We exploit the shear-melted fluid as a starting point to examine the microscopic processes underlying nonequilibrium crystallization under the influence of geometric confinement. Although the initial crystal consists of two layers of squares, the shear-generated fluid freezes first to a single-layer triangular crystal. While this intermediate crystalline state is well ordered in the plane, it is subject to a significant out-of-plane buckling distortion. The system regains its equilibrium configuration by undergoing a martensitic transition from this buckled triangular layer. The major stages in this process can be seen in Fig. 2.
Both layers of spheres are visible in Figs. 2(a-f) because the microscope is focused at the mid-plane of the sample volume. The true unit cells in the ordered stages actually are superlattices of the apparent structures, as can be seen when the focal plane is adjusted in Figs. 2(g) and (h). Focusing at mid-plane allows us to track all of the spheres in the sample volume at all stages of freezing and to identify correlated behavior in the two layers.
We identify each sphere in the plane as a local maximum in the image's brightness field. A brightness-weighted interpolation scheme allows us to locate centers to within 50 nm in each coordinate, or better than 1/6 of a sphere diameter. The raw data in this study, then, is the distribution of locations for the spheres in the mm field of view:
We use the set of sphere locations to calculate measures of order with `atomic' resolution.
The apparent areal density, , of spheres in the field of view appears as a function of time in Fig. 3. If all spheres were visible throughout the experiment, then we would expect this curve to be constant at the equilibrium density m except for noise due to spheres entering and leaving the field of view at the edges. Instead, the density is some 10 percent smaller than for as long as 30 seconds after shearing. In the disordered state, some spheres become occluded by others and are missed by the particle tracking algorithm. For a uniformly disordered suspension at volume fraction whose accessible thickness is sphere radii, the fraction of spheres visible at any time should be no greater than
This result could be modified to reflect the nonuniform vertical density profile induced in the fluid by the wall potential  and by layering in the fluid  although these corrections should be small. The value for our experimental conditions (, ) appears as the lower dashed line in Fig. 3 and accounts reasonably well for the observed degree of occlusion in our images. This agreement supports the contention that the volume fraction of spheres in the field of view is constant throughout the experiment.
Several of the most useful measures of local order require calculating the network of nearest-neighbors for the set of particle locations. Although such networks may be calculated in many ways, the Delaunay triangulation  possesses some properties, including uniqueness and an efficient implementation , which render it particularly attractive. The dual network to the Delaunay triangulation of a set of points is its Voronoi diagram, examples of which appear in Figs. 2(b,d,f,and h). Polygons in a Voronoi diagram are commonly known as Wigner-Seitz cells or nearest-neighborhoods. An examination of the Voronoi polygons in Fig. 2 makes clear the distinct changes which take place in the transitions from the shear-disordered state back to the two-layer system with square symmetry.
One way to quantify the evolution of symmetry in this system is to count the number of sides in the Voronoi polygons. Such a straightforward construction, however, can be misleading. Infinitesimally shearing a square lattice, for example, causes the typical Voronoi polygon to have six sides, two of which are infinitesimally short, rather than the empirically more satisfactory four sides. Similarly, very small displacements of individual particles can cause large changes in apparent coordination in their neighborhoods. To work around this problem we define the fractional coordination number
where A is the area of the polygon, R is the mean distance from its vertices to its center, and P is its perimeter. Eqn. (7) provides the exact result for a regular n-gon and a smaller value, , for a distorted convex n-gon. In practice, slightly clipped squares such as those discussed above yield values of only slightly greater than 4, whereas slightly sheared hexagons yield values slightly smaller than 6, as desired.
Eqn. (7) is useful for verifying the otherwise subjective observation that the ensemble of spheres achieves triangular order before regaining its equilibrium structure. The time evolution of the fractional coordination number distribution appears in Fig. 4. The most common coordination number shifts fairly rapidly but smoothly from a value near to one near starting at about sec.
While Fig. 4 demonstrates that 4-fold and 6-fold coordinated sites predominate in this system and that their relative populations vary continuously, it does not lend itself to measurements of these relative populations. To this end, we introduce the complex 6-fold and 8-fold bond orientational order parameters for the i-th particle :
where is the angle made with respect to a fixed direction by the segment connecting particle i to its j-th nearest neighbor. The angle brackets indicate an average over nearest neighbors and m is either 6 or 8. The magnitude of the m-fold bond orientational order parameter is 1 for a perfectly m-fold coordinated site and smaller otherwise. Nearest neighbor identifications for square lattices include diagonal bonds and so mandate the use of rather than to quantify the degree of local 4-fold order. Because 8-fold coordinated sites are rare, the bias introduced by this exigency should be quite small. We define a site to be ordered if either or and disordered otherwise. An ordered site is defined to be 6-fold coordinated if and 4-fold otherwise. Using these classifications, we calculate the population fractions for the 6-fold and 4-fold sites ( and , respectively) which appear in Fig. 5. The simultaneous decrease in and increase in at sec suggests that this time can be taken as the onset of the symmetry-changing martensitic transition.