Maximal Air Bubble Entrainment at Liquid-Drop Impact

At impact of a liquid drop on a solid surface, an air bubble can be entrapped. Here, we show that two competing effects minimize the (relative) size of this entrained air bubble: for large drop impact velocity and large droplets, the inertia of the liquid flattens the entrained bubble, whereas for small impact velocity and small droplets, capillary forces minimize the entrained bubble. However, we demonstrate experimentally, theoretically, and numerically that in between there is an optimum, leading to maximal air bubble entrapment. For a 1.8 mm diameter ethanol droplet, this optimum is achieved at an impact velocity of $0.25\mathrm{m}/\mathrm{s}$. Our results have a strong bearing on various applications in printing technology, microelectronics, immersion lithography, diagnostics, or agriculture.

Figure 1

Experimental characterization of air bubble entrapment. (a) Sketch of dimple formation (not drawn to scale) just prior to impact. (b) Schematic of the experimental setup used to study droplet impact on smooth surfaces. An ethanol droplet of typical radius $R=0.9\mathrm{mm}$ falls on a glass slide of average roughness 10 nm. The impact velocity is varied by varying the falling height of the droplet. For very small velocities below $0.31\mathrm{m}/\mathrm{s}$, the droplet is fixed at the tip of 0.4 mm—diameter capillary that is vertically translated downwards at a constant velocity. The bottom view is captured by a high-speed color camera (SA2, Photron Inc.). The camera is connected to a long working-distance microscope and a $5\times$ objective to obtain a 2 mm field of view. (c) An example of an interference pattern and the extracted air thickness profile. Note the difference in horizontal and vertical length scales. The exposure time was $1/15000\mathrm{s}$ and the typical frame rate of the recordings is 5000 frames per second.

Figure 2

Maximum entrapment of air bubbles. (a) Dimple height $H_d$ and (b) entrained bubble volume $V_b$ as functions of the impact velocity $U$ (upper axes) and Stokes number St (lower axes). The shape of the air layer can be characterized by the dimple height $H_d$ and the lateral extension $L$. Blue circles correspond to high-speed color interferometry measurements, red squares correspond to numerical simulations. The straight lines correspond to the derived scaling laws in the capillary regime (solid) and inertial regime (dashed) with the respective scaling exponents.

Figure 3

Comparison of experimental (blue, solid) and numerical (red, dashed) dimple profiles for two different impact velocities; $U=0.2\mathrm{m}/\mathrm{s}$ ($\mathrm{St}=7.8·{10}^3$; crossover regime) and $U=0.7\mathrm{m}/\mathrm{s}$ ($\mathrm{St}=2.7·{10}^4$; inertial regime).