Polydisperse systems

Glatter, O.

doi:10.1107/97809553602060000581

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 99

Section 2.6.1.4. Polydisperse systems

O. Glatter^a

2.6.1.4. Polydisperse systems

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In this subsection, we give a short survey of the problem of polydispersity. It is most important that there is no way to decide from small-angle scattering data whether the sample is mono- or polydisperse. Every data set can be evaluated in terms of monodisperse or polydisperse structures. Independent a priori information is necessary to make this decision. It has been shown analytically that a certain size distribution of spheres gives the same scattering function as a monodisperse ellipsoid with axes a, b and c (Mittelbach & Porod, 1962).

The scattering function of a polydisperse system is determined by the shape of the particles and by the size distribution. As mentioned above, we can assume a certain size distribution and can determine the shape, or, more frequently, we assume the shape and determine the size distribution. In order to do this we have to assume that the scattered intensity results from an ensemble of particles of the same shape whose size distribution can be described by [D_n(R)] , where R is a size parameter and denotes the number of particles of size R. Let us further assume that there are no interparticle interferences or multiple scattering effects. Then the scattering function I(h) is given by $[I(h)=c_n\textstyle \int\limits ^\infty _0 D_n(R)R^6i_0(hR){\,{\rm d}}R,\eqno (2.6.1.54)]$ where [c_n] is a constant, the factor [R^6] takes into account the fact that the particle volume is proportional to [R^3] , and [i_0(hR)] is the normalized form factor of a particle size R. In many cases, one is interested in the mass distribution [D_m(R)] [sometimes called volume distribution [D_c(R)] ]. In this case, we have $[I(h)=c_m\textstyle \int\limits^\infty_0D_m(R)R^3i_0(hR){\,{\rm d}}R.\eqno (2.6.1.55)]$

The solution of these integral equations, i.e. the computation of [D_n(R)] or [D_m(R)] from I(h), needs rather sophisticated numerical or analytical methods and will be discussed later.

The problems of interparticle interference and multiple scattering in the case of polydisperse systems cannot be described analytically and have not been investigated in detail up to now. In general, interference effects start to influence data from small-angle scattering experiments much earlier, i.e. at lower concentration, than multiple scattering. Multiple scattering becomes more important with increasing size and contrast and is therefore dominant in light-scattering experiments in higher concentrations.

A concentration series and extrapolation to zero concentration as in monodisperse systems should be performed to eliminate these effects.

References

Mittelbach, P. & Porod, G. (1962). Zur Röntgenkleinwinkelstreuung verdünnter kolloider Systeme. VII. Die Berechnung der Streukurven von dreiachsigen Ellipsoiden. Acta Phys. Austriaca, 15, 122–147.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 99