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We use emulsion droplets as models for granular media.  In particular, we confine oil droplets (in water) between parallel two glass plates, deforming the droplets into quasi-two-dimensional pancake shapes.  At high droplet concentration, the quasi-2D droplets contact and deform each other, and from their deformations we infer the forces they exert on each other.  One major advantage of this system is that there is no static friction; thus, we are able to study how the properties of this frictionless system match up with frictionless simulations and how the results differ from prior work with frictional granular systems.  During the course of our Petroleum Research Fund grant (the past two years), we developed this model system and obtained results in three distinct experiments.

1. Experimental parameters

Our experiments have many possible parameters and in the first year we identified which parameters “work.”  A sample chamber thickness of 100 microns works well.  Alberto Fernandez-Nieves (Georgia Tech) taught us the microfluidic technique we use to produce monodisperse hexane droplets in water, using SDS for the surfactant.  Hexane works well for the droplets as it doesn’t evaporate too quickly, and has a large density difference from water which is useful for our experiments (described below).  We can produce droplet sizes in the range of 100 – 400 microns.  We typically mix two different sized batches of droplets together so that the droplets don’t pack into crystalline arrays.  We use optical microscopy and a low magnification objective lens to image our samples.  

2. Analysis to find inter-droplet forces

We worked out a procedure using the observed droplet outlines to determine the pair-wise contact forces between touching droplets.  We calibrate these forces with an experiment using a monodisperse sample of droplets, and tilting the slide so that the oil droplets (less dense than the water) float up to the top of the slide.  This sets up a known pressure gradient throughout the sample chamber, and the forces acting on each droplet can be inferred from the known local hydrostatic pressure due to the weight of the droplets beneath.  Graduate student Ken Desmond developed a model connecting the observed droplet contours to the forces, and used the data from the tilted sample chamber to validate the force model.  At this point we believe our uncertainty in these forces is no greater than 10% for any given force.

3. Change in forces and contacts near the jamming transition

Ken Desmond used his force model to investigate binary samples as a function of area fraction.  For a variety of observable variables, we see a transition to a jammed state at an area fraction of f₀ ~ 0.85.  For example, having identified droplets that are touching (with contact lines), we can count the number of neighbors each droplet has – the “coordination number” Z.  As a function of area fraction, we see critical scaling, (Z - 4) ~ (f - f₀)^1/2.  This agrees with simulations of frictionless 2D particles (for example, O’Hern et al, Phys. Rev. E 68, 011306 [2003]).  The distribution of forces P(F) also varies for area fractions near f₀ in agreement with simulations.

We also study the spatial distribution of the large forces, which form chains throughout the sample.  These are the “force chains” that have been well studied in regular granular media, and our observations are similar to those seen in experiments with friction.

We are currently writing these results up for publication.  We plan to submit a short paper to Phys. Rev. Lett. focusing on the jamming transition and the force chain structure, and then hopefully a longer paper going into more detail about our methods (section 2).

4. Flow and rearrangement of droplets in hoppers

We study the driven flow of these quasi-2D emulsions; this is being done by graduate student Dandan Chen, which nicely complements prior work she did at Emory University studying the shear flow of 3D colloidal suspensions.  She studies the samples as they flow through a hopper-like geometry, where the flow constricts gradually down to a small channel width (only a few droplet diameters across).  The flow is driven at constant volume flow rate.  Similar to prior experiments with regular granular particles, she finds that the forces exerted on the sample chamber boundaries fluctuate significantly as the sample flows through.  Intriguingly, these force fluctuations are primarily due to nearby rearrangements inside the sample, no more than 4 droplet diameters away from the wall.  We are currently writing these results up for publication, probably Phys. Rev. E - Rapid.

5. Free drainage through hoppers

A new graduate student at Emory, Xia Hong, is conducting a complementary experiment to the previous one.  She also makes 2D hopper chambers, but she uses gravity to let the droplets freely pass through the hopper.  This is analogous to prior experiments by K. To et al., which studied 2D steel disks flowing due to gravity through a 2D hopper [PRL 86, 71 (2001)].  They found arches of 3-4 disks would form at narrow hopper openings and thus jam the flow.  In our work without static friction, we find that the flow never jams as long as the hopper opening is at least one droplet diameter wide.  We are continuing these experiments to find if it’s possible to reduce the gravitational force to the point where we can see larger arches.

Summary

The Petroleum Research Fund is funding substantially new directions for researchers rather than funding ongoing projects.  I feel we have been very successful.  We now know how to use quasi-two-dimensional emulsions for a variety of experiments, which required a fair bit of learning on our part.  We have publication-quality data for two experiments (inter-droplet forces, driven flow through a hopper) and promising preliminary data for a third experiment (free drainage through a hopper).    This PRF grant nucleated an exciting new direction of research for my laboratory.  The future publications will acknowledge PRF funding and I will make sure to go to the PRF website to list the publications there as requested.