The frost that lifts by drinking

The ice that grows by stealing from the droplets set supercooled water beside ice in a cloud and watched water move from the liquid to the solid through the air, because below the melting point the liquid stands higher in chemical potential. The two never touched. This essay puts ice and liquid water in contact, inside soil, and finds the same difference doing something nobody would guess from a cloud: lifting roads, cracking foundations and splitting rock, by drawing water upward through the ground and freezing it into layers.

The phenomenon is frost heave, and it is familiar to anyone who has driven a northern road in March. Pavements rise in winter, unevenly, and break up in the thaw. Fence posts are jacked out of the ground a few millimetres a year until they lean. Pipes laid too shallow are lifted and snapped. On a cold, damp morning in a garden the soil surface can be found standing on a fringe of ice needles a few centimetres tall, each one grown from the bottom, with a crumb of earth balanced on its tip. Every one of these is the same physics, and almost everyone, including many engineers for a long time, has explained it in a way that does not survive a simple experiment.

The explanation everyone gives #

Water expands by about nine per cent when it freezes. Soil holds water. Freeze the soil and the water expands, so the soil must rise. The argument is short and almost entirely wrong, and the arithmetic says so before any experiment does. A metre of silt at forty per cent porosity, saturated, contains forty centimetres of water in each square metre of ground. Nine per cent of that is 3.6 centimetres, the most the layer could lift by freezing in place even if every grain stayed put. Field heaves on susceptible soils are commonly larger, and when frozen ground is dug up it does not look like soil with ice in its pores. It looks like soil interleaved with sheets of clear, almost stone-free ice — lenses, millimetres to centimetres thick, stacked one above another and parallel to the frost line. The total thickness of the lenses is roughly the heave.

That ice came from somewhere. The water in the lenses is far more than was ever in the pores where the lenses sit, and the only place it could have come from is below: from the unfrozen soil beneath the frost line and from the water table. Frozen ground that heaves is ground that drinks.

The decisive experiment was done by Stephen Taber at the University of South Carolina in the late 1920s. He froze cylinders of clay from the top, in the laboratory, and watched lenses grow; that much reproduced the field. Then he saturated the clay not with water but with benzene and with nitrobenzene, liquids that contract when they freeze, as nearly every substance does. The clay heaved anyway, with lenses of the frozen liquid, and the expansion explanation was finished. Whatever lifts the ground does not need the solid to be bulkier than the liquid. It needs the solid to grow and the liquid to be able to reach it.

Two pressures for one substance #

The rule that decides when ice and water can sit in contact is the one every argument about chemical potential rests on: matter moves from where its chemical potential is higher to where it is lower, and two phases are in equilibrium when their potentials are equal. The new ingredient is that nothing requires the two phases to be at the same pressure. In a beaker of ice water they are, because the ice floats free and the whole system feels the air above it, which is why the triple point is a point and the ordinary melting point a single temperature. In soil they need not be. The ice can be bearing the weight of the ground above, and pressing against it; the liquid in the pores below can be connected to the water table, at whatever pressure the water table supplies.

The chemical potential of each phase rises with pressure, at a rate equal to its volume per unit mass, and falls with temperature, at a rate equal to its entropy. Expanding about the melting point $T_m$ at atmospheric pressure, and writing $\Delta T$ for how far the temperature is below it, the condition that ice at pressure $p_i$ and water at $p_w$ coexist is, to first order,

$$\frac{p_i}{\rho_i} - \frac{p_w}{\rho_w} = \frac{L\,\Delta T}{T_m},$$

with both pressures measured above atmospheric, $\rho_i$ and $\rho_w$ the two densities and $L$ the latent heat of melting per kilogram. The right-hand side is the chemical-potential advantage that ice has gained by being cold. The left is how that advantage can be spent: by squeezing the ice, or by putting the water under tension, or by some combination of the two.

Set the water’s pressure to zero, as it is for pore water connected to a water table near the surface, and the equation gives the pressure the ice can bear: $p_i = \rho_i L\,\Delta T/T_m$. With 917 kilograms per cubic metre, 333.5 kilojoules per kilogram and 273.15 kelvin, that is 1.12 megapascals for every kelvin of supercooling. At a tenth of a degree below zero it is 112 kilopascals, more than the bearing pressure under the footing of an ordinary house. At one degree it is the weight of fifty metres of soil. Ice that is cold enough can hold up anything above it and still sit in equilibrium with the liquid at its base, and if it is slightly colder than that, it is not in equilibrium: water at the base has the higher chemical potential, freezes onto the ice, and pushes the ice and everything on it upward.

Set the ice’s pressure to zero instead and the same equation says the water must be under tension, $p_w = -\rho_w L\,\Delta T/T_m$, minus 1.22 megapascals per kelvin. That is the face of the effect the soil sees. An ice lens at the top of a column of wet soil acts on the water beneath it like a pump, pulling it upward with a suction proportional to how cold the lens is — the same kind of tension that the pore that fills from dry air found in water held in nanometre pores by curvature, produced here by a phase boundary instead of a meniscus. Soil scientists call it cryosuction.

Twelve times weaker, and the one that matters #

The familiar version of the same relation is the slope of the melting line. The melting curve that leans the wrong way derived it as the Clausius–Clapeyron equation and found that, because ice is less dense than water, raising the pressure lowers the melting point, by about 0.074 kelvin per megapascal. Read the other way it says that ice and water squeezed together at equal pressure coexist below 0 °C only if the pressure rises by about 13.5 megapascals for every kelvin of cooling. That is the pressure that bursts pipes, and it is the solid–liquid counterpart of the curve that a boiling point is a pressure traced between liquid and vapour.

The two curves on that figure are the same equation with different pressures held fixed. The red one sets $p_i = p_w$: a sealed vessel full of water, freezing, where the only way the ice can grow is by compressing everything in the box, and the nine per cent expansion is the whole of the mechanism. It climbs steeply and reaches 210 megapascals at about −22 °C, where a denser form of ice, ice III, becomes stable and the curve ends. Freezing water in a perfectly rigid container cannot produce more than that, but it does not need to: a copper pipe or a glass bottle fails at a small fraction of it.

The blue curve sets $p_w = 0$, and climbs only a twelfth as fast. The ratio is not a coincidence of numbers. Dividing one slope by the other gives $\rho_w/(\rho_w - \rho_i)$, which is about 12 because ice is only about eight per cent less dense than water. The sealed pressure is large precisely because the density difference is small: a little expansion has to be resisted by a lot of pressure. The open pressure owes nothing to the density difference at all. It would be the same for a substance that contracted on freezing — benzene, nitrobenzene, almost anything — which is Taber’s experiment written as an equation.

So the weaker curve is the one that lifts the ground, and its weakness hardly matters. A twelfth of an enormous number is still a large one: 25 megapascals at −22 °C is more than the tensile strength of most rock. And the open mechanism has a property the sealed one lacks entirely. A sealed vessel exerts its pressure once, by converting a fixed volume of water to ice. An open lens keeps drawing water as long as the cold persists and water is available, so the displacement it produces is not limited by how much water was there to begin with. It is limited by how fast water can reach it, which is where the soil’s own structure takes over.

The ice that cannot get into the pores #

If cold ice simply drew water and froze it, the frost line would advance downward through wet soil like any freezing front, and the ice would form in the pores as the cold arrived. Lenses would not appear. They appear because ice finds it difficult to enter a small pore — a cousin of the difficulty the barrier a new phase has to climb described for an ice embryo, where surface energy also stood in the way — for the same reason that the small bubble blows up the big one: a curved interface carries a pressure jump, $2\gamma/r$ for a surface of energy $\gamma$ bent to radius $r$.

To grow from a lens into the space between grains, an ice front has to bend into a tongue no wider than the pore throat, and an ice–water interface bent that tightly costs energy. The ice tongue is at a higher chemical potential than flat ice by $2\gamma/(\rho_i r)$ per kilogram, and it can grow only where the supercooling pays for that. This is the Gibbs–Thomson effect, the melting-point version of the curvature correction that lowered the vapour pressure in a nanometre pore. Setting $L\,\Delta T/T_m$ equal to $2\gamma/(\rho_i r)$ gives the supercooling at which ice can invade a pore of radius $r$.

With an ice–water interfacial energy of about 0.029 joules per square metre, the product of supercooling and pore radius is about 52 nanometre-kelvins. Taking the throat between grains as roughly a tenth of their diameter, which is a fair rule for packed spheres, ice enters the pores of coarse sand at half a millikelvin below zero: at once, for practical purposes. Between ten-micrometre silt grains it needs five hundredths of a degree, and between half-micrometre clay platelets a full degree. The figure covers four decades of grain size and four of supercooling, along a single straight line of slope minus one.

Below a lens, then, there is a range of temperatures in which the lens is cold enough to pull water but not cold enough to invade the pores beneath it. In that range the ice stays where it is, as a sheet, and the water beneath stays liquid and continuous with the water table, and the sheet thickens from below as water is drawn up and frozen onto it. The arrangement is the one the ice needles in the garden make visible, each needle a column of ice growing from its base out of wet soil it cannot enter.

As the cold penetrates, the lens’s base eventually gets cold enough for ice to break into the pores below it, the supply of water to the old lens is cut off, and the front jumps downward through a frozen zone until it finds a place where a new lens can start. The result is the stack of lenses seen in frozen ground, thin near the surface where the frost advanced quickly and thicker deeper down where it slowed, each recording a pause in the freezing front.

Why silt and not sand or clay #

The same Gibbs–Thomson relation sets a ceiling on cryosuction. A lens can sustain a pressure difference no larger than the curvature pressure at which ice would break through into the pores below; once the supercooling at its base exceeds the entry value, ice invades the soil and the lens stops. In coarse soil that ceiling is tiny. The entry pressure for millimetre sand is half a kilopascal, the weight of a few centimetres of soil, so a lens in sand can never hold up the ground above it. Ice simply grows into the pores as the cold arrives, the water freezes in place and expands its nine per cent, and the heave is negligible.

Fine soil has the opposite problem. Clay’s pores are so narrow that the lens can sustain the full suction its supercooling allows, megapascals if it is cold enough, but water flows through clay extremely slowly. Permeability, the ease with which a porous medium passes fluid under a pressure gradient, scales roughly as the square of the grain size, so a clay a hundred times finer than a silt passes water ten thousand times more slowly. A clay lens pulls hard and drinks little.

The rate of heave is roughly the product of the two: permeability times the suction left over once the load above has been paid. The figure computes exactly that, in the simplest version of the theory — Douglas Everett’s capillary model of 1961 — with a supercooling of two degrees at the lens. The rate is zero for coarse grains, where the entry pressure cannot beat the load, rises steeply as the grains get finer and the suction grows, and falls again as permeability collapses. Under half a metre of soil, ten kilopascals, the fastest heave is at 29 micrometres. Under a hundred kilopascals the peak is ten times smaller and sits at 2.9 micrometres. Every peak lies in the silt range, between two and sixty micrometres.

That is the shape engineers found empirically long before the theory. Arthur Casagrande proposed in 1931 that a soil was frost-susceptible if more than about three per cent of its grains were finer than 0.02 millimetres, and the criterion, refined but recognisable, still governs how road bases and railway beds are specified in cold countries: drain them, and build them from gravel and sand clean of fines, so that the cold arrives at pores ice can enter at once. The trade-off is the same one the pore that lifts highest fills slowest found for a capillary rising in a tube: narrowness raises the pressure that drives the water and throttles the flow that delivers it, and the best performer is the middle size.

Water that stays liquid below zero #

The capillary picture has a gap. In clay the lens is pulling water through soil that is itself partly frozen, because the frost line passes it on the way down. If freezing were all-or-nothing, the frozen soil between the lens and the unfrozen ground would be a plug, and heave would stop as soon as the front passed the first lens. Measured heave does not stop. It continues, slowly, behind the front, and the reason is that frozen ground is not frozen through.

Every pore freezes at its own Gibbs–Thomson temperature, and a real soil has a spread of pore sizes, so freezing is gradual. In the figure a sand has frozen essentially completely by a hundredth of a degree below zero. The silt keeps half its pore water liquid at −0.1 °C, and the clay keeps most of its liquid at −1 °C and some even at −10. That liquid is connected, a network of narrow channels and films threading through the frozen soil, and it is the pathway along which water continues to reach a lens after the front has passed it.

The figure omits a second source of liquid that dominates in the finest soils: films of water a few molecules thick that remain unfrozen against mineral surfaces, because the arrangement ice–water–mineral has a lower energy than ice pressed directly against the mineral. The phenomenon is called premelting. It means that ice touching a grain is separated from it by a liquid layer whose thickness shrinks as the temperature falls but does not vanish, and those layers carry water too. John Wettlaufer, Alan Rempel and Grae Worster worked out in the early 2000s how premelted films let ice lenses push on soil grains with a force set by the same $\rho_i L\,\Delta T/T_m$, which removes the capillary model’s ceiling and accounts for heave pressures well above anything Everett’s model allows. Robert Miller had argued for something like it in 1972, under the name secondary heave, with the frozen fringe between a lens and the unfrozen soil doing the work.

A heater in disguise #

A lens grows only as fast as its latent heat can be carried away. Every kilogram of water that freezes onto its base releases 333.5 kilojoules, and that heat has to be conducted upward through the frozen ground to the cold surface; if it is not, the base warms to the melting point and the suction disappears. A heave of one centimetre a day, about as fast as field heave ever runs, means freezing 9.2 kilograms of water per square metre per day, which releases about 35 watts per square metre — a substantial fraction of the heat a winter surface loses on a cold night.

This is why heave is fastest when the frost line is advancing slowly. A rapid cold snap drives the front downward too fast for water to arrive, the soil freezes in place, and little heave occurs; a long spell of steady, moderate cold holds the front almost still, and the lens has time to drink. The same soil can heave hardly at all in one winter and badly in the next, according to the timing of the cold rather than its depth.

The same pump in rock, concrete and cells #

The relation $\rho_i L\,\Delta T/T_m$ does not know that it is in soil. Rock with fine cracks in it holds water in exactly the arrangement a lens needs, and Joseph Walder and Bernard Hallet showed in 1985 that ice segregating in a crack, fed by water drawn through the surrounding rock, wedges it apart far more effectively than freezing in place could. Their model predicted, and laboratory and field measurements support, that frost cracking is most intense not in the deepest cold but between about −4 and −15 °C, held for a long time: cold enough for large suctions, warm enough for the liquid to flow. Much of the shattered rock on mountainsides was broken by lenses rather than by expansion.

Concrete suffers the same way, which is why it is made with deliberate air bubbles — entrained voids a few tenths of a millimetre apart that give ice somewhere to grow without wedging the paste. The bubble is a pore large enough for ice to enter at once, a sand grain’s worth of space inside the cement, offered so that cryosuction has nothing to pull against.

And the effect turns up in biology, in reverse. When tissue freezes slowly, ice forms first outside the cells, and the ice and the cell contents are then two forms of water at different chemical potentials separated by a membrane. Water leaves the cells and joins the external ice, the cells shrink and their contents concentrate, exactly as the soil water migrates to a lens. Freeze-tolerant frogs and insects survive by controlling where that ice grows and what dissolves in their cells, using the degree a spoonful of solute buys in the other direction.

Where the picture stops #

The figures are a deliberately simple version of a hard problem. The pressure relation is linearised about the melting point and assumes constant latent heat and densities, which is accurate to a few per cent over the first ten degrees of supercooling and drifts beyond; the sealed curve uses the empirical Simon fit instead, which is good all the way to the ice III boundary. The pore-entry calculation treats pores as spheres with a single throat radius a tenth of the grain size, when real pore spaces are irregular and their throats depend on how the grains are packed. The susceptibility curve is a capillary model with permeability simply proportional to grain size squared and a fixed supercooling at the lens; it gets the shape right — a peak in silt that moves finer under load — and it gets the size of heave pressures wrong, because it ignores the premelted films that let real lenses push much harder.

Nothing in the pictures carries time. Real heave depends on the rate at which cold arrives, on how heat moves through ground whose conductivity changes as it freezes, on solutes that lower the freezing point and that ice rejects into the remaining liquid, and on the load and stiffness of whatever is being lifted. The domain is saturated or nearly saturated fine-grained soil with water available from below: dry soil, and soil sealed off from any water table, cannot heave by this mechanism however susceptible its grain size, because there is nothing for the lens to drink.

The damage a road suffers is also not what these figures show. The heave itself is often survivable; the harm comes in the thaw, when the lenses melt from the top down into soil that is still frozen beneath and cannot drain. The released water saturates the upper layer, which loses almost all its strength until it dries, and that is when traffic breaks a road surface into potholes and seasonal weight limits go up.

Still open: when a new lens begins #

The conditions for a lens to stop are well understood: it stops when ice breaks into the pores below it. The conditions for a new one to start are not. Why a lens initiates at a particular depth, and so what sets the spacing of the stack in a frozen soil, depend on how stress concentrates in the frozen fringe, on the distribution of pore sizes and on small heterogeneities in the soil, and the models that reproduce observed spacings do so with parameters that are hard to measure independently. Predicting the heave pressure of a particular soil from its grain-size distribution alone is still not possible to better than a factor of a few, and engineering design still rests on frost-susceptibility tests in the laboratory rather than on calculation. The same gap matters for permafrost, where the ground ice built up by segregation over thousands of years is now thawing and the ground subsiding, and where how much excess ice a slope contains decides whether it settles gently or slumps.

The habit worth carrying away is to ask, whenever two phases of one substance meet, whether they are really at the same pressure. Ice in contact with water it can draw on is in equilibrium when $p_i/\rho_i - p_w/\rho_w = L\,\Delta T/T_m$, so a lens a degree below the melting point can bear 1.12 megapascals while drinking water at atmospheric pressure — twelve times less than freezing water in a sealed vessel, but needing no expansion at all. Gibbs–Thomson curvature keeps the ice out of the pores beneath it, permeability decides how fast it can drink, and silt, between the two, is where the ground lifts.