The sandcastle a spoonful of water holds

The outlet that jams by chance took dry grains and showed that arches form across an outlet for reasons of geometry and chance, with friction doing the holding. A dry heap has no strength of its own beyond that friction: it will stand at an angle that does not know the size of the heap, and any steeper face simply flows. Dry sand has no tensile strength at all. It can be pushed on and it can be sheared against pressure, but it cannot be pulled, and nothing about a pile of dry grains holds a vertical surface up.

Pour a little water on it, and all of this changes. On a beach the sand just above the waterline holds a footprint with sharp edges and vertical sides, and a child can carve a wall and a tower. The change is out of all proportion to what was added. A typical sandcastle sand has only a few per cent of water by mass, the same mass could not cover the grains in anything like a film, and adding more does little until suddenly the wall collapses into soup. A trained sand sculptor will use a large proportion of water while compacting, and then lets most of it drain. So the question is what the water is doing, and why so little of it does the whole job.

Two pulls in one bridge #

Water wets sand. At each point where two grains touch or nearly touch, surface tension draws any available water into a small collar around the contact, with a neck at the middle and a meniscus curving inward. This is called a capillary bridge, and the wet plates that hold on when pulled dealt with the same kind of film between flat surfaces. A bridge between two spheres has a simple geometry: seen from the side its profile is close to an arc of a circle.

Such a bridge pulls the two spheres together in two ways. The surface of the water around the neck runs along a circle of radius $r_2$, and surface tension along that line pulls axially with a force $2\pi r_2\gamma$. Then the pressure inside the water is lower than outside, by the amount Laplace’s formula gives for a curved surface:

$$\Delta p = \gamma\left(\frac{1}{r_1} - \frac{1}{r_2}\right),$$

where $r_1$ is the small radius of the meniscus’s curve and $r_2$ the neck. Since $r_1$ is much smaller than $r_2$ for a small bridge, this is a suction, and it acts over the neck’s area to give $\pi r_2^2\Delta p$. Atmospheric pressure from outside then squeezes the grains together. As how high water will climb already observed for a tube, the holding force comes from the air, in the sense that this suction is what atmospheric pressure acts against.

For a bridge of half-angle $\beta$ between touching spheres of radius $R$, geometry gives both radii in terms of $\beta$ and the contact angle. Summing the two forces and letting the bridge shrink yields a surprising result. The surface-tension part goes to nothing as the neck narrows, and the suction part grows, because $r_1$ shrinks. Their sum approaches a limit:

$$F \to 2\pi R\gamma\cos\theta.$$

The size of the bridge has disappeared. The force is set by the radius of the grain, the surface tension and the contact angle. A bridge the size of a bacterium between two grains of sand pulls about as hard as one a thousand times larger.

A force that does not count the water #

This independence of volume is the main result. On the plot, the force sits on a plateau from the smallest volume calculated, a millionth of $R^3$, out to a thousandth, with a change of only 7 per cent. It starts to fall significantly only as the bridge grows to several per cent of a grain’s volume, where the neck is no longer small and the geometry begins to differ. At that point the bridges start to touch neighbouring bridges anyway.

The physical reason is that the two contributions to the force trade against each other. A smaller bridge has more curved surfaces and hence stronger suction, over a smaller area. Since the suction grows as the area shrinks, the product stays about constant. There is also a route to the result through energy. A bridge’s force is the rate of change of surface energy with separation, and when a sphere is moved by a small distance, the surface exchanges air-solid area for water-solid area. It is a geometric fact that this exchange depends only on $R$ and $\gamma\cos\theta$, and not at all on the volume, in the limit of small separation.

The same calculation also explains something that seems at odds with the plateau. Real grains are not smooth, and bridges with very small volumes first fill only the small gaps between asperities. So at very low water content, the effective radius of contact is the radius of a roughness feature, far smaller than that of the grain, and the force is correspondingly lower. As water increases, the bridge grows past the roughness and sees the entire grain, and the force jumps towards its plateau. Measurements by Halsey and Levine in 1998 and others found the strength in this regime rising with water content until a plateau is reached. In very humid air, a similar capillary effect from condensed water occurs in fine powders, as the pore that fills from dry air discusses, which explains why flour and salt clump in a damp kitchen.

