High-Resolution Small-Angle X-Ray Diffraction Study of Long-Range Order in Hard-Sphere Colloidal Crystals

Abstract

The long-range order parameters in single crystals of hard colloidal spheres grown in sediments of colloid-polymer mixtures are determined using synchrotron small-angle x-ray diffraction with a resolution of 10 26 of the wave vector. The interplanar positional order derived from the width of lattice reflections extends over at least 500 lattice planes. The lattice planes are orientationally correlated within ϳ0.1 ± throughout the crystals, whereas the stacking of hexagonal planes remains random. DOI: 10.1103/PhysRevLett.88.208301 PACS numbers: 82.70.Dd, 42.70.Qs, 61.10. -i Suspensions of hard-sphere colloids display an entropydriven fluid-crystal transition. This remarkable phenomenon widely serves as a simple model of crystallization in atomic systems [1][2][3]. Recently, colloidal crystals are also attracting significant attention for their high potential as photonic band gap materials [4]. For these applications the emphasis is on the preparation of large single crystals possessing long-range order, with a low density of defects and small fluctuations of the particle positions. The growth of millimeter-large colloidal crystals has been reported by many researchers [1,2], while the actual order parameters are hardly ever addressed. The elongated shape of the reflections in an earlier synchrotron x-ray diffraction experiment [5] on charge-stabilized single colloidal crystals clearly indicates that these crystals have a mosaic structure consisting of many orientationally correlated but positionally independent fragments. In this Letter the characterization of long-range order parameters in hardsphere crystals is performed using high-resolution smallangle x-ray diffraction.The challenge of x-ray diffraction to characterize longrange order in colloidal crystals stems from the very large difference between the typical lattice period a and the x-ray wavelength l; their ratio e l͞a is typically of the order 10 23 ø 1. The length L over which the positional order is preserved manifests itself in the width dq of the reciprocal lattice reflections [6,7]. For L as large as ϳ10 3 a, the width dq is of the order of 10 26 of the x-ray wave vector k 0 2p͞l. To resolve it, one needs to fulfill conditions of coherent interference of diffracted waves over distances larger than L ϳ 10 6 l.In the transverse direction the coherence length l tr l͞q [8] is mostly governed by the angular size q of the radiation source seen from the position of the sample. Parts of the crystal separated in the transverse direction by more than l tr are irradiated by waves with different and uncorrelated phases. The positional order in the transverse direction can thus be resolved only on a distance of N tr l tr ͞a e͞q (1) lattice periods.In the longitudinal direction the coherence length l long l 2 ͞Dl is mostly restricted by the spectral width Dl of the radiation source. The waves diffracted at the front and back faces of the sample are coherent if the length of their path from the source to the detector differs by less than l long . This conditi...

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