Application of an extended Porod law to the study of the ionic aggregates in telechelic ionomers

Porod's law extended to the sixth-order term can be written I = (K_p/s⁴ + K₆/s⁶)U²(s,σ) where I is the scattered intensity, s = 2(sinθ)/λ, θ being half the scattering angle and λ the wavelength used; U²(s, σ) describes the interphase profile and σ is a measure of the width of the interphase transition zone. K_p and K₆ are two constants. In the same way as K_p can be related to the specific area, K₆ is related to a pure number θ called here `angulosity'. For an angulous body, θ 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 that K₆/K_p = θ/2π²S so that, in a two-phase system, the ratio K₆/K_p 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 K_p, K₆ and σ. 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.