Small, Uncorrelated Defects

What happens if we are truly dealing with small, dense, uncorrelated defects, rather than with large blemishes Theorists have pondered the question. Their current view is captured in the next problem. 

Problem Estimate the roughness of a triple line in the presence of small, dense defects, assuming that the line minimizes its energy. 

Solution It is convenient to describe the blemished surface in terms of small squares of lateral dimension Tq to the side (the size of a defect). Each square is a defect (of area Tq) whose interfacial energy with the liquid is modulated by an amount A7, and hence by an energy 1 = A7rg. 

Consider an arc of length L moving across a distance X. The number of squares swept is N XL/tq. The energy Ei gained during the sweep is not proportional to N (since the average is zero) but, rather, to y/N  

At the same time, energy is lost because of the line s elasticity. The pertinent wavevector is g L , and equation (3.10) provides an elastic energy 7gX per unit length. Over a length L, this translates to 

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The existence of large blemishes is hinted at by yet another observation contained in Figure 3.7. We get virtually the identical profile after displacing the wet region by a distance of 500 pm (for instance by tilting the substrate). This outcome would be impossible if we were dealing with small, uncorrelated defects. The size of the blemishes would have to be at least 500 pm. 

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