MATE 664 Lecture 15

Heterogeneous Nucleation

Author

Dr. Tian Tian

Published

March 2, 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:

Compare heterogeneous nucleation with homogeneous nucleation
Analyze the driving-force terms in both cases
Derive the ratio between heterogeneous and homogeneous nucleation barriers and rates
Recall methods for determining the equilibrium nucleation shape

Recall: key results from homogeneous nucleation

Homogeneous (spherical) nucleus gives the following results:

Critical nucleus size

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

Nucleation free energy barrier

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

Nucleation rate

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

Recall: homogeneous nucleation implications

$\Delta G_c \propto \gamma^3$: very sensitive to the interfacial free energy
Zeldovich factor $Z$ 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

What’s the missing picture?

Can we really treat the nuclei as spheres?

Practical considerations: heterogeneous nucleation

General idea: can we have smaller $\Delta G_c$ if other competing energies exist in the system?
Analog: nucleation of crystals on a beaker wall
Need to consider the free energy change before and after wall surface is covered by the nucleus

Triple interface balance: the Young’s equation

Analogous to the classical wetting theory, the “contact angle” on a droplet can be described by the Young’s equation

\[ \gamma_{\mathrm{S}} = \gamma_{\mathrm{L}} \cos \theta + \gamma_{\mathrm{SL}} \]

Heterogeneous nucleation on a wall: volume vs surface

The geometry of the droplet gives:

Volume of nucleus: $V_{\mathrm{n}} = \frac{\pi R^{3}}{3} (2 - 3 \cos \theta + \cos^{3} \theta)$
Interfacial area of nucleus with solution: $A_{\mathrm{s}} = 2 \pi R^{2} (1 - \cos \theta)$
Interfacial area of nucleus with wall: $A_{\mathrm{c}} = \pi R^{2} \sin^{2} \theta$

Final solution to heterogeneous nucleation barrier:

\[ \Delta G_c^{\text{het}} = V_{\mathrm{n}} \Delta G_{\mathrm{V}} + \gamma_{\mathrm{sn}} A_{\mathrm{s}} + (\gamma_{\mathrm{nw}} - \gamma_{\mathrm{sw}}) A_{\mathrm{c}} \]

Heterogeneous nucleation on a wall: results

We can compare the hetero- and homogeneous barriers:

\[ \frac{\Delta G_{\mathrm{het}}}{\Delta G_{\mathrm{homo}}} = \frac{2 - 3 \cos \theta + \cos^{3} \theta}{4} = f \]

Plot of $f$ as function of contact angle $\theta$

Heterogeneous in binary alloys: geometry

At triple-interface, we have the balance

\[ \gamma_{\alpha\alpha} = 2 \gamma_{\alpha \beta} cos \psi \]

Grain boundary $\gamma_{\alpha\alpha} \neq 0$
What does $\gamma_{\alpha\alpha} = 0$ mean? Homogeneous nucleation!

Heterogeneous in binary alloys: $\Delta G_c^{\text{het}}$ results

Volume $V = \frac{2 \pi R^3}{3} (2 - 3 \cos \psi + \cos^3 \psi)$
Area of cap $A_c = 4 \pi R^2 (1 - \cos \psi)$
Area below the cap: $A_b = \pi r^2 = \pi R^2 (1 - \cos^2 \psi)$

Overall heterogeneous nucleation barrier:

\[ \Delta G_c^{B} = (\frac{2 \pi R^3}{3} \Delta G_m + 2\pi R^2 \gamma_{\alpha\beta})(2 - 3 \cos \psi + \cos^3 \psi) \]

Ratio between hetero- and homogeneous nucleation energy barriers

Compare the two barriers, they are quite similar

\[\begin{align} \Delta G_c^{H} &= (\frac{4 \pi R^3}{3} \Delta G_m + 4\pi R^2 \gamma_{\alpha\beta}) \\ \Delta G_c^{B} &= (\frac{2 \pi R^3}{3} \Delta G_m + 2\pi R^2 \gamma_{\alpha\beta})(2 - 3 \cos \psi + \cos^3 \psi) \end{align}\]

Ratio:

\[\begin{align} \frac{\Delta G_c^{B}}{\Delta G_c^{H}} = \frac{1}{2}(2 - 3 \cos \psi + \cos^3 \psi) \end{align}\]

This is similar to our case of heterogeneous nucleation on a wall, but with different coefficient (why?)!

