Supercooling: Why Pure Water Stays Liquid at −40 °C

The thermodynamic setup: metastable, not stable

Below the melting point T_m, ice is the stable phase — its molar Gibbs free energy is lower than the liquid's. The difference per unit volume, the volumetric driving force Δg_v, grows with how far you undercool. To good approximation near T_m:

• Δg_v ≈ L_v·ΔT / T_m, where L_v is the latent heat of fusion per unit volume and ΔT = T_m − T is the undercooling.
• For water, the latent heat of fusion is L = 334 kJ/kg; with ρ_ice ≈ 917 kg/m³ that is L_v ≈ 3.06 × 10⁸ J/m³.
• At ΔT = 10 K, Δg_v ≈ (3.06 × 10⁸ · 10)/273 ≈ 1.1 × 10⁷ J/m³ — a strong thermodynamic push toward ice.

So why doesn't the liquid freeze the instant T drops below 0 °C? Because being allowed to freeze is not the same as having a pathway. A macroscopic block of ice cannot appear at once; the transition must start from a microscopic seed, and building that seed costs surface energy that the tiny volume cannot yet repay. The supercooled liquid is metastable: locally stable, globally not, and separated from the stable state by an activation barrier — exactly the structure of a chemical reaction over a transition state.

Classical nucleation theory and the critical radius

Consider a spherical ice embryo of radius r forming inside supercooled water. Its free-energy change relative to pure liquid has a competition of two terms — the volume gain and the surface cost:

• ΔG(r) = −(4/3)πr³·Δg_v + 4πr²·γ
• The volume term −(4/3)πr³·Δg_v is negative: making ice lowers free energy, and it scales as r³.
• The surface term +4πr²·γ is positive: the new ice–water interface has energy γ (the interfacial free energy, ≈ 28–32 mJ/m² for ice–water), and it scales as r².

For small r the r² surface term dominates, so ΔG rises — small embryos are unstable and tend to redissolve. Only past a threshold does the r³ term win. Setting dΔG/dr = 0 gives the critical radius:

• r* = 2γ / Δg_v
• The barrier height at r* is ΔG* = 16πγ³ / (3·Δg_v²).

Note ΔG* ∝ γ³/Δg_v² ∝ γ³·T_m²/(L_v²·ΔT²): the barrier collapses as 1/ΔT² as you cool harder, which is why there is a fairly sharp onset temperature rather than a gentle drift. An embryo that reaches r* sits on a knife-edge; add one more molecule and it grows spontaneously, shedding latent heat all the way.

Putting numbers on the barrier

Take water at ΔT = 40 K (T ≈ 233 K), near the homogeneous limit, with γ ≈ 30 mJ/m² and Δg_v ≈ 3.06 × 10⁸ · 40/273 ≈ 4.5 × 10⁷ J/m³.

• Critical radius: r* = 2γ/Δg_v = 2·0.030 / 4.5 × 10⁷ ≈ 1.3 × 10⁻⁹ m ≈ 1.3 nm. That embryo holds only a few hundred water molecules.
• Barrier height: ΔG* = 16π·(0.030)³ / (3·(4.5 × 10⁷)²) ≈ 2.2 × 10⁻¹⁹ J ≈ 1.4 eV ≈ 70 k_BT at 233 K.

Now compare with a modest undercooling of ΔT = 5 K: Δg_v falls by 8×, so r* grows to ≈ 10 nm and ΔG* balloons by 64× to ≈ 4500 k_BT. A barrier of thousands of k_BT is astronomically improbable to surmount by thermal fluctuation, which is exactly why a clean sample sits happily supercooled at −5 °C for hours yet freezes within microseconds once ΔT reaches ~38 K. The 1/ΔT² law makes the transition switch-like.

The nucleation rate: an Arrhenius switch

The probability per unit volume per unit time of forming a supercritical nucleus is the nucleation rate, and it has an Arrhenius (Boltzmann) form set by the barrier:

• J = J₀ · exp(−ΔG* / k_BT), with a kinetic prefactor J₀ ≈ 10³³–10³⁶ m⁻³s⁻¹ that bundles the molecular attempt frequency and the density of molecules at the embryo surface.
• Because ΔG* ∝ 1/ΔT², the exponent changes enormously over a few degrees. Near T_hom, J typically jumps by several orders of magnitude per kelvin of extra undercooling.

This steepness is why nucleation looks like a threshold even though it is really a rate. Above some undercooling the expected waiting time in your droplet is centuries; a degree or two colder and it drops below a microsecond. The observed T_hom ≈ −38 °C for water is simply the temperature where J becomes large enough to freeze a micron-scale droplet within the experimental observation time — it is mildly volume- and time-dependent, not a fundamental constant like T_m. Smaller droplets, having fewer molecules and less volume to sample, tolerate slightly deeper undercooling.

