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Figure 3: Holographic particle-image velocimetry. (a) Measured three-dimensional trajectories of 500 colloidal spheres travelling down a microfluidic channel in a pressure-driven flow. Each sphere represents the position of a particle in one field of a holographic snapshot. Features from a sequence of fields are linked into trajectories that are colored by the particle's measured speed. (b) Poiseuille flow profile along the vertical direction obtained from the data in (a). Particles are excluded from the shaded regions by their interactions with the upper and lower glass walls of the channel. The dashed curve is a fit to the anticipated parabolic flow profile.

Nanometer-resolution 3D particle-image velocimetry

Holographic particle tracking has immediate applications for three-dimensional particle-image velocimetry. Figure 3(a) shows the superimposed trajectories of 500 individual one-micrometer-diameter polystyrene spheres (Duke Scientific, catalog number 5100A) travelling down a 2 cm long microfluidic channel of 100  $ \unit{\mu m}$ width and 17  $ \unit{\mu m}$ depth. The spheres were dispersed in water at a volume fraction of $ 10^{-5}$ , and were advected by a pressure-driven flow of water created by raising a reservoir against gravity. Images were obtained in a $ 50 \times 70~\ensuremath{\unit{\mu m}}\xspace ^2$ area near the middle of the channel, with the focal plane set roughly 5  $ \unit{\mu m}$ below the lower glass-water interface. Spheres' locations in each snapshot are linked with a maximum-likelihood algorithm (7) into single-particle trajectories, $ \vec{r}_p(t)$ , sampled at 1/60 s intervals. Not every time step is represented in each particle's trace because faster-moving particles near the mid-plane of the flow occasionally obscure slower-moving particles near the walls. Figure 3(a) presents only those particle positions that were identified unambiguously. Even such incomplete time series can be used to estimate the particles' instantaneous velocities. The traces in Fig. 3(a) are colored according to the trajectory-averaged speed.

These trajectories also are useful for mapping the three-dimensional flow field. Each point in Fig. 3(b) represents one particle's speed as a function of its mean height, $ z$ , in the microfluidic channel. The superimposed results of 1000 such trajectories clearly show the parabolic flow profile expected for Poiseuille flow down a channel, the width of the cluster of data reflecting spatial variations across the long horizontal axis of the channel. The limits of the vertical axis indicate the positions of the channel's upper and lower walls, with heights being reported relative to the microscope's focal plane. The dashed horizontal lines represent the region of the flow into which particles cannot wander because of their hard-sphere interaction with the glass walls. The fit parabola shows the flow vanishing at the channel's boundaries.

Holographically characterizing fast-moving particles

Figure 4: Holographic characterization of streaming particles. (a) Trajectory-averaged radii $ a_p$ and refractive indexes $ n_p$ for a sample of commercial polystyrene spheres in water. Histograms show the distributions of observed sizes and refractive indexes, together with Gaussian fits. (b) Trajectory-averaged radius and refractive index as a function of mean speed.

Each trajectory also yields trajectory-averaged measurements of the radius and refractive index for each particle individually. Combining multiple measurements on a single particle minimizes systematic errors due to inevitable position-dependent variations in the illumination. The results in Fig. 4(a) show the radii and refractive indexes of the spheres in a commercial sample of polystyrene microspheres dispersed in water. The mean radius of $ a_p = 0.4995~\ensuremath{\unit{\mu m}}\xspace $ agrees with the manufacturer's specification obtained by conventional light scattering, as does the measured 2.5 percent polydispersity in the radius. The mean refractive index of $ n_p = 1.595$ is consistent with independent measurements on polystyrene spheres (15).

Single-particle characterization is a substantial benefit of holographic characterization compared with bulk light-scattering measurements, which are the usual basis for analyzing particle dispersions. Building up distributions such as the example in Fig. 4(a) from single-particle measurements eliminates the need for population models, and thus affords more general insights into a sample's composition. For example, the anticorrelation between the particles' size and refractive index evident in Fig. 4(a) would not be apparent in light scattering data. No such anticorrelation is apparent in holographic analyses of homogeneous fluid droplets (10). One interpretation of this observation is that the larger spheres in the emulsion polymerized sample are more porous, and consequently have lower refractive indexes.

Simultaneously tracking and characterizing individual particles enables us to confirm our results' freedom from motion-based artifacts. Colloidal particles' images become blurred if they move during the period that the camera's shutter is open. This blurring introduces substantial artifacts into conventional bright-field video microscopy data (4,5). As the results in Fig. 4(b) demonstrate, however, motion blurring has no discernible influence on values for the radii and refractive indexes obtained by holographic analysis for speeds as high as 500  $ \unit{\mu m}$ /s. Additional measurements reveal deviations from the population average values only for peak flow speeds exceeding 700  $ \unit{\mu m}$ /s.

