Measuring Measurement Errors

Unlike artifacts arising from actual properties of the physical system, imaging artifacts result from mis-identification of the spheres' centroids. At first blush, the bright-field image of a colloidal sphere appears as a bright region on a darker background. The centroid then may be identified with sub-pixel resolution as the brightness weighted center of brightness (17). In fact, a sphere's image is a projection of its far-field Mie scattering pattern (36), consisting of alternating dark and bright rings encircling the central intensity maximum. This more complicated pattern may be analyzed in the conventional manner (17) provided one sphere's image does not overlap with those of its neighbors. Distortions arising from overlapping scattering patterns lead to systematic deviations in the particles' apparent positions (31). These distortions, in turn, distort estimates for the pair potential derived from the measured particles positions (31). Because the errors are in the particle locations themselves, the resulting distortion of the pair potential is not detected by methods based on thermodynamic self-consistency (21,29,30).

Fortunately, distortions due to imaging artifacts can be corrected if the artifact's dependence on interparticle separation is known. Figure 2 shows two complementary methods for measuring this, one of which can be applied a posteriori to archival data without requiring additional calibration measurements.

We explicitly measure overlap distortions by using holographic optical tweezers (37,38) to hold two spheres at specified separations while a third is held far enough away to use as an undistorted reference. For each separation, we measure the apparent distance $\tilde r$ between the closely spaced pair (Fig. 2(a)), and then independently measure their separations, $r_1$ and $r_2$ , from the reference sphere with the other sphere absent (Fig. 2(b,c)). The first measure is skewed by the artifact. The two reference measurements are not. Consequently, their difference, $r = r_2 - r_1$ , is an unbiased measurement of the real separation. The difference, $\Delta(r) = \tilde r - r$ , is a measurement of the artifact, whose separation dependence is plotted as circles in Fig. 2. As previously reported (31), these systematic deviations exceed the instrumental error bound for single-particle tracking (17) at separations relevant for interaction measurements. The data in Fig. 2 were obtained with a $100\times$ NA 1.4 oil immersion objective, yielding an effective magnification of $135~\unit{nm/pixel}$ . Comparable results can be obtained with the $60\times$ objective used for interaction measurements, and with one or two metal-coated surfaces.

This approach is accurate, but somewhat painstaking, and requires samples with prohibitively low areal densities (24). We therefore introduce an alternative way to measure $\Delta (r)$ that relies on information already contained in the imaging data used to estimate the pair potential.

Some spheres in a given image will be far enough from all of their neighbors that their images are unaffected by overlap distortions. The image of such a sphere can be clipped from the larger field of view, duplicated, and used to construct composite two-sphere images at known separations, $r$ . Examples of such composite images created from displaced copies of a single sphere's image are shown in Figs. 2(d), (e) and (f). The apparent separation $\tilde r$ in each composite image is then measured to obtain the difference $\Delta(r) = \tilde r - r$ , whose separation dependence is plotted as squares in Fig. 2. This method is based on the assumption that overlap artifacts result from the linear superposition of neighboring spheres' images. Its quantitative agreement with results obtained by explicit measurement justifies this assumption. Consequently, we use composite images to measure and correct for $\Delta (r)$ in each of our measurements of colloidal interactions.

Figure 2: (Color online) Measured apparent displacement $\Delta (r)$ from true separation $r$ due to imaging artifact. Circles were obtained with holographic optical tweezer measurements (a), (b) and (c), and squares with composite images including (d), (e) and (f).