Abstract
As is well known, polymers commonly form lamellar crystals, and these assemble further into lamellar stacks and spherulites during quiescent crystallization. Fifty years ago, Vonk and Kortleve constructed the classical small-angle X-ray scattering theory (SAXS) for a lamellar system, in which it was assumed that the lamellar stack had an infinite lateral size [Vonk & Kortleve (1967 ▸), Kolloid Z. Z. Polym. 220, 19-24]. Under this assumption, only crystal planes satisfying the Bragg condition can form strong scattering, and the scattering from the lamellar stack arises from the difference between the scattering intensities in the amorphous and crystalline layers, induced by the incident X-ray beam. This assumption is now deemed unreasonable. In a real polymer spherulite, the lamellar crystal commonly has dimensions of only a few hundred nanometres. At such a limited lateral size, lamellar stacks in a broad orientation have similar scattering, so interference between these lamellar stacks must be considered. Scattering from lamellar stacks parallel to the incident X-ray beam also needs to be considered when total reflection occurs. In this study, various scattering contributions from lamellar stacks in a spherulite are determined. It is found that, for a limited lateral size, the scattering induced by the incident X-ray beam is not the main origin of SAXS. It forms double peaks, which are not observed in real scattering because of destructive interference between the lamellar stacks. The scattering induced by the evanescent wave is the main origin. It can form a similar interference pattern to that observed in a real SAXS measurement: a Guinier region in the small-q range, a signal region in the intermediate-q range and a Porod region in the high-q range. It is estimated that, to avoid destructive interference, the lateral size needs to be greater than 11 µm, which cannot be satisfied in a real lamellar system. Therefore, SAXS in a real polymer system arises largely from the scattering induced by the evanescent wave. Evidence for the existence of the evanescent wave was identified in the scattering of isotactic polypropyl-ene. This study corrects a long-term misunderstanding of SAXS in a polymer lamellar system.
Highlights
In recent decades polymers have been extensively employed in numerous fields, for example mobile phones, computers, cars, aircraft and so on
IUCrJ (2019). 6, 968–983 research papers accurate determinations have been key to revealing the relationship between structure and properties. They can be determined by atomic force microscopy (AFM) (Mullin & Hobbs, 2011; Savage et al, 2015; Wang et al, 2018) and transmission electron microscopy (TEM) (Rastogi et al, 1997; Maiti et al, 2000; Yamada et al, 2003) in real space, or smallangle X-ray scattering (SAXS) in reciprocal space (Strobl & Schneider, 1980; Hashida et al, 2010; Wang et al, 2014)
At a finite lamellar lateral size, the scattering induced by evanescent waves, especially the scattering from interfacial electrons, is most likely to be the main origin of SAXS in polymer lamellar systems
Summary
In recent decades polymers have been extensively employed in numerous fields, for example mobile phones, computers, cars, aircraft and so on. This equation describes the statistical correlation between the electron densities at two arbitrary points separated by a fixed distance, carrying structural information on the lamellar stack The latter assumption is reasonable for an infinite lateral size, which can be seen from the contributions of crystal planes with different intersection angles to the scattering at a given wavevector. For the same crystal plane with a lateral size of 250 nm, only the crystal plane with i = 15 makes a significant contribution to the scattering at 2 = 30 [see Fig. 1(c)] This means that it is not necessary to consider the interference between lamellar stacks and the scattering induced by the evanescent wave. It has been reported that, at a lower temperature, iPP can form a lamellar two-phase structure (Zhu et al, 2001; Crist & Schultz, 2016)
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