The drop that runs to the narrow end

How high water will climb found that water rises in a narrow glass tube until the weight of the raised column balances the pull of the curved surface at its top, and that the height goes inversely as the tube’s radius. The essential step was that a curved liquid surface has a pressure difference across it, the Laplace pressure, equal to the surface tension times its curvature. A meniscus curving into the air, as water’s does in clean glass, leaves the water just beneath it below atmospheric pressure, and in a vertical tube the atmosphere pushes the column up until its weight makes up the difference.

Lay the tube on its side and the column has no weight to balance. A slug of water in a horizontal tube of uniform width has a meniscus at each end, both curving the same way by the same amount, both lowering the pressure inside by the same amount; the water feels equal suction at both ends and stays where it is. That is why liquids sit still in level tubes and why a drinking straw held horizontally keeps its contents.

Now let the tube narrow towards one end. In 1712 Francis Hauksbee, the Royal Society’s demonstrator of experiments, showed its Fellows what happens: a drop of water placed in a glass tube shaped like a narrow cone moves, by itself, towards the narrow end, and keeps going until it reaches the tip. A drop of mercury in the same tube moves the other way.

Hauksbee’s demonstrations were part of a deliberate programme. Isaac Newton, then president of the Society, was collecting evidence for short-range attractive forces between particles of matter, and capillary effects were the best evidence available. In the 1717 edition of the Opticks, in the long speculative Query 31, Newton described a drop of oil of oranges placed between two glass plates that touch along one edge and open slightly along the other: the drop “will begin to move towards the concourse of the glasses”, and move faster as it approaches. He took it as proof that the attraction between glass and liquid, acting only across very short distances, grows as the gap narrows. The explanation in terms of curved surfaces and pressures came a century later from Young and Laplace, and the skin that is not a skin found what lies behind it: a surface tension is the cost of bringing molecules from the interior to the surface, and the short-range forces Newton was looking for are what set it.

Two menisci, two suctions #

The explanation is in the two menisci. A meniscus in a tube of radius $r$, meeting the wall at a contact angle $\theta$, is close to a spherical cap of radius $r/\cos\theta$, and the pressure of the liquid just inside it is

$$p = p_0 - \frac{2\gamma\cos\theta}{r},$$

below the atmospheric pressure $p_0$ by an amount inversely proportional to the tube’s radius at that point. In a tapered tube the meniscus at the narrow end has the smaller radius, so the water just inside it is at the lower pressure. The pressure in the water is therefore lower at the narrow end than at the wide end, and water flows from high pressure to low: towards the tip.

For a two-microlitre plug of water near the middle of a gentle glass taper, the two suctions are 269 and 252 pascals, a difference of 17 — small, about the pressure at the bottom of two millimetres of water, but acting on nothing heavier than the plug itself and resisted only by the plug’s viscosity. Nothing outside the tube does anything; the tube does not move; there is no gravity in the problem. The drop is pulled along by the shape of its own container.

Mercury reverses the argument. It does not wet glass: its contact angle is about 140°, $\cos\theta$ is negative, and its menisci curve the other way, bulging into the empty tube and raising the pressure of the mercury behind them. The narrow end raises it more. Mercury flows from high pressure to low, towards the wide end, and runs out of the mouth of the tube. The same reasoning explains why mercury is depressed rather than raised in a narrow vertical tube.

The angle that decides #

The direction depends on the sign of $\cos\theta$, and so it changes as the contact angle passes some value. Naively that value is 90°, where the menisci are flat. It is not, and the reason is worth having.

The wall of a tapered tube is not parallel to the axis. It is tilted by the cone’s half-angle $\alpha$, inward towards the narrow end. The contact angle is measured from the wall, so the angle the meniscus makes with the tube’s cross-section is not $\theta$ but $\theta + \alpha$ at the narrow end, where the wall leans in, and $\theta - \alpha$ at the wide end, where it leans out. The pressures inside are

$$p_{\text{narrow}} = p_0 - \frac{2\gamma\cos(\theta + \alpha)}{r_{\text{narrow}}}, \qquad p_{\text{wide}} = p_0 - \frac{2\gamma\cos(\theta - \alpha)}{r_{\text{wide}}}.$$

The tilt works against the narrowing: it makes the narrow-end meniscus flatter and the wide-end one more curved. For a long plug the difference in radii wins easily. For a short one the two effects are comparable, because both grow in proportion to the half-angle — the radii of the two ends differ by the plug’s length times $\tan\alpha$, and the tilt is $\alpha$ itself. For a gentle taper, setting the two pressures equal gives a critical contact angle

$$\tan\theta_c \approx \frac{L}{2r},$$

with $L$ the plug’s length and $r$ its radius, and no trace of the half-angle at all.

