Drop Formation by Thermal Fluctuations at an Ultralow Surface Tension

We present experimental evidence that drop breakup is caused by thermal noise in a system with a surface tension that is more than ${10}^6$ times smaller than that of water. We observe that at very small scales classical hydrodynamics breaks down and the characteristic signatures of pinch-off due to thermal noise are observed. Surprisingly, the noise makes the drop size distribution more uniform, by suppressing the formation of satellite droplets of the smallest sizes. The crossover between deterministic hydrodynamic motion and stochastic thermally driven motion has repercussions for our understanding of small-scale hydrodynamics, important in many problems such as micro- or nanofluidics and interfacial singularities.

Figure 1

Image sequences of the breakup events in systems (a) $S1$ and (b) $S2$. The breakup of the primary drop is taken as a time reference for the sequences. (a) A large primary drop of nearly $15\mu \mathrm{m}$ in diameter moves downwards, stretching behind itself a neck of fluid. When the neck size becomes of order $L_T$, it breaks at a random point, typically a few micrometers away from the base of the drop (indicated by the arrow). After the primary drop has detached, secondary instabilities develop. The scale bar represents $10\mu \mathrm{m}$. In (b), the snapoff is clearly different than in (a). A long filament attached to a nearly spherical drop is formed. After the first pinch-off, the liquid filament breaks up in the top. The image height is $154\mu \mathrm{m}$. Note the differences in time scales between the two events. See Ref. 24 for accompanying movies of the two events.

Figure 2

A typical secondary breakup event for system $S1$. (a) Image sequence showing the symmetry of the profile. The scale bar represents $5\mu \mathrm{m}$. (b) Corresponding neck profiles taken at 3 (solid curve), 15 (dashed curve), 27 (dashed-dotted curve), and 39 s (dotted curve) before breakup. The minimum of each profile has been centered on $z=0$ for clarity.

Figure 3

Satellite drop size distribution for systems $S1$ (blue) and $S2$ (red) after breakup of a primary drop. For the system with the lower surface tension, 18 primary drop events were analyzed, whereas only 5 in the case of the higher surface tension, but the total number of satellite drops was in both cases $\sim 113$. The number fraction is with respect to the total number of drops, and the drop radius is normalized by the radius of the primary drop.

Figure 4

Time dependence of the minimal neck diameter before breakup. (a) System $S1$: averages of several events (primary and secondary) overlapping within experimental scatter. The uncertainty in $t_p-t\le 1\mathrm{s}$. Dashed lines are fits from (4) with $\alpha =0.42$ fixed. Inset: A single secondary event where gravity and recoil turned out to be unimportant. The solid line follows from the Eggers prediction with $\alpha =0.42$ [10]. (b) Secondary events. Two events are shown for system $S2$ (open symbols) for which gravity was not important (detachment of the filament at its base). The uncertainty in $t_p-t\le 0.04\mathrm{s}$. Best fits yield exponents $0.43\pm 0.01$ (triangles) and $0.47\pm 0.02$ (circles). Inset: The same data on a log-log plot ($x$ and $y$ axes, bottom and left). For comparison, crosses correspond to the inset in (a) for $S1$ with an exponent of 0.42 ($x$ and $y$ axes, top and right).