From one bridge to a wall #

To go from one bridge to the strength of the bulk requires a count. Horst Rumpf gave one in 1958: at a packing fraction $\phi$ with $k$ contacts per grain, a plane through the packing cuts bridges at a density that makes the tensile strength

$$\sigma = \frac{\phi k}{\pi d^2}F,$$

and with $F = \pi d\gamma$ for grains of diameter $d$ this comes to $\phi k\gamma/d$. A packing fraction of 0.6 and six contacts per grain gives $\sigma = 3.6\gamma/d$, and for 0.3 mm beach sand that means 864 Pa.

That sounds small, and it is about a hundredth of an atmosphere. It is, however, much more than the weight of the sand above: one centimetre of sand presses with about 160 Pa. That is enough, since all a wall really needs is to hold up its own material. The plateau is the crucial feature. The tensile strength does not grow as water is added, since bridges are at their plateau, and it starts to fall only once bridges have joined up. Experiments on real sand with X-ray tomography have followed the sequence in fine detail: bridges at low water content, clusters of filled pores as it increases, then eventually fully saturated sand where no air surface remains anywhere to do any pulling. Mario Scheel and colleagues reported in 2008 that real strength stays nearly constant from a few per cent up to a surprisingly high water content, a broad plateau of the sort the figure shows. They could explain it by noting that pools of water bounded by menisci at the same pressure exert about the same cohesion that separate bridges do.

The final drop is sudden in comparison. Pore water fully connected has no menisci left, and so has no suction, and as soon as that happens the strength vanishes. The sand is then a slurry, with friction reduced by the water pressure between grains. It is a rule that any sculptor learns by feel, and it marks a sharp difference from the behaviour of a material whose strength increases with its binder, such as cement or clay.

How tall a wall #

The tallest vertical face that a material can make is one of the oldest results in soil mechanics. A cut bank fails along a slip plane, at about 45 degrees plus half the friction angle, when the weight of the wedge above overcomes the cohesion along the plane. Coulomb’s 1773 analysis gives a critical height of $H = 4c\tan(45° + \phi/2)/\rho g$, with $c$ the cohesion, here taken as the tensile strength times the tangent of the friction angle. For beach sand, this comes to about 22 cm.

That is about the height of a child’s sandcastle wall. Real sand sculptures go a great deal higher, and they do so by defeating the assumptions behind this estimate. Compacting sand in wooden forms by vigorous pounding raises its packing fraction and increases the number of contacts, and its friction angle goes up considerably, so the wedge has more to overcome. World-record sculptures stand above fifteen metres, with carefully selected sand that includes angular grains and silty fines. Their shape also matters, since their width at the base provides a thick, nearly self-supporting mass. Studying sand columns, Møller and Bonn, and Pakpour and colleagues in 2012, showed that real columns of moderate height fail by elastic buckling under their own weight, with a critical height growing as the base radius to the two-thirds power, and with Young’s modulus depending on the water and the compaction. The closest analogue is Euler’s classic problem of a tall column under its own weight.

Grain size matters directly, since strength goes as $1/d$. A finer sand holds a taller wall until very small sizes, where clay-like forces take over, and it also holds water better, since narrower pores retain the water against gravity. That is why the fine sand near a retreating waterline carves so much better than the coarse sand at the top of a beach.

The size at which water stops mattering #

For a single grain, the question becomes whether the bridge is strong enough to matter against the grain’s own weight. Bridge force goes as the diameter. Weight goes as its cube. The ratio is

$$\frac{F}{mg} = \frac{\pi d\gamma}{\tfrac{\pi}{6}\rho g d^3} = \frac{6\gamma}{\rho g d^2},$$

which falls to 1 at a diameter of about 4 mm. That length is close to the capillary length of water, which is set by the same quantities with water’s density in place of quartz’s. The size at which a body becomes round shows a similar balance of two scalings over planetary sizes, between gravity and material strength. A grain below this size is dominated by cohesion when it is damp. A pebble above it barely notices the water, and a wet gravel path crumbles just like a dry one, as a child at the top of a shingle beach discovers.

For beach sand the ratio is 185, so a damp grain is held on by nearly two hundred times its own weight. A silt grain at 20 µm is held by about forty thousand times its weight, though at that scale its other surface forces matter as much as water’s. This also tells why water makes such a large difference for sand and none for marbles: not because of anything in the material but because of the length scale. Filling a bowl with marbles and adding water does not make them stick, although the same forces are at work, with the same size of bridge force per unit of grain radius.

