MATE 664 Lecture 16

Continuous Phase Transformation: Spinodal Decomposition

Author

Dr. Tian Tian

Published

March 4, 2026

Note

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Learning outcomes

After this lecture, you will be able to:

Recall the difference between continuous and discontinuous phase transformations
Describe spinodal decomposition as a continuous phase transformation process
Identify the chemical-potential driving force in spinodal decomposition
Analyze the Cahn-Hilliard equation and its gradient driving force

Recall: general picture of phase transformation

Stability regions in $T-X_{B}$ and $G-X_{B}$ plots - spinodal: “spine-like” shape in the free energy diagram

Stability of phase transformation: second derivative of free energy

Taylor expansion of the Gibbs free energy

\[\begin{align} G\!\left(x_B^{0}+\delta x_B\right) &= G\!\left(x_B^{0}\right) + \left.\frac{\partial G}{\partial x_B}\right|_{x_B^{0}}\,\delta x_B + \frac{1}{2}\left.\frac{\partial^{2}G}{\partial x_B^{2}}\right|_{x_B^{0}}(\delta x_B)^2 \nonumber\\ &= G\!\left(x_B^{0}\right) + \frac{1}{2}\left.\frac{\partial^{2}G}{\partial x_B^{2}}\right|_{x_B^{0}}(\delta x_B)^2 \end{align}\]

Key questions to be answered for continuous transformation

Where: which region in a phase diagram does continuous phase transformation occur?
How: how does the continuous transformation differ from discontinuous one (e.g. nucleation)?
Why: why do we see the continuous behaviour?

Comparison: nucleation vs spinodal decomposition

Continuous transformation: irreversible thermodynamics view (1)

From Lecture 08 we know the fluxes between an A-B binary mixture follows:

C-frame: $J_A^C$ & $J_B^C$
V-frame: $J_A^V$ & $J_B^V$
Conservation equations

Continuous transformation: irreversible thermodynamics view (2)

Expanding the flux conservation and Gibbs-Duhem equation we get the $J_B^V$ as

\[\begin{align} J_B^V &= -\Omega^2 (c_A^2 L_B + c_B^2 L_A)(\nabla \mu_B - \nabla \mu_A) \\ &= -\Omega^2 (c_A^2 L_B + c_B^2 L_A)\nabla(\mu_B - \mu_A) \\ &= - M \nabla(\mu_B - \mu_A) \end{align}\]

So far this is just a formal diffusion equation using chemical potential driving force

Meaning of $J_B^V$ equation

Mobility $M>0$ satisfies the $\dot{\sigma}\geq 0$ postulate
What is $\mu_B - \mu_A$? 👉 Slope $\partial G/\partial X_B$ on the free energy diagram!
What is $\nabla(\mu_B - \mu_A)$? Spatial variation of the $\partial G/\partial X_B$ driving force

\[\begin{align} J_B^V &= - M \nabla(\mu_B - \mu_A) \\ &= - \frac{M\Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2} \nabla c_B \\ &= -\tilde{D} \nabla c_B \end{align}\]

Negative interdiffusivity?

From previous analysis we see that the apparent interdiffusivity $\tilde{D}$ follows:

\[ \tilde{D} = \frac{M\Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2} \]

$\tilde{D} < 0$ 👉 uphill diffusion
In continuous transformation, we see the local concentration difference amplifies!
In contrast, normal diffusion $\tilde{D} > 0$ tries to smoothen the concentration curvature.

Diffusion (homogenization) vs phase separation

Very rough estimation of the composition change using Fick’s second law

\[ \frac{\partial c_B}{\partial t} = \tilde{D}\nabla^2 c_B \]

Summary for “where” and “why” questions

Continuous phase transformation occurs where the second derivative of free energy is negative
The continuous phase separation is dominated by barrierless diffusion
The process is triggered by infinitesimal spatial fluctuation
Amplification of concentration fluctuation is enhanced by negative interdiffusivity $\tilde{D} = \frac{M\Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2} < 0$

The “how” question: evolution of patterns in spinodal decomposition

Let’s still use the 1D Fick’s second law, assuming $\tilde{D}$ is constant everywhere

\[ \frac{\partial c_B}{\partial t} = \tilde{D}\frac{\partial^2 c_B}{\partial x^2} \]

From Lecture 07 we know that the diffusion equation can be decomposed into spatial and temporal parts. Use a wave form $c_B(x, t) = \overline{c_B} + \exp(i \beta x) A(t)$, we can solve the $c_B(x, t)$ profile.

