Classical nucleation theory

Nucleation is the first step in the formation of either a new thermodynamic phase or a new structure with lower free energy via self-assembly or self-organisation. Nucleation is typically defined to be the process that determines how long we have to wait before the new phase or self-organised structure appears. Classical nucleation theory (CNT) is the most common theoretical model used to understand why nucleation may take hours, years, or never happen at all.

This is the standard simple theory for nucleation of a new thermodynamic phase, such as a liquid or a crystal. It should be borne in mind that it is approximate. The basic CNT nucleation of a new phase provides an approximate but physically reasonable prediction for the rate at which nuclei of a new phase form, via nucleation on a set of identical nucleation sites. This rate, R is the number of, for example, water droplets nucleating in a uniform volume of air supersaturated with water vapour, per unit time. So if a 100 droplets nucleate in a volume of 0.1m³ in 1s, then the rate R=1000/s. The description here follows modern standard CNT. The prediction for the rate R is

where

This expression for the rate can be thought of as a product of two factors: The first, ${\displaystyle N_{S}\exp \left(-\Delta G^{*}/k_{B}T\right)}$, is the number of nucleation sites multiplied by the probability that a nucleus of critical size has grown around it. It can be interpreted as the average, instantaneous number of nuclei at the top of the nucleation barrier. Free energies and probabilities are closely related in general, by definition. The probability of a nucleus forming at a site is proportional to ${\displaystyle \exp[-\Delta G^{*}/kT]}$. So if ${\displaystyle \Delta G^{*}}$ is large and positive the probability of forming a nucleus is very low and nucleation will be slow. Then the average number will be much less than one, i.e., it is likely that at any given time none of the sites has a nucleus.

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