Small-angle scattering behavior of thread-like and film-like systems

A film-like or a thread-like system is a system such that one of its constituting homogeneous phases has a constant thickness $\delta$ or a constant normal section of largest diameter $\delta$. The stick probability function of this phase, in the limit $\delta \to 0$, naturally leads to the definition of the correlation function (CF) of a surface or a curve. This CF fairly approximates the generating stick probability function in the range of distances larger than $\delta$. The surface and the curve CFs respectively behave as $1/r$ and $1/r^2$ as $r \to 0$. In the two cases, this result implies that small-angle scattering intensities of the relevant samples respectively behave as $1/q^2$ and $1/q$ in an intermediate range of the scattering vector $q$ and as $1/q^4$ in the outermost one. One reports the analytic expressions of the pre-factors of these behaviors. It may happen that a sample looks thread-like at large scale resolution and film-like at smaller one. The surface and the curve CFs have explicitly been evaluated for some simple geometrical shapes. Besides, it is also reported the algebraic expression of the circular cylinder CF in terms of two elliptic integral functions, and it is shown that the limits of this CF, as the height or the radius of the cylinder approaches to zero, coincide with the CF of a disk or a linear segment, respectively.Comment: 37 pages, 18 figure

Diffuse Interfaces and Small-Angle Scattering Intensity Behaviour

The contributions corresponding to the Porod, the oscillatory O(h−4) and the Kirste–Porod O(h−6) terms, present in the asymptotic expansion of the small-angle scattering (SAS) intensities, are numerically evaluated, in the presence of diffuse interfaces generated by different smoothing functions (Gaussian, spherical or Helfand–Tagami). It is shown that SAS experiments are generally unable to distinguish among different profiles, because any smoothing can be made to coincide with another type by scaling its thickness parameter. The oscillatory deviations are observable in the Porod plot of the intensities when the typical distance between parallel diffuse interfaces is greater than 50 A and the ratio of the thickness to this distance is less than 1/4. The same conclusion applies to the infinite-slit intensities