Classical Nucleation Theory

Two competing energies

When a new phase — a crystal in a supersaturated solution, a solid nucleus in an undercooled melt — starts to form as a small spherical cluster of radius r, its total free-energy change has two terms of opposite sign. A bulk (volume) term releases energy proportional to the volume of the new phase, because the new phase is thermodynamically more stable than the old one under these conditions. A surface term costs energy proportional to the area of the new interface being created, because forming any new surface has an unavoidable energetic penalty.

ΔG(r) = (4/3)π r³ Δg_v + 4π r² γ

  Δg_v  < 0   bulk free-energy change per unit volume (drives growth)
  γ           surface (interfacial) energy per unit area (resists growth)

The critical radius r*

The volume term grows as r³ and the surface term as r², so for small r the surface term dominates — tiny nuclei are net energetically unfavourable — while for large r the volume term wins. Setting the derivative dΔG/dr to zero locates the peak of that competition, the critical radius:

r*  = -2γ / Δg_v
ΔG* = 16π γ³ / (3 Δg_v²)      (the nucleation barrier, ΔG at r = r*)

Below and above r*: dissolve or grow

A nucleus smaller than r* still sits on the rising part of the ΔG curve, so shrinking it back to nothing lowers the system's free energy — sub-critical nuclei are statistically far more likely to redissolve than to keep growing. A nucleus that has, by a lucky sequence of random additions, exceeded r* sits past the peak, where further growth continually lowers ΔG; from that point on, growth is thermodynamically downhill and essentially guaranteed to continue.

live demo · a nucleus randomly growing and shrinking near r*● LIVE

Nucleation rate: an exponential bottleneck

The rate at which viable nuclei appear per unit volume per unit time follows an Arrhenius-like form, J = A · exp(-ΔG* / k_B T), where A is a kinetic prefactor. Because ΔG* itself scales as 1/Δg_v², and Δg_v grows roughly linearly with undercooling or supersaturation, small changes in how far a system is pushed from equilibrium translate into enormous changes in nucleation rate through that squared term sitting inside an exponential. This extreme sensitivity is exactly why a barely-undercooled liquid can sit for a very long time without freezing, while pushing it just a little further past equilibrium can trigger nucleation almost immediately.

Homogeneous vs heterogeneous nucleation

Everything above describes homogeneous nucleation, forming spontaneously in a uniform bulk phase and paying the full surface-energy cost. In practice, most real nucleation is heterogeneous: it happens on a pre-existing surface — a dust particle, a container wall, a scratch — which lets part of the new nucleus's interface replace an existing one instead of creating fresh surface from scratch. That reduces the effective surface energy term, lowers ΔG*, and is why boiling chips prevent bumping, why clouds need condensation nuclei to form raindrops, and why most freezing in nature starts at a surface rather than in open bulk liquid.

Frequently asked questions

What happens to a nucleus exactly at the critical radius?

At exactly r*, the free-energy curve sits at its maximum (ΔG*), and the nucleus is in an unstable equilibrium: an infinitesimal random fluctuation in either direction determines whether it shrinks back to nothing or grows into a stable phase. In practice this exact balance point is never occupied for long — thermal fluctuations push nuclei past it constantly, and only a tiny fraction happen to cross toward growth.

Why can liquids be supercooled well below their freezing point?

The nucleation rate depends exponentially on the barrier height ΔG*, which itself depends on the inverse square of the undercooling. Close to the equilibrium freezing point, undercooling is small, ΔG* is enormous, and the exponential factor makes homogeneous nucleation astronomically rare — so a very clean liquid with no dust particles or surfaces to nucleate on can persist as a liquid tens of degrees below its nominal freezing point before it randomly nucleates.

What is the difference between homogeneous and heterogeneous nucleation?

Homogeneous nucleation forms a new-phase nucleus spontaneously within a uniform bulk phase, paying the full surface-energy cost of the classical theory above. Heterogeneous nucleation instead forms on a pre-existing surface — a dust particle, a container wall, a scratch — which lets part of the new nucleus's surface replace a pre-existing interface, lowering the effective surface energy and dramatically reducing the nucleation barrier. It's why real-world freezing and boiling almost always start at a surface, not in the open bulk.

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