Soft matter / 04
Round blobs and flat puddles
Small drops of water are round. Large ones are flat. The crossover is not gradual and arbitrary; it happens at a specific size you can calculate from two properties of the liquid.
Two forces, pulling different ways
Surface tension wants to minimise surface area, and the shape with the least surface area for a given volume is a sphere. Gravity wants to lower the centre of mass, and the way to do that is to spread out flat.
Which wins depends on size, and it depends on size because the two effects scale differently. Surface energy grows with area, so with the square of the length. Gravitational energy grows with volume times height, so with the fourth power. Make something bigger and gravity gains faster. There must therefore be a size where they balance.
The capillary length
Setting the two against each other gives a length built from the surface tension, the density and the strength of gravity:
ℓc = √( γ / (ρ g) )For water at room temperature, with a surface tension around 0.072 newtons per metre, a density of a thousand kilograms per cubic metre and gravity near 9.81, this works out at roughly two and a half to three millimetres.
That number is worth holding onto, because you have been observing its consequences your whole life. Dew drops, a few millimetres across, are convincingly round. A spilled glass of water is a flat film. Mercury, which has a much higher surface tension, has a larger capillary length and stays in visibly round beads at sizes where water would have spread.
Nothing about water changes between the dew drop and the puddle. Only which of two competing energies happens to be larger at that size.
Why a stream of water breaks into drops
The same preference for less surface area explains something you can watch at any tap. A smooth cylinder of falling water does not stay a cylinder. It develops a wobble along its length, the wobble grows, and the stream pinches off into separate drops.
This is the Plateau-Rayleigh instability. A cylinder of liquid has more surface area than the same volume divided into spheres, so surface tension actively drives the breakup. The wobble grows fastest at a particular wavelength related to the diameter of the jet, which is why the drops come out fairly evenly sized rather than at random.
It is also why a thin jet breaks up sooner than a thick one, and why the neck between two forming drops thins to nothing rather than settling at some intermediate width.
What changes when the blob is elastic
A jelly is not a liquid, and it does not obey any of this in the same way.
A liquid drop resists gravity only through its surface. A gel resists through its whole volume, because the network has a shear modulus. That is an enormously stronger effect at any size above the very small, which is why a jelly turned out of its mould holds a shape with corners on it, while the same volume of water would not hold a shape at all.
Comparing surface tension with elasticity gives another length, sometimes called the elastocapillary length, roughly the surface tension divided by the shear modulus. For a centimetre-scale jelly with a shear modulus in the kilopascals, that length comes out in the micrometres. Surface tension is therefore completely irrelevant to the shape of a dessert jelly. Elasticity does all of it.
This is why the interactive specimen has no surface tension term at all. At the scale being simulated it would change nothing you could see. What matters is the network resisting shear and refusing to change volume, and those are the two rules the model enforces.
Where the two regimes meet
Very soft gels at very small scales sit between the two descriptions, and that is an active area of research. Surface tension can measurably round off the sharp corners of an extremely soft gel, and the softer and smaller it is, the more liquid-like its shape becomes. The boundary between a soft solid and a liquid, it turns out, is a matter of which length scale you are looking at.