Theories Cates model

Buzza and Gates (102) also addressed the question whether disorder or the increased dimensionality from two to three dimensions is responsible for the observed experimental behavior of the shear modulus. In particular, they explored the lack of the sudden jump in G from zero to a finite value at 0 = 0Q that is predicted by the perfectly ordered 2-D model. We have seen above that disorder appears to remove that abrupt jump in two dimensions (90). For drops on a simple cubic lattice, Buzza and Cates analyzed the drop deformation in uniaxial strain close to 0 = 0q, first using the model of truncated spheres . (For reasons given above, we believe this to be a very poor model.) They showed that this model did not eliminate the discontinuous jump in G. An exact model, based on a theory by Morse and Witten (103) for weakly deformed drops, led to G a 1/ In (0 - 0q), which eliminates the discontinuity, but still shows an unrealistically sharp rise at 0 = 0q and is qualitatively very different from the experimentally observed linear dependence of G on (0 - 0q). Similar conclusions were reached by Lacasse and coworkers (49, 104). A simulation of a disordered 3-D model (104) indicated that the droplet coordination number increased from 6 at to 10 at 0 = 0.84, qualitatively similar to what is seen in disordered 2-D systems (90). Combined with a suitable (anharmonic) interdroplet force potential, the results of the simulation were in close agreement with experimental shear modulus and osmotic pressure data. It therefore appears again that disor-... 

So, as we can see, the relationship between syntactic and prosodic structure is comph-cated and a general theory that Unks syntax and prosody has yet to be developed. Some have even argued that the difficulty really hes with our models of syntax, so that if we developed syntactic models that took more account of the ways sentences were spoken some of these problems might be resolved [1]. 

In the following we consider vesiculation, in particular spontaneous vesiculation, and we concentrate on theoretical aspects. Apart from referring to existing theory (a fair review would require more time and space), we will propose a new theoretical model for onions . Simons and Cates [2], who introduced this graphic name for multilamellar vesicles, were the first to deal with onions in thermodynamic equilibrium. Unlike these authors, we will assume the bending energies of the spheres to be negative. 

A further difference with polymers arises from the fact that the chain length f of the worm-like micelles depends on the surfactant concentration because aggregation is an equilibrium process. The polymer scaling laws which deal both with polymer chain length and concentration can be tested here only for surfactant concentration c. Moreover, the relation between f and c is not well known according to different models, f scales with c or c2. In the Cates theory... 

Elastic models lead to p=3 if the stress is uniformly distributed through the thickness of a bent sheet e.g. L. D. Landau and E. M. Lifshitz, "Theory of Elasticity," Pergamon, New York (1970) S. T. Milner and T. A. Witten, M. E. Cates, Europhysics, Lett. 5, 413... 

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