MATE 664 Lecture 13

Phase Diagram and Phase Transformation

Author

Dr. Tian Tian

Published

February 24, 2026

Note

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Kinetics of Materials Part II

What topics have we learned so far?

How to describe kinetic process: irreversible thermodynamics
What causes nonequilibrium mass transfer: chemical potential as driving force
Diffusion in materials: Fick’s equations
Determine diffusivity $D$: Macroscopic and Microscopic models

What do we study in Part II?

Insights from assignments 1 and 2

Topic 1: phase transformation

Key question: how do materials evolve when chemical potentials are not at equilibrium?

In-depth study of phase diagrams
Continuous phase transformation: spinodal decomposition
Phase transformation with barrier: nucleation

MIT course *Fundamentals of Materials Sciece*

Topic 2: interfacial phenomena

Key question: how do material interfaces change in non-equilibrium process?

Solidification: heat / mass-transfer at interfaces:
Sintering: surface-energy-mediated transformation:

Topic 3: analysis of kinetic process in materials

Key question: how do the competition between different effects change the material behaviour?

Aggregation phenomena: diffusion & reaction of soft materials
Dendrite formation: morphological control in battery electrodes
Carrier transport: designing optical materials

Topic 4: simulating kinetic problems

Key question: what methods can we use to simulate kinetic systems, and how good are they?

Macroscopic transport: phase-field method
Kinetic Monte-Carlo (KMC)
Molecular dynamics (MD)
Thermodynamic parameters from first principles and machine learning

Introduction to phase transformation

Learning outcomes

After this lecture, you will be able to:

Recall key components of a phase diagram
Distinguish the meaning of axes in a phase diagram
Identify key thermodynamic relations from a phase diagram
Analyze phase transformation regions in a phase diagram

This lecture is adapted from Porter et al. Phase Transformations in Metals and Alloys

What is phase transformation?

Before introducing any fancy terminologies, let’s define a few concepts:

Phase:
- a region with uniform structure and properties
Transformation
- a kinetic process to reduce total free energy by turning phases α –> β
Interface
- a boundary separating two phases

What can be considered as a phase?

Mathematically (Landau & Lifshitz 1963), different phases are distinguished by order parameter $\xi$
The molar free energy $F(T, \xi)$ can be written as a series expansion of $\xi$
$F$ can be Helmholtz or Gibbs free energy

\[\begin{align} F(T, \xi) = a_0(T) + a_1(T)\xi + a_2(T)\xi^2 + \cdots{} \end{align}\]

1st order phase transition
- $F$ is continuous at phase interfaces,
- $\xi$ has abrupt jump

General picture of phase transformation (1st-order)

Example of solid-liquid transformation (KOM book)
Order parameter $\xi$ can be arbitrary
$\xi$ always lower on the high-T phase

What quantity can be an order parameter?

Examples of order parameters $\xi$ (what changes across a phase boundary)

structure: e.g., fcc $\leftrightarrow$ bcc
molar volume $V_m$: liquid $\leftrightarrow$ solid
magnetization $M$: paramagnetic $\leftrightarrow$ ferromagnetic
superconducting order parameter $\psi$: normal $\leftrightarrow$ superconducting

Magnetization as order parameter (second-order transition) Nat. Commun. 15, 3294 (2024)

Dissecting the solid-liquid free energy diagram

Typical $H$ and $G$ diagrams for the single-component system
$G$ is continuous while $H$ is discontinuous
Difference in liquid-solid enthalpy: latent heat $L$

How do we get here?

Enthalpy from specific heat $C_p$: $C_p = \left(\dfrac{\partial H}{\partial T}\right)\vert_{p}$
Entropy from specific heat: $\dfrac{C_p}{T} = \left(\dfrac{\partial S}{\partial T}\right)\vert_{p}$

Single-component system: including pressure

Left to right: increasing $T$ (melting / evaporation)
Lower to upper: increasing $p$ (condensation)
How do free energy profile look like?

Single-component system: slope of phase boundary

Phase boundary equilibrium $G^{\alpha} = G^{\beta}$
Slope of $p-T$ diagram: Clausius-Clapeyron equation

\[\begin{align} \left(\frac{\partial p}{\partial T}\right)\vert_{\text{eq}} &= \frac{\Delta H}{T_{\text{eq}} \Delta V_m} \end{align}\]

Reading single-component phase diagram (Fe)

What can we say about the lattice structure about Fe allotropes?

