The walls a rinse pulls together

How high water will climb balanced a curved surface pulling on a circumference against gravity pulling on a column, and got Jurin’s law. Every later argument about capillarity kept one thing fixed: the walls. Tubes, pores, gaps between plates and the vessels of a tree were all treated as rigid, so that the liquid could do whatever the surface tension required without the container noticing.

The walls notice. The water standing in a capillary gap is below atmospheric pressure — that is what holds it up — and the same suction that lifts the column pulls on whatever bounds it. For glass tubes the pull is negligible. For thin, tall or soft walls it is not, and past a certain point it wins outright: the walls bend towards each other, the gap narrows, the suction grows because narrower gaps suck harder, and the walls snap together. That instability explains a paintbrush’s point and a clump of wet hair. It also sets a hard limit on how tall and narrow a structure a chip factory can make, and the arithmetic of that limit is unforgiving.

A pull that grows as the gap closes #

Take two neighbouring lines in a regular pattern — thin walls of resist on a silicon wafer, say, standing in the rinse water after development. Each line is a cantilever fixed at its base. While the pattern is wet all over, nothing happens. As it dries, the water level drops into the gaps and forms a meniscus in each, and the pressure in the water there is below the air’s by the Laplace pressure, $2\gamma\cos\theta/s$ for a gap of width $s$ and a contact angle $\theta$.

If every gap is equal, each line is pulled equally from both sides and stays upright. Now let neighbouring lines lean towards each other in pairs by a small amount $\delta$ each. The gap between the pair narrows to $s - 2\delta$ and its suction rises; the gaps on their outer sides widen to $s + 2\delta$ and their suction falls. Each line feels a net pull towards its partner, $2\gamma\cos\theta\,[1/(s-2\delta) - 1/(s+2\delta)]$, which for small leans is proportional to the lean. The line’s bending stiffness pushes back, also in proportion to the lean.

The figure draws the two against each other. The pull is the heavy curve. It starts as a straight line and bends upwards, because narrowing the gap raises its suction faster than linearly, and it diverges as the gap closes. The stiffness is a straight line through the origin whose slope depends on how tall the line is. A cantilever of height $H$ under a uniform load deflects by $qH^4/8B$ at its tip, where $B = Ew^3/12$ is the bending stiffness of a line of width $w$ per unit depth, so its restoring pressure per unit lean falls as the fourth power of its height.

Everything turns on which line is steeper at the origin. For a short line the stiffness is steeper, the straight state is stable, and a small lean is pushed back. At 0.8 of the critical height the two lines cross again at a lean of 0.77 of the half-gap, marked with a dot. That point is an unstable balance: a pair pushed that far by some disturbance goes on to close. For a line exactly at the critical height the two start with the same slope. For a taller one the pull wins from the start and the pair collapses spontaneously.

This is a buckling problem with the load supplied by a liquid, and it has the structure of every stability threshold in mechanics: an equilibrium whose curvature, in the sense of every minimum is a parabola, passes through zero as a parameter is changed. The critical height is where the capillary stiffness, $8\gamma\cos\theta/s^2$, equals the bending stiffness, $8B/H^4$:

$$H_c^4 = \frac{E w^3 s^2}{12\,\gamma\cos\theta}.$$

The model treats both loads as uniform over the height, which is an idealisation — the meniscus sits at a particular level and moves as the pattern dries — and more careful treatments change the numerical constant. They do not change the powers, and the powers are what matter.

The squatter the smaller #

For equal lines and spaces, $s = w$, the threshold becomes a limit on the aspect ratio:

$$\frac{H_c}{w} = \left(\frac{E\,w}{12\,\gamma\cos\theta}\right)^{1/4}.$$

The width is still in it. The critical height grows as the three-quarter power of the width, faster than the width falls, and so the aspect ratio that survives drying shrinks as the features shrink.

The figure plots it for a typical polymer resist, with a Young’s modulus of 2.5 GPa, dried from water with a zero contact angle. At 100 nanometres a line can stand 4.1 times as tall as it is wide. At 30 nanometres the limit is 3.1, and at 10 nanometres 2.3. These are the numbers the semiconductor industry met as it shrank its patterns. Lines of resist taller than about three times their width collapsed during the final rinse and dry, not because the resist was weak — the lines stood perfectly well when dry — but because for a few seconds during drying they were loaded by a meniscus in every gap.

The reason small features suffer is a matter of scaling. Surface tension is a force per unit length, and the Laplace pressure it creates grows as one over the gap. Bending stiffness grows as the cube of the line’s width. Shrink everything by the same factor and the pressure rises in inverse proportion while the stiffness per unit load falls much faster, so the ratio of capillary force to elastic force grows as the pattern shrinks. It is the same scaling that lets insects walk on water and makes surface tension irrelevant to ships: capillary forces win at small scales and lose at large ones, and a nanometre pattern is about as far into capillarity’s territory as any engineered object goes.

