Films on Slightly Rough Substrates The Healing Length

1 Films on Slightly Rough Substrates The Healing Length 

FIGURE 4.7. Thin film on a rough substrate. The wavelength A is the characteristic pitch of the roughness to be compared with the healing length e- 

We describe the modulation of the free surface by a displacement u x, y). The local thickness of the film is e + — C, and the energy W has the form  

For small deformations, we expand P e - -u — ( ) out to second order in (u — C)- The zero-order term gives a constant. The first-order term drops out since the spatial average of (u — C) is zero if the volume of the liquid is conserved. The second-order term yields the following equation for the optimum profile  

These properties have a direct bearing on varnishes used as a finish on rough surfaces (such as wood or plastics). If the refractive index of the substrate is not too different from that of the varnish, it is possible to achieve a good surface polish provided that is greater than the horizontal pitch of the surface roughness. 

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