Free energy diagrams of binary mixture

Mixing of two materials A and B causes free energy to change
Before mixing, molar free energies $G_A$, $G_B$
After mixing, molar free energy becomes

\[\begin{align} G = X_A G_A + X_B G_B + \Delta G_{\text{mix}} \end{align}\]

Ideal solution: mixing entropy

In ideal solution $\Delta G_{\text{mix}} = -T \Delta S_{\text{mix}}$
Ideal mixing entropy

\[ \Delta S_{\text{mix}} = -R (X_A \ln X_A + X_B \ln X_B) \]

Chemical potential on molar free energy diagram

Chemical potential from $G$: $\mu_A = \left(\frac{\partial G}{\partial X_A}\right)\vert_{T, p, X_B}$
Ideal solution: $\mu_A = G_A + RT \ln X_A$

Graphical explanation of chemical potential

Regular solution: mixing enthalpy

Generally $\Delta H_{\text{mix}} \neq 0$
Can be expressed by “likeliness” between A-B

\[ \Delta H_{\text{mix}} = N_A z (\varepsilon_{AB} - \frac{1}{2} (\varepsilon_{AA} + \varepsilon_{BB})) X_A X_B \]

Enthalpic and entropic contributions to mixing

Heterogeneous phase diagram

At each $T$, molar free energy of 2 phases are calculated separately
Imaginary “unstable lattice” for incompatible crystals
What is the most stable phase at each $X$?

Heterogeneous system: equilibrium

Equilibrium condition (common tangent): $\mu_A^\alpha = \mu_A^\beta$ & $\mu_B^\alpha = \mu_B^\beta$
Lever rule: graphical explanation

Phase diagram example 1: completely miscible solid & liquid

E.g. Cu–Ni alloy (fcc)

Phase diagram example 2: miscibility gap

Solid state $\Delta H_{\text{mix}} > 0$
Will be our example for spinodal decomposition

Phase diagram example 3: eutectic alloy (same lattice)

Eutetic formation due to dominant solid-phase

Additional effects for phase stability

Phase diagram does not tell everything!
Stability of a phase depends on
- Molar free energy (what phase diagram tell)
- Interfacial energy (extra work to stablize interface)
- Kinetic stability (metastable & unstable regions)
Actual important issue for kinetic course
- Interface influence: nucleation theory
- Kinetic stability: spinodal decomposition

Preview: influence of interfacial energy

Preview: second-order stability in spinodal decomposition

Summary

In today’s lecture, we overviewed the origin of phase transformation and phase diagram.

Phase transformation can be described by free energy as function of an ordered parameter
Equilibrium phase diagram is constructed by the lowest free energy phase
Free energy of mixing –> chemical potential –> constructing phase diagram
Basic literacy in phase diagram reading!

