The pore that fills from dry air

A sachet of silica gel sits in a box of new shoes and takes the water out of the air. So does a jar of flour that cakes in a damp kitchen, a lump of clay that swells after rain, and a plank of wood that grows in summer and shrinks in winter. None of those is anywhere near a hundred per cent humidity, and in every one of them liquid water appears inside a solid, out of air that on a cold window would not have produced a single drop.

The mechanism is capillarity, but not in the form the rest of this subject has met it. How high water will climb and the column that is pulled, not pushed both treat a meniscus that is already sitting in liquid, and ask what pressure it holds. Here the liquid does not exist yet. It condenses into the pore from the vapour, and whether it does depends on how the pore’s curvature changes the vapour pressure the liquid is in equilibrium with.

A drop and a pore are one equation #

The vapour pressure of water is usually quoted as a single number at a given temperature, and that number belongs to a flat surface. A curved surface has a pressure jump across it — the Laplace pressure, $2\gamma/r$ for a spherical surface of radius $r$ — and the small bubble blows up the big one because that jump is larger for a smaller radius. The same jump changes how readily molecules leave the liquid.

The argument is two lines of thermodynamics. Raising the pressure of a nearly incompressible liquid by $\Delta P$ raises the free energy of each mole of it by $V_m\Delta P$, with $V_m$ the molar volume. The vapour in equilibrium with it must have its free energy raised by the same amount, and for a vapour that means its pressure multiplied by $\exp(V_m\Delta P/RT)$. Put in the Laplace pressure and

$$\ln\frac{p}{p_\text{sat}} = \pm\,\frac{2\gamma V_m}{rRT},$$

positive for a convex surface, where the liquid is squeezed, and negative for a concave one, where it is stretched. That is the Kelvin relation, from 1871.

Everything in the formula except the radius is a property of water and a temperature, and it collects into one length. $\gamma V_m/RT$ is 0.53 nanometres at room temperature — about two water molecules — so the effect is enormous at a nanometre, a ten per cent correction at ten, and a tenth of a per cent at a hundred. It is a fact about surfaces a few molecules across, which is why it is invisible on a window and decisive in a pore.

The two branches have two familiar consequences. The raised vapour pressure over a small drop is why a mist coarsens, small drops evaporating onto large ones — Ostwald ripening in a vapour — and why a cloud cannot form in perfectly clean air: a droplet of pure water one nanometre in radius would need 286 per cent humidity to survive, and the barrier a new phase has to climb is the reason clouds form on dust and salt instead. The lowered vapour pressure over a concave meniscus is the one this essay is about.

The radius the humidity decides #

Air at a relative humidity below one has a vapour pressure below the flat saturated value. A pore whose walls are wet by water — contact angle near zero, which silica and clay and wood all give — can hold a liquid plug with a concave meniscus, and that meniscus is in equilibrium with a vapour pressure below saturation. Set the two equal and there is a radius: the Kelvin radius, below which condensed water is stable in air of that humidity and above which it evaporates.

The numbers make the phenomenon concrete. A room at fifty per cent humidity — a comfortable room — is in equilibrium with liquid water in every hydrophilic pore narrower than about three nanometres across. A damp cellar at ninety per cent fills pores twenty nanometres across. The last few per cent of humidity are where most of the change happens: going from 90 to 99 per cent multiplies the Kelvin radius by ten, which is why materials that seem dry can absorb water suddenly and heavily on a humid day.

It also explains why the materials that do this are the ones they are. Silica gel is made to be almost all pore, with pores a couple of nanometres wide, so at ordinary humidities a large fraction of its internal volume sits below the Kelvin radius and fills. Clay minerals are stacks of sheets a nanometre or so apart. Wood’s cell walls are a mesh of cellulose fibrils with gaps of a few nanometres. Each is a solid built, by manufacture or by growth, at the length scale where the Kelvin relation bites.

One curvature on the way in, two on the way out #

The figure has two curves, and the gap between them is the most interesting thing on it. A pore does not fill and empty through the same meniscus.

An empty cylindrical pore in humid air first acquires a thin adsorbed film of water on its walls. The surface of that film is a cylinder: curved around the pore, straight along it. A cylinder has one curvature, so its Laplace pressure is $\gamma/r$ rather than $2\gamma/r$, and it lowers the vapour pressure by only half as much in the logarithm. The pore therefore fills — the film thickens until it closes across the middle — when the humidity reaches the condition for a cylindrical meniscus of the pore’s radius.

A full pore empties from its open end, where the liquid surface is a hemisphere: curved in both directions, with the full $2\gamma/r$. It stays full until the humidity falls to the condition for a hemispherical meniscus of the same radius, which is lower. The same pore fills at one humidity and empties at another, with a factor of two in the logarithm between them.

