Compressive stress prior to island coalescence

Mays et al. (1968) observed that the average lattice constant varied from 0.4075 nm to 0.4063 nm as the radius of curvature of the nano-crystal varied from very large values down to about the two nm. Using (1.20), they estimated the surface stress / to be 1.175 N/m. The surface energy of gold has been estimated from independent measurements to be about 1.4 J/m see the review by Cammarata (1994), for example. Presumably, the difference between these values arises from the strain dependence of surface stress. 

An elementary model leading to a stress estimate due to this effect can be constructed in the following way. Suppose that a representative crystallite becomes attached to the substrate, or becomes locked-down against further relaxation, at some radius Rid- The internal equi-triaxial strain in the grain at this point is 

As the crystallite continues to grow to larger radius R i ia, the strain is prevented from relaxing further. In other words, as the grain becomes larger, it is subject to a mismatch strain 

This quantity is representative of the stress magnitude that can be expected to arise in the film according to the proposed mechanism. Note that it has the feature that it continues to increase as R becomes very large, approaching an asymptotic limit of —2//i ia as R/R a oo. 

In the interpretation of observational data in the context of the type of model represented by (1.23), an issue that is commonly overlooked is the importance of surface area coverage or density of islands on the growth surface. The stress field associated with an isolated island on the substrate surface is self-equilibrating overall, the stress field has no resultant force or moment and its magnitude decays to very small values within a short distance from the island. The prospect that a dilute distribution of tiny islands on a relatively thick substrate of comparable stiffness could result in a significant curvature is remote, no matter how large the mismatch stress might be. For a distribution of islands to lead to a perceptible curvature of the substrate, the distribution must be sufficiently dense so that the fields of the islands 

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