MATE 664 Lecture 17

Growth Phenomena: Coarsening

Author

Dr. Tian Tian

Published

March 9, 2026

Note

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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)

Another angle: Ostwald ripening

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

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:

Eq. concentration at interface: increase on smaller particles
B atoms: small particle –> matrix –> large particle
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

--- title: "MATE 664 Lecture 17" subtitle: "Growth Phenomena: Coarsening" author: "Dr. Tian Tian" date: "2026-03-09" format: html: {} revealjs: output-file: slides.html pdf: output-file: L17.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L17.pdf) - Handwritten notes 👉 [Open in pdf](./public/L17_annotated.pdf) ::: ::: ## Learning outcomes {.center} 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 ![](./public/mauro-separation.png) ## Recap: growth barrier comparison ![](./public/nucleation-spinodal-compare.png) ## Recap: growth stages in a nucleation process - What happens to stages III and IV? 👉 Coarsening process ![](./public/stages-growth.svg) ## 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](./public/kom-coarsening.png) ## Another angle: Ostwald ripening Experimental observation of small Pt particles are "absorbed" by the larger particles (Stage IV) ![_Chem Sci._ 2015, 6, 5197-5203](./public/ostwald-rippening.png) ## 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](https://www.youtube.com/watch?v=1ftn09RBv74) ## 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 $$ p_{\mathrm{A}} - p_{\mathrm{B}} + 2 H \gamma = 0 $$ ![](./public/curvature-AB.svg) 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: ```{=tex} \begin{align} |H| &= \frac{1}{2}(\frac{1}{R_{1}} + \frac{1}{R_{2}}) \\ &= \frac{1}{2}(\kappa_1 + \kappa_2) \end{align} ``` ![](./public/curvature_radius.svg) ## Capillary force: water droplet situation Nanoscale droplet will have excessive pressure! ![](./public/pressure-water-bubble.svg) ## 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 ```{=tex} \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 (**L**ifshitz-**S**lyozov-**W**agner, 1961) - Spherical $\beta$ particles embedded in $\alpha$ matrix in A-B mixture :::{.columns} :::{.column width="50%"} 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 ::: :::{.column width="50%"} ![](./public/LSW-theory.png) ::: ::: ## 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 $c_B^{\text{eq}}(\infty)$ ```{=tex} \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! ![](./public/gibbs-thompson.png) ## 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](../L14). The R has no cutoff compared with the QSS treatment in constrained growth. ![](./public/lsw-fRt.png) ## 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. ![](./public/lsw-diffusion.png) ## 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: ```{=tex} \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) ![](./public/normalized-diff-control.png) ## Diffusion-controlled regime rate law **Power-of-3** law: particle size growth rate ```{=tex} \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. ![](./public/rate-limited.png) ## 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 ```{=tex} \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