That is a hysteresis with no friction in it. The angle a liquid makes found contact-angle hysteresis in roughness and chemical patchiness — a contact line that sticks. This one would survive in a perfectly smooth, perfectly wetting pore with no contact line at all, because it comes from the geometry of the meniscus the liquid happens to have on the way in and the way out.

A real material has a spread of pore sizes, and each pore contributes its own pair of thresholds, so the sharp steps smear into two smooth branches enclosing a loop. At any humidity inside the loop the solid holds more water if it has come down from wet than if it has come up from dry. This is the hysteresis the closing section of the column that is pulled, not pushed pointed at — the reason a soil moisture reading has to know its own history — and it shows up here in a model that contains nothing but a distribution of cylinders.

The bridge that forms at every contact #

A pore is not the only place a concave gap occurs. Wherever two solid surfaces touch — two grains of sand, a grain and a wall, the tip of a probe and the surface it rests on — the gap between them narrows smoothly to zero at the point of contact. Somewhere in that gap it is narrower than the Kelvin radius, whatever the humidity, so a small ring of liquid always condenses around the contact, with a meniscus of exactly the Kelvin radius at its edge.

The ring is a capillary bridge, and it pulls the two surfaces together: its meniscus is at a pressure below the air’s, and that low pressure acts over the area the bridge wets. The surprising part is how the pull depends on humidity. In dry air the Kelvin radius is small, the bridge is small, and its pressure deficit is large; in humid air the bridge is large and its pressure deficit small. The two compensate, and for a sphere of radius $R$ on a flat surface the force comes out at $4\pi R\gamma\cos\theta$ with neither the humidity nor the size of the bridge in it.

So humidity decides whether a bridge exists and not how hard it pulls — which is why a powder that is free-flowing in dry air cakes abruptly above some humidity rather than gradually. Below the threshold the Kelvin radius is smaller than the roughness of the grains, the bridges cannot span the real gaps between asperities, and the grains touch dry. Above it the bridges span them, each pulls with its full humidity-independent force, and a heap that poured a moment ago holds a shape. Damp sand is the same bridge with a liquid supplied rather than condensed, and the same force per contact.

Measuring pores by what they condense #

The loop is not only a nuisance. Its position and shape are how the pore sizes of a material are measured, and the method is one of the standard characterisations of any porous solid.

The solid is cooled and exposed to a vapour at a controlled fraction of its saturation pressure, usually nitrogen at its boiling point rather than water, because nitrogen wets almost everything and does not react. The amount condensed is weighed or measured volumetrically as the pressure is raised to saturation and lowered again. Each step in pressure corresponds to a Kelvin radius, and the amount that condensed at that step is the volume of the pores of that size. The loop becomes a pore-size distribution, obtained without a microscope and averaged over the whole sample rather than over whatever a microscope happened to image.

The shape of the loop says something about the shape of the pores, too, because the argument above assumed straight cylinders open at the ends. A pore with a wide body and a narrow neck — an ink bottle — cannot empty through its body’s meniscus until its neck has emptied, so it stays full down to the neck’s humidity and then empties all at once, and the drying branch drops steeply. A slit between two plates has a meniscus curved in only one direction on both branches, and the loop narrows. Reading the loop for geometry is a whole craft with a classification of its own.

Put the two isotherms side by side and the measurement is visible. The loop for pores two and a half times larger sits higher on the humidity axis and is pressed against saturation: half their volume fills at 94.9 per cent and empties at 90 per cent, where for the 4 nm pores the same halves came at 87.7 and 76.9. The Kelvin relation makes the shift exact. The logarithm of the humidity at which a meniscus forms is inversely proportional to its radius, so multiplying every radius by 2.5 divides every logarithm by 2.5, and 87.7 per cent, whose logarithm is −0.131, becomes the humidity whose logarithm is −0.052, which is 94.9 per cent.

The loop narrows as the pores grow, in humidity if not in logarithm. The factor of two between the cylindrical and hemispherical conditions is a factor of two in the logarithm at every size, and close to saturation the logarithms themselves are small. That sets the method’s limit at the large end: pores much wider than a few tens of nanometres fill and empty so close to saturation that the condensation in them cannot be told from the liquid gathering on the outer surfaces of the grains. They are measured instead by forcing mercury into them under pressure, which is the same Laplace pressure worked the other way, for a liquid that refuses to wet. At the small end the limit is the one the model has been ignoring all along: a pore a nanometre or two across is only a few molecules wide, its meniscus is not a smooth surface, and the adsorbed film left out of the figures is most of what is in it.

The water in the pore is stretched #

The Kelvin relation can be read for the liquid instead of the vapour, and read that way it says something startling.

The condensed water is in equilibrium with vapour at a pressure below saturation, and a liquid in equilibrium with a sub-saturated vapour must itself have a lowered free energy. For a liquid the only way to lower it at fixed temperature is to lower its pressure. The pressure required is

$$P_\text{liquid} - P_\text{air} = \frac{RT}{V_m}\ln\text{RH},$$

which is negative for any humidity below one, and which is exactly the Laplace pressure of the meniscus holding the liquid in the pore. The figure checks that the two agree.

