The determination of diffuse-boundary thicknesses of polymers by small-angle X-ray scattering

Abstract

Thermal density fluctuations within phases and finite interphase widths lead to systematic deviations from Porod's law. The validity of present methods used to analyze these deviations and determine diffuse‐boundary widths is determined. In view of the inadequacies found in these methods, a simple yet accurate method is proposed to determine the diffuse‐boundary width from direct graphical analysis of slit‐smeared intensity data. The diffuse interface is modelled by a sigmoidal‐gradient model which is justified on thermodynamic grounds, with the interphase thickness as a function of the Flory–Huggins interaction parameter.

Diffuse Interfaces and Small-Angle Scattering Intensity Behaviour

“…In contrast, the second contribution Zint(h ) is an interference (int) term which cannot be interpreted as an intensity because it can be negative for some h values. Different ways of approximating Zbck(h ) and their implications for the relevant best-fit results have been thoroughly discussed by Luzzati et al (1961), Ruland (1971), Vonk (1973), Koberstein et al (1980) and Roe (1982). A further approximation, based on rather general arguments, will be discussed later.…”

Examination of errors in the determination of phase boundary thickness by small-angle X-ray scattering

“…The simplest is to expand H2(s) in a series and retain the first two terms only. Koberstein, Morra & Stein (1980) showed that such a procedure is likely to lead to an underestimate of a when the analysis is performed over a range of s in which truncation of higher-order terms is not justified. They instead proposed a procedure, applicable when H2(s) is given by (2), which is valid even for s greatly exceeding 1/a.…”

“…Deviations from this Porod law can occur in practice since the boundaries between the phases may not be perfectly sharp, and the electron densities within the phases contain local fluctuations. The potential utility of determining the phase boundary thickness, by examination of the deviation of observed SAXS intensity from Porod's law, was first pointed out by Ruland (1971) and has since been examined by a number of workers (Vonk, 1973;Ruland, 1974;Hashimoto, Todo, Itoi & Kawai, 1977;Todo, Hashimoto & Kawai, 1978;Hashimoto, Shibayama & Kawai, 1980;Koberstein, Morra & Stein, 1980;.…”

Mechanical Properties versus Morphology of Ordered Polymers. Volume III. Part I

“…The work done in bending to produce the elastica'is given by: w f ds (75) where M is the bending moment. Equating expressio'ns (74) and (75) Considering any element of arc length the expression under the integral sign in equation (76) must therefore be equal to zero and equation (76) If we neglect the effect due to shear, equation (77) …”

“…This is termed the liquid-like scatter, and may be obtained from the slope of a Porod plot [75], a plot of 16W4 Is4 and 16W4 S4. DF micrograph of PBT fragment suspended over a gold decorated, perforated support film (S).…”