Critical packing state

Starburst/cascade dendrimer dense-packing critical branching state, nanoscopic steric effects. 

Starburst (de Gennes) Dense Packing — A Critical Branched State ... 

In the first paper in 1952 we calculated the y-V relationship for the linear packing type. In this case, as criticized by Emmett in a private communication and later by Barrer et al., an approximation using (1) can lead to serious error. In addition, this type of packing is useless in actual applications, and so it was abandoned. In cases of the denser packing states considered here, errors due to (1) are not so large because a true spherical meniscus soon appears which encloses most of the sorbate liquid. 

The critical packing parameter, CPP, is described by eq 2, where v is the average volume of the amphiphile, a is the effective head group area, and / is the effective chain length of the surfactant in the molten state. The CPP can be used to predict the aggregate structures and to correlate stmctural clmges of the surfactant (or PIL) with changes to the self-assembly phases. 

It can be observed that the constant B is basically the same as that for Halpin-Tsai equation. Constant A is related to the graieralized Einstein coefficient kg. The value of kg is not a constant but depends on the state of agglomeration of the particles. It has the value of 2.5 for perfect dispersion of spherical particles and perfect matrix-particle adhesion. The value will decrease if there is dewetting and slippage at the matrix-particle interface but will increase if there is agglomeration [74]. However, kg is applicable only to matrix with Poisson s ratio of 0.5. For other Poisson ratios, a conversion is required and can be done easily with a list of relative Einstein coefficients for various Poisson s ratios presented in the works of Nielsen and co-workers [59, 63, 71]. Finally, i f depends on the maximum or critical packing fraction of the filler in the matrix. 

The critical radius at Tg is a multiple of Droplets of size N > N are thermodynamically unstable and will break up into smaller droplets, in contrast to that prescribed by F N), if used naively beyond size N. This is because N = 0 and N = N represent thermodynamically equivalent states of the liquid in which every packing typical of the temperature T is accessible to the liquid on the experimental time scale, as already mentioned. In view of this symmetry between points N = 0 and N, it may seem somewhat odd that the F N) profile is not symmetric about. Droplet size N, as a one-dimensional order parameter, is not a complete description. The profile F N) is a projection onto a single coordinate of a transition that must be described by order parameters—the... 

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