So a long plug moves inward for almost any wetting liquid, and a short one only for a liquid that wets well. A droplet as long as the tube’s radius needs a contact angle below 27°; one ten times as long moves inward up to 79°. The plug in the first figure is about four radii long, which is why its push changes sign at 62°. The rule is a geometrical one, about the shape of the plug, and it explains a practical observation that otherwise seems arbitrary: in a tapered pipette tip, small drops of a moderately wetting liquid stay put while long slugs of the same liquid run to the tip.

The same answer from the energy #

The pressure argument has a twin that needs no pressures at all. A wetting liquid lowers the system’s energy every time it covers a piece of dry wall, because the solid–liquid interface it creates costs less than the solid–air interface it destroys, by $\gamma\cos\theta$ per unit area — the balance the angle a liquid makes found Young writing down in 1805. A plug of fixed volume covers more wall in a narrow tube than in a wide one, since a thinner cylinder of the same volume is longer and has more surface. Moving towards the narrow end, it trades a short, fat shape for a long, thin one, wets more glass, and lowers its energy; the rate at which its energy falls with distance is the force on it, and that force, divided by the cross-section, is the pressure difference of the menisci.

The same bookkeeping makes the mercury go the other way: a non-wetting liquid raises the energy for each area of wall it covers, and it lowers its energy by retreating to where it covers less. And it explains why a wetting liquid condenses first in the narrowest parts of a porous solid, as the pore that fills from dry air found: a tapered pore fills from its tip, where each molecule of liquid wets the most wall.

How fast #

The push is resisted by viscosity. A plug moving through a tube is a short stretch of the Poiseuille flow that the fourth power in a pipe found, with resistance growing as the plug’s length and as the inverse fourth power of its radius. In a taper both change as the plug moves: it gets longer as it enters the narrower part, since its volume is fixed, and the radius it moves through shrinks.

Silicone oil, which wets glass completely and is a hundred times as viscous as water, keeps the flow slow enough for the steady viscous description to hold. Released near the wide end, a two-microlitre plug sets off at under a millimetre a second and arrives at the tip in under a minute, speeding up slightly as it goes: the difference between the two suctions grows as the inverse square of the radius, a little faster than the resistance of the lengthening plug. Every such speed is a fraction of one natural velocity, the ratio of surface tension to viscosity, $\gamma/\mu$: about 70 metres a second for water and 0.2 for this oil. The geometry multiplies it by the taper’s half-angle and by ratios of radii and lengths, all small, which is why the plug creeps at a millimetre a second rather than racing at the capillary speed. A steeper taper or a shorter plug moves faster in proportion; a liquid with the same surface tension and ten times the viscosity moves ten times slower, and arrives at the same place. Water, less viscous, would move a hundred times faster in the model — fast enough that the model is no longer trustworthy, for reasons that come below.

Between two plates #

The tube is not essential. Two flat plates meeting along a line, like the pages of a slightly open book, make a channel that narrows towards the line, and a drop of water placed between them runs towards the line of contact.

The geometry is two-dimensional — each meniscus is curved in one direction only, so the Laplace pressure is $\gamma\cos\theta/h$ for a gap of half-width $h$ rather than $2\gamma\cos\theta/r$ — but the mechanism is the same, and it is easy to demonstrate: rest one microscope slide on another with a hair under one edge, put a drop of water at the open end, and watch it run in. The push grows steeply as the drop approaches the line, because the gap there is narrow and both menisci pull hard; the narrower wedge pushes harder at the same distance because its gap is narrower everywhere.

That wedge is also what the corner a liquid never stops climbing found sending water up the inside corners of a square tube without limit: a liquid that wets well enough cannot resist a narrowing gap, and keeps entering it.

