Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry

Abstract

Micelles are the simplest example of self-assembly found in nature. As many other colloids, they can self-assemble in aqueous solution to form ordered periodic structures. These structures so far all exhibited classical crystallographic symmetries. Here we report that micelles in solution can self-assemble into quasicrystalline phases. We observe phases with 12-fold and 18-fold diffraction symmetry. Colloidal water-based quasicrystals are physically and chemically very simple systems. Macroscopic monodomain samples of centimeter dimension can be easily prepared. Phase transitions between the fcc phase and the two quasicrystalline phases can be easily followed in situ by time-resolved diffraction experiments. The discovery of quasicrystalline colloidal solutions advances the theoretical understanding of quasicrystals considerably, as for these systems the stability of quasicrystalline states has been theoretically predicted for the concentration and temperature range, where they are experimentally observed. Also for the use of quasicrystals in advanced materials this discovery is of particular importance, as it opens the route to quasicrystalline photonic band gap materials via established water-based colloidal self-assembly techniques.

Fig. 1.

Synchrotron SAXS (A, B, and C) and SANS (D and E) diffraction patterns of the fcc phase (A) and the quasicrystalline phases Q12 (B, D, and E) and Q18 (C). The SANS patterns have been recorded parallel (D) and normal (E) to the 12-fold rotation axis The circles and lines indicate the position of the reflections that are compared to the calculated positions for a quasicrystal in Fig. 6. The scattering vector q is given in nm^-1; the intensity is displayed on a logarithmic scale.

Fig. 6.

Diffraction patterns of the dodecagonal phase (A and D), the 0°/30°-rotated multidomain fcc phase (B and E), the enneagonal phase (C), and the 0°/20°/40°-rotated multidomain fcc phase (F). A and C are calculated for the h₁h₂h₃h₄0 reciprocal space layer, D perpendicular to it. The encircled reflections are observed experimentally. The reflections encircled in white occur for both the quasicrystalline and the mixed fcc structure. The reflections encircled in yellow occur only for the quasicrystalline structures and are observed experimentally. The reflections for the quasicrystals are indexed based on a set of five reciprocal base vectors for the Q12 phase and on a set of seven reciprocal base vectors for the Q18 phase.

Fig. 2.

Series of synchrotron small-angle X-ray scattering curves for isotropic samples in a concentration range between 14 and 20%. At concentrations of 18 and 20%, the observed peak positions are in good agreement with an fcc structure (space group ). The scattering curves at lower concentrations are characterized by a suppression of the intensity of the 110 reflection and the development of a peak at q^∗ = 0.45 nm^-1, which for the dodecagonal structure corresponds to the 20100 reflection.

Fig. 3.

Time-resolved microfocus synchrotron SAXS experiment showing the temperature-induced transition from the fcc phase to the Q12 and Q18 phases. The diffraction patterns were recorded in 10-s frames. The dotted line indicates that the involved length scale during the phase transition remains constant. The transition into the Q12 phase proceeds by appearance of an additional set of six reflections after 10 s and subsequently into the Q18 phase by the splitting into further six reflections after 50 s, as indicated by the arrows.

Fig. 4.

Micellar packing in the fcc phase (A and D), in a multidomain 0°/30°-rotated, twinned fcc phase (n = 2) with 12-fold symmetry (B and E), and in a multidomain 0°/20°/40°-rotated twinned fcc phase (n = 1) with 18-fold symmetry (C and F). (Upper) A view normal to the micellar layers; (Lower) a view perpendicular to the layers.

Fig. 5.

Schematic representation of a layer of hexagonally packed polymer micelles (A), an overlay showing the structural relation between an fcc (111) layer and the corresponding layers of the enneagonal and dodecagonal quasicrystals (B and C), a periodic tiling and the tile representing the [111] projection of the fcc phase (D), the tiling and the tiles of the dodecagonal Q12 phase (F) and the enneagonal Q18 phase (G), and an overlay showing the structural relation between the fcc crystal and the enneagonal and dodecagonal quasicrystals with their respective PAS (E). Different colors represent different layers.