Fluid pinch-off dynamics at nanometer length scales.

PHYSICA L R EVIEW LET T ERS V OLUME 92, N UMBER 24 week ending 18 JUNE 2004 Fluid Pinch-Off Dynamics at Nanometer Length Scales J. C. Burton, J. E. Rutledge, and P. Taborek Department of Physics and Astronomy, University of California, Irvine, California 92697, USA (Received 28 January 2004; published 18 June 2004) The breakup of a drop of inviscid fluid into two smaller drops is determined by a competition between surface and inertial forces. This process forms a thin filament of fluid with a connecting neck that shrinks to zero diameter at a finite time singularity. We present measurements of the electrical resistance of a liquid bridge of mercury as it undergoes pinch off. The electrical measurements allow us to probe the region of the singularity down to nanosecond times and nanometer lengths. Near pinch off, the resistance of the liquid bridge diverges as t 2=3 , as expected for inviscid flow. DOI: 10.1103/PhysRevLett.92.244505 PACS numbers: 47.55.Dz, 47.20.Dr Capillary pinch off in fluids has been of recent interest because of its technological importance and because it is a simple example of a system that displays a spontaneous finite time singularity [1–5]. When gravitational forces are negligible, a fluid droplet in static equilibrium forms either a sphere or a spherical cap if it is attached to a surface. For a sufficiently strong perturbation of the droplet shape, two separate droplets with spherical sur- faces form. The dynamical process by which the fluid changes its topology and transforms into two distinct pieces typically involves the formation of a long slender filament of fluid that connects the two nascent droplets. Surface tension drives flows that shrink the diameter of the filament in a nonuniform way and eventually lead to pinch off in a neck region. The dynamics of this process is governed by the nonlinear Navier-Stokes equations. Near the singularity in the neck region, the solutions have a relatively simple self-similar form that can be character- ized by power laws. There are two types of generic behavior distinguished by the value of the dimensionless p ratio Oh = d , where is the surface tension, is the density, is the viscosity, and d is the neck diameter. For Oh > 1, inertia is negligible, and the flow results from a balance between surface tension and viscous forces, which leads to an asymptotic linear relationship between d and , the time remaining until pinch off: d / Oh is initially small, as d ! 0 there will be a transition to a viscously dominated regime near 0. This transition has been observed using viscous mixtures, but for simple fluids such as helium [12], water [1], or mercury, the crossover to viscous behavior is not expected to occur until d reaches molecular dimensions where the validity of the continuous fluid description breaks down. In par- ticular, pinch off in mercury droplets should be described by the potential flow equations analyzed in [13,14] and are expected to obey the scaling law of Eq. (2), even for neck dimensions d of molecular size. Pinch off in the potential flow regime has typically been studied using strobe photography and high-speed video [1,4,5,12,15,16]. The spatial resolution of these imaging techniques is limited by the conventional con- straints of optical instrumentation, but, even more im- portantly, both experiment [1] and computer simulation [13] show that the profile of an inviscid pinch is reentrant, and resembles the region near the stem of an apple, as shown in Fig. 1. This means that the singularity in the This is known as the viscous thread regime and has been studied by numerous authors [4,6 –11]. For Oh < 1, the effects of viscosity are negligible, and the flow is a balance between surface tension and inertia. For this case of inviscid potential flow, dimensional analysis yields a different relationship between and d: The characteristic neck size d is a function of , with d ! 0 as ! 0, so Oh can span a wide range of values during a pinch-off flow. This analysis suggests that even if FIG. 1. A schematic diagram showing a cross section of the neck region close to pinch off for an inviscid fluid. The liquid filament attaches to the lower droplet at a point below the horizon formed by the droplet, and is not visible using standard projection imaging techniques. The electrical resistance of the filament is determined by the minimum neck diameter d and the cone angle . Although is not directly measured in our experiment, both theory and observation indicate that as described in the text.  2004 The American Physical Society d /