MATE 664 Lecture 15

Heterogeneous Nucleation

Dr. Tian Tian

2026-03-02

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!

Nucleation At Grain Boundary

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

Nucleation At Grain Boundary

  • 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