Pamela T. Korda and David G. Grier
Dept. of Physics, James Franck Institute, and
Institute for Biophysical Dynamics
The University of Chicago, Chicago, IL 60637
Date: January 9, 2001
Uniform colloidal microspheres dispersed in a fluid medium have a natural tendency to organize themselves into regular three-dimensional arrays known as colloidal crystals. Colloidal self-assembly offers a means of distributing materials in complex three-dimensional structures over length scales ranging from nanometers to micrometers and with order extending, in principle, over macroscopic dimensions. Consequently, colloidal crystals have potential applications as components of photonic , optoelectronic , and sensor  devices. Furthermore, they are of fundamental interest as model systems for studying the microscopic mechanisms of crystallization and melting .
Many potential applications for colloidal crystals require samples which are free from defects over macroscopic length scales . Unfortunately, colloidal crystals are subject to many of the same defects which mar conventional crystalline materials, such as edge dislocations, stacking faults, grain boundaries, vacancies, interstitials and impurities. This Letter describes how optical gradient forces, generated with optical tweezers, can be used to correct some classes of lattice defects in thin colloidal crystals. Our approach is best suited to removing local defects from large crystalline domains and might be used in conjunction with other pre- and post-processing steps to achieve microscopic perfection in colloidal crystals over very large dimensions.
Several approaches have been proposed for minimizing defect formation during colloidal crystallization. Temperature gradients have been used to control the deposition rate of spheres onto a growing crystal, through the Soret effect . Although the resulting crystals have macroscopically large domains, light scattering reveals random close packed ordering rather than the face-centered cubic (FCC) symmetry expected for repulsive spheres[7,8]. Epitaxial growth by sedimentation onto lithographically defined templates  has been very successful in creating well-oriented crystals of exceptionally high quality. Even templated crystals, however, can suffer from microscopic defects whose relaxation can be prohibitively slow.
An alternative approach to achieving microscopic order is to remove defects after the crystal has self-assembled. Thermal cycling is commonly used to anneal atomic and molecular solids. Unfortunately, this approach cannot be applied to colloidal crystals in general. The free-energy barriers to collective rearrangements are often large, and the associated relaxation rates are slow [8,10,11]. Furthermore, the temperature range over which a desired phase is stable can be relatively small. Heating the entire suspension enough to enhance defect recombination can drive undesirable phase transitions, induce disruptive convection currents in the supporting fluid, or even damage the colloidal particles.
Mechanical shearing has been shown to induce crystalline order in otherwise fluid suspensions  and to iron out stacking faults and other large-scale topological defects in face-centered cubic colloidal crystals [10,13]. However, shearing leaves behind twin boundaries and other smaller-scale defects .
In any annealing process, the goal is to provide the activation energy needed to relax existing defects without generating new ones. By applying controlled forces to individual colloidal particles, optical tweezers offer one route to this goal. Optical tweezers  use forces exerted by intensity gradients in strongly focussed laser beams to trap and move colloidal particles. We use a rapidly translating optical tweezer to introduce elastic strain locally within a thin colloidal crystal, thereby delivering kinetic energy to the lattice without heating the surrounding fluid.
Our samples consist of -diameter silica spheres (Bangs Labs lot number 4258) dispersed in deionized water (ionic concentration on the order of M) at a volume fraction . The particles are confined between two glass plates separated by . Under these conditions, the spheres tend to form three to four layers of close-packed crystal with a nearest-neighbor separation of . Densest packing under these conditions varies from three layers of FCC-like triangular symmetry to four layers of body-centered cubic (BCC) symmetry [15,16].
The sample is mounted on the stage of an inverted optical microscope, as shown schematically in Fig. 1, and allowed to equilibrate overnight. This is enough time for crystal domains to grow to roughly 1 mm in extent. The heterogeneously nucleated crystals tend to have many imperfections, including random stacking faults and grain boundaries. A typical field of view appears in Fig. 2 and reveals a high degree of microscopic order marred by a concentration of topological defects.
We project an optical tweezer into the crystal by directing a laser beam (wavelength 532 nm) through the microscope's objective lens (Olympus S-Plan Apo NA 1.4 oil immersion), by means of a dichroic mirror. White light illumination passes through this mirror and forms images of the particles on an attached CCD camera. Laser light, on the other hand, is brought to a diffraction-limited focus in the focal plane where it forms a three-dimensional optical trap. By tilting a galvanometer-driven mirror (Cambridge Technology Model 6350) centered at a conjugate point to the back aperture, as shown in Fig. 1, we can move the optical trap across the field of view at up to 400 Hz.
