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How sticky droplets behave when there's a lot of them

I've been writing my dissertation, so updates have been scarce, but I can report some interesting papers I've read along the way.

There is a great paper by Ryotaro Shimizu and Hajime Tanaka that explains a new mechanism for how immiscible droplets within a fluid merge into bigger droplets, and has universal applications from emulsions to cosmetics and foods.  Immiscible means two fluids cannot mix.  The models described here work are applicable for low viscosities.  This work is based on computer simulations, but the authors claim there are experiments to back up their claims.  The paper has been recently published in Nature Communications, so time will tell on what further peer review will reveal. 

Shimizu and Tanaka focus on phase separation, which is a fundamental physical phenomenon of multiple bodies collectively interacting.  Most familiar encounter I can think in everyday life is olive oil droplets merging into a single mass in a cooking pot.  The process is called coarsening, and the oil droplets can be thought of as domains in a continuous medium called water.  The way by which droplets coalesce is dependent on their initial concentration.  The concentration changes the mechanism by which they interact.  

When the concentration is low, droplets merge via a mechanism called Ostwald ripening.  This is as old as the hills (actually since the 1890s), and works by smaller droplets steadily diffusing into large droplets at the expense of the smaller droplets.  

Figure 1.  Small droplets spontaneously dissolve due to their instability.  The material then diffuses into larger droplets, which are more energetically stable.    Source: Wikipedia

Another mechanism is works when there are enough droplets to occupy about 50% of the total volume in a given object.  There, hydrodynamic forces coarsen the droplets into a so-called bicontinuous pattern, as seen in Figure 2.  Bicontinuous refers to two continuous phases.   You can see the black and white patterns continuing beyond the box in the moments of the animation.  The merging is so rapid that normal diffusion is dwarfed by the swift currents that result in the final structure of the animation. 

Figure 2.  Emergence of bicontinuous structure.  Note the rapid emergence of two continuous phases; that is indicative of rapid flow.

The third mechanism accounts for an intermediate droplet concentration, somewhere between 21 and 35% of the volume occupied by the droplets.  This is the most difficult mechanism and has been the subject of various researchers for decades.  That is because neither diffusion nor hydrodyanmcs can be neglected and thus makes it difficult to make approximations.  Hydrodynamic forces stem from thermally-induced Brownian motion, the technical term for the random collisions between particles in a fluid, as seen in Figure 3.    Diffusion occurs during the collision, and thus the hydrodynamics and diffusion are coupled.

Figure 3.  Animation of Brownian motion without coalescence.   The blue path traces the trajectory taken by the yellow particle from  incessant collisions.

However, what Shimizu and Tanaka claim is that the collisions are not random, but are driven by a so-called composition Marangoni force, which is induced by long-range interfacial tension gradients on the droplets.  In other words, the interfacial energy on a droplet varies because of other droplets near it.   This is illustrated from the paper in Figure 4.

Figure 4.  Snapshot of droplets with interfacial energy gradients, with blue color being the minimum, and red as the maximum.  Velocity arrows are drawn on and in the droplets of b to indicate preferred velocities on a given droplet region.  Thus, the droplet in the low left is drawn toward the small droplet next to it due to the largest arrows pointing toward it from the further end of this droplet.

A droplet much smaller than its neighbors draws neighboring droplets toward it and eventually induce them to collide.  In short, a larger droplet tends to eat a smaller droplet by direct collision.   The attraction from droplets stems from the different gradients of surface tension between the droplets.  The merging then lowers the overall energy of the system, since equilibrium just means the system has fallen to its lowest energy state.  

Source:  R. Shimizu, H. Tanaka, A novel coarsening mechanism of droplets in immiscible fluid mixtures, Nature Communications, 6 (2015).