L4: Precipitation nucleation

Learning Objectives:

1. Derive the equation controlling the free energy changes upon nucleation

   Phase change negative; strain and surface energies positive

2. Discuss the critical size of a nucleus and the energy barrier to be overcome.

   Found by taking derivative of delta G and setting to zero. Depends on undercooling.

3. Develop a simple prediction for the nucleation rate and explain its variation with temperature

   With increasing undercooling there is a trade-off between (i) high driving force, and (ii) slower diffusion.

4. Show that heterogeneous nucleation, e.g. on a grain boundary, is much easier than homogeneous nucleation

   determined by relative energy of boundary (or defect) and precipitate. High undercooling = more homogenous nucleation

(1) Derive the equation controlling the free energy changes upon nucleation:

The free energy change (ΔG) upon nucleation is influenced by three main factors:

• Volume free energy change (ΔGv): This is the driving force for nucleation, which is negative as it represents the energy reduction when a more stable phase forms.

• Surface energy (γ⋅A): This is the energy required to create the interface between the new phase and the matrix, which is positive.

• Misfit strain energy (ΔGe​): This is the energy associated with the volume mismatch between the new phase and the matrix, also positive.

  

The overall free energy change for a spherical nucleus is given by:

where r is the radius of the nucleus.

(2) Discuss the critical size of a nucleus and the energy barrier to be overcome:

The critical nucleus size (r∗) is the size at which the nucleus can grow rather than shrink. It is determined by setting the derivative of ΔG with respect to r to zero:

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The energy barrier (ΔG∗) for nucleation is the free energy change at the critical radius:

This barrier must be overcome for nucleation to occur.

(3) Develop a simple prediction for the nucleation rate and explain its variation with temperature:

The nucleation rate (Nhom​) depends on the concentration of critical nuclei and the rate at which atoms can diffuse to form these nuclei. The concentration of critical nuclei is given by:

where C0​ is the number of atoms per unit volume, kk is Boltzmann's constant, and T is temperature. The nucleation rate is then:

where f0 is a pre-exponential factor and ΔGm​ is the activation energy for diffusion.

The nucleation rate initially increases with undercooling (lower temperature) due to a higher driving force (ΔGv), but at very low temperatures, diffusion slows down, reducing the nucleation rate. This results in a peak nucleation rate at an intermediate temperature.

(4) Show that heterogeneous nucleation, e.g. on a grain boundary, is much easier than homogeneous nucleation:

Heterogeneous nucleation occurs at defects like grain boundaries, dislocations, or impurities, which reduce the energy barrier for nucleation. The energy change for heterogeneous nucleation (ΔGhet) is modified by a shape factor S(θ):

θ is the contact angle between the nucleus and the grain boundary. Since S(θ)<1S(θ)<1, the energy barrier for heterogeneous nucleation is lower than for homogeneous nucleation, making it easier.

• Green curves (C∗): Critical nucleation concentration for heterogeneous (solid) and homogeneous (dashed) nucleation.

• Red & Black curves (N): Nucleation rates; heterogeneous (Nhet) peaks earlier and higher than homogeneous (Nhom), meaning it's easier to nucleate on surfaces.

• Blue dashed curve: Diffusion rate increases with temperature.

• Key takeaway: At low temperatures, nucleation is favoured; at high temperatures, diffusion dominates. The balance between these controls phase transformations

In summary:

• Heterogeneous nucleation is favoured because defects reduce the energy barrier.

• The nucleation rate peaks at an intermediate temperature due to a trade-off between driving force and diffusion rate.

• The critical nucleus size and energy barrier are determined by balancing the driving force, surface energy, and misfit strain energy.