MATE 664 Lecture 14

Introduction To Nucleation Theory

Author

Dr. Tian Tian

Published

February 25, 2026

Note

Slides 👉 Open presentation🗒️
PDF version of course note 👉 Open in pdf
Handwritten notes 👉 Open in pdf

Learning outcomes

After this lecture, you will be able to:

Identify the driving force when a system becomes supercooled or supersaturated
Describe the difference between discontinuous and continuous phase transformations
Analyze the nucleation free energy barrier $\Delta G_c$
Describe the role of surface and interfacial energy in nucleation
Describe the pseudo-steady-state kinetic model for nucleation

Phase-diagram in non-equilibrium region

How to induce phase transformation from a phase diagram?
Going low in temperature –> (super)cooling

Transformation driving force in single-component phase diagram

Cooling from liquid to solid provides driving force

\[ \Delta G^{\text{L->S}} = \frac{\Delta H^{\text{L->S}}(T_m - T )}{T_m} < 0 \]

Why doesn’t ice form spontaneously?

Phase transformation in binary phase diagram ($T-X_B$)

Ways to introduce driving force:

(Super)saturation
(Super)cooling

Phase transformation in binary phase diagram ($G-X_B$)

Metastable regions in binary phase diagram (nucleation)
Unstable regions (spinodal decomposition)

Phase transformation in binary phase diagram ($G-X_B$)

Metastable regions in binary phase diagram
Tangent-to-line method: driving force $\Delta G_B$ (molar)

Nucleation theory in a nutshell

Crystal growth can be divided in 4 regions
Nucleation theory deals with regions I and II (incubation & pseudo-steady-state)

Introducing interfacial energy (1)

Creating interfaces between different atoms causes energy to change!
Surface energy $\gamma_A$ (vacuum, unit J/m$^2$ or N/m)

\[ \gamma_A = \frac{1}{2 a_0} w_{AA} (Z_s - Z_b) \]

Interatomic energy in bulk

Creation of surface

Introducing interfacial energy (2)

Interfacial energy $\gamma_{AB}$ can be calculated using $\gamma_A$ $\gamma_B$ and $\Delta W_{AB}$
If adhesion between A and B are not strong, interface unlikely to form!

\[ \gamma_{AB} = \gamma_A + \gamma_B - \Delta W_{AB} \]

Nucleation theory: overall nucleation free energy

Nucleation free energy has bulk and interface parts

\[\begin{align} \Delta G_N &= \Delta G_N^{\text{bulk}} + \Delta G_N^{\text{interfacial}} \\ &= n (\mu^\beta - \mu^\alpha) + \eta n^{2/3} \gamma_{\alpha \beta} \\ \end{align}\]

Shape factor $\eta = (36 \pi)^{1/3} \Omega^{2/3}$

Different scaling between bulk & interfacial F.E.

$\Delta G^{\text{bulk}} \propto -n^{1.0}$
$\Delta G^{\text{interfacial}} \propto +n^{2/3}$
Maximum $\Delta G_c$ at $n_c$.
$\partial G_c/\partial n |_{n=n_c} = 0$

The critical nucleus size $n_c$

Classical homogeneous nucleation gives

Critical nucleus size

\[ n_c = -\frac{8}{27} \left[\frac{\eta \gamma_{\alpha \beta}}{\mu_\beta - \mu_\alpha} \right]^3 \]

Nucleation free energy barrier

\[ \Delta G_c = \frac{4}{27} \frac{(\eta \gamma)^3}{(\mu_\beta - \mu_\alpha)^2} \]

Pseudo-steady-state nucleation theory

Consider the “quasi-diffusion” in $n$-landscape!

Q.S.S. governing equations

“Quasi flux” of $n \rightarrow n+1$: $J_n$
Can also model $\partial N_n/\partial t$ from assignment 2
At equilibrium, detailed balance follows:

\[ J_n(t) = \beta_n N_n(t) - \alpha_{n+1} N_{n+1}(t) = 0 \]

The nucleation rate at quasi-pseudo-state can be computed using a known $N_n$ distribution

Q.S.S. assumptions

In a constrained equilibrium system, $N_n$ follows the Boltzmann distribution

\[ \frac{N^{\text{ceq}}_n}{N_t} \approx \exp \left( -\frac{\Delta G_n}{k_B T} \right) \]

Result:

\[ J_n(t) = - \beta_n \left[ \frac{\partial N_n}{\partial n} + \frac{N_n}{k_B T} \frac{\partial \Delta G_n}{\partial n} \right] \]

