The barrier a new phase has to climb

The pressure inside a small bubble is larger than inside a big one, and the same fact governs whether the bubble can exist at all. This essay is about the first moment of a phase transition, which is the only difficult part of it.

Two terms, opposite signs #

A droplet of radius $r$ forming from supersaturated vapour changes the free energy by

$$\Delta G(r) = \frac{4}{3}\pi r^3 \Delta g_v + 4\pi r^2\gamma$$

The first term is the gain from having the molecules in the liquid rather than in the vapour, and it is negative when the vapour is supersaturated: $\Delta g_v = -(kT/v_m)\ln S$, where $S$ is the ratio of the actual vapour pressure to the equilibrium one.

The second is the cost of the new interface, and it is always positive. Surface tension is an energy per unit area, and a droplet that did not exist a moment ago has made some.

At small $r$ the $r^2$ term dominates and the sum rises. At large $r$ the $r^3$ term takes over and it falls. Between them is a maximum, and it is the whole problem.

The two terms have opposite signs and different powers of the radius, which is what produces a barrier rather than a slope. The pressure inside a curved surface rises as the surface tightens, so a droplet below the critical radius has a vapour pressure above the surrounding vapour’s — and evaporates even in an atmosphere that is already supersaturated. A small droplet is not a small success. It is a failure that has to be got past.

The maximum is at $r^* = 2\gamma/|\Delta g_v|$, and the barrier height is $\Delta G^* = 16\pi\gamma^3/3\Delta g_v^2$. Both are found here by searching the drawn curve and checked against those expressions, which agree to a part in a thousand.

A droplet smaller than $r^*$ shrinks; one larger grows. So a fluctuation must produce, by chance, a droplet already past the critical size before anything can happen.

The rate, which is a threshold in disguise #

How often does chance manage it? The probability of a fluctuation costing $\Delta G^*$ is the Boltzmann factor, and the rate is that times a kinetic prefactor counting how often molecules arrive:

$$J = J_0\,e^{-\Delta G^*/kT}$$

Nothing about that function is discontinuous. It is smooth, finite and computable at every supersaturation — and it is so steep that no experiment can see it as anything but a threshold. Below the onset the rate is unmeasurably small; a few per cent above it, the vapour condenses instantly.

Two consequences follow that are worth separating.

The onset is sharp and reproducible. A quantity that varies by ten decades per ten per cent has a well-defined crossing of any measurable level, and the crossing is insensitive to everything except what sits in the exponent.

The prefactor barely matters. $J_0$ is uncertain by several orders of magnitude and the predicted onset moves by less than one per cent when it is changed by $10^5$. That is why a theory with a badly known prefactor makes usable predictions, and it is a general feature of any exponentially sensitive rate — the same insensitivity that makes tunnelling rates predictable despite crude models of the barrier.

The rate is a threshold in disguise, and the disguise is exponential. A rate depending exponentially on a barrier is steep enough that the barrier looks like a wall with a sharp top: nothing happens, nothing happens, and then everything does. What makes nucleation steeper still is that the barrier itself depends on the driving force, so the two dependences compound — and the onset is sharper than any ordinary Arrhenius process.

The same barrier for a bubble #

Everything above is written for a droplet condensing from vapour, and the identical argument runs for a bubble forming in a liquid — with one change of sign that makes the practical situation much worse.

A bubble of radius $r$ in a superheated liquid gains volume free energy from the vapour being the stable phase and pays the same $4\pi r^2\gamma$ for its surface. The critical radius is again $2\gamma/|\Delta g_v|$, and the driving force is now the difference between the actual pressure and the equilibrium vapour pressure at that temperature.

The difficulty is that the barrier for a bubble in water at atmospheric pressure is enormous until the liquid is very hot indeed. Superheating water to 280 °C is possible, which is 180 degrees past its boiling point, and it is why a smooth clean container can hold liquid water far above 100 °C and then flash to steam all at once.

The asymmetry between boiling and freezing has the same origin. Water supercools by 38 degrees and superheats by 180, because the interface between liquid and vapour is expensive — 72 millinewtons per metre — and the interface between liquid and ice is cheap, about 30, and enters the barrier cubed.

Counting the molecules in a critical cluster #

It is worth converting the critical radius into a number of molecules, because the number is what makes the theory’s difficulties concrete.

