Mechanism of Contact between a Droplet and an Atomically Smooth Substrate

When a droplet gently lands on an atomically smooth substrate, it will most likely contact the underlying surface in about 0.1 s. However, theoretical estimation from fluid mechanics predicts a contact time of 10–100 s. What causes this large discrepancy, and how does nature speed up contact by 2 orders of magnitude? To probe this fundamental question, we prepare atomically smooth substrates by either coating a liquid film on glass or using a freshly cleaved mica surface, and visualize the droplet contact dynamics with 30-nm resolution. Interestingly, we discover two distinct speed-up approaches: (1) droplet skidding due to even minute perturbations breaks rotational symmetry and produces early contact at the thinnest gap location, and (2) for the unperturbed situation with rotational symmetry, a previously unnoticed boundary flow around only $0.1\mathrm{mm}/\mathrm{s}$ expedites air drainage by over 1 order of magnitude. Together, these two mechanisms universally explain general contact phenomena on smooth substrates. The fundamental discoveries shed new light on contact and drainage research.

Popular Summary

When a droplet gently lands on a smooth surface, a thin layer of air is always trapped under the droplet, and it then drains away. Simple as it may seem, there is a large discrepancy between theory and observation. Basic fluid mechanics predicts that the droplet takes about 10 to 100 s to contact the surface. Almost all contacts, however, actually occur in about one-tenth of a second—a mystery that lacks a satisfactory explanation. Understanding this discrepancy can impact a wide range of applications, including inkjet printing, protection of surfaces from freezing rain, and cleanups of large-scale oil spills. We combine advanced optical measurements, surface profile measurements, and high-speed photography to reveal two mechanisms responsible for speeding up contact.

We gently dropped silicone droplets at very low speeds onto atomically smooth surfaces and recorded impacts from the bottom and side using two high-speed cameras. We also used dual-wavelength interferometry to monitor the size of the air gap and confocal profilometry to measure surface deformation. Small perturbations from the environment break the rotational symmetry and produce a tiny height difference of $0.5\mu \mathrm{m}$ between front and back, which causes a large thickness asymmetry right before contact. This asymmetry makes the thinnest region contact the surface early, before draining most of the air out of the gap. We also discover a small boundary velocity around 0.1 mm/s, neglected by previous research, which plays a dominant role in the air drainage for the symmetric situation.

Together, these two mechanisms universally explain general contact phenomena on smooth substrates. Our discoveries could have an important impact on both fundamental droplet physics and the efficacy of many practical applications.

Figure 1

Short and long lifetimes on inclined and leveled oil-film-coated substrates. All scale bars are $200\mu \mathrm{m}$. (a),(b) Side view images of a droplet landing on inclined and leveled substrates, respectively, with velocity $V_0=0.32\mathrm{m}/\mathrm{s}$, droplet diameter $d=1.7\mathrm{mm}$, drop viscosity 50 cSt, and liquid film thickness $30\pm 5\mu \mathrm{m}$. The lifetime is 10 times longer on the leveled substrate. (c) Schematic illustration of the impact. (d),(e) Bottom view of the impact shown in (a) and (b). The green arrow line indicates the direction of skidding, with droplet center moving from the red to the blue spot. Apparently (d) skids much longer and (e) has a more symmetric pattern. (f),(g) Summarizing the entire process of (d) and (e) by plotting the gray value along the green line versus time. We divide it into three stages: (1) rebounding, (2) oscillatory, and (3) stable stages. Panel (g) shows a remarkably long stable stage. (h) Droplet lifetime versus skidding distance at different tilt angles. There exist two distinct states on a liquid surface (solid symbols), while only the short-lifetime state appears on a solid mica surface (open symbols). Inset: Short-lifetime data on a zoomed-in scale.

Figure 2

Underlying mechanism for short and long lifetimes. (a)–(c) Initial contact at large, small, and zero tilt angles. Panels (a) and (b) have short lifetimes and (c) has a long lifetime. In (a) and (b) the droplet center moves from the red to the blue spot along the green arrow line, which is also the axis of symmetry. For large $\theta$ in (a), two contacts locate symmetrically beside the green line, but they converge into one as $\theta$ decreases in (b). For the leveled situation in (c), a single contact occurs randomly on the perimeter. (d)–(f) Air gap profiles corresponding to (a)–(c). They are measured along the green arrow line for (a) and (b), and along an arbitrary diameter for (c). They decrease with time and the lowest curve is right before the contact moment. Apparently, (d) and (e) have asymmetric profiles whose narrow region at the front produces early contact and a short lifetime, while the symmetric air gap in (f) leads to the long lifetime. (g)–(i) Liquid film deformation below the drop. Panel (g) illustrates a typical short-lifetime experiment: the surface sinks at the center with the magnitude $8\mu \mathrm{m}$ but rises around the edge. Panels (h) and (i) show the long-lifetime situation. Because of limited dynamic range, they come from two identical experiments with focal planes $20\mu \mathrm{m}$ apart. The surface sinks much deeper to $25\mu \mathrm{m}$, which prevents the droplet from skidding and causes long lifetime. Insets are snapshots by confocal microscopy, and averaging the red region yields one typical curve in the main panel.

