In-Plane Stacking Disorder in Polydisperse Hard Sphere Crystals Janne-Mieke Meijer, Volkert W. A. de Villeneuve, and Andrei V. Petukhov* Van’t Hoff Laboratory for Physical and Colloid Chemistry, Department of Chemistry, Faculty of Science, Utrecht UniVersity, Padualaan 8, 3584 CH Utrecht, The Netherlands ReceiVed October 9, 2006. In Final Form: NoVember 22, 2006 We demonstrate that in random-stacking hard-sphere colloidal crystals the stacking disorder not only exists in the direction perpendicular to the close-packed layers, but also extends in the lateral direction. The existence of such in-plane stacking disorder is suggested by a recent observation of lateral broadening of the Bragg scattering rods in microradian X-ray diffraction and is further confirmed here by real-space confocal microscopy in two hard-sphere colloidal systems with different relative gravity effects. Due to the in-plane stacking disorder, the hexagonal planes consist of islands with different lateral A, B, and C positions with characteristic line defects in between them. The real-space layer-by-layer stacks of images also reveal the 3-D structure of the defects. The chance  to find another line-defect above a line-defect in the layer below turns out to be close to 1/2sindependent of relative gravityswhich can be explained by the two different stacking options above a defect. The stacking of a few sets of several line defects situated on top of each other turns out to be predominantly FCC-like. I. Introduction Inherent to the natural ordering process of colloidal crystal- lization is the formation of defect structures, such as stacking faults, grain boundaries, and vacancies. The most efficient packing of spheres can be achieved by arranging them into stacked hexagonal close packed layers. 1 Regular stacking sequences of ABCABC and ABABAB types lead to face-centered cubic (FCC) and hexagonal close-packed (HCP) structures, respectively. Although the FCC structure was proven to provide a little bit more space for particle fluctuations, the free-energy difference between FCC and HCP is extremely tiny, 2-4 less than 10 -3 k B T per particle, where k B T is the thermal energy. As a result, a so-called random-hexagonal close-packed (RHCP) crystal struc- ture is often found. 5-9 Moreover, an additional gain of entropy due to the variety of possible RHCP configurations can stabilize the RHCP structure 10 for sufficiently small crystals, although a flat wall seems to promote the FCC structure. 11 Since crystal nuclei have RHCP structure and the reorganization from RHCP to FCC structure is expected to take months to years for experimental systems, 8,12 RHCP structures are quite commonly observed in hard-sphere systems. The typical quantity of defects encountered is therefore much larger than that expected for equilibrium conditions. The crystal structure can be conveniently characterized using diffraction techniques. 9,13 In the reciprocal lattice of RHCP crystals, the reflections with h-k not divisible by 3 are smeared out into so-called Bragg rods along the l direction as a result of the stacking disorder. Here we use the usual RHCP basis, described in more detail in, for example, ref 9. However, since only three distinct lateral positions are involved, the stacking- independent reflections with h-k divisible by 3 remain sharp. They correspond to periodicities common to all layers and will be referred to as Bragg spots. Figure 1a displays an example of a high-resolution small-angle X-ray scattering (SAXS) pattern obtained at the beam line BM-26 ‘DUBBLE’ 14 of the European Synchrotron Radiation Facility (Grenoble, France) using mi- croradian diffraction setup. 13 The pattern is measured in a single colloidal crystal with an X-ray beam orthogonal to the hexagonal planes (i.e., l ) 0). The intensity of the diffraction peaks strongly decay with increasing length of the diffraction vector q mostly due to the decay of the form factor. In addition, the structure factor of the Bragg spots (e.g., 110, 220, 300) is much stronger than that of the reflections originating from the Bragg rods (e.g., 100, 210). Recently, microradian-resolution small-angle X-ray diffraction of a sedimented colloidal single-crystal revealed that the Bragg rods and Bragg spots have different widths also in the direction orthogonal to the l. 13 Usually, one assumes that every layer has a unique lateral position and the stacking disorder is present only in the direction perpendicular to the close-packed layers. In that case both the stacking-dependent Bragg rods and the stacking- independent Bragg spots should be equally sharp in the lateral direction. The observed additional broadening of the Bragg rods within the (h,k) plane can be understood if one assumes that the close-packed hexagonal (111) planes consist of islands with different A, B, or C lateral positions of the spheres with characteristic line-defects in between them as illustrated in Figure 1b. The stacking-independent Bragg spots, such as the 110 reflection, do not “notice” the difference between the islands, and therefore, they are not broadened by the island structure of the layers. In contrast, switching from one island to the other leads to an additional phase shift of (120° of their contributions * Corresponding author: a.v.petukhov@chem.uu.nl. (1) Sloane, N. J. A. Nature 1998, 395, 435-436. (2) Bolhuis, P. G.; Frenkel, D.; Mau, S. C.; Huse, D. A. Nature 1997, 388, 235. (3) Bruce, A. D.; Wilding, N. B.; Ackland, G. J. Phys. ReV. Lett. 1997, 79, 3002. (4) Mau, S. C.; Huse, D. A. Phys. ReV.E 1999, 59, 4396. (5) Pusey, P. N.; van Megen, W.; Bartlett, P.; Ackerson, B. 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N.; Ryan, A. J.; Heeley, E. J. Appl. Cryst. 2003, 36, 791-794. 3554 Langmuir 2007, 23, 3554-3560 10.1021/la062966f CCC: $37.00 © 2007 American Chemical Society Published on Web 02/20/2007