At S = 1.5 the critical droplet has a radius of 2.66 nm and contains about 2,500 water molecules. At S = 3 it is 0.98 nm and about 130. At S = 5 it is 0.67 nm and about 40, of which more than half are in the surface layer.

So the theory is being asked to treat a 40-molecule cluster as an object with a bulk free energy and a macroscopic surface tension. That it works as well as it does is genuinely surprising, and the usual explanation is a cancellation: the errors in the two terms are of the same sign and partially cancel in the barrier, which depends on the ratio $\gamma^3/\Delta g_v^2$.

A free energy is the logarithm of a count of arrangements, and putting numbers to that count is what makes the barrier vivid. A critical cluster of forty molecules is a fluctuation whose probability is the exponential of the barrier — about one in $10^{31}$ at a modest supersaturation, which is why pure water can be cooled far below freezing and sit there. The fluctuation is not forbidden. It is merely not going to happen in the time available.

The reason such an improbable fluctuation happens at all is the number of attempts. A cubic centimetre of vapour contains 10¹⁹ molecules colliding 10⁹ times a second each, so the number of trials per second is around 10²⁸ — which is why a barrier of 73 kT, with a Boltzmann factor of 10⁻³¹, gives a rate of about one per cubic centimetre per second rather than none at all.

What supersaturations are actually reached #

The theory says water vapour should sit uncondensed until about S = 3 to 4 at room temperature. That is a factor of three or four more vapour than the equilibrium amount, and it never happens in the atmosphere.

What supersaturations actually are is worth stating against the equilibrium picture. A phase boundary says where a transition is thermodynamically allowed; everything in this essay is about the region on the wrong side of that line, where the system is metastable — permitted to change and unable to start. The distance past the line is the driving force, and it has to be large before the barrier comes down far enough for anything to begin.

The reason is that the atmosphere is full of surfaces. Every dust grain, salt crystal and combustion particle offers a place for a droplet to form as a cap rather than as a sphere, and a cap has less new interface for the same curvature.

The critical radius is unchanged — it is set by the curvature, and the cap has the same curvature as the sphere — but the volume and the new area are both smaller, by the factor $f(\theta) = (2+\cos\theta)(1-\cos\theta)^2/4$. Since the barrier sits in an exponent, a factor of a few in $f$ is tens of orders of magnitude in rate.

So homogeneous nucleation is almost never what happens. Clouds form at supersaturations of a fraction of a per cent, on aerosol; bubbles form on scratches and dissolved gas; ice forms on bacteria that are astonishingly effective ice nucleators and are sold commercially for making snow.

A surface changes the arithmetic by changing the geometry. The barrier for a cluster forming on a wall is the homogeneous barrier multiplied by a factor that depends only on the contact angle, so how well the new phase wets the surface decides how effective that surface is as a nucleation site. A well-wetted wall can cut the barrier by orders of magnitude — which is why almost nothing in practice nucleates in the bulk, and why cloud seeding works at all.

Where the pure case can be reached #

Removing the surfaces is difficult and the results are worth stating, because they are the test of the theory.

Water supercools to −38 °C if it is clean and divided into droplets small enough that most contain no nucleating particle. That temperature is a homogeneous nucleation limit and is reproducible; below it, ice forms in microseconds. Cloud droplets in the atmosphere routinely reach −30 °C and freeze at −38 °C for exactly this reason.

Superheated water reaches about 280 °C at atmospheric pressure before it boils, if it is degassed and in a smooth container. The bumping of superheated liquid in a laboratory flask is that barrier being overcome all at once, and the boiling chips added to prevent it are heterogeneous nucleation sites deliberately supplied.

And a cloud chamber is a nucleation detector. Vapour is expanded to a supersaturation just below the homogeneous threshold, so nothing condenses — until an ion passes through and provides a nucleation site, whereupon a droplet forms on it. The track of a charged particle is a row of nucleated droplets, and the whole instrument is an amplifier that turns one ion into something visible.

The equilibrium picture shows every transition as a plateau: heat going in, temperature not moving. Every one of those plateaus in practice requires a nucleus, and the metastable excursions this essay is about are invisible on such a curve — they are the reason the plateau starts late, and they are erased by the time the system has settled.