Heterogeneous nucleation: why real water freezes near 0 °C

Homogeneous nucleation is the exception, not the rule. Ordinary water is riddled with impurities, dust, container walls, and dissolved particles, and these foreign surfaces catalyse freezing by letting the embryo form as a cap on the substrate rather than a full sphere. The barrier is multiplied by a geometric factor that depends only on the ice–substrate contact angle θ:

• ΔG*_het = ΔG*_hom · f(θ), with f(θ) = (2 + cosθ)(1 − cosθ)² / 4.
• For θ = 180° (non-wetting, useless catalyst) f = 1: no help. For θ → 0 (ice wets the surface perfectly) f → 0: the barrier essentially vanishes.
• A good ice nucleus like silver iodide (AgI) has a crystal lattice almost matching ice (mismatch ≈ 1.5%), giving a small θ and letting it seed ice at only a few degrees of undercooling — the basis of cloud seeding.

Biology exploits this too. Bacteria such as Pseudomonas syringae display the InaZ ice-nucleation protein, which templates the water lattice so effectively it triggers freezing at −2 °C; it is the active ingredient in artificial snow (Snomax). Conversely, antifreeze proteins in Arctic fish and freeze-tolerant insects do the opposite — they bind to nascent ice surfaces and raise the effective barrier, extending supercooling and depressing the freezing temperature far below the equilibrium melting point (thermal hysteresis).

Where supercooling shows up — and bites

Supercooling is everywhere once you know to look:

• Clouds: the vast majority of mid-level clouds contain supercooled droplets between 0 °C and −38 °C. This is central to the Wegener–Bergeron–Findeisen process, where ice crystals grow at the expense of supercooled droplets (ice has a lower saturation vapour pressure) and produce most mid-latitude rain and snow.
• Aircraft icing: a plane flying through a supercooled cloud provides exactly the foreign surface the droplets were waiting for; they nucleate on contact and glaze the wings — a serious flight hazard that drives de-icing systems.
• Hand-warmers: the click-to-activate sodium-acetate packet is supercooled molten salt held ~40 °C below its 58 °C freezing point. Flexing a metal disc injects seed crystals, triggering an avalanche of crystallisation that dumps its latent heat as warmth.
• Cryopreservation: deep supercooling followed by vitrification (freezing to a glass without crystals) lets organs and cells be stored without ice damage — a booming field for transplant logistics.
• Instant slush: tap the bottle of commercially super-purified soda that's been chilled below 0 °C, and the shock nucleates ice through the whole bottle in seconds.

Subtleties and common misconceptions

"Supercooled water is really cold ice." No — it is genuinely liquid, just metastable. Its structure is disordered like ordinary water; it simply hasn't crossed the nucleation barrier.

"Freezing a supercooled sample makes it all ice instantly." Not quite. When a −10 °C supercooled sample nucleates, only a fraction actually freezes — the released latent heat warms the mixture straight back up toward 0 °C, and it settles as a slush of ice + water at the equilibrium point. Energy conservation caps the ice fraction at roughly (c·ΔT)/L; for water at ΔT = 10 K that is only about (4186·10)/334000 ≈ 12.5%.

"Classical nucleation theory is exact." It is a powerful sketch but treats a nanometre embryo with a macroscopic surface tension γ and a sharp interface — dubious when r* holds only a few hundred molecules. CNT routinely mispredicts absolute rates for water by many orders of magnitude, so γ is often used as an effective fit parameter. Modern work invokes two-step nucleation (a dense amorphous precursor forms first) and the debated existence of a liquid–liquid critical point in deep-supercooled "no-man's land" water below ≈ −40 °C, where the sample crystallises faster than it can be probed.

"Undercooling and superheating are unrelated." They are mirror images. A superheated liquid (water above 100 °C with no bubble sites) is the same metastable-with-a-barrier physics, only the embryo is a vapour bubble and the barrier resists boiling instead of freezing — bumping in a microwave is superheating's nucleation catastrophe.

Homogeneous vs heterogeneous nucleation of ice from supercooled water

Property	Homogeneous	Heterogeneous
Trigger	Spontaneous density/order fluctuation in the bulk	A foreign surface (dust, wall, AgI, protein)
Onset undercooling	ΔT ≈ 38–40 K (pure μm droplets)	ΔT ≈ 1–15 K (ordinary tap water)
Energy barrier	ΔG* full, no reduction	ΔG*·f(θ), f(θ) = (2+cosθ)(1−cosθ)²/4 ≤ 1
Contact-angle role	None (no substrate)	Small θ (wetting) → f≈0 → almost no barrier
Where it dominates	Cloud droplets, clean lab vials	Nearly all everyday freezing
Rate sensitivity	J changes ~orders/degree near T_hom	Set by best available catalyst site