Figure 3

Boundary velocities significantly influence air leakage. (a) Velocity profiles inside air gap. The total velocity $v_r$ is composed by two parts: a parabola $v_p$ from pressure gradient plus a linear profile $v_c$ caused by boundary velocity. $v_{ra}$ and $v_{rb}$ are upper and lower boundary velocities, respectively. (b) Side view image illustrating the flow inside droplet (also see Movie 4). The bright streaks are tracer particles’ trajectories moving from the red end to the purple end. This flow provides the outward boundary velocity $v_{ra}$ from the droplet side. (c) Bottom view image illustrating the flow inside liquid film (also see Movie 5). Again, the bright streaks are tracer particles’ trajectories moving from the red end to purple. This flow provides the outward boundary velocity $v_{rb}$ from the liquid film side. (d) Obtaining $v_{ra}$, $v_{rb}$, and ${\overline{v}}_c$ experimentally. At time $t=698\mathrm{ms}$, we obtain $v_{ra}$ and $v_{rb}$ from tracking particles in droplet (solid symbols) and liquid film (open symbols), respectively. Because particles’ velocities decrease as their locations are further away from boundary, we use the upper bound of their velocities as the boundary velocities $v_{ra}$ (dotted line) and $v_{rb}$ (dashed curve). Their average gives ${\overline{v}}_c=(v_{ra}+v_{rb})/2$ (dash-dotted curve). (e) ${\overline{v}}_r$ (solid curve) and ${\overline{v}}_c$ (dash-dotted curve) decrease with time, and ${\overline{v}}_c$ takes a major fraction in ${\overline{v}}_r$. (f) Comparison of lubrication force $F$ calculated from our method and a previous method (Appendix pp1-s4). Our method (red symbols) agrees with the droplet weight (blue line) while the previous method (black symbols) does not.

Figure 4

(a) Experimental setup for measuring the evolution of air film. (b) Schematic illustration of the RICM setup. (c) A sample plot of Eq. (a1) with $\lambda =434\mathrm{nm}$ (red curve) and 546 nm (blue curve); $n_0=n_2=1.404$, which is the index of silicone oil, $n_1=1$ for air and $\mathrm{NA}=0.4$.

Figure 5

A typical example of intensity along one fixed line (vertical axis) with respect to time (horizontal axis), for $\lambda =434\mathrm{nm}$ (a) and 546 nm (b).

Figure 6

(a) Measured intensity evolution at the center, i.e., the red line in Fig. 5, for $\lambda =434\mathrm{nm}$ (yellow dots) and 546 nm (blue dots), and the corresponding curve fittings (central red and blue curves). (b) Corresponding height evolution at center calculated from (a).

Figure 7

(a) Measured intensity along a fixed line, i.e., the green line in Fig. 5, for $\lambda =434\mathrm{nm}$ (yellow dots) and 546 nm (blue dots), and the corresponding curve fittings (central red and blue curves). (b) Corresponding height profile calculated from (a).

Figure 8

Configuration of confocal imaging.

Figure 9

(a) Experiment: constant focal plane, surface profile is changing. (b) Calibration: focal plane is changing (acquiring a $z$ stack), surface profile is unchanged. (c) Experiment with the focal plane located deep inside the liquid film. The setting is equivalent to (a). In principle, the measured fluorescence intensity in each column should be the same.

Figure 10

A typical example of calibration data. The $x$ axis represents the position of the focal plane. $z>0$ means focal plane inside the liquid film and $z<0$ means it inside the air film. Black curve: Fluorescence light intensity. Blue curve: Reflection intensity of laser light. Red line: Calibration which passes through the fluorescence data at $z=\{-8,6\}\mu \mathrm{m}$. Shaded area: Dynamic range. Dotted cyan line: Asymptote of the fluorescence at large $z$.

Figure 11

Upper panel: Plotting the liquid film interface (the black curve and symbols) and the droplet interface (the red curve) together in one plot. The difference in between is the air gap. Clearly the air gap thickness is much smaller than the liquid film deformation. Also note that the black curve is measured by confocal profilometry and the red curve is obtained by dual-wavelength interferometry: due to equipment limitations, they can be measured only in two separate experiments with identical conditions instead of in one single experiment. The lower panel magnifies this air gap in a zoomed-in scale.

Figure 12

Tracer particles and interference fringes for wavelength (a) 434 nm and (b) 546 nm. The tiny dark spots which appear in both channels are the tracer particles in liquid film, while the bright spots which only appear in (b) are the tracer particles in droplet.

Figure 13

Intensity line profile for different times. From left to right, the corresponding time of the red lines are 291, 331, 658, 738, 1023, 1183 ms. They form three time intervals: 291–331 ms, 658–738 ms, and 1023–1183 ms.

Figure 14

Air film profiles at six different times indicated by the red lines in Fig. 13. Note that they form three time intervals and the length of each interval is different.

Figure 15

Dots: Velocity of tracked particles in droplet (left-hand column) and in liquid film (right-hand column). All the graphs are plotted in the same scale. Green dots: Selected fast particles. Red line: Boundary velocity $v_{ra}$ (left-hand column) and $v_{rb}$ (right-hand column) obtained by fitting the fast particles (green dots).

Figure 16

Evaluation of the integral Eq. (a4) from experimental data with different upper bond $\psi R$, as defined in Eq. (a5). The red dashed line indicates the weight of drop.

Figure 17

Droplet lifetime versus skidding distance for various droplet diameters. All droplets have the same viscosity (50 cSt) and similar impact velocities (0.36, 0.33, and $0.32\mathrm{m}/\mathrm{s}$ from top to bottom), with only the diameter significantly varied from 3.2 to 1.7 mm (from top to bottom). Similar results are obtained as in Fig. 1: there always exist two distinct states with long and short lifetimes.

Figure 18

Droplet lifetime versus skidding distance for various liquids with different viscosities, surface tensions, and impact velocities. The same result reproducibly appears: there always exist two distinct states with long and short lifetimes. The impact velocities are, from top to bottom, 0.32, 0.13, 0.13, 0.36, and $0.10\mathrm{m}/\mathrm{s}$, respectively.

Figure 19

Bottom view of drop skidding on mica with (a) large and (b) small tilt angles. The droplet center moves from the red to the blue spot along the green arrow line, which is also the axis of symmetry.