Lifshitz scale/length

As Bergman has pointed out (Bergman and Stroud 1992), and discussed earlier (in Section 1.2.2(c)), this behaviour is expected when the Lifshitz scale is greater than the percolation correlation length. It is not clear however if the Lifshitz scale is given by /max InL or simply by InL near pc-... 

Nano-structures comments on an example of extreme microstructure In a chapter entitled Materials in Extreme States , Cahn (2001) dedicated several comments to the extreme microstructures and summed up principles and technology of nano-structured materials. Historical remarks were cited starting from the early recognition that working at the nano-scale is truly different from traditional material science. The chemical behaviour and electronic structure change when dimensions are comparable to the length scale of electronic wave functions. Quantum effects do become important at this scale, as predicted by Lifshitz and Kosevich (1953). As for their nomenclature, notice that a piece of semiconductor which is very small in one, two- or three-dimensions, that is a confined structure, is called a quantum well, a quantum wire or a quantum dot, respectively. 

We have studied the the fracture properties of such elastic networks, under large stresses, with initial random voids or cracks of different shapes and sizes given by the percolation statistics. In particular, we have studied the cumulative failure distribution F a) of such a solid and found that it is given by the Gumbel or the Weibull form (3.18), similar to the electrical breakdown cases discussed in the previous chapter. Extensive numerical and experimental studies, as discussed in Section 3.4.2, support the theoretical expectations. Again, similar to the case of electrical breakdown, the nature of the competition between the percolation and extreme statistics (competition between the Lifshitz length scale and the percolation correlation length) is not very clear yet near the percolation threshold of disorder. 

A first attempt at this was accomplished in [16] where the Lifshitz-Slyozov theory was adopted to allow for spatial variations including the concept of the local particle size distribution. In that work, however, undue emphasis was placed on the diffusion of the particles and the concept of forming macroscopic length scale patterns via the svegliabile mechanism was not realized (see a discussion in the other paper of the author in these proceedings). 

---