The same shape in other places #

The structure — a cost that goes as the boundary and a gain that goes as the bulk — recurs wherever a new region of something has to start.

In magnetism. A reversed domain in a magnetised material pays a domain-wall energy proportional to its area and gains Zeeman energy proportional to its volume, so it too has a critical size and a barrier. That is why a magnet has a coercive field rather than reversing the moment a reversed field is applied, and why the reversal is sudden when it comes.

In crystallisation and in precipitation, where the barrier decides which polymorph appears: a metastable form with a lower interfacial energy can nucleate faster than the stable one and appear first, which is Ostwald’s rule of stages and a permanent nuisance in pharmaceutical manufacture.

In the early universe, where a first-order phase transition proceeds by the nucleation of bubbles of the new vacuum, with a rate governed by an action rather than a free energy but with the identical barrier structure.

And in cavitation, where a liquid under tension tears. Water in a tree is under negative pressure and does not boil, because the barrier to forming a bubble is large — and when it does fail, the click of the cavitation event is audible with a microphone against the trunk.

The same shape turns up as hysteresis. A magnetisation curve lags the applied field because nothing happens until reversed domains can form, and once they can, the change runs away. That lag is a nucleation barrier read as a graph, and the coercivity of a permanent magnet is the field needed to get past it — which is why a magnet’s strength is a statement about barriers rather than about how much magnetisation it contains.

Why a cloud forms at a fraction of a per cent #

The heterogeneous argument above works by wetting: a droplet forms as a cap on a solid and pays for less interface. Atmospheric nucleation mostly works a different way, and it is worth doing because it produces a number that can be checked against every cloud in the sky.

Most atmospheric aerosol is soluble — sea salt, ammonium sulphate, nitrates — so a droplet forming on one is not pure water but a solution. That changes the arithmetic twice, in opposite directions.

Curvature raises the vapour pressure over a droplet, by the Kelvin factor, and the effect goes as $1/r$: a small droplet evaporates in air that a flat surface would be in equilibrium with. Dissolved solute lowers the vapour pressure, and since the amount of solute is fixed while the volume grows, the lowering goes as $1/r^3$. Adding the two,

$$\ln S \approx \frac{A}{r} - \frac{B}{r^3},$$

which is negative at small $r$, positive at large $r$, and has a maximum in between.

That maximum is the whole of cloud formation. Below it, a particle sits in stable equilibrium as a submicron haze droplet: pushed larger it wants to evaporate, pushed smaller it wants to grow, and it stays where it is. Above it, the droplet is past the peak and grows without limit — it has activated, and it becomes a cloud droplet.

The peak’s height is the critical supersaturation, and for a typical soluble particle a tenth of a micrometre across it comes out at about one part in a thousand — a supersaturation of 0.1 per cent, not the 200 per cent homogeneous nucleation would demand. That is why clouds form on the slightest lifting, and it is the quantitative version of the claim made above that the atmosphere never reaches the pure-vapour threshold.

Two consequences follow that matter beyond meteorology. Larger and more soluble particles have lower critical supersaturations, so as an air parcel rises and its supersaturation climbs, the particles activate in order of size — and the number that activate before the growing droplets consume the excess vapour is what sets the number of droplets in the cloud. And a cloud with many small droplets is brighter than one with few large ones carrying the same water, because scattering goes with total cross-sectional area. So the population of soluble particles in the air below a cloud decides how much sunlight it reflects, through a chain of reasoning that begins with a curvature term and a solute term added together.

What happens after the first drop #

Nucleation is the beginning of a phase transition and not the end of it, and the same curvature argument that made the barrier governs what happens next — in a direction that is usually a nuisance.

Once several droplets exist, they are not in equilibrium with each other. A small droplet has a higher vapour pressure than a large one, by the same Kelvin factor as above, so vapour leaves the small ones and arrives at the large ones. The small shrink, the large grow, and eventually the small vanish.

That is Ostwald ripening, and its signature is a mean size growing as the cube root of time in the diffusion-limited case — slow, unstoppable, and requiring no motion, no collision and no coalescence. The droplets never touch. They communicate through the vapour, and the direction is fixed by which of them is more curved.

The same process runs in every two-phase mixture where the minority phase can dissolve and re-deposit.