The rinse that almost helps #

The formula suggests an obvious fix: dry from a liquid with a lower surface tension. It works, but less well than it seems.

The figure plots the critical height of 30-nanometre lines against the surface tension of the liquid they are dried from. Water, at 72 millinewtons per metre, allows 92 nanometres. Isopropanol, at 21.7, allows 124. Hexane allows 129, and a fluorinated rinse at 12 millinewtons per metre allows 143. The gains are real, and they are small, because the height depends on the surface tension only through its fourth root. Halving the tension raises the lines by 19 per cent. Doubling their height would need a liquid with a sixteenth of water’s surface tension, about 4.5 millinewtons per metre, which no ordinary liquid has.

The contact angle offers the same weak lever. The angle a liquid makes with a solid belongs to all three interfaces, the pull is proportional to its cosine, and treating the resist’s surface so that the rinse meets it at 70 degrees rather than zero cuts the pull by a factor of three and raises the critical height by a third. Surfactants added to the rinse lower both the tension and, often, the cosine, and they were one of the industry’s first responses.

The remedy that works completely is to have no meniscus at all. Carbon dioxide above 31 °C and 74 bar is a supercritical fluid, with no distinction between liquid and gas and so no surface between them and no surface tension. Replace the rinse with liquid carbon dioxide, take it past its critical point, and vent it as a gas, and the pattern goes from wet to dry without ever passing through the state in which the figure above applies. The same trick makes aerogels, whose fragile silica networks would be crushed by the capillary forces of ordinary drying, and it releases the moving parts of micromachines, which would otherwise stick to the substrate as the last film of water between them evaporates. This is the capillary counterpart of the column that is pulled, not pushed: a tree survives water under tension because its vessel walls are thick and lignified, and a nanometre pattern survives only if the tension is never allowed to form.

Jurin’s law with walls that give #

The same interaction changes the oldest result in capillarity. Hold two thin sheets vertically, clamped at their bottoms, a millimetre apart, with their lower edges in a bath of water. Jurin’s law says the water climbs between them to $2\gamma/\rho g d$, 14.7 millimetres for a gap of one millimetre. But the column of water between the sheets is below atmospheric pressure by $\rho g z$ at height $z$, which is exactly what holds it up, and the air outside is at atmospheric pressure. So each sheet is pushed inwards by a load that grows from zero at the bath to $\rho g h$ at the meniscus.

The sheets bend inward, the gap at the meniscus narrows, and the meniscus, obeying Jurin’s law for the narrower gap, climbs higher. That loads the sheets over a longer length with a larger suction, which bends them further. The figure computes where this feedback settles, by integrating each sheet’s deflection from its bending moment and iterating Jurin’s law on the gap at the meniscus, for plastic sheets with a modulus of 3 GPa.

At 900 micrometres thick the sheets barely give, and the water stands 1.01 times Jurin’s height. At 600 micrometres it stands 1.03 times, and at 450 micrometres 1.09. Then the curve turns steeply upwards, and at about 420 micrometres there is no longer a height at which the climb stops. The water rises, the sheets close behind it, and they zip shut from the bottom up. The last height the water holds before the zip, 1.18 times Jurin’s, is the fold at which the stable balance and the unstable one meet and annihilate, the same structure the first figure showed for resist lines.

The largest correction to Jurin’s law before the instability is only eighteen per cent. Flexibility does not change capillary rise gradually; it leaves it almost untouched and then removes it. That is the signature of an instability rather than of a correction, and it is why capillary rise between flexible walls is used as a method: wet sheets of paper, the pages of a book dropped in water and the fibres of a paintbrush are pulled together by the same rise feeding back on itself, and they either stay nearly as they were or close completely.

The length below which water wins #

Every result above compares a bending stiffness with a surface tension, and the comparison can be packaged into a single length. For a sheet of thickness $t$ and modulus $E$, with bending stiffness $B = Et^3/12(1-\nu^2)$ per unit width, the elastocapillary length is

$$L_{ec} = \sqrt{B/\gamma}.$$

A sheet much longer than this is bent easily by the forces of a meniscus; a sheet much shorter is effectively rigid to them.

The figure plots it for four materials. It grows as the three-halves power of the thickness, so it spans an enormous range. A steel sheet a micrometre thick has an elastocapillary length of half a millimetre; a plastic film of the same thickness, 62 micrometres; a rubber one, a single micrometre. A rubber sheet fifty micrometres thick has a length of 0.40 millimetres, and a drop of water a few millimetres across placed on a square of it will fold the square up around itself. Experiments in the 2000s used exactly that to make small three-dimensional shapes, with the sheet’s outline cut so that the folding drop produced a cube, a pyramid or a cylinder.

