MATE 664 Lecture 17

Growth Phenomena: Coarsening

Dr. Tian Tian

2026-03-09

Learning outcomes

After this lecture, you will be able to:

  • Recall coarsening as a surface-energy-induced growth phenomenon
  • Recall the main assumptions of coarsening theory
  • Identify the competition between diffusion and reaction-rate control
  • Analyze particle size distribution functions during coarsening

Recap: continuous and discontinuous phase transformation

Recap: growth barrier comparison

Recap: growth stages in a nucleation process

  • What happens to stages III and IV? 👉 Coarsening process

Coarsening: growth mechanism involving particle size distribution

Pb-Sn alloy coarsening experiment shows that particle distribution remains almost constant (Stage III)

Metall. Trans., 1988, 19A, 2713-2721

Another angle: Ostwald ripening

Experimental observation of small Pt particles are “absorbed” by the larger particles (Stage IV)

Chem Sci. 2015, 6, 5197-5203

Analog: two-balloon experiment

  • The effect of curvature on the total free energy is analogous to the two-balloon experiment
  • The driving force is capillarity
  • YT Video

Capillarity as a driving force

The term “capillarity” refers to a broad variety of phenomena involving the interface

  • Excess free energy caused by surface energy + curvature
  • Driving force: minimizing interfacial area
  • Morphology change: interfaces will move during the optimization process
    • Coarsening: large particles “absorb” small particles; particles do not touch
    • Coalescence/sintering: interface between close-contact particles disappear; particles touch and merge

Crash course: interfacial coordinate system

For the interface between A and B that has no parallel movement, the curvature \(H\) depends on the direction of the normal vector \(\mathbf{n}\), so that

\[ p_{\mathrm{A}} - p_{\mathrm{B}} + 2 H \gamma = 0 \]

in either definition, \(p_A > p_B\) (makes sense?)

Crash course: interfacial curvature

The surface curvature (H) can be expressed using the two principle radii (R_{1}) and (R_{2}) of the surface:

\[\begin{align} |H| &= \frac{1}{2}(\frac{1}{R_{1}} + \frac{1}{R_{2}}) \\ &= \frac{1}{2}(\kappa_1 + \kappa_2) \end{align} \]

Capillary force: water droplet situation

Nanoscale droplet will have excessive pressure!

Capillarity as driving force: high level description

The driving force from capillarity \(\delta f\) is generally the energy change caused by the volume swept out by the interface, so that

\[\begin{align} \delta f = \gamma (\kappa_1 + \kappa_2) \end{align} \]

The driving force has 2 factors:

  • non-zero surface energy \(\gamma\):
  • curvatures \(\kappa_1\), \(\kappa_2\) (\(\propto 1/R\))

Coarsening (Ostwald ripening)

The evolution of an inhomogeneous structure in a solid solution or a colloidal system (stages III and IV)

  • Feature: small particulates dissolve, and redeposit onto large particulates.
  • Driving force: minimization of total interfacial energy.
  • Mass transport: driven by curvature-dependent surface potential.
  • Size and number of particles change with time.

Classical mean-field theory of coarsening

  • Most prominent theory LSW theory (Lifshitz-Slyozov-Wagner, 1961)
  • Spherical \(\beta\) particles embedded in \(\alpha\) matrix in A-B mixture

Key take-aways:

  1. Eq. concentration at interface: increase on smaller particles
  2. B atoms: small particle –> matrix –> large particle
  3. Smaller particles shrink; larger particles grow

Influence of curvature on free energy: Gibbs-Thompson effect

From our balloon analog, curvature-induced pressure for an isotropic sphere is:

\[ \Delta p = p_A - p_B = \gamma(\kappa_1 + \kappa_2) = 2 \frac{\gamma}{R} \]

By adding an B particle in to the \(\beta\) phase, the change of volume is \(\Omega_B\) and there is an increase of free energy \(2 \frac{\gamma \Omega_B}{R}\). The interfacial concentration \(c_B^{\text{eq}}(R)\) is then higher than \(c_B^{\text{eq}}(\infty)\)

\[\begin{align} c_B^{\text{eq}}(R) &= c_B^{\text{eq}}(\infty) \exp(\frac{2 \gamma\Omega}{k_B T R}) \\ &\approx c_B^{\text{eq}}(\infty) \left[ \exp(\frac{2 \gamma\Omega}{k_B T R})\right] \end{align} \]

This is known as the Gibbs-Thompson effect

Gibbs-Thompson effect in a phase diagram

  • Gibbs-Thompson effect will shift \(\mu_B\) to higher values when particles are smaller
  • Difference between interfacial concentration creates a diffusion field!

