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

Abstract

The tails of the smeared small-angle X-ray intensities, scattered by two triblock copolymers of poly(methyl methacrylate-b-butadiene-b-methyl methacrylate) are analysed by using the smeared leading terms of the intensity asymptotic expansion. In this way, it is possible to determine the particle shape and size as well as the area and the Kirste±Porod curvosity of the sample interfaces.

Small-angle scattering intensity behaviours of cylindrical, spherical and planar lamellae

“…where R 1 and R 2 are two generic radii, the overbar denotes the complex conjugate and`denotes the real part. This average was explicitly calculated for the case R 1 = R 2 = R, where it yields the intensity scattered by a homogeneous cylinder of radius R and height H. The resulting expression, ®rst obtained by Miller & Schmidt (1962) [see also the work of Ciccariello (1989) and a note on page 511 of the paper by Ciccariello (1991a)], involves elliptic functions and is therefore of poor practical usefulness. Quite similarly, a closed form of (1) will also involve elliptic functions, so as to be practically useless.…”

“…1) and partly inside (this happens on the plane through the axis of the CL and the centre of the sphere) the region bounded by the lateral surface with radius (D À t) [see also Fig. 1 of Ciccariello (1989)]. By the same considerations, it appears evident that AE is hyperbolically parallel to all the lateral surfaces of the considered CL grain for = (m + 1)D À 2t, (m + 1)D À t, (m + 2)D À 2t, F F F , (N + m)D À t. It is noted that the former values are the sums of the radius of AE with those of all the lateral surfaces.…”

“…where m and n range between 1 and N. After determining the set of parameters that have to be considered in the two sums on the right-hand side of (2), we simply need to recall that the expressions of g { and | for two coaxial cylindrical surfaces of radii R 1 and R 2 and height H can immediately be obtained from equation (3.3) of Ciccariello (1991a). They respectively are…”

Reconstruction of the form of a particle from its three-dimensional asymptotic form factor

“…In the realm of small-angle scattering (SAS), the scattering density is assumed to be a two-value function (Debye et al, 1957). If we con®ne ourselves to the case of a dilute sample consisting of a monodisperse collection of identical particles equally oriented inside the sample, the observed scattering intensity I q takes the form (Guinier & Fournet, 1955) I q x p Án 2 F q Y 1 where q is the scattering vector, Án 2 the contrast, x p the particle number and F q the geometrical form factor of the particle.…”

“…It is now observed that samples obeying condition (i) have recently been observed . Moreover, the asymptotic analysis of some isotropic intensities (Ciccariello & Sobry, 1999) indicates that condition (ii) holds practically true even in the presence of a polydispersity smaller than 20% and one can con®dently expect that (ii) applies also in the presence of a small particle misalignment. Finally, condition (iii) is no longer necessary when the mean particle size exceeds 200 A Ê because, in this case, the SAS behaviour will be fairly independent of interparticle interference in the outer q range 0X1 À 0X3 A Ê À1 .…”