Heterogeneous nucleation barrier

Comparison between nucleations along defects

Heterogeneous nucleation in alloys: other nucleus dimensionalities

From previous figure we see that $\Delta G_c$ on defect becomes smaller for lower-dimensionality defects
But do low dimensional defects always win? Not essentially.
Number of sites available also decreases.

Assume the average grain size is $L$, with grain boundary thickness $\delta$, available sites follows

\[ n^{\text{defect}} \propto n_t (\frac{\delta}{L})^{3 - d} \]

Competition between hetero- and homogeneous nucleation rates

When considering the rates, two factors matter in the overall $J$ equation:

Free energy barrier $\exp( - \Delta G_c/k_B T)$ (hetero > homo)
Total available sites $n_t$ (hetero < homo)

\[ \ln\!(\frac{J^B}{J_H}) = \ln\!(\frac{\delta}{L}) + \frac{\Delta G_c^H - \Delta G_c^B}{k_B T} \]

Overall nucleation regimes

$R_B = k_B T\ln(\frac{L}{\delta})$
Homogeneous nucleation favours when $R_B > \Delta G_c^H - \Delta G_c^B$

What else may be missing?

The nucleation geometry may be very different from spherical or curved!
Different facets have distinct surface free energy
Overall goal: when the volume $V$ is fixed, can we know the equilibrium shape of a crystal, so that surface energy is minimized?
Wulff construction: optimizing the geometry of equilibrium shape
Higher energy facet would have longer distance to the center!

Wulff construction for equilibrium crystal shape

Theoretical Wulff construction for elements

Crystalium demo

Tran et al. Scientific Data, 2016, 3:160080

Nucleation: example demo

Nucleation demo: how do we get the values?