Where the push is used #

Birds drink with it. Phalaropes, small shorebirds with long thin beaks, feed on drops of water containing their prey. A bird grasps a drop at the tip of its beak and opens and closes the beak rapidly; each cycle the two menisci, at the narrow and wide ends of the drop in the wedge between the upper and lower mandibles, move by different amounts, and contact-angle hysteresis lets the drop advance on each opening and not retreat on each closing. The drop ratchets up the beak into the mouth in a fraction of a second, against gravity. Manu Prakash, David Quéré and John Bush described the mechanism in 2008.

Plants and animals gather water with it. Cactus spines and the spindle-shaped knots on spider silk are tapered fibres, and on the outside of a cone the argument runs the other way: a drop wrapped round the fibre is more curved where the fibre is thinner, has a higher internal pressure there, and moves towards the thicker part. Fog droplets collected on a cactus spine’s tip run down it to its base, where the plant absorbs them; on spider silk, they gather at the knots.

Microfluidic devices pump with it. A channel whose width changes along its length, or whose walls have a gradient of wettability, moves drops without any pump, and tapered channels are used to drive and position small volumes in diagnostic chips. The pressures involved are small — tens to hundreds of pascals — but so are the volumes, and the channels can be made in any shape.

What the simple account leaves out #

Contact-angle hysteresis. A contact line resists moving: the angle it makes with the wall can vary over a range, between a receding value and an advancing value, without the line moving at all. The angle a liquid makes found that range to be ten degrees or more on ordinary glass. Within it, the menisci can adjust their angles to cancel the push from the taper, and a drop in a gentle taper, with a push of a few tens of pascals, often does not move. Hauksbee’s experiment works best in clean glass with a steep taper and a well-wetting liquid, for exactly that reason, and the phalarope’s beak exploits the hysteresis rather than fighting it.

Moving contact lines. A contact line advancing over a dry wall dissipates energy in the thin wedge of liquid next to it, and the dissipation grows logarithmically as the wedge thins towards a molecular size. For water moving at centimetres per second the dissipation at the contact lines can exceed that in the plug’s bulk, which the model’s Poiseuille resistance omits; that is the main reason its speeds for water are overestimates.

Plants do not use it to lift water. Tapering vessels are common in plants, and it is tempting to think that the push drawn here helps raise sap. It cannot do much: the column that is pulled, not pushed found that a tree’s water is held up by tension transmitted from the evaporating surfaces of its leaves, at pressures of minus several atmospheres, against which tens of pascals from a taper are negligible.

Gravity. A tube held at an angle adds the plug’s weight to the push. In a vertical tapered tube holding a plug, the plug’s weight competes with the difference between the two menisci, and a plug can hang at rest at the height where they balance — the tapered tube’s counterpart of Jurin’s height, set by both radii rather than one.

Evaporation. A drop near the open end of a tube loses liquid there, and a volatile liquid’s meniscus is cooled and its surface tension raised, setting up the flows along the surface that the surface that pulls toward the stronger side described. In a small drop moving slowly, those flows can be comparable to the drive.

Still open: how fast a contact line can be made to move #

The tapered tube is a clean way to apply a known, steady push to a liquid plug and watch its contact lines move, and it has been used for that purpose. What it shows is the part of the problem still argued about: how a contact line moves at all. A continuous liquid sliding over a solid with no slip at the wall would need infinite force to move its contact line, because the shear rate in the thin wedge next to the line diverges. Real contact lines move, so something at the molecular scale must relieve the singularity — slip of the liquid over the first molecular layer of the solid, a precursor film running ahead of the visible line, or molecules hopping across the line by thermal activation.

Each of those gives a different relation between the push and the speed, and experiments have found each of them working in different systems. Whether a single description covers them, and how the microscopic parameters it needs are set by the chemistry of the solid and the liquid, has not been settled. In a tapered tube the drive is known exactly — two Laplace pressures, set by two radii and a contact angle — and the speed is measured; the difference between the two is the contact line’s resistance, which is the quantity the theories disagree about.

The drive itself has been settled since Hauksbee’s time and was put in its present form by Laplace and Young a century later. The small bubble blows up the big one found the same pressure, $2\gamma/r$, sending air from a small soap bubble into a large one connected to it. A plug in a tapered tube is that experiment turned inside out: the tighter curvature decides again, and here the liquid flows towards it, because a meniscus curving into the air lowers the pressure behind it where a bubble raises it.