The drop that leaves in the same time however hard it lands

The drop that will not merge followed a drop that sits on a film of air and refuses to join its neighbour, and it ended by asking what finally lets the merger happen. This is a drop on a surface that refuses to let it stay at all. On a leaf of lotus or nasturtium, or on a soot-blackened plate or a modern water-repellent coating, a landing drop does not spread out and stick. It flattens into a thin pancake, gathers itself back up and leaves the surface, nearly intact, like a small rubber ball. Filmed at thousands of frames a second it is one of the prettiest things in fluid mechanics.

Denis Richard, Christophe Clanet and David Quéré filmed exactly that in 2002 and published a short letter with a strange finding in it. They dropped water on a heavily textured, water-repellent surface at speeds from around 20 cm/s to over 2 m/s and timed the contact. The drops spread by very different amounts, and it might be expected that a drop spreading further would take longer to go out and come back. But the contact time did not depend on the speed. It depended on the drop’s size only, and for a water drop of a millimetre in radius it came to about 10 milliseconds, whatever had been done to it.

A drop is a quadratic energy #

The clue is in how a drop stores energy. As the skin that is not a skin explained, a free drop is at its energy minimum as a sphere, and deforming it creates new surface at a cost of $\gamma$ per unit area. For a small deformation of the shape by an amount $x$, the extra area, and hence the extra energy, must go as the square of $x$, by the argument of every minimum is a parabola. An energy quadratic in displacement is a spring. Its stiffness has the units of force per length, which are the units of surface tension itself, and the only number around to set it is $\gamma$. So a drop’s stiffness is of order $\gamma$, with a numerical factor from geometry.

Now the argument is the one that made a pendulum’s small swings keep time. For a mass on a linear spring, the time for one half-cycle is

$$\tau = \pi\sqrt{\frac{m}{k}},$$

and speed is nowhere in it. A harder push gives more compression, in exact proportion; the restoring force also grows in exact proportion. So the motion scales in amplitude and keeps its clock. With $m \sim \rho R^3$ and $k \sim \gamma$ the contact time is then

$$\tau = C\sqrt{\frac{\rho R^3}{\gamma}},$$

in which the constant was measured at $C = 2.6$.

It is worth being clear about what this figure is and is not. The curves integrate a mass and a linear spring. The spring’s stiffness is chosen from the measured constant, so the figure does not derive 9.7 ms from anything. The result it shows is that a linear spring has one contact time at every speed, and the physical claim that a landing drop is such a spring comes from the experiments. Why it should be so at large deformation, when the pancake is far from a slightly distorted sphere, is not obvious. Theory that follows the actual flow inside the drop came later and is still not simple. The drop’s actual internal motion is a rapid outward flow in a thin sheet, with a rim, a very long way from the small-amplitude picture. Yet the law survives.

What the time is made of #

Size enters as the three-halves power. Doubling a drop’s radius makes the contact last 2.8 times as long.

The numbers invite a comparison. Lord Rayleigh found in 1879 how a free drop oscillates when disturbed, as a liquid sphere under its own surface tension, and the lowest mode, a sphere swinging between a prolate and an oblate shape, has an angular frequency $\omega^2 = 8\gamma/(\rho R^3)$. Its period is $2\pi/\omega = 2.22\sqrt{\rho R^3/\gamma}$. So the same combination of quantities appears and with a similar numerical factor. Rayleigh’s own calculation contains the scaling that the contact time would have, a hundred and twenty years before anyone timed it. But a simple comparison of the numbers gives pause. If a landing drop were half a cycle of that lowest mode, its contact would last about 1.1 in those units; what is measured is 2.6, more than a full cycle. The drop on a surface is not simply oscillating as a free drop does. It has a wall on one side, it has no lower half to squash outward, and it moves as a thin spreading sheet. The agreement of the scaling and the disagreement in the coefficient are both real, and a formula that gets one right need not get the other.

Viscosity is missing from all this and that is a real prediction. Water is about the least viscous thing most people think about. In experiments with mixtures of water and glycerol, varying the viscosity by a factor of ten leaves the contact time largely unchanged, though the drop loses more energy and leaves more slowly. Nor is the rebound purely elastic. A drop arriving at 2 m/s typically leaves at a much lower speed, with a good deal of its kinetic energy having gone into internal sloshing and vibration, the mode of oscillation it is still doing as it flies off. Those vibrations do not lie in the spring model, in which energy is conserved between a single degree of freedom and itself. They are more like the ripples it may set up on leaving, and the model says nothing of them.