This robustness is surprising at first blush because particles travelling at several hundred micrometers per second traverse several of our camera's pixels during its 1 ms shutter period. The resulting incoherent average of the oscillatory scattering pattern serves primarily to reduce the contrast in the direction of motion, however, and so has little influence on the Lorenz-Mie fit. Even this amount of blurring could be reduced through the use of a faster shutter or a pulsed laser for illumination.

Being able to characterize individual colloidal particles as they travel down a microfluidic channel provides an effective basis for detecting molecular-scale coatings on functionalized beads. If the individual spheres' radii were known to within a nanometer or so, then the presence of a molecular coating of similar refractive index could be discerned in the apparent increase in the radius. More generally, the characteristics of a treated sample can be compared with control measurements on untreated spheres.

Label-free molecular binding assays

Figure 5: Detection of avidin binding to biotinylated polystyrene spheres. (a) Yellow circles show the probability distribution for the measured particle radii in stock biotinylated polystyrene spheres. Red circles show the corresponding distribution for a sample of these spheres after incubation with neutravidin. Dashed curves are guides to the eye. (b) Equivalent distributions for particles' refractive indexes. Arrow indicates redistribution of probability from low density tail in the stock sample to the peak in the coated sample.

Figure 5 shows one such comparative study of 2  $ \unit{\mu m}$ diameter biotinylated polystyrene spheres before and after incubation with neutravidin. The biotinylated polystyrene spheres used in this study were obtained from Polysciences Inc (Warrington, PA) (catalog number 24172). Neutravidin was obtained from Invitrogen (Carlsbad, CA) (catalog number A2666). A neutravidin solution at a concentration of 1 mg/mL was prepared by adding 1 mg of neutravidin to 1 mL of phosphate buffer saline (PBS) (50 mM, [NaCl] = 50 mM). The stock sample of beads was obtained by adding $ 10~\unit{\mu L}$ of the as-delivered dispersion to $ 990~\unit{\mu L}$ of PBS. The coated sample was prepared by adding $ 10~\unit{\mu L}$ of the as-delivered dispersion to $ 990~\unit{\mu L}$ of neutravidin solution. Particles were incubated and shaken at room temperature for 1 hr before they were introduced into the microfluidic channels by capillary action. Flow was induced by introducing a slip of absorbent paper into one end of the channel and images recorded until results were obtained for 1,000 spheres from each sample. Each data set consisted of roughly 5,000 holographic measurements, which were obtained over the course of roughly 5 min.

From these measurements, we determined that the untreated sample has a population-averaged radius of $ 0.996 \pm 0.015~\ensuremath{\unit{\mu m}}\xspace $ , consistent with the manufacturer's specification. The incubated population appears to some 6 nm larger, with an average radius of $ 1.002 \pm 0.015~\ensuremath{\unit{\mu m}}\xspace $ . Even though the two size distributions plotted in Fig. 5(a) overlap substantially, a Wilcoxon rank-sum test demonstrates that their means differ with better than 99 percent certainty. This then constitutes a statistically significant detection of change in the treated sample's radius, which can reasonably be ascribed to the presence of a molecular-scale coating. The coating's thickness, in this case, is consistent with the size of a multi-domain avidin derivative.

Pronounced differences between the two samples also are evident in the measured distribution of refractive indexes, plotted in Fig. 5(b). The incubated sample's distribution is significantly sharper, presumably because protein, whose refractive index is similar to that of polystyrene, displaces water in the spheres' porous surfaces, and raises their effective refractive indexes. This would affect the more porous particles on the lower side of the refractive index distribution more than the denser particles on the high side, thereby sharpening the distribution. The arrow in Fig. 5(b) indicates this redistribution.

Similar analyses of random samples of the two data sets further confirm that the particles from the untreated sample all come from the same population, whose size and refractive index is consistent with the manufacturer's specification. The treated samples, by contrast show more variability in size, possibly because the thickness and evenness of the bound avidin layer can vary from sphere to sphere.

These results demonstrate the utility of hardware-accelerated digital video microscopy for detecting molecular-scale coatings on functionalized colloidal spheres. Unlike conventional molecular binding assays, holographic analysis does not require fluorescent or radiological markers, and so eliminates the effort and expense ordinarily required to label molecules bound to beads.

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Next: Discussion Up: Flow visualization and flow Previous: Holographic video microscopy
David Grier 2009-07-17