The transient nature of the shear-induced fluid is revealed in the dynamics of individual spheres. To perform such measurements, we follow each sphere through a sequence of consecutive video frames to build up a stop-action picture of its trajectory. We identify spheres in consecutive frames with a maximum likelihood algorithm which minimizes the total length of all trajectories at each time step. For this algorithm to work, a single sphere's thermal motion must not carry it more than half of a typical nearest-neighbor separation in the 1/30 second interval between consecutive frames. A randomly diffusing sphere travels a mean-squared distance given by the Einstein-Smoluchowsky equation:

where is the self-diffusion coefficient.
For a sphere immersed in
a fluid of viscosity at
temperature **T**, is the same as the Stokes self-diffusion
coefficient,

Viscous drag at nearby smooth walls reduces this diffusivity by an amount which diminishes with distance from the walls. This effect was calculated by Lorentz in 1907[33] and recently was measured directly [34] for submicron spheres near glass walls. An isolated sphere whose center is 4 radii above the wall should have a self-diffusion coefficient . For the spheres in this study, m, so that the wall-corrected free self-diffusion coefficient is m/sec. Over a frame period of 1/30 second, a single sphere diffuses roughly 0.4 m in the plane. This is sufficiently smaller than the typical inter-sphere separation of 1.2 m that sphere tracking is practical.

The long-time self-diffusion coefficient is further reduced by interactions with neighboring spheres through the so-called ``cage effect'' [35,36]. Recently, Löwen, Palberg, and Simon [37] (LPS) suggested on the basis of forced Rayleigh scattering measurements and molecular dynamics simulations that the cage effect reduction in the self-diffusion coefficient adopts a universal value at freezing of . We have previously observed quantitative agreement with this empirical criterion in nonequilibrium freezing through direct observations on a bulk supercooled colloidal fluid [28]. Although its applicability to a strongly confined system remains to be justified, the LPS criterion provides a useful point of reference for the present study.

To estimate the self-diffusion coefficient ,
we track particles through
10 consecutive video frames beginning at time **t**
and calculate the mean-squared displacement,
,
averaged over the field of view.
A linear least-squares fit to eqn. (9) then provides a measure
of the time-dependent self-diffusion coefficient .
Such measurements, repeated at regular 2 second intervals,
appear in Fig. 6.
The choice of 1/3 second measurement periods is a trade-off between probing
long-time behavior and avoiding system-wide variations due to the
ongoing phase transitions.
We found it necessary to subtract a secular term from the trajectory
data to null out a small residual drift in the early stages of our experiment.
The rapid decline in directly after cessation of shearing contrasts
sharply with previous measurements on an 18-layer system for which fluid-like
diffusion is observed to persist for an hour or more.
It is consistent with observations of rapid surface-mediated crystallization
in sheared colloidal suspensions with only one wall [38].
In the present case, and
the LPS criterion
suggests that the suspension has already begun to freeze
by .

The Hansen-Verlet criterion provides an independent determination of the freezing point based on instantaneous local ordering. The degree of local ordering in this case is measured by the structure factor

where the angle brackets indicate an average over angles in reciprocal space. The first peak of at wavenumber encodes information on local ordering. Hansen and Verlet [39] discovered empirically that reaches the value 2.85 when a fluid freezes to an fcc solid regardless of the details of the fluid's microscopic interactions. This result, obtained originally from simulations of Lennard-Jones systems, has been supported by scattering measurements on materials such as Ar [40,41], Rb [42,43], and other liquid metals [44,45]. For bcc solids the freezing point coincides with ; for two-dimensional systems the corresponding threshold is . We have found very good agreement with the Hansen-Verlet criterion in the freezing of bulk charged colloidal suspensions, both in equilibrium [46], and also for nonequilibrium freezing [28] and for the two-dimensional system in equilibrium [46]. In each of these cases, we calculated explicitly from the particle distribution derived from single-layer images of the suspension. Calculating is not so straightforward in the present study, however, because we do not know the correct phase factors for particles in different layers. Thus, the peaks in the derived structure factor reflect not only the in-plane correlations, but also correlations between planes. To the extent that these two planes are strongly coupled in the confined geometry, these correlations are correspondingly stronger than would be found for a single layer. We expect, therefore, that the appropriate Hansen-Verlet freezing criterion for this system would fall between the values for two-dimensional and bcc systems, that is .

Snap shots of the in-plane structure factors corresponding to the images in Fig. 2 appear in Fig. 7. The anisotropic but still continuous Debye-Scherrer ring evident at in Fig. 7(b) is characteristic of a fluid very near its freezing point. The average of Fig. 7(b) over angles in reciprocal space shown in Fig. 8 shows that at . The first peak reaches roughly 6 seconds later.

The dynamical (LPS) and static (Hansen-Verlet) freezing criteria suggest that the ensemble of microspheres is driven to a fluid state by shearing, begins to solidify within 2 seconds after cessation of shearing, and achieves solidity within 6 seconds after that. The initially formed solid is very disordered, however, as can be seen both in Fig. 7(b) and directly in Figs. 2(c) and (d), and is largely 6-fold coordinated (see Figs. 4 and 5). These 6-fold domains grow and coalesce so that the distribution of coordination numbers becomes increasingly sharply peaked around = 6. Throughout this grain coarsening process, both the local dynamics described by and the degree of ordering measured by indicate that the system is in a solid, crystalline state. The degree of ordering reflected by the well-defined 6-fold symmetric peaks in the structure factor in Figs. 7(c) and 8 is surprising given that this is not the equilibrium symmetry. Another perplexing observation is that a single triangular crystalline layer appears to fill the space originally occupied by two layers of square crystal. A two-layer triangular crystal with a correspondingly larger lattice spacing would appear to have a honeycomb of missing sites in our projection, and a three-layer crystal would not fit into the available space.

Wed Feb 15 13:32:44 CST 1995