Use of SAXS and linear correlation functions for the determination of the crystallinity and morphology of semi‐crystalline polymers. Application to linear polyethylene

Abstract

ABSTRACT:The use of correlation functions to obtain the morphological parameters of crystalline-amorphous two-phase lamellar systems is critically reviewed and extended. It is shown that processing of the experimental SAXS-patterns only significantly affects the curvature of the autocorrelation triangle and that the parameters of the corresponding ideal two-phase structure can be determined independently of the data processing procedure. The methods to be used depend on the normalization of the correlation function. The validity of the formulation is illustrated for a sample of linear polyethylene, cooled and heated at 10°C per min. Crystallite thickening during crystallization and surface melting during heating are observed. The overall crystallinity and the fraction of semi-crystalline stacks during crystallization and melting are determined quantitatively as a function of temperature using the total scattering power of the corresponding ideal two-phase structure, correlation functions, and a scaling procedure. Absolute intensities are not required. The SAXS results are confirmed by independent techniques (DSC, WAXD, and SALLS). During crystallization, amorphous regions are present outside the semi-crystalline regions because growing spherulites do not fill space completely. During melting, larger amorphous regions develop in the spherulites because of the complete melting of stacks.

X-Ray Scattering of Soft Matter

“…The general inferiority of geometrical construction methods [162,163] as compared to more involved methods which consider polydispersity has first been demonstrated by SANTA CRUZ et al [130], and later in many model calculations by CRIST [165][166][167]. Nevertheless, in particular the first-zero method is frequently used.…”

“…Obviously, the autocorrelation triangle of the ideal lattice (dotted curve) is not preserved in paracrystalline stacks of higher polydispersity. Thus, a simple linear extrapolation ("linear regression autocorrelation triangle", LRAT [162]) will only yield reliable information concerning the properties of the idealized lattice from the real data, if the polydispersity remains rather low.…”

“…Several publications describe the search for a simple graphical analysis [22,159,[162][163][164] of the 1D correlation function by means of a geometrical construction. It is the drawback of all such methods that polydispersity and heterogeneity are not considered.…”

“…(8.67) can be applied to a renormalized correlation function γ 1 (x/L app ). The method which has been proposed by Goderis et al [162] is based on the implicit assumption that the first zero, x 0 , of the real correlation function is shifted by the same factor as is the position of its first maximum,L app . The idea is already described in the first paper of VONK and KORTLEVE ( [159], p. 22) as a method to retrieve fit parameters.…”

“…If the initial part of the correlation function exhibits significant deviations from a straight line, the proposers of the first-zero method recommend to carry out a linear regression (LRAT) [162] on the autocorrelation triangle. The problem of doing so is demonstrated in Fig.…”