Aggregation self-similar solution

A quick demonstration may be made of the constant aggregation kernel, which allows exact calculation of the self-similar solution. We let a x, x ) = a, and recognize it to be homogeneous as described by (5.2.14) with m = 0 so that the scaling size h t) is proportional to t. Equation (5.2.16) may be solved by Laplace transform, which is left as an exercise to the reader. However, the reader may more readily verify that for this case by... 

Establish directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a x, x ) = the self-similar solution il/ rj) = e. (Hint Recognize the convolution on the right-hand side of (5.2.16). Letting = ij/ where ij/ is the derivative of the Laplace transform ij o ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to ij/). 

Several works have shown that the aggregation of isotactic and syndiotactic chains leads to the formation of stereoeomplexes for which the iso/syndio stoichiometry is found equal to 1/2, probably with a structure composed of a double-stranded helix of a 30/4 helicoidal isotactic chain surroxmded by a 60/4 helicoidal syndiotactic chain. Syndiotactic PMMA self-aggregates exhibit similar structures, with conformations close to extended chains. Experimental data indicate that, in self-aggregated syndiotactic PMMA in solution, some of the ester groups are close in contact, probably in a double helix slructure with solvent molecules included in the cavities of inner- and inter-helices. Isotactic PMMA self-aggregates also exhibit conformational helix structures. 

Our choice of basis functions is designed so that the trial solution (6.2.9) has the desired asymptotic properties of the aggregation frequency. Such properties are often available through knowledge of their relationship to the asymptotic properties of the self-similar distribution. What we mean by asymptotic properties and how those of the frequency are related to those of the self-similar distribution are elucidated in the discussion that follows. 

We now return to the issue of the choice of basis functions for solution of the inverse problem (6.2.8). The behavior of the aggregation frequency that relates to the small- / behavior of the function (j)(rj) is the issue of specific interest. We choose to fit with y-distributions that can accommodate either a singular or nonsingular nature of the self-similar distribution near the origin and accordingly set... 

FIGURE 6.2.4 Comparison of the aggregation frequency from the inverse problem with the actual (constant) frequency for various values of the regularization parameter when the self-similar distribution is known with 10% error. Regularization improves the quality of the inverted solution up to a certain value of (From Wright and Ramkrishna, 1992.) (Reprinted with permission from Elsevier Science.)... 

Well-defined mono end-capped C o-containing alkali-soluble PMAA (PMAA-fe-C o) polymer also self-assembled into nanoscale aggregates in aqueous solution [82]. PMAA-fe-Cso was soluble at high pH and showed pH-responsive properties. In dilute solution, the micelles coexisted with large secondary aggregates. These secondary aggregates consist of individual micelles that possess a microstructure similar to LCMs. Both the micelles and LCMs were formed via the closed association mechanism. The Rj, of the micelles increased from 6 to 10 nm with increasing... 

The UV-visible spectrum (Fig. 6) of the aggregates described earlier shows a 0.25-eV shift toward lower energy of the plasmon peak with a slight decrease in the bandwidth (0.8 eV) compared to that observed in solution (0.9 eV). As observed earlier with monolayers, by washing the support, the particles are redispersed in hexane and the absorption spectrum remains similar to that of the colloidal solution used to make the self-assemblies. 

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