Shapes and Fissility of Highly Charged and Rapidly Rotating Levitated Liquid Drops

We use diamagnetic levitation to investigate the shapes and the stability of free electrically charged and spinning liquid drops of volume $\sim 1$ ml. In addition to binary fission and Taylor cone-jet fission modes observed at low and high charge density, respectively, we also observe an unusual mode which appears to be a hybrid of the two. Measurements of the angular momentum required to fission a charged drop show that nonrotating drops become unstable to fission at the amount of charge predicted by Lord Rayleigh. This result is in contrast to the observations of most previous experiments on fissioning charged drops, which typically exhibit fission well below Rayleigh’s limit.

Figure 1

Cutaway schematic drawings showing a drop levitating near the top of the 58 mm-diameter magnet bore. Magnetic field lines are illustrated by dashed gray lines. (a) Configuration used to charge the drop, including the setup used to measure the charge on the drop. (b) Configuration used to spin a charged drop.

Figure 2

The evolution of the equilibrium shapes of charged ethanol drops with increasing angular momentum; $x\propto (\text{charge}{)}^2$ is the fissility. Drops (a)–(c): column 1 spherical, at rest; 2 oblatelike, close to the transition to a capsulelike shape; 3 capsule shaped; 4 double-lobed shape; 5 double lobed with pronounced neck (“dumbbell” shaped), prior to fission; 6 fission at the neck. Drop d (with the same $x$ as c) fissions by emitting a fine jet from a Taylor cone (see also enhanced image, Supplemental Material [31]). Drops with higher $x$ (e.g., drop e) fission by the Taylor cone mode. The fission of drop c appears to be a hybrid mode. The scale is the same in all images. See Supplemental Material for videos [31].

Figure 3

Dimensionless length of the longest axis of the drop $R^{*}$ versus dimensionless angular velocity ${\mathrm{\Omega }}^{*}$, shown for a representative set of charged and uncharged TBW drops. The fissility $x\propto (\text{charge}{)}^2$ is indicated. Arrows indicate the path taken with increasing and decreasing angular momentum. Solid line: numerical result for $x=0$, from Ref. [10]. Dashed lines are guides to the eye.

Figure 4

Shape stability as a function of the fissility, $x\propto (\text{charge}{)}^2$, and dimensionless angular momentum, $L^{*}$. Experimental data bound the lower region of stable axisymmetric shapes (shaded blue), and the region of stable dumbbell and capsule shapes (unshaded). Crossing into the upper region (shaded red), drops become unstable to fission by necking for $x<0.65$, or by formation of a Taylor cone $x>0.68$; for $0.65<x<0.68$, both Taylor cone and asymmetric dumbbell fission are observed. The boundaries of the shaded regions are quadratic fits to the data. Numerically determined boundaries for the axisymmetric shapes are indicated by the broken line for a free drop (dot-dash), and levitated drop (dashed).