Droplet Trajectories

We have devised a technique for locating polydisperse emulsion droplets in three-dimensional confocal image stacks (see section on Droplet Location). By acquiring image stacks at regular time steps and applying the droplet location scheme to each stack in turn, we can then extract three-dimensional droplet trajectories. To link together features in a stack with those in a successive stack we use the trajectory code created by Crocker and Grier.

The fact that emulsion droplets have a range of sizes makes droplet location difficult, but for the trajectory extraction it is quite helpful since the droplets are no longer identical. We factor this extra information into the trajectory extraction process to reduce the likelihood of false identification of a feature from one time step to another.

Radius information also helps us resolve another challenge. Our emulsion system is refractive index matched, giving a clear sample allowing confocal microscopy, and buoyancy matched, which allows us to “switch off” gravity. Neither matching is perfect, however. In particular, in the data described here the emulsion droplets sediment since they are denser than the surrounding continuous phase. Although in the bulk this sedimentation amounts to just a few centimetres over several months, in the confocal images the high magnification means that roughly half the droplets in the field of view drop out of sight by the end of an image stack sequence, their numbers replenished by more droplets entering the frame from above. This is one example of a system that appears quiescent on a lab bench or supermarket shelf, but which microscopically is not.

For trajectory extraction, what matters is not the absolute velocity of features so much as their relative movement between successive stacks. The observed rates of sedimentation are a problem since the distance moved as a result is easily enough to confuse the trajectory extraction process. We tackle this issue by calculating the average drift between stacks, transforming to a common coordinate frame to perform the trajectory analysis, and applying the inverse coordinate transformation to return the final full trajectories. Omitting the final step yields trajectories relative to the overall drift. The fact that our droplets are non-identical is helpful since it allows us use large droplets as drift markers, effectively labelling the sample without any chance of polluting it with alien tracer materials.

There is a crucial assumption in this prescription, which is that the average three-dimensional drift is constant across the entire image volume. In fact, we have found that although this is usually the case, in our emulsion system there are important instances where it is not. The swirls described in the Network Ripples section amount to dislocations in image stacks, since they occur on a time scale that is short (of order of the time taken to acquire four image slices) compared to the time frame of a total stack acquisition, and result in a significant and sometimes uneven relocation across the image plane. Viewing the image volume sideways on, as in the figure below, for an aggregated emulsion in which a swirl occurred during the stack acquisition shows why we term this a dislocation:

The positive z-direction is deeper into the sample: slices are taken at a succession of z values. The horizontal line across the image, the dislocation, is caused by a swirl. The image to the right shows a portion of the image in the x-y plane (corresponding to the dumbbell zone in the image above) and shows how droplets are typically relocated during a swirl. Incidentally, comparing the two panels also confirms that the image resolution in the z-direction is significantly poorer than that in the x and y directions.

Feature identification in the vicinity of a dislocation is compromised. In addition, it is not possible to determine a meaningful single drift velocity between two stacks if one of them contains a dislocation. Of the 32 stacks in the sequence presented here, 4 contained dislocations. We have limited ourselves to the extraction of tracks for stack sub-sequences that avoid dislocations.

The figure below shows a sample of extracted droplet trajectories for a 30% oil-in-water emulsion depletion flocculated with 0.9% PEO.

The tracked droplet is comparable in size to others close by, yet the feature-finding recipe is still able to resolve it over a succession of stacks. Note that the droplet motion in the x-direction is dramatically larger than in the y-direction. Lots of droplets means lots of trajectories:

These are the trajectories extracted from roughly the front (i.e. low-z) 1/3 of the image volume and projected onto the x-y plane. Note that in our experimental configuration the microscope views the sample from the side, and gravity acts in the positive x direction. The numerous approximately vertical lines, which run downwards on the plot, show individual sedimenting droplet trajectories over the first 30 minutes of the experiment.

Sedimentation is not perfectly vertical, as can be seen by plotting trajectories in the x-z plane:

The left-hand plot shows trajectories from the same system for a region embracing roughly 4% of the sample for a mid-range y-value and projected onto the x-z plane. Typically trajectories lie at an angle to the vertical, showing that sedimentation is not trivially parallel to gravity in this experiment. The right hand panel set, in which time increases from left to right, shows the corresponding image segment with some sample features identified. Not that there are fewer tracks at the top and bottom of the sample since in the former droplets are emerging from above and in the latter they are vanishing from view, and in both cases extracted tracks are short and have not been plotted.

Trajectories viewed in either plane are seen to deviate from perfectly straight paths, and these deviations, best seen using relative trajectory plots, can be analysed to extract droplet dynamics over and above “simple” average drift. Here is a sample of droplet trajectories, plotted relative to the average drift, corresponding to a region of the x-y trajectory plot above:

Though trajectories in this system are dominated by sedimentation, there is evidence of large-scale deviation from perfectly vertical movement. Evolution of the network is evidenced by deviations from perfectly linear motion, suggesting the existence of “hot” and “cold” network regions.