PMC

a, Silicone oil drop in immersion oil in zero field. b–h, Structures seen as a function of frequency spanning hydrodynamic and dipolar regimes. The electric field is perpendicular to the plane of the page.

a, At 70 Hz, each droplet in the array is a section of an oblate spheroid. The top inset shows the droplet acting as a lens for the bottom layer of droplets, see text. b, At 1 KHz, each droplet is a section of a prolate spheroid. The FFTs in the bottom insets of (a, b) display weak hexagonal ordering. The field is perpendicular to the plane of the page.

See for dynamics. a, Polygonal deformations at f = 7 Hz, E = 10 V/µm. With increasing droplet radius, the number of facets per drop increases; n = 3 to 12 are observed. The field is perpendicular to the plane of the page. b, The relationship between droplet radius normalized for n = 3 and n is roughly linear, signifying the existence of a characteristic wavelength λ. Here λ = (2π)(2 µm) ≈ 12 µm. The solid black line is obtained from Lee and McConnell.

A frequency driven transition from oblate (electrohydrodynamics dominated) to prolate (dipolar dominated) droplet deformation is observed. Shown is a droplet that is a, oblate at 70 Hz, b, spherical at 100 Hz and c, prolate at 1 KHz. In a–c, the field is parallel to the plane of the page and pointing vertically upwards. d, The scaled deformation D_scaled, plotted with frequency f for different sized drops shown by different colored circles, collapses onto a single curve, qualitatively consisent with a theoretical expression for static droplets (solid red line). Variation with f of the hydrodynamic length l_h (solid black line). A frequency quench from the hydrodynamic regime (large l_h, negative D) towards the dipolar regime (small l_h) is used to make monodisperse droplet arrays of controllable shape. Micrographs obtained in e, strong hydrodynamic regime at f = 0.5 Hz (l_h ~ 1 mm), f, weak hydrodynamic regime at f = 25 Hz (l_h ~ 25 µm) and g, dipolar regime at f = 1 KHz (l_h ~ 0.5 µm). In e–g, the field is perpendicular to the plane of the page.

a, Time evolution of the surface topology of a single drop at an electric field E = 5.1 V/µm and f = 0.5 Hz. The electric field is parallel to the plane of the page, pointing vertically upwards. b, Left: snapshot of a cloud of droplets with the timeline of each droplet overlaid. Right: the rms velocity v_rms of moving droplets plotted against applied electric field intensity E² during one period of oscillation (f = 0.03 Hz) exhibits two thresholds: one indicating onset of steady motion, and the second, the onset of chaotic motion. c, The in-plane deformation of droplets at E = 12 V/µm and f = 1 Hz, 2 Hz and 3 Hz. In b and c, the electric field is perpendicular to the plane of the page. d, The droplets become more circular (χ more narrowly distributed near unity) as the frequency is increased from 1 to 3 Hz.