Amplification in continuous transformation

The general solution to the waveform $c_B$ is:

\[\begin{align} c_B - \overline{c_B} &= A(\beta, 0) \exp(-\beta^2 \tilde{D}) exp(i \beta x) \\ &= A(\beta, 0) \exp(-R(\beta)) exp(i \beta x) \end{align}\]

Wavelength $\lambda = 2\pi / \beta$
$R(\beta) = -\beta^2 \frac{M \Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2}$: amplification factor
$\beta \rightarrow \infty$ should give maximal amplification ?! 👉 We always end up with noise ?!
Not the case in real scenario 👉 stablization by interfaces!

Example: spinodal decomposition in computer simulations

See Seol, et al. Acta Materialia 51, 5173–5185 (2003)
Final pattern always has a principal wavelength, why?

Interfacial energy as stabilization term

The term $R(\beta) = -\beta^2 \frac{M \Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2}$ uses homogeneous free energy of a mixture
In reality creating a new phase always increase the free energy by surface energy (similar to nucleation in Lecture 14)
In spinodal decomposition, surface energy acts as “stablization” for pattern formation!

Energy at gradient interfaces

Unlike in nucleation where sharp interface can be defined (thus $\gamma$), spinodal decomposition usually accompanies diffuse interface
$c_{B}$ has a finite gradient, not sharp step function

Additional term for interface gradient to Helmholtz free energy density

\[ f_{\gamma} = \kappa \left( \frac{\partial c_B}{\partial x} \right)^2 \]

$\kappa >0$ is the gradient energy coefficient that penalizes the creation of sharp interface
At interface $\left( \frac{\partial c_B}{\partial x} \right)^2$ becomes large!

Cahn-Hilliard equation for spinodal decomposition

It is just Fick’s second law with a more precise $\tilde{D}$ counting the interfacial term
Total Helmholtz free energy comes from both homogeneous free energy density $f^{\text{homo}}$ and $\kappa (\partial c_B/\partial x)^2$

Define a general “diffusion potential” for inhomogeneous system:

\[\begin{align} \Phi(x) &= \frac{\partial F}{V \partial cB} \\ &= \frac{\partial f^{\text{homo}}}{\partial c_B} - 2 \kappa \nabla^2 c_B \end{align}\]

Cahn-Hilliard equation (2)

Instead of chemical potential $\mu_B - \mu_A$, we use $\Phi$ as the overall potential

Fick’s 1st law

\[\begin{align} J_B &= -L \nabla \Phi \\ &= -\frac{\tilde{D}}{(\partial^2 f^{\text{homo}} / \partial X_B^2)} \nabla \{\frac{\partial f^{\text{homo}}}{\partial c_B} - 2 \kappa \nabla^2 c_B \} \end{align}\]

Fick’s 2nd law

\[\begin{align} \frac{\partial c_B}{\partial t} &= - \nabla\cdot J_B\\ &= \nabla \cdot \{\frac{\tilde{D}}{(\partial^2 f^{\text{homo}} / \partial X_B^2)} \nabla \{\frac{\partial f^{\text{homo}}}{\partial c_B} - 2 \kappa \nabla^2 c_B \}\} \end{align}\]

Cahn-Hilliard equation (3)

Final format of Cahn-Hilliard equation

\[\begin{align} \frac{\partial c_B}{\partial t} = M_0 \left[ \frac{\partial^2 f^{\text{homo}}}{\partial c_B^2} \nabla^2 c_B - 2 \kappa \nabla^4 c_B \right] \end{align}\]

