4.5 Nucleation and the free-energy barrier

A phase transition does not happen everywhere at once. A new phase must start as a tiny embryo — a microscopic bubble in a liquid, a droplet in a vapour, a crystallite in a melt — and that embryo carries a surface, which costs energy. The competition between the bulk free energy the new phase gains and the surface energy it must pay creates a barrier, and crossing it is nucleation.

Volume gain versus surface cost

Consider a spherical embryo of the new phase, radius $R$ , forming in the old one when the new phase is favoured by a bulk free-energy difference $\Delta p$ per unit volume. Two terms compete:

\Delta G(R) \;=\; -\tfrac43\pi R^3\,\Delta p \;+\; 4\pi R^2\,\sigma.

The volume term is negative — the embryo is the more stable phase — and grows as $R^3$ . The surface term is positive — the interface costs the surface energy $\sigma$ per unit area — and grows as $R^2$ . At small $R$ the surface term dominates and $\Delta G$ rises; at large $R$ the volume term wins and $\Delta G$ falls. Between them is a maximum.

Setting $d(\Delta G)/dR = 0$ locates the critical radius and the barrier height:

R^* \;=\; \frac{2\sigma}{\Delta p}, \qquad \Delta G^* \;=\; \frac{16\pi\sigma^3}{3(\Delta p)^2}.

An embryo smaller than $R^*$ lowers its free energy by shrinking and re-dissolves; one larger than $R^*$ lowers it by growing and runs away. The critical radius is the top of the hill, an unstable balance.

ambient pressure p_∞ = -100 atm surface tension σ = 72 mN/m

temperature = 20 °C vapour pressure p_v = 0.023 atm

A vapour bubble of radius R in a liquid under tension Δp = p_v − p_∞ has Gibbs free energy ΔG(R) = −(4/3)πR³Δp + 4πR²σ. The first term (volume × pressure difference) drives growth; the second (surface area × surface tension) opposes it. ΔG peaks at the critical radius R* = 2σ/Δp with barrier height ΔG* = 16πσ³/(3Δp²). Above R* the bubble grows spontaneously; below R* it collapses. Thermal fluctuations cross the barrier at a rate J = J₀ exp(−ΔG*/kT) — exponentially sensitive to the barrier. For pure water at room temperature, the barrier is below 100 kT only when Δp exceeds ~1000 atm, recovering the homogeneous tensile-strength estimate of Lesson 1.2. The barrier is far too high at modest tensions to explain why real water tears at 0.1 atm — the resolution is heterogeneous nucleation, next lesson.

Slide the surface energy $\sigma$ and the driving force $\Delta p$ . The downward cubic and upward quadratic sum to a barrier whose peak height $\Delta G^*$ is read off directly. Stronger driving (larger $\Delta p$ ) lowers and narrows the barrier; higher surface energy raises it steeply, since $\Delta G^* \propto \sigma^3$ .

Why pure phases are so reluctant to transform

By the theory of activated processes, the rate of forming a critical embryo is set by the Boltzmann factor $e^{-\Delta G^*/k_B T}$ . Because $\Delta G^*$ for a pure substance is typically hundreds to thousands of $k_B T$ , that factor is astronomically small — pure water can be cooled well below freezing or heated above boiling without transforming, because the homogeneous barrier is effectively uncrossable. Real transitions almost always start at heterogeneous sites — a dust mote, a scratch on a container, a pre-existing pocket of the new phase — where a foreign surface replaces part of the costly new interface and slashes the barrier. The pristine, homogeneous case derived here is the upper bound on how reluctant a transition can be; the surfaces of the real world make it happen.