Hard-Sphere Aggregation Models

In the comparison of DLCA results with experimental data on colloidal aerogels, in which the primary particles are relatively large and monodisperse, excellent agreement was obtained in nearly aU respects. Basic and neutral aerogels were not fit as well, which was hypothesized due to polydispersity and the neglect of deformations and restructuring effects. Finally, different measures for the average size of the clusters within the gel network were 

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One of the conceptually simplest representations of a molecule, also used in the construction of hard-sphere space filling models, is obtained by centring spheres of suitable Van der Waals radii at the positions of the atomic nuclei. To convey a feeling of size the aggregate is enclosed in a generating ellipsoid that circumscribes the atom most remote from the centre to define the volume of the Van der Waals body [212]. 

Whittle and Dickinson used a more elaborate model, incorporating harmonic bonds, short-ranged repulsive forces, and a shifted-center Leimard-Jones potential to study aggregates of soft particles typical of food colloids [70]. These authors also studied the failure of a colloidal gel under strain [71]. Similar studies, using either hard spheres or repulsive-core particles, have also been performed by other groups [72-75]. 

Hard-Sphere Model, in 1973, it was suggested that the ionomer peak is ascribed to the distance between ionic aggregates, with the contrast provided by the difference in the electron density between the ionic aggregates and the hydrocarbon phase (22). The scattering centers are considered to be points on a paracrys-talline lattice. In 1983, a refined model was proposed (23) in this hard-sphere model, the midtiplets have hquid-like order and the distance of closest approach... 

Figure 3 schematically illustrates the interference model where it is assumed that the ionomer peak arises from a preferred interparticle distance. Recently, Yarusso and Cooper have proposed an interpretation of the ionomer peak which is based on the liquid-like scattering from hard spheres originally described by Fournet. The Fournet model is quantitatively capable of fitting the X-ray peak from sulfonated polystyrene ionomers. In the case of zinc-neutralized material, about half of the ionic groups were found to aggregate into well ordered domains ( clusters ) with the remainder... 

Aggregated phases, formed by the addition of third components, have also been studied in ILs. Often, these phases form micelles, which can easily be modeled by Eq. (6.35), using the appropriate form factor [28] and structure factor [65]. Using this approach, one can obtain the micelle size, hard sphere radius, chain aggregation number, and chain size [66]. However, this is not always possible because of the limited -range available—a problem that is sometimes unavoidable depending on the instrument used. 

The thermodynamic model described in this section enables one to estimate the fraction of the hard phase that is microphase-separated and aggregated into nanodomains, and determine whether those nanodomains form spheres, cylinders, or lamellae. The next step is to develop an appropriate micromechanical model to enable prediction of elastic properties (Yomig s modulus). 

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