Perturbation analysis for Cahn-Hilliard equation

Toy model for the Helmholtz free energy

\[ f^{\text{homo}} = 16 \frac{f^{m}}{(c^\beta - c^\alpha)^4} \left[ (c_B - c^\alpha)(c_B - c^\beta) \right]^2 \]

How will the amplitude change?

\[ \frac{d A(t)}{dt} = \frac{M_0 \beta^2}{(c^\beta - c^\alpha)} \left[ 16 f^{m} - 2 \kappa \beta^2 (c^\beta - c^\alpha) \right] A(t) \]

Amplification factor $R(\beta)$ in Cahn-Hilliard equation

\[ R(\beta) = -M_0 \left( \frac{\partial^2 f}{\partial c_B^2} \beta^2 + 2 \kappa \beta^4 \right) \]

The $\beta^4$ term now stablizes $R(\beta)$!

Cahn-Hilliard equation: critical wavelength

Critical wavelength $\lambda_c = \frac{\pi}{2} (c^\beta - c^\alpha) \sqrt{\frac{\kappa}{f^m}}$
$\lambda > \lambda_c$ –> Growth
Fastest growing wavelength $\lambda_{\text{max}} = \sqrt{2} \lambda_c$

Applications of spinodal decomposition: pattern formation

Summary

Continuous phase transformation is caused by initial composition inside the spinodal line (unstable region)
Negative $\tilde{D}$ causes spatial fluctuation to amplify
Cahn-Hilliard equation describes the driving force
Continuous phase separation occurs in many material systems and give rise to rich phase separation patterns