The prefactor $RT/V_m$ is 137 megapascals, so modest changes in humidity are huge changes in pressure. Water condensed in the pores of a sachet of silica gel in a fifty-per-cent room is under a tension of ninety-five megapascals — nearly a thousand atmospheres — and it is stable, because the pore is too small to hold a bubble large enough to grow. The column that is pulled, not pushed found the calculated homogeneous cavitation limit at about −140 MPa and the tensions inside a tree at −0.5 to −3 MPa. Pores in ordinary room air sit a hundred times further along the same line than any tree does, and reach the cavitation limit at thirty-six per cent humidity.

This is the same relation a psychrometer uses to measure a tree’s tension, read in the opposite direction. The instrument seals a sample in a small chamber and measures the humidity it comes to equilibrium with; the humidity gives the tension through exactly this logarithm. A tree’s xylem at −1 MPa is in equilibrium with air at 99.3 per cent humidity, and a clay at −95 MPa with air at 50. The equation does not know whether the stretched water is in a vessel a hundred metres tall or in a slit a nanometre wide.

The same logarithm turns up a third time in a place that looks unrelated. The humidity of the air over a salt solution is lowered because the dissolved ions dilute the water, and the pressure that comes from counting turns that dilution into an osmotic pressure. Over a saturated solution of common salt the humidity is 75 per cent; the osmotic pressure of that solution is $-(RT/V_m)\ln 0.75$, about 39 megapascals; and the Kelvin radius at 75 per cent is 3.65 nanometres. A humid room, a saturated brine and the water in a 3.65-nanometre pore are three states of water at the same chemical potential, and a salt dish in a sealed box of silica gel is how laboratories fix a humidity to calibrate instruments that measure all three.

Where the Kelvin relation stops #

Below a couple of nanometres the liquid is not a continuum. The relation treats the condensed water as a bulk liquid with a surface tension, and a pore one nanometre in radius holds a plug a few molecules across in which there is no bulk and hardly a surface. The surface tension of a highly curved surface is not the flat value either, and the correction, of order a molecular length over the radius, is itself uncertain. The Kelvin radii quoted at fifty per cent humidity are therefore estimates of the right size rather than predictions to three figures.

The adsorbed film has been ignored. Before any pore fills, its walls carry a film of water a few molecules thick, bound by forces other than capillarity, and the open radius available to the meniscus is the pore radius minus that film. Pore-size measurements correct for it with an empirical film thickness, and a real isotherm rises gently well before the capillary step because of it.

The contact angle has been taken as zero. A pore whose walls are only partly wetted has a flatter meniscus, a smaller effect, and a Kelvin radius smaller by the cosine of the angle; a hydrophobic pore has a convex meniscus and resists filling above saturation. Activated carbons, which are hydrophobic, take up water very differently from silica for this reason.

And the solid does not move. Clay swells and wood swells because the tension in the condensed water pulls on the walls that hold it, and a pore that changes size as it fills changes its own Kelvin radius. The swelling of materials with humidity is this relation coupled to their elasticity, and nothing on this page includes it.

What the figures cannot show #

Every figure here is an equilibrium, and equilibrium in a nanometre pore can take a long time to reach. Vapour has to diffuse into a tortuous network, condense, and release its latent heat, and the heat has to go somewhere; a dry sample put into humid air takes up water over hours, and the branches of a measured isotherm depend on how long each step was held. The loop drawn here is the limit of infinite patience.

Nor do the curves show where in a material the water is. A solid with a spread of pore sizes, partly filled, has its small pores full and its large ones empty, and whether the full pores are connected to one another decides whether the material conducts water, electricity or heat — which is a percolation question about the arrangement of pores, not about their sizes, and a distribution of sizes contains no arrangement.

Still open: a contact angle that is not a property of the materials #

Every result on this page used the contact angle, took it as zero, and noted what would change if it were not. Throughout the subject that angle has been treated as a fact about three substances — a solid, a liquid and the vapour around them — fixed once they are chosen, which is how the angle a liquid makes derives it.

It is not fixed. The balance of surface energies that sets it can be altered without touching any of the three materials, by storing energy in the region under the wetted area — most simply by putting a voltage across a thin insulator beneath the drop. Whether a surface can be made to change its wettability on command, how far, and why real surfaces stop responding before the simple theory says they should, is a question about the one quantity every capillary calculation here assumed was given.

The habit worth carrying away is the sign. An equation with a curvature in it has two branches, and the familiar one is usually only half of it. The raised vapour pressure over a drop was known long before anybody used the lowered one over a pore, and the second branch turned out to hold a humidity sensor, a pore-size measurement, the tension of water in a tree and the swelling of every piece of wood in a house.