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Thermal fluctuations drive nanoscale droplet formation

21 February 2019

The stability of increasingly narrow liquid streams diverges from the classical picture.

The breakup of liquid streams into droplets is familiar to anyone who has watched water stream out of a shower head. Driven by surface tension, the phenomenon is well-described by the Rayleigh–Plateau framework developed in the late 19th century: Cylinders of water are unstable to sufficiently long-wavelength disturbances, and the mode with the fastest-growing amplitude above a critical wavelength determines the droplet size into which the cylinder will divide.

The classical framework breaks down as streams get very thin—micrometers to nanometers wide—and macroscopically negligible thermal fluctuations become relevant. To account for those fluctuations, Chengxi Zhao, James Sprittles, and Duncan Lockerby at the University of Warwick in Coventry, UK, extended current models by adding stochasticity to the usual lubrication equation. They then analyzed the stability of narrow liquid streams both theoretically and through molecular dynamics simulations (as depicted in the figure).

Animation of grouping behavior

On macroscopic length scales where thermal fluctuations are negligible, the new results match those from existing theories. However, as the streams become small and fluctuations become sizable, the predictions diverge. Classical Rayleigh–Plateau theory underpredicts the breakup droplet size, whereas lubrication models without fluctuations overpredict it. The new theory from the Warwick researchers matches the results obtained using molecular dynamics simulations.

The theory and simulations also point to something surprising: a violation of the classical stability criterion. If a stream is sufficiently short and wide, it should not exhibit a Rayleigh–Plateau instability. Only unstable modes grow and form droplets; however, if a mode’s wavelength is too long relative to the stream, it can’t break the fluid into droplets, and short wavelength modes should be stable. However, the researchers observed short-wavelength modes growing in the simulated streams when they should have been stable.

Zhao and coworkers realized that the thermal fluctuations in their model generated the short-wavelength modes that were unexpectedly growing. Their simulation confirmed that on length scales where such fluctuations are important, liquid streams eventually break up even when they are well within the classically stable regime. The next question is whether the prediction can be experimentally tested in nanofluidic systems that are sensitive to thermal fluctuations. (C. Zhao, J. E. Sprittles, D. A. Lockerby, J. Fluid Mech. 861, R3, 2019.)

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