--- title: "MATE 664 Lecture 16" subtitle: "Continuous Phase Transformation: Spinodal Decomposition" author: "Dr. Tian Tian" date: "2026-03-04" format: html: {} revealjs: output-file: slides.html pdf: output-file: L16.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L16.pdf) - Handwritten notes 👉 [Open in pdf](./public/L16_annotated.pdf) ::: ::: ## Learning outcomes {.center} After this lecture, you will be able to: - **Recall** the difference between continuous and discontinuous phase transformations - **Describe** spinodal decomposition as a continuous phase transformation process - **Identify** the chemical-potential driving force in spinodal decomposition - **Analyze** the Cahn-Hilliard equation and its gradient driving force ## Recall: general picture of phase transformation Stability regions in $T-X_{B}$ and $G-X_{B}$ plots - spinodal: "spine-like" shape in the free energy diagram ![](./public/stability-region.png) ## Stability of phase transformation: second derivative of free energy Taylor expansion of the Gibbs free energy ```{=tex} \begin{align} G\!\left(x_B^{0}+\delta x_B\right) &= G\!\left(x_B^{0}\right) + \left.\frac{\partial G}{\partial x_B}\right|_{x_B^{0}}\,\delta x_B + \frac{1}{2}\left.\frac{\partial^{2}G}{\partial x_B^{2}}\right|_{x_B^{0}}(\delta x_B)^2 \nonumber\\ &= G\!\left(x_B^{0}\right) + \frac{1}{2}\left.\frac{\partial^{2}G}{\partial x_B^{2}}\right|_{x_B^{0}}(\delta x_B)^2 \end{align} ``` ## Key questions to be answered for continuous transformation - **Where**: which region in a phase diagram does continuous phase transformation occur? - **How**: how does the continuous transformation differ from discontinuous one (e.g. nucleation)? - **Why**: why do we see the continuous behaviour? ## Comparison: nucleation vs spinodal decomposition ![](./public/mauro-separation.png) ## Continuous transformation: irreversible thermodynamics view (1) From [Lecture 08](../L08) we know the fluxes between an A-B binary mixture follows: - C-frame: $J_A^C$ & $J_B^C$ - V-frame: $J_A^V$ & $J_B^V$ - Conservation equations ## Continuous transformation: irreversible thermodynamics view (2) Expanding the flux conservation and Gibbs-Duhem equation we get the $J_B^V$ as ```{=tex} \begin{align} J_B^V &= -\Omega^2 (c_A^2 L_B + c_B^2 L_A)(\nabla \mu_B - \nabla \mu_A) \\ &= -\Omega^2 (c_A^2 L_B + c_B^2 L_A)\nabla(\mu_B - \mu_A) \\ &= - M \nabla(\mu_B - \mu_A) \end{align} ``` So far this is just a formal diffusion equation using chemical potential driving force ## Meaning of $J_B^V$ equation - Mobility $M>0$ satisfies the $\dot{\sigma}\geq 0$ postulate - What is $\mu_B - \mu_A$? 👉 Slope $\partial G/\partial X_B$ on the free energy diagram! - What is $\nabla(\mu_B - \mu_A)$? Spatial variation of the $\partial G/\partial X_B$ driving force ```{=tex} \begin{align} J_B^V &= - M \nabla(\mu_B - \mu_A) \\ &= - \frac{M\Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2} \nabla c_B \\ &= -\tilde{D} \nabla c_B \end{align} ``` ## Negative interdiffusivity? From previous analysis we see that the apparent interdiffusivity $\tilde{D}$ follows: $$ \tilde{D} = \frac{M\Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2} $$ - $\tilde{D} < 0$ 👉 uphill diffusion - In continuous transformation, we see the local concentration difference amplifies! - In contrast, normal diffusion $\tilde{D} > 0$ tries to smoothen the concentration curvature. ## Diffusion (homogenization) vs phase separation Very rough estimation of the composition change using Fick's second law $$ \frac{\partial c_B}{\partial t} = \tilde{D}\nabla^2 c_B $$ ## Summary for "where" and "why" questions - Continuous phase transformation occurs **where** the second derivative of free energy is **negative** - The continuous phase separation is dominated by **barrierless** diffusion - The process is triggered by infinitesimal spatial fluctuation - Amplification of concentration fluctuation is enhanced by **negative interdiffusivity** $\tilde{D} = \frac{M\Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2} < 0$ ## The "how" question: evolution of patterns in spinodal decomposition Let's still use the 1D Fick's second law, assuming $\tilde{D}$ is constant everywhere $$ \frac{\partial c_B}{\partial t} = \tilde{D}\frac{\partial^2 c_B}{\partial x^2} $$ From [Lecture 07](../L07) we know that the diffusion equation can be decomposed into spatial and temporal parts. Use a wave form $c_B(x, t) = \overline{c_B} + \exp(i \beta x) A(t)$, we can solve the $c_B(x, t)$ profile. ## Amplification in continuous transformation The general solution to the waveform $c_B$ is: ```{=tex} \begin{align} c_B - \overline{c_B} &= A(\beta, 0) \exp(-\beta^2 \tilde{D}) exp(i \beta x) \\ &= A(\beta, 0) \exp(-R(\beta)) exp(i \beta x) \end{align} ``` - Wavelength $\lambda = 2\pi / \beta$ - $R(\beta) = -\beta^2 \frac{M \Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2}$: amplification factor - $\beta \rightarrow \infty$ should give maximal amplification ?! 