The same length decides the paintbrush. A brush dipped in water and withdrawn gathers its hairs into a point, and hairs clump in wet fur and in the feathers of a diving bird for the same reason. Each hair is an elastic rod, the water between neighbours pulls them together, and whether they clump depends on whether their free length exceeds a length built from their stiffness, their spacing and the surface tension. Hairs shorter than it spring back apart; longer ones stick, and once stuck in pairs they can clump into larger bundles by the same argument applied to the pairs. The result is the hierarchy of clumps visible in wet hair. The stiffness that sets the length is a fourth-power function of a hair’s radius, so fine hair clumps and coarse hair does not.

The airway that a film can close #

The same competition runs inside every breath. The smallest airways of a lung, the bronchioles, are soft tubes a fraction of a millimetre across, lined with a thin film of liquid. The film’s surface tension pulls inward on the tube wall exactly as the meniscus in a resist gap pulls on its lines, and the wall’s own stiffness, together with the tethering of the surrounding lung tissue, pushes back. At the end of a deep exhalation the airways are narrowest and the film thickest relative to them, and two things can go wrong. The film can gather into a ring and then a plug that blocks the tube, an instability of the film itself; or the tube wall, if it is soft enough, can buckle inward under the film’s pull, the elastic instability drawn in the first figure. In healthy lungs neither happens, because the liquid carries a surfactant that lowers its tension several-fold as the surface is compressed.

The small bubble that blows up the big one explains why the air sacs at the ends of these airways need that surfactant to stay open at all. The airways need it for the reason worked out here: a tube whose wall is weak compared with its lining’s surface tension closes, and the closing makes the pull stronger. Premature babies born before their lungs produce enough surfactant are at risk of both failures, and the treatment, a dose of surfactant delivered into the lungs, works by moving the balance back across the threshold rather than by strengthening the walls.

The black films of soap, drawn in the film that goes black before it bursts, show the same competition from the liquid’s side. There the two surfaces of the film pull each other together through the liquid between them, and what holds them apart is not an elastic wall but a repulsion between the surfactant layers. In each case the outcome is decided by comparing a force that grows as a gap closes with one that holds it open, and in each case the answer is a threshold rather than a gradual change.

What the beam models leave out #

The figures use the simplest mechanics that captures the instability, and the simplifications are worth stating.

The load is placed by assumption. In the first three figures the capillary load is spread uniformly over the line’s height. In reality it acts where the meniscus is, which moves down as the pattern dries, and a load applied near the tip bends a cantilever more than the same load spread along it. More careful treatments move the numerical constant in the critical height and leave its powers alone.

The walls are ideal beams. Resist lines have rounded tops, slightly sloping sides and a base that is not perfectly clamped; they also adhere to one another once they touch, which is why a collapsed pattern does not recover when the liquid is gone. The adhesion adds a second threshold: lines that touched briefly during drying may or may not separate again depending on whether their elastic energy can overcome the contact.

The pattern is perfect. A real wafer has lines of slightly varying width and gaps of slightly varying spacing, and collapse starts where the pattern is weakest. Near the threshold the fraction of lines that collapse rises steeply but not discontinuously, and yield data are fitted with distributions, not with a sharp line.

Drying is quasi-static. The meniscus is assumed to sit at equilibrium at each moment. Rapid drying, evaporation that is not uniform across the wafer, and the flows that the pore that lifts highest fills slowest found in the rise itself all add dynamic loads that the static threshold does not see.

Still open: how to dry the smallest structures at all #

As patterns shrink below ten nanometres, even the remedies run out. Supercritical drying works, but it is slow and expensive for a factory processing thousands of wafers an hour; low-tension rinses gain only the fourth root; and the resists themselves are getting thinner and softer as the chemistry is pushed towards finer resolution. Current approaches include drying by sublimation — freezing the rinse and turning it directly to vapour — rinses whose surface tension is made to drop sharply during drying, and changing the process so that the tallest, thinnest structures are never exposed to a liquid at all. Which of these will work at the scale of the next generation of transistors, whose features approach the size of a few dozen atoms, is being decided by trial in production rather than by theory, because at that scale the continuum description of surface tension itself begins to be doubtful: a meniscus a few nanometres across contains only a few molecules across its width, and whether the Laplace pressure still applies there is the same question the pore that fills from dry air left open about the Kelvin equation.

The habit worth carrying away is to ask of any capillary result whether its walls are part of the problem. A liquid in a gap pulls on the gap, and the pull grows as the gap closes, so any wall flexible enough to move will move in the direction that makes the pull stronger. Rigid walls hide this completely, which is why Jurin’s law is exact in glass. Flexible ones expose it as an instability with a sharp threshold, and the threshold scales so badly with size that at the smallest scales the question is not how high the water climbs but whether anything is left standing when it has gone.