Ice cream is the everyday case. It is a suspension of small ice crystals, and it is smooth when they are small enough not to be felt. In a freezer that cycles in temperature, the crystals ripen: the small ones melt slightly and re-freeze onto the large, the mean size climbs, and the texture goes from smooth to gritty. Nothing has gone wrong chemically; the crystals have simply been given time. Commercial stabilisers work by slowing the transport rather than by changing the thermodynamics.

Emulsions fail the same way, and the standard cure is a nice inversion of the argument. Adding a small amount of a component that is insoluble in the continuous phase means that as a droplet tries to shrink, its trapped component becomes more concentrated, and the resulting osmotic term opposes the shrinking. The ripening stalls at a droplet size where the two effects balance, which is exactly the Köhler equilibrium of the previous section with the roles of the phases exchanged.

And it sets a limit on high-temperature alloys. A nickel superalloy is strong because it is full of fine precipitates that obstruct dislocations, and its service temperature is limited by how fast those precipitates ripen. The blade does not melt and does not corrode; its microstructure coarsens, and the cube-root law says when.

What the theory gets wrong #

Classical nucleation theory is one of the most-used and least-trusted calculations in physical chemistry, and being clear about which parts fail is the point of this section.

It uses a bulk surface tension for a droplet of ten molecules. The critical droplet at S = 5 has a radius of 0.67 nm and contains about thirty molecules, most of them in the surface. Whether a surface tension measured on a flat interface applies to such an object is not obvious, and the leading correction — the Tolman length — is of the same size as the radius.

It uses a bulk free energy too. The same objection applies: a thirty-molecule cluster does not have a well-defined bulk.

The prefactor is a kinetic estimate. It counts arrival rates at the critical cluster and ignores much of what happens afterwards. Measured rates for some substances depart from prediction by ten or more orders of magnitude, and the sign of the discrepancy is not consistent between substances.

And the reaction coordinate is assumed to be the radius. Real clusters are not spherical and may pass through non-compact intermediates; two-step nucleation, in which a dense disordered cluster forms first and then orders internally, is now believed to be common in proteins and in some crystals.

What survives all that is the shape of the argument — a barrier that falls as the driving force rises, a rate that is exponentially sensitive to it, and a surface that removes most of the barrier — and it survives because the exponent is what matters.

What the theory gets wrong is worst near the critical point, and the reason is that its central quantity vanishes there. Approaching the critical point the surface tension goes to zero, so the barrier does too, and a transition that requires a barrier everywhere else requires none there. The model is built on a quantity it assumes is well-defined, and in the one regime where the assumption fails it fails completely rather than gradually.

Why a metastable state can last for ever #

The last thing to extract is what “metastable” means quantitatively, because the word is used loosely and the arithmetic behind it is sharp.

A metastable state is one that is not the free-energy minimum and is separated from the minimum by a barrier. Whether it matters that it is not the minimum depends entirely on the barrier’s height in units of kT, and the dependence is exponential — so there is no such thing as a slightly metastable state in practice. A barrier of 20 kT gives a lifetime of microseconds and one of 100 kT gives a lifetime longer than the age of the universe.

That is why diamond is described as metastable with respect to graphite and nobody worries: the barrier is enormous, and the statement that graphite is the stable form has no observable content at room temperature. It is also why glass is a solid: a glass is a liquid that has been cooled past the point where it could nucleate a crystal, and the barrier that stops it doing so afterwards is what makes a window rather than a puddle.

The general habit is worth carrying. A thermodynamic statement about which state is stable is a statement about a limit that may never be approached, and the question of whether it will be is a question about a barrier and a number of attempts. Both are computable, and the answer is almost always either “immediately” or “never”.

The ladder from here #

Later rungs on this anchor: the Zeldovich factor and the proper steady-state derivation of the prefactor; the Tolman correction to the surface tension of a small droplet; spinodal decomposition, which is what happens when the driving force is so large that there is no barrier at all and the transition is not nucleated but unstable; and two-step nucleation, where the reaction coordinate is not a size.

The neighbouring ladders are the small bubble that blows up the big one, which is the same curvature argument as a pressure, the part of the curve no fluid follows, which is the metastable region this essay explains the persistence of, and the column that is pulled, not pushed, where the same barrier decides when water under tension tears.