👉 We always end up with noise ?! - Not the case in real scenario 👉 **stablization** by interfaces! ## Example: spinodal decomposition in computer simulations - See Seol, et al. _Acta Materialia_ 51, 5173–5185 (2003) - Final pattern always has a principal wavelength, why? ![](./public/seol-simulation.png) ## Interfacial energy as stabilization term - The term $R(\beta) = -\beta^2 \frac{M \Omega}{N_0} \frac{\partial^2 G}{\partial X_B^2}$ uses _homogeneous_ free energy of a mixture - In reality creating a new phase always increase the free energy by surface energy (similar to nucleation in [Lecture 14](../L14)) - In spinodal decomposition, surface energy acts as "stablization" for pattern formation! ## Energy at gradient interfaces - Unlike in nucleation where sharp interface can be defined (thus $\gamma$), spinodal decomposition usually accompanies **diffuse interface** - $c_{B}$ has a finite gradient, not sharp step function Additional term for interface gradient to Helmholtz free energy density $$ f_{\gamma} = \kappa \left( \frac{\partial c_B}{\partial x} \right)^2 $$ - $\kappa >0$ is the gradient energy coefficient that penalizes the creation of sharp interface - At interface $\left( \frac{\partial c_B}{\partial x} \right)^2$ becomes large! ## Cahn-Hilliard equation for spinodal decomposition - It is just Fick's second law with a more precise $\tilde{D}$ counting the interfacial term - Total Helmholtz free energy comes from both homogeneous free energy density $f^{\text{homo}}$ and $\kappa (\partial c_B/\partial x)^2$ Define a general "diffusion potential" for inhomogeneous system: ```{=tex} \begin{align} \Phi(x) &= \frac{\partial F}{V \partial cB} \\ &= \frac{\partial f^{\text{homo}}}{\partial c_B} - 2 \kappa \nabla^2 c_B \end{align} ``` ## Cahn-Hilliard equation (2) Instead of chemical potential $\mu_B - \mu_A$, we use $\Phi$ as the overall potential - Fick's 1st law ```{=tex} \begin{align} J_B &= -L \nabla \Phi \\ &= -\frac{\tilde{D}}{(\partial^2 f^{\text{homo}} / \partial X_B^2)} \nabla \{\frac{\partial f^{\text{homo}}}{\partial c_B} - 2 \kappa \nabla^2 c_B \} \end{align} ``` - Fick's 2nd law ```{=tex} \begin{align} \frac{\partial c_B}{\partial t} &= - \nabla\cdot J_B\\ &= \nabla \cdot \{\frac{\tilde{D}}{(\partial^2 f^{\text{homo}} / \partial X_B^2)} \nabla \{\frac{\partial f^{\text{homo}}}{\partial c_B} - 2 \kappa \nabla^2 c_B \}\} \end{align} ``` ## Cahn-Hilliard equation (3) Final format of Cahn-Hilliard equation ```{=tex} \begin{align} \frac{\partial c_B}{\partial t} = M_0 \left[ \frac{\partial^2 f^{\text{homo}}}{\partial c_B^2} \nabla^2 c_B - 2 \kappa \nabla^4 c_B \right] \end{align} ``` ## Perturbation analysis for Cahn-Hilliard equation Toy model for the Helmholtz free energy $$ f^{\text{homo}} = 16 \frac{f^{m}}{(c^\beta - c^\alpha)^4} \left[ (c_B - c^\alpha)(c_B - c^\beta) \right]^2 $$ How will the amplitude change? $$ \frac{d A(t)}{dt} = \frac{M_0 \beta^2}{(c^\beta - c^\alpha)} \left[ 16 f^{m} - 2 \kappa \beta^2 (c^\beta - c^\alpha) \right] A(t) $$ ## Amplification factor $R(\beta)$ in Cahn-Hilliard equation $$ R(\beta) = -M_0 \left( \frac{\partial^2 f}{\partial c_B^2} \beta^2 + 2 \kappa \beta^4 \right) $$ The $\beta^4$ term now stablizes $R(\beta)$! ![](./public/cahn-hilliard-rb.png) ## Cahn-Hilliard equation: critical wavelength - Critical wavelength $\lambda_c = \frac{\pi}{2} (c^\beta - c^\alpha) \sqrt{\frac{\kappa}{f^m}}$ - $\lambda > \lambda_c$ --> Growth - Fastest growing wavelength $\lambda_{\text{max}} = \sqrt{2} \lambda_c$ ![](./public/cahn-hilliard-rb.png) ## Applications of spinodal decomposition: pattern formation ![](./public/mauro-example.png) ## Summary - Continuous phase transformation is caused by initial composition inside the spinodal line (unstable region) - Negative $\tilde{D}$ causes spatial fluctuation to amplify - Cahn-Hilliard equation describes the driving force - Continuous phase separation occurs in many material systems and give rise to rich phase separation patterns