Scattering from Polydisperse Systems

Light Scattering from Polydisperse Systens.— The scattering from a polydis-perse hard-sphere system in the Percus-Yevick approximation has been considered independently by Vrij (K = 0) and Blum and Stell (all K). 

For spherical particles, the excess Rayleigh ratio AR(K) over that of the continuum solvent is given in the Rayleigh-Debye limit by  

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 

The extension of the Percus-Yevick solution for hard-sphere mixtures to the polydisperse limit yields an explicit expression for H (K), which is a complicated functon of the second and third moments of f a). 

The influence of polydispersity on the analysis of quasi-elastic lightscattering data is considerable. For non-interacting particles in the Stokes-Einstein regime, the effective diffusion coefficient is  

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This method of analysis may also be extended to the interpretation of electrophoretic light scattering from polydisperse systems. 

We shall see that measurement of the osmotic pressure of a polydisperse system permits the experimental evaluation of M (Chapter 3), and light scattering experiments enable us to measure Mw (Chapter 5). It follows from the definition of these various averages that... 

The Lagrange expansion technique can also be applied to the calculation of the particle-scattering factors Px (q) of branched or linear polymers of DP = x from the path-weight generating function of the polydisperse system. In Chap. C.I1I we have shown the equivalence... 

Shioi, A., Harada, M., and Tanabe, M. (1995), Static light scattering from oil-rich microemulsions containing polydispersed cylindrical aggregates in sodium bis(2-ethylhexyl) phosphate system,/. Phys. Chem., 99,4750 1756. 

Bending moduli can in principle be obtained for two types of systems (i) extended, flat surfaces or interfaces, the subject matter of this section, and (ii) surfaces that are already strongly curved, and for which y is zero or extremely low, such as in vesicles or micro-emulsions. For instance such moduli can be inferred from shape fluctuations, from the Kerr effect (sec. 1.7.14] or from polydispersity using some scattering technique. We repeat that this type of measurement is often ambiguous because the bending contributions to the Helmholtz energy can only be estimated when all other contributions are accurately known. 

Number-average data of a polydisperse system are of little value. It is important to use other methods, such as small-angle X-ray scattering, to determine the extent of polydispersity of the system before making conclusions from colligative-property measurements. 

For a system of homodisperse molecules, the value of a single decay constant can be determined from equation (13), a single exponential. However, scattering from a solution of polydisperse molecules results in a distribution of exponentials consequently, analysis of DLS data must be made using a probability function which accounts for the distribution of molecular sizes. 

According to Weissman, the scattering intensity I in a polydisperse system arises from size fluctuations (proportional to where a is the particle... 

The polarizability, in turn, is proportional to the particle volume, a oc and the difference between the refractive indices of the particle and the solvent, with the latter quantity being accessible from differential refractometry measurements. For homogeneous spheres of radius / , the ratio between the scattering intensity and the particle volume fraction, //0, is proportional to / , and this quantity can be used to estimate the particle size. For a polydisperse system, the signal is proportional to the following ratio of moments ... 

FTIR, NMR, and EXAFS and ex situ methodologies such as electron microscopy (SEM and TEM) are also powerful and important tools in the investigation of the mechanisms by which materials form. Combination of experimental approaches not only facilitates their interpretation but also enables cross-correlation between experimental phenomena. This is especially important because SAXS provides information on reciprocal space. The estimation of the structure of a scatterer from its scattering profiles is called the inverse scattering problem, and this problem cannot be solved uniquely [1]. Scattering profiles are complicated further when polydispersity effects are operative, which is usually to some extent the case for sol-gel systems. In practice, the interpretation of SAXS patterns therefore depends heavily on the development of hypothetical structural models and on comparison of the simulated scattering profile, which can be calculated from a given structure, with the experimental profile. Hence, additional independent structural or chemical information may aid in the interpretation of SAXS profiles. 

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