The same balance also applies to insects walking on water, to condensation on the fine hairs on a leaf, and to why powders behave so strangely when they are slightly damp. All of them share the feature that surface tension scales with length and weight with its cube, and all are dominated by surface tension at small sizes. The walls a rinse pulls together deals with the corresponding bridge force in microfabrication, where it is enough to destroy structures.

What happens to a dry-looking wall #

A sandcastle that has dried out in the sun falls apart as the last of the water leaves. It will hold together long after the sand looks dry, because a bridge’s strength does not depend on its volume, so the water can disappear for a long time with no apparent effect on the wall, until the bridges disappear altogether. That is one reason a castle stands through most of a hot afternoon and then crumbles quite quickly. The other is that, with the drying of the outer layer, a difference develops between wet interior and dry surface, and dry grains fall off the outside.

A related experiment was done on salt-water sand. Evaporating seawater from a castle leaves dissolved salt, which recrystallises at the contacts and can cement the grains together, and dry salt-cemented sand can therefore hold a shape even after it has dried. Adding fine clay or some adhesive does much the same thing. Sand-sculpture contests in fact often spray their finished work with a dilute glue solution, and that replaces what the evaporating water was doing.

A striking historical comparison was published in 2014, by Daniel Bonn’s group, on how ancient Egyptians moved large stones. A wall painting from about 1900 BCE shows a statue on a sledge, with a person pouring water in front of it. They measured the pulling force of a sledge on sand with various amounts of water, and found that the correct amount of water about halves the force needed, since a sledge on dry sand pushes up a heap in front of it and on damp sand does not. The maximum is at a few per cent water and it worsens again on adding more. It is the plateau in a different mechanical test, and it fits the same range of water content.

What these estimates leave out #

The bridge calculation assumes smooth spheres in contact, a static bridge and a circular meridian, which is a good approximation at small volumes and overestimates the force a little at large ones. Real sand has rough, irregular grains. Bridges in a packing are not independent, and Rumpf’s formula uses averages for contact count and orientation that are not measured precisely. Thus the 864 Pa figure is an estimate, right in order of magnitude and typically within a factor of two or so of measurements. The bank-failure height uses Coulomb’s approximation of a straight slip plane. The fall from the plateau, as the caption says, has no derivation here; its shape depends on how pore clusters connect, and the correct treatment is an active topic.

The bridges drawn are also static, in equilibrium with no flow. A real wet sand under shear has its bridges stretch and rupture, with energy lost at each rupture. Each rupture carries away a certain energy, and this rather than the static force decides the failure of wet granular flows, and hence the bank’s behaviour after failure. All of this concerns water and clean silica. A contaminated surface, an oily film or a change in the contact angle, reduces $\cos\theta$, and sand that has been treated to repel water does not make a castle at all, however much is poured on it.

Still open: how a wet pile flows #

The static strength of wet sand is well understood. Its behaviour when it starts to move is not. A dry granular flow has a fairly successful continuum theory, in which friction depends on a single rate variable, as the sand whose friction depends on how fast it flows describes. Wet flows add a cohesion, a rupture energy and water redistributing itself between bridges, on a slow time scale set by the viscosity of the water. Those effects make a simple law hard to find. Researchers have found that wet granular flows have a yield stress that rises with the rate in some regimes and falls in others, with hysteresis when bridges reform slowly.

This is more than a curiosity, because landslides, debris flows and the behaviour of damp powders in industrial processes all depend on it. A wet slope may hold for years and fail suddenly after rain has raised the water content just enough to remove the suction, and the warning signs of that are not obvious. How much the water content at failure can be predicted from the grain-size distribution, the initial packing and the rate of infiltration remains a subject where measurements outrun theories, and an accurate forecast of the failure of a given slope is out of reach.

A bridge of water between two grains pulls with about $2\pi R\gamma\cos\theta$, whatever its volume, because a smaller bridge has more suction over less area. So a few per cent of water does the whole job, and more makes no difference until the bridges run together and vanish. The pull goes as grain size and the weight as its cube, and they meet near 4 mm: beach sand is held by 185 times its weight, and a pebble hardly at all.