No abstract

Porod's law extended to the sixth-order term can be written I = (Ke/s 4 + K6/s6)U2 (s,o ") where I is the scattered intensity, s = 2(sin8)/,~, 0 being half the scattering angle and a the wavelength used; U2(s, tr) describes the interphase profile and tr is a measure of the width of the interphase transition zone. Kp and K6 are two constants. In the same way as Kp can be related to the specific area, K6 is related to a pure number 8 called here 'angulosity'. For an angulous body, 0 always is negative and can easily be calculated when its geometry is simple. It does not depend on the dimensions of the body. It is shown in the present paper thatso that, in a two-phase system, the ratio K6/Kp represents the angulosity per unit area S of the interface between the phases. A least-squares analysis of the experimental small-angle X-ray scattering (SAXS) curve gives the values of Kp, K6 and 0-. The method was successfully applied in the case of telechelic ionomers to characterize their ionic aggregates. These aggregates present a larger angulosity than that of a parallelepiped. Their volume is relatively small and only contains a small number of ions. The results agree with the results obtained by other techniques. It can be concluded from this that the introduction of the s -6 term into Porod's law is judicious and allows the structure of the phases to be better characterized.

Twenty seven samples of carboxylato- and sulfonato-telechelic polyisoprenes associated with various cations (Na, K, Rb, Cs, Mg, Ca, Sr, and Ba) have been investigated by small-angle X-ray scattering. The Bragg spacing characteristic of the ionic peak is directly proportional to the root-mean-square end-to-end distance (r rms) of the polyisoprene chain. In the series of sulfonato-telechelic polyisoprenes, the Bragg spacing is approximately equal to r rms, whereas in the series of carboxylato-telechelic polyisoprenes it amounts to 21/2 r rms. It also appears that the ionic aggregates are more likely distributed according to a planar hexagonal network. An original method has been used for the tail-end analysis of the SAXS profile, which is based on the general vertex contribution to the Kirste−Porod law. It results that the ionic aggregates are of an angulous shape. Four different functions have been used to account for the interphase profile between the ionic phase and the polymeric matrix. The ionic aggregates would accordingly contain an average of 10 alkali-metal cations with a tetrahedral stacking, whereas six alkaline-earth-metal cations would be organized according to an equilateral prism. The oxygen atoms of the anionic groups mainly contribute to the width of the interface. The additional peaks observed in the upturn of the curve at very low angles are the signature of a superstructure similar to that one previously observed for carboxylato-telechelic poly(tert-butyl acrylate)s.

Five interpenetrating polymer network (IPN) epoxy‐acrylate samples of the same composition synthesized with variable conditions – temperature and initiator rate – were studied by small‐angle X‐ray scattering (SAXS). The scattering curves, at first sight showing little detail, were meticulously analysed. The entire scattering curve of an IPN compound was analysed using new methods for background subtraction and desmearing. The treatment suggests a fractal behaviour of the internal surface, associated with this two‐phase system, on a scale between 100 and 10 nm and, in the tail end of the SAXS curves, reveals maxima corresponding to those of two regular spheres with radii of the order of 7 and 14 nm. Analysis of the beginning of the curves yields one or two correlation lengths according to the conditions of the synthesis, close to 100 and 20 nm. These results are consistent with the general model of IPN structures as revealed by other physico‐chemical techniques.

A preceding paper handled, by way of application, the usefulness of Porod's law extended to the second nonoscillating term. The h-6 term allows the structure of the phases to be better characterized. This paper is mainly concerned with the setting up of the main equations used in this preceding paper. The h-6 term is analysed from the correlation function 7(r) and related to the 'stick probability function'. It can be positive or negative. The positive case appears in smooth phases and has been previously analysed by Kirste & Porod. The negative case occurs in the presence of linear edges resulting from the meeting of surfaces that are planar in the vicinity of their intersection. More precisely, it is shown that the h-6 negative term results from the finite length of the edge. Its magnitude depends on the dihedral angles at the vertex defined by the limited sharp edges. The smaller the dihedral angles, the greater the h -6 term amplitude. The new concept of angulosity, 0, a pure number characterizing the geometry of the phase, is introduced. In this way, it is possible to develop similar equations for a specific surface, angularity and angulosity. Some simple-geometry examples are developed. The region where the extended Kirste-Porod law is useful in analysing small-angle scattering curves is discussed.