The entropy that depends on how fast it was cooled

The rung below this one finds that ice keeps 3.41 joules per kelvin per mole at absolute zero, and finds where the number comes from: two hydrogens near each oxygen and two far, six legal arrangements out of sixteen, $R\ln(3/2)$. It is a count.

A count has a property worth naming explicitly, because the next case does not have it. Nobody can change it. Freeze the ice quickly or slowly, in a laboratory or in a glacier — cooling it by adiabatic demagnetisation if that helps, and the residual entropy is the same number, because it is a statement about how many ground states the crystal has.

Now cool a liquid fast enough that it never crystallises at all.

Why the liquid’s entropy falls faster than the crystal’s #

The excess entropy of the liquid over the crystal starts at the entropy of fusion, at the melting point, and decreases on cooling. That direction is not obvious and it is the whole reason the problem exists.

The liquid’s heat capacity exceeds the crystal’s — it has configurational degrees of freedom the crystal has frozen out, and rearranging them absorbs heat. So on cooling, the liquid loses entropy faster than the crystal does, and the gap between them closes:

$$\Delta S(T) = \Delta S_{\text{fus}} - \Delta C_p \ln\frac{T_m}{T}.$$

That expression has a root. For the substance drawn — a melting point of 305 kelvin, a fusion entropy of 43 joules per kelvin per mole, an excess heat capacity of 62 — it reaches zero at 152 kelvin.

Kauzmann pointed this out in 1948, and the difficulty is what happens below. If the extrapolation held, the supercooled liquid would have less entropy than the crystal, which for a disordered arrangement against an ordered one is hard to accept, given that entropy is a count of arrangements and the liquid plainly has more of them; a little further down it would have negative entropy, which is not merely hard to accept but forbidden by the third law.

What actually intervenes #

Nothing so drastic happens, because the liquid stops being a liquid first.

A supercooled liquid relaxes on a timescale that grows extremely fast as it cools — faster than the exponential that decides everything would give — not as an Arrhenius exponential but faster, following a form that appears to diverge at a finite temperature. When that time reaches the timescale of the experiment, the liquid can no longer explore its configurations during the measurement, and it stops following the equilibrium curve. It has become a glass.

That figure is the answer to the obvious objection. If the trouble is that the experiment is too fast, do it more slowly.

The trouble is that the transition temperature moves logarithmically with the time available. Three and a half kelvin per decade means that going from a laboratory cooling rate to a geological one — say twelve decades — buys forty kelvin, and reaching the Kauzmann temperature from an ordinary glass transition needs far more than that once the divergence steepens. The number the figure computes is $10^{41}$ seconds per kelvin. The universe is $10^{17}$ seconds old.

So the extrapolation cannot be tested and the paradox cannot be resolved by measurement. It is a question about what would happen in an experiment nobody can do.

What the residual entropy is, then #

The consequence for the third law is worth stating precisely, because it is different from the ice case in kind rather than in size.

Ice’s residual entropy is the entropy of a degenerate ground state — the system is in equilibrium, the ground state genuinely has $\Omega$ configurations of equal energy, and the entropy $k\ln\Omega$ is a property of the substance.

A glass’s residual entropy is the entropy of a system that is not in equilibrium. There is no claim that the configurations it might have had are degenerate; the claim is only that it cannot get from the one it has to any of the others in any available time. The number is real — it is measured calorimetrically, by integrating $C_p/T$ from zero and comparing with the value from spectroscopy — and it is a property of the sample’s history.

That distinction has a practical face. Two pieces of the same glass, cooled differently, have measurably different densities, different refractive indices, different heat capacities and different residual entropies, and they will slowly converge if left alone. The convergence is called physical ageing, it takes years at room temperature for a polymer and geological times for a silicate, and it is the same relaxation that produced the glass transition, running at a rate nobody has to wait for it to finish — an approach to equilibrium that only runs forwards.

Whether a substance behaves as a solid or a liquid is decided by the ratio of its relaxation time to the observation time, and the glass transition is that ratio passing through one during a cooling run. Everything in this rung is a consequence of the same comparison, applied to entropy rather than to flow.

