Класична теорія утворення ядер: Розмір кристала, який повинен досягти для виживання

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.

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.

Heterogeneous vs Homogeneous 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.

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