flat surface

6. Etching Rate g: solid-etchant interface energy density Curvature
Strain energy difference
w0 is the strain energy of flat surface

7. Competition of Two Tendencies
Surface energy g
For a perfectly flat surface, a stress applied parallel to the surface induces a uniform stress state in the solid. For a wavy surface, however, the stress field is nonuniform: the magnitude of the stress is larger at troughs than at crests. Consequently, the solid evaporates faster at the troughs than at the crests, so that the wave amplitude grows over time, and the surface roughens. When both surface energy and stress act together, the ratio of the surface energy to the elastic energy defines a length scale: short waves decay, but long waves grow.
Residual stresss

8. Interface Energy Influence on Roughness Evolution
For a perfectly flat surface, a stress applied parallel to the surface induces a uniform stress state in the solid. For a wavy surface, however, the stress field is nonuniform: the magnitude of the stress is larger at troughs than at crests. Consequently, the solid evaporates faster at the troughs than at the crests, so that the wave amplitude grows over time, and the surface roughens. When both surface energy and stress act together, the ratio of the surface energy to the elastic energy defines a length scale: short waves decay, but long waves grow.
Surface energy g 

9. Strain Energy Influence on Roughness Evolution
For a perfectly flat surface, a stress applied parallel to the surface induces a uniform stress state in the solid. For a wavy surface, however, the stress field is nonuniform: the magnitude of the stress is larger at troughs than at crests. Consequently, the solid evaporates faster at the troughs than at the crests, so that the wave amplitude grows over time, and the surface roughens. When both surface energy and stress act together, the ratio of the surface energy to the elastic energy defines a length scale: short waves decay, but long waves grow.
Residual stresss 

10. Frequency Dependency of Interface Energy Change

11. Frequency Dependency of Strain Energy Change

12. Frequency Dependency of Etching Rate

13. Roughness Evolution Pattern
Stress state:
Material parameters: