The metal that feels colder than the wood

The equation that only runs forwards found that heat spreads by diffusion, smoothing differences of temperature at a rate set by the thermal diffusivity, and that the distance a disturbance reaches grows only as the square root of time. The summer that reaches the cellar in December followed the same equation underground, where the yearly swing of the surface temperature is carried down a few metres, shrinking and arriving late.

Both are about one body. The commonest thermal experiment anyone performs involves two: a hand laid on a surface. Its result is a sensation, cold or warm, and the sensation is so reliable that most people believe it tells them the temperature of what they touch. It does not. A steel rail and a wooden bench sitting side by side in the same shade for an hour are at exactly the same temperature, and the rail feels much colder. The question is what a hand is measuring, and the answer is one of the few exact solutions of the heat equation that can be checked with a fingertip.

Two bodies, one interface #

Take two bodies, each thick compared with how far heat can diffuse during the contact, at temperatures $T_1$ and $T_2$, and press them together along a flat face. Heat flows from the warmer to the cooler, and the heat equation, solved on each side with the condition that the temperature and the heat flux are continuous at the face, gives a pair of error-function profiles. The striking part of the solution is the temperature at the face itself. It jumps, at the moment of contact, to

$$T_c = \frac{e_1T_1 + e_2T_2}{e_1 + e_2}, \qquad e = \sqrt{k\rho c},$$

and stays there. Each body’s temperature is weighted by its thermal effusivity, the square root of its conductivity $k$ times its heat capacity per volume $\rho c$.

The profiles spread with time, each side’s as the square root of its own diffusivity times the time, but the value where they meet is fixed from the first instant. Against steel the skin’s surface drops to 21.0 °C and stays there; against pine it drops only to 30.3 °C. The nerve endings that sense temperature lie a fraction of a millimetre beneath the skin’s surface, within the region the change reaches in a fraction of a second, and they report something close to the interface temperature. The steel feels like an object at 21 °C. The wood feels like one at 30. Both are at 20.

The constancy is a consequence of the square root. Both bodies are semi-infinite and both are disturbed only by the contact, so the problem has no length scale of its own. The only length available is $\sqrt{\alpha t}$, the distance diffusion has reached — the random walker’s square root that the walk that comes home found governing how far anything diffusing gets — and the temperature profile must be a function of $x/\sqrt{\alpha t}$ alone. At $x = 0$ that ratio is zero at every time, so the temperature there cannot change. The same scaling made the ice on a lake in the ice that grows more slowly the thicker it gets thicken as the square root of time.

Why conductivity is only half of it #

The usual explanation of the cold metal is that metal conducts heat away from the hand faster. That is half right. The weighting in the formula is not the conductivity but the square root of conductivity times heat capacity per volume. A material can draw heat from the hand either because it carries heat away quickly or because it can absorb a lot of heat in the thin layer near the contact without warming much, and the effusivity counts both.

Water shows the difference. Its conductivity, 0.6 W/m·K, is a fifth of granite’s, but its heat capacity per volume, 4.2 MJ/m³·K, is twice granite’s, and its effusivity comes out within a factor of two. Skin meets water at 25.6 °C and granite at 24.3 — so a bath at room temperature feels nearly as cold as a stone slab, which conductivity alone would not predict. Polystyrene foam is the opposite case: it conducts poorly and holds almost no heat, being mostly air, so its effusivity is thirty times smaller than wood’s and skin meets it at 32.6 °C, almost its own temperature. Foam feels warm at any temperature anywhere near the hand’s.

Ice is a useful middle case. At 0 °C it is colder than room-temperature granite, yet at the same temperature it would feel slightly warmer, its effusivity being a little smaller. Skin touching ice at 0 °C meets it at about 12 °C, which is why ice can be held for a few seconds while a block of aluminium at 0 °C, meeting skin at about 2 °C, is painful at once.

A jeweller’s test #

The effusivity of a material is set by its microscopic constitution. Its heat capacity per volume counts the ways its atoms can store energy, the count half a kT for every way of moving made — about $3k$ per atom in a solid at room temperature, so it is roughly proportional to the density of atoms. Its conductivity depends on what carries the heat: free electrons in a metal, which is why metals conduct so well, and vibrations of the lattice in an insulator. Most insulators conduct poorly, and so feel warm. A few conduct better than any metal, because their lattices are stiff and light and their vibrations travel fast and far, as the gas of sound that carries a diamond’s heat found for diamond.