Analog: diffusion in external potential

\[ J = -L_{11} \nabla (\mu_1 + \phi) = -D_1 \left( \frac{\partial c}{\partial x} + \frac{c}{k_B T}\frac{\partial \phi}{\partial x} \right) \]

Q.S.S. nucleation rate: final results

Assuming the nucleation rate is determined by $J \approx J_{n_c}$
$Z$ is the Zeldovich factor (~0.1)

\[\begin{align} J &= Z \beta_c n_t \exp(-\frac{\Delta G_c}{k_B T}) \\ Z &= \sqrt{\frac{\Delta G_c}{3 \pi n_c^3 k_B T}} \end{align}\]

Implication of Q.S.S. nucleation rate

Zeldovich factor is around 0.1
Particles can shrink when they are not reaching $n_c$!
Rule of thumb: $\Delta G_c \leq 76 k_B T$, otherwise no detectable nucleation
At $T=298$ K, $\Delta G_c \leq 1.95$ eV

\[\begin{align} J &= Z \beta_c n_t \exp(-\frac{\Delta G_c}{k_B T}) \\ Z &= \sqrt{\frac{\Delta G_c}{3 \pi n_c^3 k_B T}} \end{align}\]

Summary

Nucleation is a type of discontinuous phase transformation that is triggered by the difference in free energy at supercooling / supersaturation
At unsteady-state conditions, nucleation free energy barrier is caused by the positive interfacial energy
Nucleation free energy barrier is characterized by $\Delta G_c$, giving critical nucleus size $n_c$
The evolution of particle number at each size $N_n$ can be described by a “diffusion-like” analog

What to learn next

Is homogeneous nucleation the whole picture? Maybe not. Consider the following examples