Why a rubber ball is different #

The strangeness of the result becomes clearer by comparing it with what a solid does.

Heinrich Hertz worked out in 1882 how two elastic spheres deform when pressed together, while he was on Christmas break, in a paper of remarkable completeness. A sphere pressed on a plane makes a circular contact whose radius grows with the depth, so the contact has more material to compress at larger depth, and the force grows as the $3/2$ power of the compression. It is a spring that gets stiffer the more it is pressed. Solving the collision then gives a contact time that falls with speed, slowly, as $V^{-1/5}$, and that is what the bounces that add up to a stop and similar calculations depend on. The figure has the check built into it: the ratio of the contact times at the two ends of the range agrees with that exponent to two per cent.

For a drop, the corresponding statement would be that its energy has some larger power of the deformation than two. The experiments say it does not. It stores energy as area, and its area rises nearly quadratically with the amount of squash, up to surprisingly large squash. So on the plot the two curves cross only at the one speed where they were matched. A slow ball stays longer than a drop of the same size and a fast one stays for less. In a test with no labels, the contact time at two speeds would say which was the liquid.

This comparison has more behind it than taste. A contact between a drop and a surface is a load, and engineers of turbines and aircraft need to know how a drop transmits force. The peak force follows from $m V/\tau$, so it goes as speed, with a mean over the whole contact of about $2 m V/\tau$. Rain has been measured beating on surfaces at about this scale. For 1 mm radius drops arriving at terminal speed, the force is not huge, but it arrives through a small contact area and the pressure spike at first touch, before the drop has begun to spread, is far larger than the mean. That spike is what erodes the leading edges of wind-turbine blades over the years, and neither the spring nor the mean can tell about it.

Where the rule applies #

The result was established in a narrow regime and it would be unwise to treat it as general.

The Weber number $\rho V^2 R/\gamma$ compares kinetic energy with surface energy, and for most landing drops it runs from about one up to a few hundred. Below about one, the drop deforms hardly at all. It lands like a heavy bead, and surface forces such as adhesion and any trapped air become important. Far above a hundred, it may break apart. On a real surface there is another, separate limit: surfaces that repel water do so because their roughness traps air beneath the drop, and at high enough speed the liquid is driven down into the roughness, displacing the air, and then sticks. A lotus leaf has two scales of roughness partly to resist this. So the rule that a drop leaves in the same time holds only where it leaves at all.

Above the capillary length, $\sqrt{\gamma/\rho g}$, or 2.7 mm for water, gravity starts to matter as much as surface tension. A large puddle resting on a water-repellent surface takes the shape of a flattened pancake of constant thickness, and its dynamics change. Real raindrops above about 2.5 mm have also started to flatten under their own fall. So the regime where the rule is exact occupies only part of the chart, and a good many ordinary raindrops are near its edge.

More recently it has been found that a surface can do better than the rule. Certain patterns of ridges or a slight curvature make the drop break symmetry during the spreading, so that it leaves as a flattened pancake in a much shorter time, at a fraction of the usual contact. James Bird and colleagues in 2013 reported this with a macroscopic ridge reducing contact by about 40 per cent. On a surface with suitable posts, drops have left from the pancake stage in about a quarter of the normal time, by a mechanism of the drop bouncing as a whole rather than recoiling. A butterfly wing has features of this kind, and how far it depends on them is not certain. Such results show that the coefficient 2.6 is not a law of nature. The scaling is more robust, and it holds for those cases too.

What contact is good for #

The reason to care about 10 milliseconds is in what can or cannot happen in that time. A drop on a surface colder than it can lose heat and start to freeze. Whether a supercooled drop freezes onto an aircraft wing or a power line depends on the time it stays, and how quickly heat and the ice front can move.

The picture shows that the contact is too brief to cool the drop much. Only a thin layer gets cold. Experiments with supercooled drops on textured surfaces have found them to bounce off at temperatures where they would freeze instantly on a smooth surface, and that has driven a considerable effort towards anti-icing coatings. The method is limited, because a humid cold environment will frost the texture itself before the drops arrive, and a frost-filled surface no longer repels water at all. But it is a nice case of a physical constant of contact, the brevity of a bounce, deciding an outcome.

There are biological uses as well. A leaf that sheds a drop in 10 ms has also lost little of its water-borne contamination, and a drop that leaves takes dirt with it. Cicadas and some other insects use the energy of merged droplets to jump water away from their wings, a process in which the same scaling appears for the time of the jump. Pathogenic spores can get thrown off leaves by the bouncing of rain, and a slow film on a plate would have had none of this behaviour.