Diamond therefore feels cold, colder than copper at the same temperature, despite being a transparent insulator. That is the basis of a test jewellers use to tell a diamond from glass, cubic zirconia or most other imitations: a heated metal tip is touched to the stone and the rate at which the stone draws heat from it is measured. The tip is reading the stone’s effusivity, and diamond’s is so much larger than those of the imitations that a single touch distinguishes them. Lips were the older instrument; a diamond held to the lip feels icy where glass does not.

The heat that leaves the hand #

The amount of heat crossing the contact is the other half of the sensation. It is largest at the first touch and falls with time, as the cooled layers on each side thicken and the temperature gradients across them flatten.

The flux falls as $1/\sqrt t$, and its size is fixed by the two effusivities: $q = (T_1 - T_2)\,e_1e_2/(e_1 + e_2)/\sqrt{\pi t}$. At one second after touching steel, eight kilowatts per square metre leave the skin — about a watt through a fingertip. A minute later it has fallen eightfold.

The curves for copper and steel nearly coincide, although copper’s effusivity is almost three times steel’s. Once an object’s effusivity is much larger than the skin’s, the weighted average in the formula is dominated by the object and the interface sits at almost the object’s own temperature; the heat flow is then limited by how fast the skin can supply heat, not by how fast the object can take it. All metals feel about equally cold for that reason, and the sensitive comparison is among materials whose effusivities are close to skin’s: glass, stone, water, wood.

The sensation tracks the flux as well as the temperature. Cold receptors respond most strongly to a rapid fall in temperature and adapt within seconds, which is why the cold of a metal surface is sharpest at the moment of contact and fades as the hand is held there, even though the metal is still drawing heat from it. That fading is the receptors’ adaptation; the physics says the surface temperature has not changed.

The bench that does not burn #

The same formula works for objects warmer than skin, and there it decides what burns.

Skin is damaged within minutes at a surface temperature above about 45 °C, and within seconds above 55 or 60. A metal object meets skin at very nearly its own temperature, so a metal surface at 50 °C is already uncomfortable and one at 60 °C burns. Glass and water, with effusivities close to skin’s, meet it at about the average of the two temperatures: water at 54 °C brings the skin to 45. Pine meets skin at a temperature weighted heavily towards the skin’s own, and a wooden bench in a sauna at 90 °C brings the skin only to 45 °C — at the edge of what skin tolerates, and harmless for the minutes a sitter spends on it — while a steel fitting in the same sauna, at the same temperature, meets the skin at 85 °C. Saunas are built of wood, with recessed or wrapped metal, for that reason.

Air is the extreme. Its effusivity is nearly three hundred times smaller than water’s, so air at 90 °C barely changes the skin’s surface temperature on contact; what limits the time a person can spend in a hot sauna is the slower heating of the body as a whole, and the scalding sensation that comes when water is thrown on the stones is the latent heat of steam condensing on the skin, delivered far faster than air can deliver its own.

An impedance for heat #

The weighted average has a familiar shape. What happens where the medium changes found that a wave meeting the boundary between two media is partly reflected and partly transmitted by amounts fixed by the ratio of their impedances, and that the field at the boundary is set by the same ratio. Heat has a counterpart. The summer that reaches the cellar in December found that a periodic temperature at a surface travels into the ground as a damped thermal wave, and when such a wave meets a boundary between two materials its reflection is governed by the ratio of their effusivities: the reflection coefficient is $(e_1 - e_2)/(e_1 + e_2)$, exactly the form for sound meeting a change of acoustic impedance.

So the effusivity is the thermal impedance of a material, the ratio of the heat flux it accepts to the temperature swing that drives it, and the contact temperature is the voltage at the junction of two impedances driven by two sources. A hand on steel is a high-impedance source shorted by a low-impedance load: the junction sits near the load’s value, and heat flows freely. A hand on foam is the reverse: the junction stays near the hand’s value, and almost nothing flows. Infrared thermography uses the same reflection to find flaws hidden under surfaces — a void or a delamination is a change of effusivity, and a brief heat pulse applied to the surface comes back from it early.

