Scattering Curves of Ordered Mesoscopic Materials

Abstract

Analytical expressions for the scattering functions of ordered mesoscopic materials are derived and compared to experimentally determined scattering curves. Ordered structures comprising spheres (fcc, bcc, hcp, sc), cylinders (hex, sq), and lamellar structures are considered. The expressions take into account particle size distributions and lattice point deviations, domain size, core/shell structures, as well as peak shapes varying analytically between Lorentzian and Gaussian functions. The expressions allow one to quantitatively describe high-resolution synchrotron small-angle X-ray (SAXS) and neutron scattering (SANS) curves from lipid and block copolymer lyotropic phases, core/shell nanoparticle superstructures, ordered nanocomposites, and ordered mesoporous materials. In addition, the diffuse out-of-plane scattering of grazing incidence GISAXS and GISANS experiments of laterally ordered thin films can be quantitatively analyzed.

Structural and Mechanical Hysteresis at the Order-Order Transition of Block Copolymer Micellar Crystals

“…The horizontal axis is normalized in this way simply to highlight the relative peak positions. These peak positions are consistent with the hexagonally packed-cylindrical micelle phase (1: 3 :2) [18,35]. The peak at 3 is not observed in SANS due to the wavelength smearing and resolution issues that are inherent with neutron scattering experiments, but this was visible in the SAXS shown in Figure 5.…”

Structure and optical function of amorphous photonic nanostructures from avian feather barbs: a comparative small angle X-ray scattering (SAXS) analysis of 230 bird species

“…We used a Porod background (q 24 dependence; see §3 and figures 4-5) and the splitPearson VII function with a Levenberg -Marquardt least square method to fit all the observed scattering features ( peaks and shoulders) present in the azimuthal profiles. The Pearson VII function is a combination of Gaussian and Lorentzian (Cauchy) type peak profiles that is generally used to closely approximate X-ray scattering peaks [53,54]. The split-Pearson VII accommodates any asymmetry in peak shapes.…”

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

“…For this phase, the expressions of the intensity given in §2 have to be slightly modified because the lattice has two dimensions instead of three. In the following, using a simple model, we derive these expressions by assuming that the two-dimensional order is a true long-range order, and not a quasi-two-dimensional order like in the more sophisticated para-crystal model [14]. A two-dimensional hexagonal phase consists of the two-dimensional packing of very long particles having a cylindrical symmetry.…”

“…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].…”