LSW theory: the particle distribution picture

Similar to the nucleation theory where \(J_n\) measures the flux of particle size distribution function \(N(n, t)\), we’re also interested in the particle size distribution over time, \(f(R, t)\), with following components

  • distribution (density) function of particle size \(R\): \(f(R, t)\)
  • radial particle density between \(R \to R+dR\): \(dN(R\to R+dR, t) = f(R, t)dR\)
  • conservation of volume: \(\sum_i R_i^2 \dfrac{dR_i}{dt} = 0\)

The radial distribution function

Compare with the quasi-steady state picture of nucleation in Lecture 14. The R has no cutoff compared with the QSS treatment in constrained growth.

Diffusion-controlled coarsening kinetics

In the diffusion-controlled regime of coarsening, the rate of growth for particles is associated with the surface flux from excess concentration to the bulk.

Rate equations in diffusion-controlled regime

  • Growth rate from flux onto a sphere

\[ \frac{dR}{dt} = - \tilde{D} \frac{(c^{\text{eq}}(R) - <c>)}{R}\omega_B \]

  • Excess surface concentration:

\[ c_B^{\text{eq}}(R) \approx c_B^{\text{eq}}(\infty) \left[ \exp(\frac{2 \gamma\Omega}{k_B T R})\right] \]

  • Conservation

\[ \sum_i R(c^{\text{eq}}(R) - <c>) = 0 \]

Diffusion-controlled regime: final results

The growth rate at each \(R\) is:

\[\begin{align} \frac{d R}{d t} &= \frac{2 \tilde{D} \gamma \Omega_B^2 c^{\text{eq}}(\infty)}{k_B T R} \left(\frac{1}{<R>} - \frac{1}{R}\right) \end{align}\]
  • \(<R>\): average radius of particles. \(f(<R>) = 0\)
  • \(R < <R>\): \(dR/dt<0\) 👉 shrink!
  • \(R_{\text{max}} = 2 <R>\)

The steady-state particle size distribution

The radius distribution has a very nice feature that even if \(<R>\) grows over time, at steady state, the normalized radius \(R/<R>\) has the same distribution:

  • most frequent size \(R \approx 1.13 <R>\)
  • no particle larger than \(1.5<R>\) (cutoff)

Diffusion-controlled regime rate law

Power-of-3 law: particle size growth rate

\[\begin{align} <R(t)>^3 - <R(0)>^3 = \frac{8 \tilde{D} \gamma \Omega^2 c^{\text{eq}}(\infty)}{9 kB T} = K_D t \end{align}\]

In experiment the measured growth follows \(\langle R \rangle \propto t^{1/3}\). See previous example of Pb-Sn alloy

Source-limited growth regime

Another regime is the source-limited coarsening. Diffusion in matrix is very fast and rate limiting step is the source / sink at interface.

Source-limited growth: change of formula

Again, we have curvature-dependent interfacial concentration difference, but the growth rate is purely controlled by the concentration difference!

\[ \frac{d R}{d t} = \frac{2 K c^{\text{eq}}(\infty) \Omega^2 \gamma}{k_B T} (\frac{<R>}{<R^2>} - \frac{1}{R}) \]

  • Particle will shrink if \(R < <R^2>/<R>\)

Source-limited growth: change of power law

For source-limited growth, we will have the radius grow in a power-of-2 fashion

\[\begin{align} \langle R^{2}(t) \rangle - \langle R^{2}(0) \rangle &= \frac{64 K c^{\text{eq}}(\infty) \Omega^2}{81 k_B T} \\ &= K_s t \end{align}\]

In experiments you will measure that \(\sqrt{\langle R^{2}(t) \rangle} \propto t^{1/2}\), a different power law than the diffusion-controlled growth!

Summary

  • Driving force for coarsening: capillarity (surface energy + curvature)
  • Key take away from coarsening: particle growth kinetics & size distribution
  • Diffusion and rate-limit regimes