Sugar crystal formation

Heterogeneous nucleation

Snow formation

Diffusion-controlled growth

--- title: "MATE 664 Lecture 14" subtitle: "Introduction To Nucleation Theory" author: "Dr. Tian Tian" date: "2026-02-25" format: html: {} revealjs: output-file: slides.html pdf: output-file: L14.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L14.pdf) - Handwritten notes 👉 [Open in pdf](./public/L14_annotated.pdf) ::: ::: ## Learning outcomes {.center} After this lecture, you will be able to: - **Identify** the driving force when a system becomes supercooled or supersaturated - **Describe** the difference between discontinuous and continuous phase transformations - **Analyze** the nucleation free energy barrier $\Delta G_c$ - **Describe** the role of surface and interfacial energy in nucleation - **Describe** the pseudo-steady-state kinetic model for nucleation ## Phase-diagram in non-equilibrium region - How to induce phase transformation from a phase diagram? - Going low in temperature --> (super)cooling ![](./public/phase-diagram-shift.png) ## Transformation driving force in single-component phase diagram - Cooling from liquid to solid provides driving force $$ \Delta G^{\text{L->S}} = \frac{\Delta H^{\text{L->S}}(T_m - T )}{T_m} < 0 $$ - Why doesn't ice form spontaneously? ![](./public/free-energy-SL.svg) ## Phase transformation in binary phase diagram ($T-X_B$) **Ways to introduce driving force**: - (Super)saturation - (Super)cooling ![](./public/binary-phase-sep.png) ## Phase transformation in binary phase diagram ($G-X_B$) - Metastable regions in binary phase diagram (nucleation) - Unstable regions (spinodal decomposition) ![](./public/binary-diagram-small-change.png) ## Phase transformation in binary phase diagram ($G-X_B$) - Metastable regions in binary phase diagram - _Tangent-to-line_ method: driving force $\Delta G_B$ (molar) ![](./public/binary-diagram-tangent-df.png) ## Nucleation theory in a nutshell - Crystal growth can be divided in 4 regions - Nucleation theory deals with regions I and II (incubation & pseudo-steady-state) ![](./public/stages-growth.svg) ## Introducing interfacial energy (1) - Creating interfaces between different atoms causes energy to change! - Surface energy $\gamma_A$ (vacuum, unit J/m$^2$ or N/m) $$ \gamma_A = \frac{1}{2 a_0} w_{AA} (Z_s - Z_b) $$ :::{.columns} :::{.column width="50%"} Interatomic energy in bulk ![](./public/E-AA.svg) ::: :::{.column width="50%"} Creation of surface ![](./public/demo-surface.svg) ::: ::: ## Introducing interfacial energy (2) - Interfacial energy $\gamma_{AB}$ can be calculated using $\gamma_A$ $\gamma_B$ and $\Delta W_{AB}$ - If adhesion between A and B are not strong, interface unlikely to form! $$ \gamma_{AB} = \gamma_A + \gamma_B - \Delta W_{AB} $$ ![](./public/WAB.svg) ## Nucleation theory: overall nucleation free energy - Nucleation free energy has bulk and interface parts ```{=tex} \begin{align} \Delta G_N &= \Delta G_N^{\text{bulk}} + \Delta G_N^{\text{interfacial}} \\ &= n (\mu^\beta - \mu^\alpha) + \eta n^{2/3} \gamma_{\alpha \beta} \\ \end{align} ``` - Shape factor $\eta = (36 \pi)^{1/3} \Omega^{2/3}$ ## Different scaling between bulk & interfacial F.E. :::{.columns} :::{.column width="50%"} - $\Delta G^{\text{bulk}} \propto -n^{1.0}$ - $\Delta G^{\text{interfacial}} \propto +n^{2/3}$ - Maximum $\Delta G_c$ at $n_c$. - $\partial G_c/\partial n |_{n=n_c} = 0$ ::: :::{.column width="50%"} ![](./public/delta-g-homo.svg) ::: ::: ## The critical nucleus size $n_c$ Classical homogeneous nucleation gives - Critical nucleus size $$ n_c = -\frac{8}{27} \left[\frac{\eta \gamma_{\alpha \beta}}{\mu_\beta - \mu_\alpha} \right]^3 $$ - Nucleation free energy barrier $$ \Delta G_c = \frac{4}{27} \frac{(\eta \gamma)^3}{(\mu_\beta - \mu_\alpha)^2} $$ ## Pseudo-steady-state nucleation theory - Consider the "quasi-diffusion" in $n$-landscape! ![](./public/qss-flux.svg) ## Q.S.S. governing equations - "Quasi flux" of $n \rightarrow n+1$: $J_n$ - Can also model $\partial N_n/\partial t$ from assignment 2 - At equilibrium, detailed balance follows: $$ J_n(t) = \beta_n N_n(t) - \alpha_{n+1} N_{n+1}(t) = 0 $$ - The nucleation rate at quasi-pseudo-state can be computed using a known $N_n$ distribution ## Q.S.S. assumptions - In a constrained equilibrium system, $N_n$ follows the Boltzmann distribution $$ \frac{N^{\text{ceq}}_n}{N_t} \approx \exp \left( -\frac{\Delta G_n}{k_B T} \right) $$ - Result: $$ J_n(t) = - \beta_n \left[ \frac{\partial N_n}{\partial n} + \frac{N_n}{k_B T} \frac{\partial \Delta G_n}{\partial n} \right] $$ - Analog: diffusion in external potential $$ J = -L_{11} \nabla (\mu_1 + \phi) = -D_1 \left( \frac{\partial c}{\partial x} + \frac{c}{k_B T}\frac{\partial \phi}{\partial x} \right) $$ ## Q.S.S. nucleation rate: final results - Assuming the nucleation rate is determined by $J \approx J_{n_c}$ - $Z$ is the Zeldovich factor (~0.1) ```{=tex} \begin{align} J &= Z \beta_c n_t \exp(-\frac{\Delta G_c}{k_B T}) \\ Z &= \sqrt{\frac{\Delta G_c}{3 \pi n_c^3 k_B T}} \end{align} ``` ## Implication of Q.S.S. nucleation rate - Zeldovich factor is around 0.1 - Particles can shrink when they are not reaching $n_c$! - Rule of thumb: $\Delta G_c \leq 76 k_B T$, otherwise no detectable nucleation - At $T=298$ K, $\Delta G_c \leq 1.95$ eV ```{=tex} \begin{align} J &= Z \beta_c n_t \exp(-\frac{\Delta G_c}{k_B T}) \\ Z &= \sqrt{\frac{\Delta G_c}{3 \pi n_c^3 k_B T}} \end{align} ``` ## Summary - Nucleation is a type of discontinuous phase transformation that is triggered by the difference in free energy at supercooling / supersaturation - At unsteady-state conditions, nucleation free energy barrier is caused by the positive interfacial energy - Nucleation free energy barrier is characterized by $\Delta G_c$, giving critical nucleus size $n_c$ - The evolution of particle number at each size $N_n$ can be described by a "diffusion-like" analog ## What to learn next Is homogeneous nucleation the whole picture? Maybe not. Consider the following examples :::{.columns} :::{.column width="50%"} **Sugar crystal formation** - _Heterogeneous nucleation_ ![](./public/rock-crystal.png) ::: :::{.column width="50%"} **Snow formation** - _Diffusion-controlled growth_ ![](./public/snow-form.png) ::: :::