The water the dent displaces

The weight of the water that is not there derived Archimedes’ principle from the pressures on a submerged body’s faces: they do not cancel, and what survives is exactly the weight of the fluid the body has pushed aside. The block the water does not lift found that the principle is really a statement about what the body’s underside feels, and that a face the water cannot reach is not pushed. Through eight arguments the floating body was always large, and the surface it floated in was flat right up to its side.

A sewing needle laid carefully on water breaks that last assumption. It is seven times denser than water and floats anyway, sitting in a visible dent. The usual explanation hands the job to a different force: buoyancy for boats, surface tension for needles. Keller’s theorem, proved by Joseph Keller in 1998, says that the division is a matter of bookkeeping. The pull of the surface on the needle is exactly the weight of the water missing from the dent, and a needle floats, like a ship, by displacing its own weight of water. Most of what it displaces is just beside it rather than under it.

The two ways water holds a needle up #

A long cylinder lying on water is touched by the water along two lines, one on each side, and below those lines its surface is wetted. Two forces act. The water’s pressure on the wetted underside pushes up, and because the underside sits below the undisturbed level, that pressure is greater than the air’s. And at each contact line the surface, which is not a skin but behaves like one, pulls along its own direction with a force $\gamma$ per unit length, 72.8 millinewtons per metre for clean water. Where the surface slopes away from the needle at an angle $\psi$, the vertical part of that pull is $\gamma\sin\psi$ on each side.

The shape of the surface is fixed by a balance of its own. Inside the dent the water is below the undisturbed level, so the pressure just under the surface is lower than the air’s by $\rho g|h|$, and the surface must curve to hold that difference, as the small bubble blows up the big one found every curved surface does. The balance between curvature and depth sets a length, the capillary length

$$\ell = \sqrt{\frac{\gamma}{\rho g}} = 2.73\ \text{mm for water},$$

over which a disturbance of the surface dies away — the same length that sets how high water will climb up a wall — and for a long straight object it has an exact solution: the depth at any point and the slope there are tied by $|h| = 2\ell\sin(\phi/2)$. The steeper the surface meets the needle, the deeper the dent at the contact line, up to two capillary lengths for a vertical wall.

Where the surface meets the needle is set by chemistry. Water meets a given solid at a fixed contact angle $\theta$ — the angle a liquid makes with what it sits on, decided by the three surface energies at the line — about 100° for a slightly greasy steel needle, so the needle’s position in the dent, the depth of the dent and the slope at the contact line are all decided together by the needle’s weight.

The needle in the figure sits with its centre nearly a millimetre and a half below the level, in a dent several millimetres wide. The water meets it below its widest point, sloping at 37°.

The dent weighs what the surface carries #

Keller’s observation is short. Take the water in one dent — the region between the curved surface and the level it would have if the needle were not there — and ask what holds it in place. It is pulled sideways and down by its own weight and pressure, and its curved upper surface is held by the tension along that surface. Following the force balance along the whole meniscus, from the contact line out to where the surface is flat again, every bit of curvature is matched by a pressure deficit, and the total comes out exactly:

$$\gamma\sin\psi = \rho g\int |h|\,dx.$$

The vertical pull of surface tension at the contact line is the weight of the water missing from the dent. It is an identity, not an approximation, and the figures check it by computing both sides: the surface-tension force at the contact line, and the area of the computed dent, integrated point by point.

So the total upward force on the needle is the weight of water that would fill the region under it down to the wetted surface, plus the weight of water that would fill the two dents. Both are water displaced from where it would otherwise be. A floating needle displaces its own weight of water, exactly as Archimedes requires; it is just that four-fifths of that water is beside it.

That restatement has a consequence that is easy to check at a kitchen table. A glass of water with a needle floating in it weighs the sum of the two, obviously, and the water level rises by the needle’s mass divided by the density of water — the same rise a boat of that mass would cause. The water displaced into the rest of the glass by the dent is real water that has moved.

Pressing it deeper #

Push the needle down and both forces change. The surface tension’s vertical part grows as the dent deepens and the surface at the contact line steepens, until near a vertical meniscus it can grow no more. The pressure under the needle grows too, at first, as the wetted underside sinks deeper, and then shrinks as the contact lines climb the needle’s sides and the region below the level, under the needle, becomes a narrower strip. The total peaks — for this needle at 160 millinewtons per metre, with its centre 2.78 millimetres down — and falls beyond.

