Examples of the microscopic shapes whose diffusion was studied; these images were taken in situ using differential interference contrast microscopy as described in the text. The elementary particle diameter is the same in each image, , and the scale varies between images.
Left ordinate: the gravitational scale height computed from the Boltzmann distribution giving the average distance from the bottom wall vs the number of particles in the cluster. As the number of elemental particles and hence mass of the cluster increases, the expected mean separation of the bottom of the cluster from the wall decreases (squares). Right ordinate: the computed ratio between near-surface translational diffusion and translational diffusion in the bulk, for spheres described by Eq. (5) (circles).
Probability of deviation from the planar configuration was considered for the lightest anisotropic cluster, a two-sphere dumbell, in order to inspect the extent to which the systems studied in this paper can be considered as quasi-2D. The plot shows the fractional time that the axis of the dumbell deviates from the 2D plane by at least the angle specified in the abscissa. For this worst-case cluster, the deviation exceeds less than 1% of the time.
Examples of raw dynamic data. (a) Mean-squared translational displacement (squares) and rotational displacement (circles) plotted against time for a linear cluster of four particles [Fig. 1(d)]. Lines through the data are linear regressions. (b) Distribution of step size of translational displacement (known as the van Hove distribution), plotted here regarding the time interval of 60 ms. (c) Distribution of step size of rotational displacement, plotted regarding the time interval of 60 ms. Lines through the data in (b) and (c) are fits to the Gaussian distribution expected for Fickian motion.
Translational diffusion coefficient (squares) and rotational diffusion coefficient (circles) are shown for a series of clusters in which spheres are sequentially added to a central sphere until six have been added to form a hexagon. One observes that whereas decreases smoothly and nearly exponentially, displays three distinct zones: Precipitous drop upon adding the first sphere, steady decline as the next three spheres are added, then plateau for the final two spheres. As a result, changes in the effective hydrodynamic radius for translation and rotation are significantly decoupled.
Data presented in a fashion to display semi-independently the consequences of changing the number of spheres in the cluster and its shape. (a): Along the downward-sloping diagonal, the cluster grows in size with minimal change in shape, as denoted by a line. The image is symmetric across this axis. Leaving this diagonal, the number of spheres in the cluster is constant but the conformation is increasingly extended, as denoted by the upward-sloping line. Data for all of these cases, except for the semitransparent ones, are tabulated in Table I. (b): Following a path parallel to the diagonal in (a), the ratio decreases. (c): Following a path perpendicular to the diagonal in (a), the ratio decreases. (b) and (c) are presented as to summarize qualitative trends evident from close inspection of situations tabulated in Table I.
Tabulated translational and rotational diffusion coefficients ( and , respectively) for the shapes depicted in Figs. 1 and 6(a). The uncertainty (“Unc”) column is the standard deviation from fits of the mean-squared displacement to time elapsed. For each shape, the inertial radius of gyration in units of micrometers was determined. There was a strong inverse correlation (0.97) between the radius of gyration and the rotational diffusion coefficient. However, the radius of gyration fails to capture the full picture, since at low Reynolds numbers, inertia is not the proper representation of hydrodynamic interaction. The aspect ratio, here chosen to be the maximum ratio of one axis to the axis orthogonal to it within the plane, was also determined, but was not found to strongly correlate with diffusion constants.
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