Intraparticle interference

If the actual intensity u is replaced by the corrected intensity which would be observed if intraparticle interference (see below) were of negligible consequence, then on substituting from Eq. (18) for //o and performing the integration we obtain for the similarly corrected turbidity... 

Measurements at low angles are subject to considerable error, and for this reason it is often preferred to apply appropriate corrections to scattering intensities measured at larger angles. The observed intensity ie in a direction 0 will be reduced on account of intraparticle interference by a factor cusomarily designated by P(0), which depends on the size and shape of the particle as well as on the angle 0. Thus, by definition... 

Before scattering intensity measurements can be converted to molecular weights, the two corrections previously discussed—the dissymmetry correction for intraparticle interference and the extrapolation to zero concentration—must be introduced, or established to be negligible. The relationships given in the preceding sections unfortunately account rigorously for either only in the absence of the other. The theory of the concentration dependence of the scattered intensity applies to the turbidity corrected for dissymmetry, and the treatment of dissymmetry is strictly valid only at zero concentration (where interference of radiation scattered by different polymer molecules vanishes). 

The form factor depends only on intraparticle interferences and is independent of concentration as long as the particles remain unchanged. For example, for a uniform sphere of radius a comparable to Xq ... 

EXTENSION TO LARGER PARTICLES AND TO INTRAPARTICLE INTERFERENCE EFFECTS... 

The facts that we have explicitly included the intraparticle interference function P[Q) in the analysis of scattering intensities and that it is accessible experimentally allow us to characterize colloidal dispersions structurally in more detail than we have been able to so far. In order to understand this, we need to understand clearly what we mean by small or large values of 6 or s and how they affect the behavior of P(6). This will also help us to understand how (and why) it is possible to combine light scattering with x-ray or neutron scattering to study structures of particles and their aggregates. 

The next region, for Q s larger than the above limit but still small, signifies the onset of intraparticle interference. As we have already seen from Equations (63) and (65), the function P(Q) here is of the form... 

Let us skip the next region temporarily and consider the range Q > 1. As mentioned above, intraparticle interference within the primary particles determines the function P(Q) in this case, and the functional form of P(Q) is determined by the shape of the primary particle (assumed here to be spherical). For spheres, one can show that... 

The situation becomes more complex if light scattering measurements are made at larger angles where there is intraparticle interference in the scattered light. The angular variation of intensity is related to the radius of gyration RG of the polymer. 

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. 

The "shell-core" model(13) originally proposed 1n 1974 and later mod1fied(l4,l5) 1s representative of the intraparticle interference models. It postulates that in the dry state a cluster of ca. 0.1 nm 1n radius is shielded from surrounding matrix Ions not incorporated Into clusters by a shell of hydrocarbon chains, Figure 2. The surrounding matrix ions that cannot approach the cluster more closely than the outside of the hydrocarbon shell are attracted to the cluster by dipole-dipole interactions. This mechanism establishes a preferred distance between the cluster and the matrix ions a distance of the order of 2 nm accounts for the spacing of the SAXS ionic peak. 

In equation (67) F is the excess zero-angle scattering amplitude (over the solvent amplitude) for a particle of diameter (T , and Bj is the intraparticle interference factor defined by... 

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