The needle’s weight, 112 millinewtons per metre, crosses the rising part of the curve at 1.43 millimetres, and that is where it floats, stably: push it down and the support exceeds the weight and pushes it back. The peak is the needle’s ceiling. Load it beyond — with a heavier needle of the same thickness, or by pressing on it — and there is no position at which the water can hold it.

The floating depth grows steadily with density until the fold, where the stable branch meets an unstable one and both end. A cylinder of this thickness and density 11 — lead — would float, barely, in a dent nearly three millimetres deep; at 11.2 it would not. There is no gradual sinking past the fold. The surface has nowhere further to go, the contact lines slide up and over the top, and the needle drops through. Anybody who has floated a needle and then tapped the glass has seen the fold from the unstable side: a small disturbance pushes it past the peak and it is simply gone.

Where the needle’s weight ends up #

Keller’s identity also answers a question the surface-tension picture leaves hanging: if the needle is held up by the surface, what holds up the surface? The answer is the water under it, and the water under the dent is held up by the bottom of the glass. The pressure that only knows depth found that the pressure at the bottom of a vessel is set by the height of the water above it, and with a needle floating the water stands higher everywhere outside the dent by just enough to make up the volume the dent and the needle displaced. Spread over the bottom, that extra height weighs exactly what the needle weighs.

So the needle’s weight reaches the floor as it would for a boat, through a slight rise of the whole water surface, and the surface’s tension is not a separate support standing on nothing. It is the link that lets the needle push water aside without the water closing over it — the mechanism by which a body denser than water can displace more water than its own volume. The tension holds the shape of the dent; the shape of the dent decides how much water is displaced; and the displaced water carries the weight, as it always has.

Two dents that meet #

Two needles floating near each other share their dents. Each sits a little lower on the side facing the other, where the two dents overlap and the surface is lowered by both, and each therefore lies on a slope that tips it towards its neighbour. The cereal that gathers at the edge of the bowl found that like menisci attract and unlike ones repel, with a reach of a few capillary lengths, and floating needles are the textbook example: laid parallel a few millimetres apart, they slide together and touch. Keller’s bookkeeping says the same thing in terms of energy. Two dents that overlap together displace the needles’ combined weight of water with less surface raised and lowered, and the pair falls into the lower state, releasing the difference as the motion that brings them together.

The same counting also covers a body floating at a boundary between two liquids, oil over water, where what holds it is the difference between the two densities and the tension of the oil–water interface takes the place of the air–water one. The dent is then an indentation in the interface, filled with oil where there would have been water, and its support is the weight of the water it displaced less the weight of the oil that replaced it.

Who is held up by which #

The split between dent and underside depends on size, and the capillary length is the dividing line. The dent’s contribution is a pull along two lines, so it grows only in proportion to the object’s width, while the water displaced under the object grows with its cross-sectional area, as the square. A cylinder a tenth of a millimetre across is held almost entirely by its dents. One a millimetre across, 92 per cent. At a capillary length in radius the two contributions are equal, and for anything much larger the dent is a small correction to Archimedes in the form every schoolchild learns.

That division is the reason the world of floating insects exists. A water strider weighing ten milligrams stands on six legs, each covered in fine wax-coated hairs that make the water meet it at about 167°, and each pressing a long dimple into the surface. By Keller’s theorem the dimples together hold ten microlitres of displaced water — the insect’s own weight. Its legs, a fraction of a millimetre thick, are far below the capillary length, and the dents carry essentially all of it; experiments pressing a single strider leg into water find it can carry about fifteen times the insect’s weight before breaking through, a margin that lets the strider stand still on the surface while rowing across it.

How dense a thing can float #

The same calculation, searched for the heaviest load each cylinder can carry, gives the densest material it can be made of.

For very small cylinders the limit is simple. The dents can carry at most $2\gamma$ per unit length — two contact lines, each pulling straight up when the meniscus there is vertical — and a cylinder of radius $R$ and density $\rho_s$ weighs $\pi R^2\rho_s g$ per unit length, so the allowed density goes as $1/R^2$: a cylinder ten times thinner may be a hundred times denser. A steel needle of the kind sold for sewing, half a millimetre thick, floats with a large margin; a darning needle of one and a half millimetres floats only if it is clean and dry and laid down gently; a steel rod of two millimetres sinks whatever is done to it.