Given sufficient laser power (100 mW, in this case), scanning the tweezer back and forth through the crystal effectively melts a swath roughly 3 lattice constants across. At lower power, the tweezer cannot pull a sphere out of the potential well formed by its neighbors, but still can still elastically strain the sphere's neighborhood. In either case, the tweezer imparts activation energy needed to relax many-particle topological defects embedded in an otherwise well-ordered domain. Translating the crystal past the scanning tweezer leads to defect reduction over large areas through a process analogous to zone refining in conventional materials. For the present study, we used a tweezer scan range of at a repetition rate of 3 Hz and focused on comparably sized defects of three different types for which excitation-induced relaxation is particularly effective: misoriented domains, phase slips, and stacking faults.
Fig. 2(a) shows one face of a three-layer self-assembled colloidal crystal. It contains many defects, most notably numerous phase slips and a small misoriented domain, outlined in white. The defects can be seen more clearly in the Voronoi diagram, Fig. 2(d), corresponding to Fig. 2(a), whose domains are shaded to emphasize orientation.
Ordinarily, the colloidal lattice is in thermal equilibrium with the surrounding fluid. The kinetic energy imparted to the spheres by the scanning tweezer disrupts this equilibrium, effectively melting the lattice locally while leaving the fluid unaffected. Spheres in the zone-melted region have an opportunity to explore new configurations. Once the tweezer moves away from the region, they rapidly return to thermal equilibrium with the suspending medium and recrystallize, adopting the orientation of the largest and most rigid domain in their vicinity. The latent heat of melting released in this process is small compared with the heat capacity of the fluid, and thus the system's temperature does not change perceptibly.
Figures 2(b) and 2(e) show the same crystal after part of it has passed through the scanning tweezer, whose path is indicated by a dashed line. The left half of the previously-misaligned domain has been absorbed into the surrounding large domain, while the right half, which has not yet been locally melted, retains its original orientation.
Once the entire region has passed through the scanning tweezer, as shown in Figs. 2(c) and 2(f), the misoriented domain has been completely incorporated into its larger neighbor. Furthermore, many, although not all, of the smaller topological defects have been ironed out. The remaining smaller defects can be annealed out by repeated passes.
Moving this particular crystal through the field of view exposes a neighboring region of three-layer triangular order. The interface between square and triangular domains moves little under the influence of the scanned tweezer, possibly because far more particles comprise the triangular domain, and possibly because variation in the sample cell's thickness across the field of view stabilizes the three-layered structure and pins the interface between the domains .
The same technique used to eliminate the misaligned grain can be used to remove smaller defects, such as linear phase slips. Figure 3(a) shows the Voronoi diagram for a region containing one such defect. The Wigner-Seitz cells corresponding to the defect are shaded. Figure 3(b) shows the same region, after the crystal has passed through the scanning tweezer twice. As was the case for the fairly large defect shown in Figure 2, local recrystallization allows the spheres to explore their configuration space and find the lowest-energy state.
To remove small self-contained defects, it is not even necessary to melt the crystal locally, as is demonstrated in the sequence of images in Fig. 4. Here a well-localized stacking fault in the  face of the FCC domain, shown in Fig. 4(a), is stressed indirectly by an offset low-power scanning optical tweezer. The resulting elastic distortion rapidly drives the collective rearrangements needed to remove the defect.
The foregoing examples demonstrate that a scanning optical tweezer can be used to correct a variety of structural faults in self-assembled colloidal crystals without requiring particular attention the defects' detailed structures. While optical annealing alone would be a time-consuming approach to creating perfect macroscopic crystals, it could be a useful post-processing step for a controlled-growth technique, such as colloidal epitaxy, which naturally results in high-quality crystals. Furthermore, this technique could be applied to an entire sample without specifically identifying defective regions. This would make it possible to use optical zone refining in an automated system to reduce defect densities in colloidal crystals. One limitation of this technique is that optical zone refining requires optically thin crystals into which the tweezer can penetrate.
The present study focuses on qualitative features of defect remediation in colloidal crystals by scanned optical tweezers. Identifying the tweezer-induced local disruption with local heating, however, suggests a broader role for the techniques described. Controlled local heating should be useful for probing colloidal suspensions' thermodynamic properties, including energy barriers to defect formations in colloidal crystals, barriers to crystallization in colloidal glasses, and viscoelastic properties of the broadest range of suspensions. Such investigations are ongoing.
This work was funded by the National Science Foundation
through grant number DMR-9730189 and through the MRSEC
program of the NSF through grant DMR-9880595.