X-ray scattering by bicontinuous cubic phases

Abstract

Ia3d bicontinuous cubic phases are interpreted as constituted of a fluid film of constant thickness supported by the gyroid minimal surface. We first compute the gyroid structure factor. Then we decorate the surface by the film. We compute the diffracted intensities and compare to X-ray intensities measured in some lipid systems

Three-dimensional periodic complex structures in soft matter: investigation using scattering methods

“…We calculated the G-surface structure factors f G hkl using the exact G-surface IPMS [48] and their values are given in electronic supplementary material, table S1 along with more recent calculations [15,47]. Using the approximation of the G-surface by Von Schnering & Nesper [53] by a nodal surface of equation sinð2pxÞ cosð2pyÞ þ sinð2pyÞ cosð2pzÞ þ sinð2pzÞ cosð2pxÞ ¼ 0;…”

“…To take into account the thickness of the film, L, we introduced a simple method where the points on the G-surface are convoluted in real space by an infinitely thin spherical shell of radius R¼L/2 (figure 5) [48]. The main advantage of this method is that the form factor of this spherical shell (4pR 2 ðsinðqRÞ=qRÞ) appears simply as a product in the expression of Figure 5.…”

“…The cubic unit cell for the Ia 3d bi-continuous phase for the rods model and the G-surface model. Using the G-surface model, different methods can be used [14] to take into account the thickness of the film L (a) a linear profile along the normal at each point of the surface [47] (b) a convolution by an infinitely thin shell [48] (c) a lattice method with the calculation of the distance to the G-surface [49] (d ) the iso-density surfaces using the nodal surfaces approximation [50].…”

“…3 ) [48]. Taking into account the thickness L ¼ 2R, the final expression reads This model is valid only for sufficiently small values of R with respect to the mean curvature radius of the G-surface, typically for an R/a ratio less than 0.25 [48].…”

X-ray studies of the self-organized structure formed by 1,2-bis(4'-n-alkoxybenzoyl)hydrazine (BABH-n) homologues. 1. Ia3d-gyroid structure

“…One of the reasons is that due to their spatially and dynamically disordered nature only a few peaks are observed, which are not enough to reconstruct the three-dimensional electron-density map. One also encounters another complicated situation: as Luzzati et al (1968) and Clerc & Dubois-Violette (1994) reported, the SAXS patterns of some lyotropic Cub phases were reproduced by using a structural model where the aliphatic tails are placed on the TPMS (inverse phase), but the reverse situation, i.e. where the polar group is placed on the TPMS (direct phase), is equally plausible for intensity fitting.…”