The same clock in other jobs #

The combination $\sqrt{\rho R^3/\gamma}$ turns up throughout the behaviour of small liquid bodies, which is a sign that the quantity is more general than a bounce. It is called the capillary–inertial time, and for water at a millimetre it is a few milliseconds. A jet of water breaking into drops, which the thread that cannot stay a thread analysed, pinches at a rate set by just this time for the thread’s own radius, and an inkjet printer’s design begins from it. Two drops that touch and merge pull together at a speed set by it, with the bridge between them growing as the square root of time. A soap film punctured at one point opens with a speed set by the same balance, as Taylor and Culick found.

In all of those, the fluid’s inertia is resisted by surface tension and nothing else is involved. Viscosity does not appear until the size has become so small that it dominates, at a length called the viscous–capillary length, about 14 nm for water. So in the middle of a very large range of sizes, water is effectively an inviscid fluid with a skin, and all the events in this range keep this time. A chart of contact time against radius is thus also a chart of how fast a raindrop can oscillate, or a jet can pinch, within a modest factor.

A further example makes for good viewing. Two small drops on a water-repellent surface, on merging, release surface energy, and the merged drop can jump off the surface unaided. The speed it reaches scales with a capillary–inertial velocity, $\sqrt{\gamma/\rho R}$, about 0.2 m/s for a drop of 100 µm radius. It has been shown that condensers can use that effect to clear their surfaces and improve heat transfer. Cicadas appear to keep their wings clean by it, and a fungus uses a similar mechanism to launch its spores, in an elegant biological use of a small asymmetric merger, as Anne Pringle and colleagues showed. The amount of energy available is tiny and its use is exact.

This raises the question of what a drop remembers when it leaves. The ring left by the ring the drop leaves behind belongs to a drop that stayed. On a good water-repellent surface, a bouncing drop leaves nothing, and any solute comes with it, which is why such surfaces clean themselves.

What is not in the spring #

The spring picture has a single degree of freedom, and the drop has thousands. It does not follow the actual flow, the thin lamella, the rim, the pressure at the moment of first contact, the shape at maximum spread, or the possibility of a thin jet being fired upward at the end of the recoil, which sometimes happens from a tiny trapped air bubble. It has no energy loss. It has been given its one parameter by experiment, and the spreading it predicts is linear in speed, whereas the measured spread goes more like the speed to about the half power, so the spring is plainly a wrong description of the size of the spread. That it nevertheless predicts the time correctly is a reminder that a correct time need not imply correct kinematics.

A deeper account has been slow in coming. Soto, Clanet, Quéré and others have explained parts of the spreading using a scaling for a thin sheet with effective gravity from the drop’s deceleration. Numerical solutions of the full equations do reproduce the 2.6, but they do not explain it in a line. Figures in this essay use one number that came from the experiment, and the whole reason the constant is not 1.1 has not been given here.

Finally the bounce itself relies on a trapped layer of air. On a flat, smooth, hydrophobic surface in vacuum, things go differently, and a drop landing at low enough speed on any surface may float for a moment on the air that cannot escape in time, and bounce without touching, as the drop that will not merge described on its own liquid. A figure of a contact time does not say what, at a molecular level, is meant by contact.

Still open: how much shorter a contact can be made #

The rule has become a floor to be beaten. Since a shorter contact means less heat transferred, less ice and less dirt taken on, there is an industrial interest in surfaces that shorten it. Researchers have found ways, using ridges, posts, curvature and flexible surfaces. A flexible surface may be the most interesting of these: a thin flexible sheet that recoils out of phase with the drop can reduce the contact, a result that depends on the two oscillators being tuned, as the mass that makes another stand still would suggest. How short a contact can theoretically be is not known. There is no reason to expect it can be reduced to zero, and the various designs have so far not been pushed beyond a factor of around four. Whether such surfaces can keep their structure under years of rain, and survive abrasion, is the practical question, and it is the reason that nearly all demonstration coatings have stayed in the laboratory.

A drop’s energy is quadratic in its deformation, so it behaves as a linear spring with a stiffness set by γ, and its contact time, $2.6\sqrt{\rho R^3/\gamma}$ or 9.7 ms for a 1 mm water drop, contains no speed: harder landings go further and come back faster, unlike a Hertz ball, whose contact falls as $V^{-1/5}$. Water behaves as a better spring than rubber does. The habit worth keeping is to look for what is absent from a formula: when a quantity that everyone expected to matter is simply not there, the simple scaling behind its absence is usually more durable than any calculation of the coefficient.