How deep the touch reaches #

The formula assumes each body is thick compared with $\sqrt{\alpha t}$. That depth depends on the material’s diffusivity, $\alpha = k/\rho c$ — not its effusivity — and it is small.

In one second a touch changes the temperature of a piece of wood to a depth of less than half a millimetre, of glass to under a millimetre, of steel to a few millimetres. Only that layer takes part in the contact, and anything beneath it is invisible to the hand for that long. A wooden veneer a millimetre thick glued over a steel plate feels like wood for the first few seconds, then increasingly like steel as the cooled layer reaches through the veneer; a coat of paint a tenth of a millimetre thick on a metal railing changes almost nothing, since in a fraction of a second the touch has reached through it. Materials meant to feel warm — the handles of tools, the rims of steering wheels, the seats of public benches in cold climates — are made of low-effusivity materials at least a few millimetres thick.

Foam is a curiosity on this chart. It diffuses heat nearly as deeply as granite, because its tiny heat capacity lets a small amount of conducted heat warm a deep layer; yet it draws almost nothing from the hand, because its effusivity is tiny. Diffusivity says how far the disturbance goes; effusivity says how much heat it takes. The two combine $k$ and $\rho c$ differently, as a ratio and as a product, and confusing them is the most common error in reasoning about which materials feel warm.

What the hand’s arithmetic leaves out #

Contact is not perfect. Real surfaces touch at a few high points, with air in between, and a thin layer of air or sweat at the interface adds a resistance that keeps the skin’s surface warmer than the formula says for the first fraction of a second. Pressing harder improves the contact; a cold metal object feels colder when gripped than when brushed.

Skin is not uniform. Its outer layer is dead, dry and less conductive than the living tissue beneath, and blood flow in the dermis supplies heat from below, which the semi-infinite solid assumes away. For contacts longer than tens of seconds the blood supply matters, and the skin’s temperature recovers rather than staying at the interface value.

The perception adapts. As above, cold receptors respond to change and adapt to steady states. The same physics determines the stimulus; the nervous system decides how loudly it reports it, and the sensation of an object’s temperature is a judgement made from both.

Diffusion is assumed instantaneous. The heat equation lets a disturbance reach everywhere at once, falling off as an error function. At times shorter than the collision time of whatever carries heat — picoseconds in a solid — that is wrong, and the heat that arrives as a wave described the correction. For a hand on a railing the error is many orders of magnitude below anything a nerve could feel.

The formula is for flat faces of thick bodies. A thin sheet of metal — a sheet of aluminium foil — has too little heat capacity to keep its surface cool, and once the touch has reached through it, it warms rapidly to the skin’s temperature. Foil from an oven can be picked up by its edges within a second or two for that reason, while the dish it covered cannot.

Still open: how the skin’s sensors turn heat into a sensation #

The physical stimulus is well understood: an interface temperature and a heat flux, both computable from the effusivities. What happens next is less so. The sensors responsible for cool and cold are ion channels in the membranes of nerve endings, and the best known, TRPM8 — the receptor that menthol also activates, which is why mint feels cool — opens as the temperature falls below about 25 °C. Warm and heat are sensed by others. How a population of such channels, in nerve endings at different depths with different thresholds and different rates of adaptation, combines to produce the judgement that an object “is cold”, and why that judgement weights the initial rate of cooling so heavily, is being worked out.

One practical form of the question is the design of materials that feel warm or cool on purpose — fabrics that feel cool against the skin in summer, surfaces in cars and phones whose apparent temperature is chosen. Textile scientists measure a fabric’s “cool touch” by the peak heat flux in the first fraction of a second of contact, which is the effusivity argument made into a standard test. How well that peak predicts what people report is a question for experiments with people, and the answers vary with the fabric’s texture as well as its thermal properties.

The heat equation’s part is settled, and it is more surprising than it looks. When two bodies touch, their common surface takes a temperature that is decided at the first instant and does not change, a weighted average with weights that are neither conductivities nor heat capacities but the square root of their product. A hand on a railing reads that weighted average, and it is one of the few measurements of a diffusion problem that everybody makes every day.