The contact angle matters less than expected for the thinnest objects, since the meniscus can always steepen to vertical before the contact line slides over the top, and more for thicker ones, where how far the water can be kept from climbing the side decides how much water is displaced underneath. A water-repellent coating lets a thicker rod float; a clean, wetting surface, which water climbs eagerly, makes it harder. A drop of detergent sinks a floating needle twice over, after first, if dropped to one side, pulling it towards the stronger surface: it halves the surface tension, which halves the dents’ capacity and shrinks the capillary length, and it lowers the contact angle so that the water climbs the needle and over it.

A coin and a razor blade #

Keller’s theorem holds for bodies of any shape, not only long cylinders, with the dent’s volume measured all round the contact line. An aluminium coin of the Japanese one-yen denomination, a gram in mass and two centimetres across, floats on water if laid flat. Its contact line is its rim, about six centimetres long, and the surface tension along it, even pulling straight up all the way round, could hold up less than half the coin’s weight. What holds the rest is the dent: a disc of displaced water spread under and around the coin, a few millimetres deep, whose total volume is a millilitre — the coin’s weight in water.

Coins and razor blades also exploit a freedom the figures leave out. A contact line sitting on a sharp edge is not obliged to meet the surface at the material’s contact angle. It can rest at any angle between the contact angle measured on the face above and the one measured on the face below, a range that a right-angled edge widens by ninety degrees — a rule Gibbs stated and that experiments on pinned drops confirm. So the meniscus at a coin’s rim can steepen past what a smooth cylinder would allow before the line slides over, and flat, sharp-edged objects float at loads that the cylinder calculation forbids. A razor blade floats far better than a needle of the same weight.

Where the two-dimensional dent stops #

The figures treat an infinitely long cylinder in clean, still water, with a single contact angle and a meniscus that is a perfect two-dimensional curve. A real needle has ends, and around them the dent is three-dimensional and shallower; for a needle many capillary lengths long the ends are a small correction, for a short pin a large one. Real surfaces show contact-angle hysteresis — the angle at which a line advances is larger than the angle at which it recedes, by tens of degrees for an ordinary needle — so a needle’s floating depth depends on how it was put down, and the fold can arrive earlier or later than the figure says. Dust, grease and dissolved surfactants change the surface tension locally and drift with any current. And the figures are static. A needle dropped rather than laid down arrives with momentum, and whether it floats depends on whether the impact pushes it past the fold before the surface can push back; water striders, which stand still on the surface, must also walk and leap on it, and the forces involved in doing so are dynamic and far exceed the static ones.

What the figures can say without qualification is the identity they check. Whatever the shape, the contact angle or the history, at rest the vertical pull of the surface on a floating body equals the weight of the water its dent displaces. The domain of the rest of the argument is a body small compared with the capillary length, clean, and laid down slowly, and within it the limits computed here are the limits.

Still open: how much a floating thing can be made to carry #

The question of how heavy a load small objects can carry on water is being asked in earnest now, by engineers designing tiny robots that stand and walk on water like insects, and by materials scientists making superhydrophobic surfaces that keep water at contact angles above 160°. Leg shapes that bend to put more contact line in the surface, hairs that pin the line at many sharp edges, and flexible bodies that let the surface wrap them have all raised the load per unit length beyond the rigid cylinder’s. How far a soft, structured body can push the dent’s capacity — and whether the dynamic forces of walking and jumping, rather than the static load, are what really bound the size of a water-walker — are being worked out one design and one insect at a time.

The principle underneath them is fixed. The vertical pull of surface tension on a floating body equals the weight of the water displaced by the dent in the surface, so a floating needle — 1.36 mm of steel, its centre 1.43 mm below the level — displaces its own weight of water, 79 per cent of it beside the needle; the dent’s share falls from all of it for small bodies to half at a capillary length, 2.73 mm, and the heaviest load is reached when the meniscus stands vertical at the contact line. Archimedes never needed amending. He needed the dent counted.