# A Review of Classical and Nonclassical Nucleation Theories

[Cryst. Growth Des. 2016, **16**, 11, 6663–6681](https://pubs.acs.org/doi/10.1021/acs.cgd.6b00794)

## Abstract
Classical Nucleation Theory $\leftrightarrow$ nonclassical two-step pathway approach (diffuse interface theory and density functional theory)

## 1. Introduction
Nucleation : Formation of a new thermodynamic phase from an od phase with high free energy to low
## 2. Nucleation Theories
Homogeneous nucleation : No rule for foreign surfaces. Large supersaturations are needed.($\leftrightarrow$Heterogeneous nucleation)
CNT : phenomenological approach.
Monte Carlo and MD : using first principals.
### 2.1. Classical Nucleation Theory
CNT : Prediction of nucleation rates.
Volmer, Weber, Becker, Doring, Frenkel.

\begin{eqnarray}
\Delta G = \frac{-4\pi r^3}{3\nu}k_BT \ln S + 4\pi r^2\sigma \tag{1}
\end{eqnarray}

> Definition
$\Delta G$ : The change in the free energy of the system
$r$ : Spherical nucleus of radius
$\nu$ : The volume of single molecule
$\sigma$ : The surface energy of interface
$S = P/P^*$ : The vapor saturation ratio
$k_B$ : The Boltzmann constant
$T$ : The temperature

> Derivation
> $\Delta G$ is the change in Gibbs free energy when vapor turns to ice. Thus
\begin{eqnarray}
\Delta G =&& G_f - G_i\\
=&& N(\mu_{xs}^f-\mu_{xg}^i) + 4\pi r^2\sigma \tag{A1}
\end{eqnarray}
where
\begin{eqnarray}
G_f =&& \mu_{xs}^f N + 4\pi r^2\sigma,\\
G_i =&& \mu_{xg}^i N.
\end{eqnarray}
Here, $f,i$ represents final and initial, and we define $\mu_{xg},\mu_{xs}$ as chemical potential of $x$ in gas and solid phase respectively, $\sigma$ as surface tension, and $N$ as number of molucules. When in the isoterm condition and the final state is at equibrium, the relation between $\mu_{ig},\mu_{is}$ will be
\begin{eqnarray}
&&\mu_{xs}^f = \mu_{xg}^f = \mu_{xg}^i + k_BT \ln \frac{p^* }{p}\\
&&\therefore \mu_{xs}^f-\mu_{xg}^i = k_BT \ln \frac{p^* }{p} = -k_BT \ln \frac{p}{p^* }\tag{A2}
\end{eqnarray}
By equation(A1) and (A2),
\begin{eqnarray}
\Delta G = -Nk_BT \ln \frac{p}{p^* } + 4\pi r^2\sigma.
\end{eqnarray}

$\Delta G$ passes through a maximum at $r^*$.
\begin{eqnarray}
\Delta G^* =\frac{16\pi \sigma^3\nu^2}{3k_B^2T^2 \ln S^2}\tag{2}
\end{eqnarray}

The nuclation rate, $J$ is expressed as follows. ($J \leftrightarrow \sigma$)
\begin{eqnarray}
J =&& A\exp\left(-\frac{\Delta G^* }{k_B T}\right)\\
=&& A\exp\left(-\frac{16\pi \sigma^3\nu^2}{3k_B^3T^3 \ln S^2}\right)\tag{3}
\end{eqnarray}
Critical nuclei : 10-1000 molecules.(1000 : phenol, naphtthalene, azobenzene)

Assumtions of CNT are
$\mathrm{(a)}$ the nucleus can be described with the same macroscopic properties(density, structure, composition) of the stable phase,
$\mathrm{(b)}$ the nucleus is spherical and the interface between the nucleus and the solution is a sharp boundary,
$\mathrm{(c)}$ vapor-liquid interface is approximated as planar, <u>regardless of critical cluster size</u>.

### 2.2. Predictions of CNT in Vapor−Liquid Systems
### 2.3. Predictions of CNT in Metallic Vapor Systems
### 2.4. Predictions of CNT in Glass Systems
### 2.5. Predictions of CNT in Supercooled Liquid Metals and Alloy Systems

## 3. Fusion of extended modified liquid drop model with dynamic nucleation theory (EMLD-DNT) model

## 4. Density functional theory

## 5. Diffuse interface theory

## 6. Non-nucleation theories pathways

## 7. Single steps or multisteps?

## 8. Thermodynamic treatment

## 9. Conclusion