Self-similarity lattice structural models

For structures with a high curvature (e.g., small micelles) or situations where orientational interactions become important (e.g., the gel phase of a membrane) lattice-based models might be inappropriate. Off-lattice models for amphiphiles, which are quite similar to their counterparts in polymeric systems, have been used to study the self-assembly into micelles [ ], or to explore the phase behaviour of Langmuir monolayers [ ] and bilayers. In those systems, various phases with a nematic ordering of the hydrophobic tails occur. 

A percolation model, that has been introduced in connection with vulcanization of chains is the case in which two different species of bonds, say A and B, are placed on a lattice with concentrations Ca and Cb, respectively. Species A has the same properties as the usual bonds in random percolation whereas on species B is imposed the restriction that no more than two bonds of the same species B can be formed on the same site. Thus, species B forms polymer chains while species A acts as a crosslink. In the limit Ca = 0, Cb 0, the system reduces to self-avoiding chains, described by exponents different fi-om percolation. The opposite limit, Ca 0, Cb = 0, is the usual random-bond percolation. In the intermediate case, it was foimd - that percolation in which clusters are composed of sites connected by bonds of either species belongs to the same universality class as random percolation, unless the particular situation is realized in which percolation occurs when the typical size of chains made out of B bonds only diverges. In this case, there is a crossover from random percolation exponents to selfavoiding walk exponents , similar to the situation in lattice-gas correlated percolation. (These chains of B atoms must be distinguished from the sometimes chain-like structures formed randomly in the usual percolation process.)... 

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