Bragg Rods and Multiple X-Ray Scattering in Random-Stacking Colloidal Crystals

Abstract

Synchrotron small-angle x-ray diffraction images of random-stacking-induced Bragg scattering rods are obtained in a wide range of wave vectors from a single colloidal crystal. The results reveal a strong multiple scattering effect, which leads to new features in the diffraction pattern -secondary Bragg rods. We argue that dynamic x-ray diffraction is rather common for high-quality colloidal photonic crystals and should be taken into account. DOI: 10.1103/PhysRevLett.90.028304 PACS numbers: 82.70.Dd, 42.70.Qs, 61.10.-i The spontaneous formation of crystals in dispersions of colloidal spheres [1,2] remains an intriguing phenomenon, which has aroused renewed interest because of its photonic applications [3]. Modern microscopy [4 -7] allows 3D imaging of colloidal crystals giving mainly information on local structure and ordering. Scattering techniques are more appropriate for the study of longrange order in colloidal crystals. Until recently this involved primarily light scattering [2,[8][9][10][11], which limits the range of scattering vectors severely and is known to be extremely prone to multiple scattering. In the x-ray field a new era has spawned by the advent of synchrotron radiation, which offers high-quality x-ray beams and has led to the development of 2D x-ray detectors. These now permit small-angle x-ray diffraction (SAXD) on colloidal single crystals [12] and allow us to achieve a resolution sufficient to determine long-range order parameters in high-quality crystals [13]. So far, the so-called kinematic theory, where it is assumed that diffraction is weak, has been used [12 -15] to describe x-ray diffraction patterns from colloidal crystals. However, already long ago it was realized that in perfect crystals with long-range periodic positional order, the kinematic approach fails and one has to use the more complicated dynamic theory [16], which accounts for mutual interactions between the incident and all the diffracted waves. While for SAXD in colloidal crystals the applicability of the kinematic approach is hardly ever questioned, our results indicate a significant effect of dynamic x-ray diffraction, which reveals itself in our crystals in an unusual way as novel features in the diffraction pattern.Calculations [17] indicate that colloidal hard spheres should favor a face centered cubic crystal as a closepacked structure, but in practice a random hexagonal close-packing (rhcp) structure is rather more common [5,[9][10][11]13,14] either as a transient state or possibly even permanently (factors such as polydispersity may readily influence the very small free energy differences involved). Because of the irregular stacking of hexagonal layers in three possible lateral positions, some features in the reciprocal lattice of rhcp are smeared out into Bragg scattering rods [10,13,14,18] in the direction perpendicular to the hexagonal layers [ Fig. 1(a)]. Periodicities common to all layers lead to sharp Bragg spots.We have been able to image Bragg scattering rods directly in a single diffraction patter...

Periodic order and defects in Ni-based inverse opal-like crystals on the mesoscopic and atomic scale

“…4 by dark circles. This is characteristic for the layered structure with a considerable amount of stacking disorder [36]. This finding confirms the more significant impact of the stacking defects than effect of limited thickness of the sample.…”

“…Within the usual Wilson theory [12] the probabilities (5) are then used to derive the structure factor assuming that the crystal has an infinite number of layers. The result [12,14,36,38,39] proves that some of the reflections are stacking-independent and remain sharp for any values of α. The other reflections smear out into diffuse Bragg rods along the normal to close-packed hexagonal planes.…”

“…Let us consider a layered structure formed by the hexagonal close-packed layers arbitrary stacked in the growth direction. For such a structure there are two types of reflections, the stacking-independent and stacking-dependent with the different structure factors [36,37]. If the sequence of layers is not perfect, i.e., possesses some deviations from ideal fcc or hcp, some additional features, namely diffuse Bragg rods passing through the stacking-dependent reflections, appear [36].…”

Evidence of the hexagonal columnar liquid-crystal phase of hard colloidal platelets by high-resolution SAXS

“…Here, we use sterically stabilised colloidal gibbsite platelets with non-adsorbing polymer as a depleting agent. We have been able to successfully prepare large crystals of hard colloidal spheres for SAXS studies [55,56] using such non-adsorbing polymer before. In addition, in the case of platelets, the depletion interaction is expected to favour a perpendicular orientation of the columns with respect to the sample walls.…”

A simple model for dynamic small-angle X-ray diffraction in colloidal crystals

“…structure (Sanders, 1968;Liu et al, 1993;Verhaegh et al, 1995;Zhu et al, 1997;Dux & Versmold, 1997;Pronk & Frenkel, 1999;Kegel & Dhont, 2000;Martelozzo et al, 2002;Petukhov et al, 2002Petukhov et al, , 2003Versmold et al, 2002), even though calculations indicate that colloidal hard spheres should prefer a face-centered cubic (f.c.c.) structure (Bolhuis et al, 1997).…”