Cates theory

Although motional averaging might occur in ways other than that envisioned by Cates, temperature-jump experiments have yielded values of Tbr that indicate Tbr < in the region where the relaxation is nearly monoexponential, in agreement with Cates theory. In addition, Cates theory offers distinctive predictions for the concentration-dependencies of the viscoelastic behavior these allow the theory to be tested rather stringently. To obtain these predictions, we note that in the semi-dilute regime, the mean-field reptation time is L 4>, where 0 is the volume fraction of surfactant. Hence, from Eqs. (12-31) and... 

At lower ionic strength, 0.1 M, Candau et al. (1988) found a steeper power law than predicted by Cates theory, namely 17 0. This deviation has been attributed to incompletely... 

The flow properties of disordered micellar phases are now reasonably well understood. For spherical micelles the viscosity can be estimated from modified hard-sphere-suspension theories, while for disordered semidilute cylindrical micelles the Cates theory of entangled living polymers provides at least a good starting point, and in some cases nearly quantitative prediction of rheological properties. 

Fig. 7.—A cated theory described above, with soluble com- the two end portions of the antibody forming first, one (or both) then separating from the antigen, and the central part of the antibody then assuming its shape and holding the active ends in position for attachment to two antigen molecules.

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... 

The second step in Ten Cate s two-step approach was to focus on crystal-crystal interaction by means of an explicit two-phase DNS of the turbulent suspension that completely resolves the translational and rotational motions and collisions of the spherical particles plus the turbulence of the liquid between the particles. The particle motions are driven by the turbulent flow and the particles affect the turbulent flow of the liquid in between. When particles approach one another down to a distance smaller than the grid spacing, lubrication theory is exploited to bridge the gap between them. 

Experiments on asymmetric diblocks reveal similar behaviour. An initially disordered /PEp = 0.77 PEP-PEE polymer forming a hex phase was observed to order on application of reciprocating shear (Almdal et al. 1996). The ODT was found to increase, with the same scaling as for lamellae, as anticipated by theory. The limit of stability of the disordered phase obtained by Almdal et al. (1996) was fitted with a functional form, A B yY w + 1), that is in agreement with the theory of Marques and Cates (1990) together with the assumption that... 

Rhoades, D.F. and Cates, R.G., Toward a general theory of plant antiherbivore chemistry, Recent Adv. Phytochem., 10, 168, 1976. 

As remarked earlier, the nonlinear viscoelastic behavior of entangled wormy micellar solutions is similar to that of entangled flexible polymer molecules. Cates and coworkers (Cates 1990 Spenley et al. 1993, 1996) derived a full constitutive equation for entangled wormy micellar solutions, based on suitably modified reptation ideas. The stress tensor obtained from this theory is (Spenley et al. 1993)... 

Fuchs M, Cates ME (2009) A mode coupling theory for Brownian particles in homogeneous... 

For t/j = 1 (linear chains). Equation (11.9a) provides the correct value, d = 2, corresponding to a macromolecular coil at the 0-point (see Table 11.2). As noted previously, d = 4/3 for a percolation cluster, irrespective of the dimension of the Euclidean space (see Table 11.1) therefore, from Equation (11.9a), we obtain df= 4, which is consistent with the Flory-Stockmayer theory [60] for phantom chains. For three-dimensional space, d > 3 has no physical meaning because the object cannot be packed more densely than an object having a Euclidean dimension. It is evident that this discrepancy is due to the phantom nature of the polymer chains postulated by Cates [56] it is therefore, necessary to take into account self-interactions of chains due to which the dimension of a polymer fractal assumes a value that has a physical meaning. 

Some possible approximations have been considered by Cates [56], who concentrated attention on macromolecular entanglements, which play an important role in the description of the behaviour of block polymers [86-89]. Cates believes that the fact that the concept of polymer fractal neglects the effects of macromolecular entanglements is the main drawback of this theory. Nevertheless, Cates [56] introduced several simplifications that make it possible to ignore these effects for dilute solutions and relatively low molecular masses. However, in the opinion of Cates, even in the case of predominant influence of entanglements, theoretical interpretation of this phenomenon is impossible without preliminary investigation of the properties of the system in terms of Rouse-Zimm dynamics, which can serve as the basis for a more complex theory. It was assumed [56] that the effects of entanglement can be due to the substantially enhanced local friction of macromolecules. 

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