How the residual entropy is actually obtained #

The numbers in this rung are calorimetric, and the method deserves stating because it is one of the more indirect measurements in thermodynamics and it is what makes the residual entropy a fact rather than an inference.

The third law’s usable content is that a perfect crystal has zero entropy at zero temperature. So the absolute entropy of a substance at any temperature can be obtained by cooling it as far as possible, measuring $C_p$ all the way up, and integrating $C_p/T$ — adding the latent heats at each transition. That gives a number with no arbitrary constant in it, which is unusual and is the whole reason the law is useful.

The same substance’s entropy can be obtained a second way, from statistical mechanics, using spectroscopic data: the vibrational frequencies, the rotational constants, the electronic states. That calculation assumes the system reaches its ground state and counts from there.

Where the two agree, everything is consistent. Where the calorimetric value is lower, the difference is the residual entropy — the part the integration never saw, because the system was already stuck when the cryostat reached its lowest temperature. Giauque and his collaborators established the method in the 1930s and found ice’s 3.4, carbon monoxide’s 4.6 and nitrous oxide’s 5.8 that way.

For a glass the same comparison gives the same kind of answer with a different meaning attached. The calorimetric integration is short by an amount that depends on the cooling history, and the two samples of the same substance that differ can be compared directly — which is how the rate dependence in the first figure is established as a measurement rather than as a model output.

What is thought to be going on #

The status of the paradox is that it has several candidate resolutions and no agreed one, and it is worth laying them out because the disagreement is substantive rather than terminological.

There is a genuine transition at the crossing. In this reading there is an ideal glass — a thermodynamic phase, reached only in infinitely slow cooling, with an entropy equal to the crystal’s and a genuinely unique configuration. The observed glass transition is then a kinetic shadow of a real transition sitting a few tens of kelvin below. Random first-order transition theory develops this and predicts specific relations between the kinetic and thermodynamic temperatures.

The extrapolation is simply wrong. The excess heat capacity is measured over a limited range and extrapolated logarithmically; if it falls off below the glass transition, the entropy curve flattens and never crosses. There is some evidence for exactly that in a few systems, from measurements on ultrastable glasses made by vapour deposition, which reach states equivalent to millions of years of ageing.

The crossing is real and unremarkable. Some argue that a supercooled liquid having less entropy than a crystal is not actually forbidden — it is only surprising — and that the third law constrains the entropy at zero rather than the ordering of two entropies above it.

None of the three has been eliminated. The measurements that would separate them are exactly the ones the figure above says are impossible for ordinary glasses, which is why the vapour-deposited ultrastable materials matter so much: they reach further down the equilibrium curve than any cooling can.

Why an ideal glass would be strange #

It is worth saying what the first of the three candidate resolutions actually asserts, because “there is a genuine transition at the crossing” sounds mild and is not.

An ideal glass would be a phase with the entropy of a crystal and none of its order — a single configuration, or an exponentially small number of them, picked out of the astronomically many the liquid was exploring, with no symmetry distinguishing it and no way to say in advance which one. The transition to it would be a thermodynamic transition of a kind nothing else in physics offers: a change of phase with no order parameter that anybody can write down.

That is not obviously incoherent, and models exist in which it happens. Mean-field spin glasses have exactly such a transition, provably, and the random first-order theory is an attempt to carry the result to real liquids in three dimensions. What is missing is any way to see it, since by construction it is reached only in infinite time.

The competing view is that the mean-field result does not survive in three dimensions, that activated processes always find a way down, and that the entropy curve simply bends. Simulations can be run in the relevant range and give conflicting answers, because the relevant range is precisely where equilibration takes longer than any simulation.

So the argument is between two positions that both explain the data and differ about an unobservable, which is an uncomfortable place for a physical question to sit and is where this one has been for seventy years.

The one development that has moved it is experimental rather than theoretical, and it is worth ending the section on. Depositing a glass molecule by molecule onto a substrate held just below the transition lets each arriving molecule find a good position before it is buried, which produces a material equivalent to one aged for thousands or millions of years. Those ultrastable glasses sit measurably further down the equilibrium curve than any cooled sample, with residual entropies a substantial fraction lower and transition temperatures tens of kelvin higher.

