Under dimensioned Linear Systems

Monodisperse spheres are not only uniquely easy to characterize, but also very rarely encountered. Polymerization under carefully controlled conditions allows the preparation of the polystyrene latex shown in Figure 1.8. Latexes of this sort are used as standards for the size calibration of optical and electron micrographs (also see Section 1.5a.3). However, in the majority of colloidal systems, the particles are neither spherical nor monodisperse, but it is often useful to define convenient effective linear dimensions that are representative of the sizes and shapes of the particles. There are many ways of doing this, and whether they are appropriate or not depends on the use of such dimensions in practice. There are excellent books devoted to this topic (see, for example, Allen 1990) and, therefore, we consider only a few examples here for the purpose of illustration. 

Cyclohexene is oxidized very slowly in the presence of TS-1 little if any epoxide could be obtained under conditions of rapid oxidation of 1- and 2-alkenes to the corresponding epoxides. This low reactivity has been ascribed to the molecular dimensions of cyclohexene, which cannot enter the channel system of TS-1. Evidence for this suggestion was obtained by elution chromatography when TS-1 was loaded in a chromatographic column and a mixture of cyclohexene and 2-hexene injected, the retention time for cyclohexene was much less than that of linear 2-hexenes, despite the higher boiling point of cyclohexene (Tatsumi et al., 1990a). 

The systems studied by these authors are poly-L-glutamate in aqueous 0.3 ikf sodium phosphate at pH 7.85 and 37°, poly-L-lysine in aqueous 1.0 M sodium bromide at pH 4.54 and 37°, (and poly- -benzyl-E-aspartate in w-cresol at 100°). The tendency for these polypeptides to form ordered structures limits the choice of solvent systems in which random coil dimensions may be studied. These solvents, furthermore, cannot be solvents for the randomly coiled form of the polypeptides. However, if conditions are achieved (like those under which Brant and Flory carried out their investigation), such that the linear expansion of... 

The combination of spermine and uranyl acetate, on the other hand, yields well-defined toroids of electron microscopic dimensions. Usually one finds only a few percent of such compacted DNA molecules within a large population of linear or circular DNA strands, but under carefully controlled conditions quantitative compaction to toroids of quite uniform size has been achieved (Fig. 8.6.21) (Bottcher, 1998). Chromium and cobalt ammonium complexes have similar effects, but compaction yields are usually low. The least poisonous system to compact DNA toroids consists of a hydrophobic tripeptide, namely a (yai derivative with a terminal naphthalene sulfonate headgroup (Vengerov et al., 1985). 

In the evolution of solids from solution, a wide spectrum of structures can be formed. In Fig. 4, a simple schematic representation of the structural boundary condition for gel formation is presented. At one extreme of the conditions, linear or nearly linear polymeric networks are formed. For these systems, the functionality of polymerization /, is nearly 2. This means there is little branching or cross-linking. The degree of cross-linking is nearly 0. In silica, gels of this type can be readily formed by catalysis with HCl or HNO3 under conditions of low water content (less than 4 mol water to 1 mol silicon alkoxide). The ideal fractal dimension for such a linear chain structure is 1. The phe-... 

The theoretical analysis of the IR spectra of ultrathin films on various substrates and at interfaces will involve two assumptions (1) the problem is linear and (2) the system under investigation is macroscopic that is, one can use the macroscopic Maxwell formulas containing the local permittivity. The first assumption is valid only for weak fields. The second assumption means that the volume considered for averaging, a (the volume in which the local permittivity is formed), is lower than the parameter of inhomogeneity of the medium, d (e.g., the effective thickness of the film, the size of islands, or an effective dimension of polariton), a < d.ln this case, the response of the medium to the external electromagnetic field is essentially the response of a continuum. The... 

The electrokinetic properties of porous media, suspensions, and isolated particles can be determined within the same theoretical framework, which essentially assumes that the basic governing equations can be linearized. The resulting system can be solved numerically when the double layer thickness is not too small compared to a characteristic dimension of the medium under consideration. Hence, these results are applicable to finely divided media of submicrometric dimensions. 

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