Nucleation demo: supersaturation limit

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

--- title: "MATE 664 Lecture 15" subtitle: "Heterogeneous Nucleation" author: "Dr. Tian Tian" date: "2026-03-02" format: html: {} revealjs: output-file: slides.html pdf: output-file: L15.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L15.pdf) - Handwritten notes 👉 [Open in pdf](./public/L15_annotated.pdf) ::: ::: ## Learning outcomes {.center} After this lecture, you will be able to: - **Compare** heterogeneous nucleation with homogeneous nucleation - **Analyze** the driving-force terms in both cases - **Derive** the ratio between heterogeneous and homogeneous nucleation barriers and rates - **Recall** methods for determining the equilibrium nucleation shape ## Recall: key results from homogeneous nucleation Homogeneous (spherical) nucleus gives the following results: - Critical nucleus size $$ n_c = -\frac{8}{27} \left[\frac{\eta \gamma_{\alpha \beta}}{\mu_\beta - \mu_\alpha} \right]^3 $$ - Nucleation free energy barrier $$ \Delta G_c = \frac{4}{27} \frac{(\eta \gamma)^3}{(\mu_\beta - \mu_\alpha)^2} $$ - Nucleation rate ```{=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} ``` ## Recall: homogeneous nucleation implications - $\Delta G_c \propto \gamma^3$: very sensitive to the interfacial free energy - Zeldovich factor $Z$ 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 ## What's the missing picture? _Can we really treat the nuclei as spheres?_ ![](./public/sphere-cow.png) ## Practical considerations: heterogeneous nucleation - General idea: can we have smaller $\Delta G_c$ if other competing energies exist in the system? - Analog: nucleation of crystals on a beaker wall - Need to consider the free energy change before and after wall surface is covered by the nucleus ![](./public/hetero-nucleation.svg) ## Triple interface balance: the Young's equation Analogous to the classical wetting theory, the "contact angle" on a droplet can be described by the Young's equation ![](./public/young-dupre.svg) $$ \gamma_{\mathrm{S}} = \gamma_{\mathrm{L}} \cos \theta + \gamma_{\mathrm{SL}} $$ ## Heterogeneous nucleation on a wall: volume vs surface The geometry of the droplet gives: - Volume of nucleus: $V_{\mathrm{n}} = \frac{\pi R^{3}}{3} (2 - 3 \cos \theta + \cos^{3} \theta)$ - Interfacial area of nucleus with solution: $A_{\mathrm{s}} = 2 \pi R^{2} (1 - \cos \theta)$ - Interfacial area of nucleus with wall: $A_{\mathrm{c}} = \pi R^{2} \sin^{2} \theta$ Final solution to heterogeneous nucleation barrier: $$ \Delta G_c^{\text{het}} = V_{\mathrm{n}} \Delta G_{\mathrm{V}} + \gamma_{\mathrm{sn}} A_{\mathrm{s}} + (\gamma_{\mathrm{nw}} - \gamma_{\mathrm{sw}}) A_{\mathrm{c}} $$ ## Heterogeneous nucleation on a wall: results We can compare the hetero- and homogeneous barriers: $$ \frac{\Delta G_{\mathrm{het}}}{\Delta G_{\mathrm{homo}}} = \frac{2 - 3 \cos \theta + \cos^{3} \theta}{4} = f $$ ![Plot of $f$ as function of contact angle $\theta$](./public/delta-g-hetero.svg) ## Heterogeneous in binary alloys: geometry - At triple-interface, we have the balance $$ \gamma_{\alpha\alpha} = 2 \gamma_{\alpha \beta} cos \psi $$ - Grain boundary $\gamma_{\alpha\alpha} \neq 0$ - What does $\gamma_{\alpha\alpha} = 0$ mean? Homogeneous nucleation! ![Nucleation At Grain Boundary](./public/nucleation-gb.png) ## Heterogeneous in binary alloys: $\Delta G_c^{\text{het}}$ results ![Nucleation At Grain Boundary](./public/nucleation-gb.png) - Volume $V = \frac{2 \pi R^3}{3} (2 - 3 \cos \psi + \cos^3 \psi)$ - Area of cap $A_c = 4 \pi R^2 (1 - \cos \psi)$ - Area below the cap: $A_b = \pi r^2 = \pi R^2 (1 - \cos^2 \psi)$ Overall heterogeneous nucleation barrier: $$ \Delta G_c^{B} = (\frac{2 \pi R^3}{3} \Delta G_m + 2\pi R^2 \gamma_{\alpha\beta})(2 - 3 \cos \psi + \cos^3 \psi) $$ ## Ratio between hetero- and homogeneous nucleation energy barriers Compare the two barriers, they are quite similar ```{=tex} \begin{align} \Delta G_c^{H} &= (\frac{4 \pi R^3}{3} \Delta G_m + 4\pi R^2 \gamma_{\alpha\beta}) \\ \Delta G_c^{B} &= (\frac{2 \pi R^3}{3} \Delta G_m + 2\pi R^2 \gamma_{\alpha\beta})(2 - 3 \cos \psi + \cos^3 \psi) \end{align} ``` Ratio: ```{=tex} \begin{align} \frac{\Delta G_c^{B}}{\Delta G_c^{H}} = \frac{1}{2}(2 - 3 \cos \psi + \cos^3 \psi) \end{align} ``` This is similar to our case of heterogeneous nucleation on a wall, but with different coefficient (why?)! ## Heterogeneous nucleation barrier ![Comparison between nucleations along defects](./public/gc-diff-boundaries.png) ## Heterogeneous nucleation in alloys: other nucleus dimensionalities - From previous figure we see that $\Delta G_c$ on defect becomes smaller for lower-dimensionality defects - But do low dimensional defects always win? Not essentially. - Number of sites available also decreases. Assume the average grain size is $L$, with grain boundary thickness $\delta$, available sites follows $$ n^{\text{defect}} \propto n_t (\frac{\delta}{L})^{3 - d} $$ ## Competition between hetero- and homogeneous nucleation rates When considering the rates, **two** factors matter in the overall $J$ equation: - Free energy barrier $\exp( - \Delta G_c/k_B T)$ (hetero > homo) - Total available sites $n_t$ (hetero < homo) $$ \ln\!(\frac{J^B}{J_H}) = \ln\!(\frac{\delta}{L}) + \frac{\Delta G_c^H - \Delta G_c^B}{k_B T} $$ ## Overall nucleation regimes - $R_B = k_B T\ln(\frac{L}{\delta})$ - Homogeneous nucleation favours when $R_B > \Delta G_c^H - \Delta G_c^B$ ![](./public/nucleation-regimes.png) ## What else may be missing? - The nucleation geometry may be very different from spherical or curved! - Different facets have distinct surface free energy - Overall goal: when the volume $V$ is fixed, can we know the equilibrium shape of a crystal, so that surface energy is minimized? - Wulff construction: optimizing the geometry of equilibrium shape - Higher energy facet would have longer distance to the center! ## Wulff construction for equilibrium crystal shape ![](./public/wulff.png) ## Theoretical Wulff construction for elements [Crystalium demo](http://crystalium.materialsvirtuallab.org) _Tran et al. Scientific Data, 2016, 3:160080_ ## Nucleation: example demo ![](./public/kom-alloy.png) ## Nucleation demo: how do we get the values? ![](./public/kom-solution-1.png) ## Nucleation demo: supersaturation limit ![](./public/kom-supersat.png) ## 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