What they have not shown is a transition. They reach further and the curve continues, which is consistent with both remaining candidates and rules out only the most extreme versions of each. That is progress of a modest kind, and it is the kind available: the question is about a limit, and an experiment can only ever get closer to one.

Where else a frozen degree of freedom appears #

The pattern in this rung — a degree of freedom that stops equilibrating and keeps whatever it had — is not confined to glasses, and recognising it elsewhere is what makes it more than a curiosity about window panes.

A spin glass does the same thing with magnetic moments rather than with positions, and its residual entropy is likewise history-dependent. The theory is better developed there, because the disorder is fixed rather than self-generated, and it is the source of most of what is believed about the structural case.

A protein folds into one of many conformations and can be trapped in the wrong one, which is why the energy landscape language of glasses is the standard language of protein folding as well.

And the early universe froze out several quantities in the same sense: the neutron-to-proton ratio stopped equilibrating when the weak interaction rate fell below the expansion rate, which is a rate comparison of exactly this shape, and what remains is whatever the ratio was at that moment. The primordial helium abundance is a residual, in exactly this rung’s sense, of a reaction that ran out of time.

In each case the comparison is between two rates — how fast the system relaxes, and how fast the conditions change — and the frozen value is set where they cross. That comparison is worth having as a habit, because it identifies which quantities in a problem are equilibrium properties and which are fossils.

Where this stops being right #

The relaxation law used here diverges at a finite temperature by construction. The Vogel–Fulcher form is a fit rather than a derivation, and whether the divergence is real or whether the relaxation time merely becomes very large is one of the open questions. Using a form that does not diverge changes the arithmetic of the last figure by many orders of magnitude and changes none of its conclusions.

The excess heat capacity was taken as constant. It is not; it varies with temperature, and the logarithm in the entropy expression is a consequence of assuming it does not. Real extrapolations use measured $C_p(T)$ and the crossing temperature moves by tens of kelvin depending on the functional form assumed.

A single relaxation time is a fiction. A supercooled liquid relaxes with a broad distribution of times, so “the” transition is a range rather than a point, and different measurements — calorimetric, dielectric, mechanical — locate it at slightly different temperatures.

And a glass is not a state. Everything above treats the glass as having a definite entropy, and it has one only in the sense that a snapshot does. The system continues to relax, so the entropy continues to fall, at a rate that itself falls — which is why “the residual entropy” needs a stated timescale to mean anything at all.

What the pictures cannot show #

The entropy curves are drawn as continuous functions of temperature, and entropy is not measured that way. What is measured is a heat capacity, over a range, and the entropy is the integral of $C_p/T$ from as low as the cryostat reaches. Everything below that is an extrapolation, and the residual entropy is a difference between two such integrals — so the quantity the whole essay is about is inferred rather than read.

Nor can the figures show that the flat portions are not flat. A glass below its transition is still relaxing, and its entropy is still falling, immeasurably slowly. Drawing a horizontal line is drawing the limit of a process that has not stopped, and the difference between “has stopped” and “is too slow to see” is precisely the difference this rung is about.

Where this ladder goes next #

Four rungs stand on third-law. The first found that cooling proceeds by fractions and never reaches the floor. The second found that a degenerate ground state keeps an entropy that is a count. The third found a system whose bounded spectrum lets its temperature pass through infinity. This one finds a residual entropy that belongs to the experiment rather than to the substance.

The habit worth carrying away is about quantities that depend on the observation. When a measured number changes with how the measurement was made, the honest response is not to find the true value but to ask what the number is a property of. A glass’s residual entropy is a property of a history, and saying so is more informative than any single value would be. The same reframing applies to a yield stress, to a coercive field, and to a critical current: each is a threshold measured on a timescale, and each has a literature about the timescale rather than about the threshold.

What is left on this ladder is what the law is a statement about, once the counting is taken seriously. Ice keeps an entropy because its ground state is degenerate, and a glass keeps one because it has not reached its ground state — so both cases turn on the structure of a spectrum rather than on anything about heat. Restating the third law that way explains why nuclear spins carry a large entropy down to microkelvin temperatures and violate nothing.