Interparticle interference model

The viscosity of some fluids (particle solutions or suspensions) measured at a fixed shear rate that places the fluid in the non-Newtonian regime increases with time as schematically shown by curve C of Figure 13.39. This behavior can be explained by assuming that in the Newtonian region the particles pack in an orderly manner, so flow can proceed with minimum interference between particles. However, high shear rates facilitate a more random arrangement for the particles, which leads to interparticle interference and thus to an increase in viscosity. Models that illustrate the thixotropic and rheopectic behavior of structural liquids can be found elsewhere (58,59). 

A molecular interpretation of scattering data is model dependent, and several models for the distribution of salt groups in ionomers have been proposed to explain the ionic peak. They consist mainly of two approaches (1) that the peak arises from structure within the scattering entity, 1.e., from intraparticle Interference, and (2) that the peak arises from interparticle interference. 

Figure 8. Two models describing the spatial organization of the ionic sites, a Two-phase model composed of ionic clusters (ion-rich regions) dispersed in a matrix of the intermediate ionic phase, which is composed of fluorocarbon chains and nonclustered ions. The ionic scattering maximum arises from an interparticle interference effect, reflecting an average intercluster distance S. b Core-shell model in which the ion-rich core is surrounded by an ion-poor shell composed mostly of perfluorocarbon chains. The core-shell particles are dispersed in the intermediate ionic phase. The scattering maximum arises from an interparticle interference effect, reflecting a short-range order distance S of the core-shell particle. Note that the crystalline region was not drawn in the model for the sake of simplification and that the shape of the core-shell particle may not necessarily be spherical.

At high sample concentrations, the intensities in I Q) are reduced at small Q, and this can lead to artefacts in Guinier plots that exhibit reduced or even negative / o values. The orientations of the particles are correlated with one another, and leads to interparticle interference phenomena. The hard sphere model can be used to calculate /hs(6) as a first approximation to the experimental curves [6,7] ... 

The spatial correlations between crystallins, however, reduce 1(0) to 2.5% of that for the independent scatterers model. Thus interparticle interference accounts for the transparency of the eye lens, which otherwise would be turbid [145]. 

The first uses a model of the scattering system to evaluate equation 4.121. This approach works best for a system composed of a dilute, randomly dispersed arrangement of discrete scattering particles that have a definite shape. Also, if the distribution of interparticle distance can be stated, interparticle interference effects can be included in this approach. The model approach is used in deriving equations 4.51 and 4.55. More specifically, this approach is the basis of the electron and the x-ray scattering techniques for determining the structure of small molecules. Van de Hulst [30] and Chu [31] have covered in detail the effects of... 

What has been said so far applies to dilute systems, but densely packed colloidal particles, such as highly filled nanocomposites, require to take into account the interparticle interference effects. Another assumption that is not always valid is that the particles are homogeneous and monodisperse in size. Particle anisotropy and polydispersity are very common factors that bring about severe deviations of the system from ideality. A distribution of sizes must therefore usually be included in the theoretical models used to reproduce the experimental SAXS patterns. 

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... 

This factor is analogous to the electron density contrast in SAKS theory. For the two-phase model it consists of discrete particles embedded in a continuous matrix. If there are Alp particles in volume Vp, then taking into accoxmt the interparticle interference, characterized by an interference factor S(Q), the following equation can be obtained ... 

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