Diffuse Interfaces and Small-Angle Scattering Intensity Behaviour

Abstract

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 HelfandTagami). 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.

Direct analysis of small-angle smeared intensity tails. II. Applications

“…On physical grounds it is expected that ' 10 A Ê and that the lengths which are unambiguously resolved by the asymptotic analysis of the intensity tails are not smaller than 50 A Ê (Ciccariello & Sobry, 1997). In these cases it will result that " 'a i 0X2, whatever i.…”

“…In general, it depends on the (local) dihedral angle along the edges and can be both positive and negative. Its general expression is fully known only in the case of vertices (Ciccariello & Sobry, 1995). Assume now that the particles inside our sample have a well de®ned shape and that they are monodisperse (this condition will be relaxed later).…”

“…For coaxial cylinders the leading oscillatory pinhole contribution is [see equation (3.4) of Ciccariello (1991) and substitute therein 2H i with…”

“…The expressions of parameters A i and B i are simply obtained from the asymptotic expansion (Ciccariello, 1991) of the pinhole intensities scattered from concentric spheres and coaxial cylinders. For concentric spheres, the leading oscillatory contribution is [see equation (3.1) of Ciccariello (1991)]…”

“…An application of the results obtained in the previous paper (Ciccariello & Sobry, 1999), hereinafter referred to as paper I, will now be illustrated. In fact, we shall analyse the outer parts of the small-angle X-ray intensities scattered by two samples of the triblock copolymer (pMMA-block-pBD-block-pMMA), consisting of methyl methacrylate (MMA) and butadiene (BD).…”