--- title: "MATE 664 Lecture 13" subtitle: "Phase Diagram and Phase Transformation" author: "Dr. Tian Tian" date: "2026-02-24" format: html: {} revealjs: output-file: slides.html pdf: output-file: L13.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L13.pdf) - Handwritten notes 👉 [Open in pdf](./public/L13_annotated.pdf) ::: ::: # Kinetics of Materials Part II ## What topics have we learned so far? - How to describe kinetic process: **irreversible thermodynamics** - What causes nonequilibrium mass transfer: **chemical potential** as driving force - Diffusion in materials: **Fick's equations** - Determine diffusivity $D$: **Macroscopic** and **Microscopic** models ## What do we study in Part II? Insights from assignments 1 and 2 ![](./public/assignment-comparison.svg){.center} ## Topic 1: phase transformation **Key question**: how do materials evolve when chemical potentials are not at equilibrium? :::{.columns} :::{.column width="50%"} - In-depth study of **phase diagrams** - Continuous phase transformation: **spinodal decomposition** - Phase transformation with barrier: **nucleation** ::: :::{.column width="50%"} ![MIT course _Fundamentals of Materials Sciece_](./public/mit-phase-diagram.png) ::: ::: ## Topic 2: interfacial phenomena **Key question**: how do material interfaces change in non-equilibrium process? :::{.columns} :::{.column width="50%"} - **Solidification**: heat / mass-transfer at interfaces: - **Sintering**: surface-energy-mediated transformation: ::: :::{.column width="50%"} ![Copyright: kintek.com](./public/sintering-process.png) ::: ::: ## Topic 3: analysis of kinetic process in materials **Key question**: how do the competition between different effects change the material behaviour? :::{.columns} :::{.column width="50%"} - **Aggregation phenomena**: diffusion & reaction of soft materials - **Dendrite formation**: morphological control in battery electrodes - **Carrier transport**: designing optical materials ::: :::{.column width="50%"} ::: {.content-visible when-format="html"} ![Artificial dendrite growth](./public/Artificial_dendrites_growth.gif) ::: ::: ::: ## Topic 4: simulating kinetic problems **Key question**: what methods can we use to simulate kinetic systems, and how good are they? - Macroscopic transport: **phase-field method** - Kinetic Monte-Carlo (**KMC**) - Molecular dynamics (**MD**) - Thermodynamic parameters from **first principles** and **machine learning** # Introduction to phase transformation ## Learning outcomes {.center} After this lecture, you will be able to: - **Recall** key components of a phase diagram - **Distinguish** the meaning of axes in a phase diagram - **Identify** key thermodynamic relations from a phase diagram - **Analyze** phase transformation regions in a phase diagram This lecture is adapted from Porter et al. _Phase Transformations in Metals and Alloys_ ## What is phase transformation? Before introducing any fancy terminologies, let's define a few concepts: :::{.columns} :::{.column width="50%"} - **Phase**: - a region with uniform structure and properties - **Transformation** - a kinetic process to reduce total free energy by turning phases α --> β - **Interface** - a boundary separating two phases ::: :::{.column width="50%"} ![Phase transformation scheme](./public/phase-transformation-scheme.svg){width="85%"} ::: ::: ## What can be considered as a phase? - Mathematically (Landau & Lifshitz 1963), different phases are distinguished by _order parameter_ $\xi$ - The molar free energy $F(T, \xi)$ can be written as a series expansion of $\xi$ - $F$ can be Helmholtz or Gibbs free energy ```{=tex} \begin{align} F(T, \xi) = a_0(T) + a_1(T)\xi + a_2(T)\xi^2 + \cdots{} \end{align} ``` - **1st order** phase transition - $F$ is continuous at phase interfaces, - $\xi$ has abrupt jump ## General picture of phase transformation (1st-order) - Example of solid-liquid transformation (KOM book) - Order parameter $\xi$ can be arbitrary - $\xi$ always lower on the high-T phase ![](./public/order-parameter.png) ## What quantity can be an order parameter? :::{.columns} :::{.column width="50%"} Examples of order parameters $\xi$ (what changes across a phase boundary) - **structure**: e.g., fcc $\leftrightarrow$ bcc - **molar volume** $V_m$: liquid $\leftrightarrow$ solid - **magnetization** $M$: paramagnetic $\leftrightarrow$ ferromagnetic - **superconducting** order parameter $\psi$: normal $\leftrightarrow$ superconducting ::: :::{.column width="50%"} Magnetization as order parameter (second-order transition) ![_Nat. Commun. 15, 3294 (2024)_](./public/ncomm-order.png) ::: ::: ## Dissecting the solid-liquid free energy diagram - Typical $H$ and $G$ diagrams for the single-component system - $G$ is continuous while $H$ is discontinuous - Difference in liquid-solid enthalpy: latent heat $L$ ![](./public/porter-free-energy.png) ## How do we get here? - Enthalpy from specific heat $C_p$: $C_p = \left(\dfrac{\partial H}{\partial T}\right)\vert_{p}$ - Entropy from specific heat: $\dfrac{C_p}{T} = \left(\dfrac{\partial S}{\partial T}\right)\vert_{p}$ ## Single-component system: including pressure - Left to right: increasing $T$ (melting / evaporation) - Lower to upper: increasing $p$ (condensation) - How do free energy profile look like? ![](./public/single-comp-pT.png) ## Single-component system: slope of phase boundary - Phase boundary equilibrium $G^{\alpha} = G^{\beta}$ - Slope of $p-T$ diagram: Clausius-Clapeyron equation ```{=tex} \begin{align} \left(\frac{\partial p}{\partial T}\right)\vert_{\text{eq}} &= \frac{\Delta H}{T_{\text{eq}} \Delta V_m} \end{align} ``` ## Reading single-component phase diagram (Fe) - What can we say about the lattice structure about Fe allotropes? ![](./public/Fe-phase-diagram.png){width="55%"} ## Free energy diagrams of binary mixture - Mixing of two materials A and B causes free energy to change - Before mixing, **molar** free energies $G_A$, $G_B$ - After mixing, molar free energy becomes ```{=tex} \begin{align} G = X_A G_A + X_B G_B + \Delta G_{\text{mix}} \end{align} ``` ![](./public/g-mix.png) ## Ideal solution: mixing entropy - In ideal solution $\Delta G_{\text{mix}} = -T \Delta S_{\text{mix}}$ - Ideal mixing entropy $$ \Delta S_{\text{mix}} = -R (X_A \ln X_A + X_B \ln X_B) $$ ## Chemical potential on molar free energy diagram - Chemical potential from $G$: $\mu_A = \left(\frac{\partial G}{\partial X_A}\right)\vert_{T, p, X_B}$ - Ideal solution: $\mu_A = G_A + RT \ln X_A$ ![Graphical explanation of chemical potential](./public/chem-pot-phase-diag.png){width="50%"} ## Regular solution: mixing enthalpy - Generally $\Delta H_{\text{mix}} \neq 0$ - Can be expressed by "likeliness" between A-B $$ \Delta H_{\text{mix}} = N_A z (\varepsilon_{AB} - \frac{1}{2} (\varepsilon_{AA} + \varepsilon_{BB})) X_A X_B $$ ![](./public/AB-likeliness.png){width="60%"} ## Enthalpic and entropic contributions to mixing ![](./public/delta-g-mix-figure.png){width="60%"} ## Heterogeneous phase diagram - At each $T$, molar free energy of 2 phases are calculated separately - Imaginary "unstable lattice" for incompatible crystals - What is the most stable phase at each $X$? ![](./public/hetero-free-energy-diagram.png) ## Heterogeneous system: equilibrium - Equilibrium condition (common tangent): $\mu_A^\alpha = \mu_A^\beta$ & $\mu_B^\alpha = \mu_B^\beta$ - Lever rule: graphical explanation ![](./public/hetero-lever-rule.png) ## Phase diagram example 1: completely miscible solid & liquid - E.g. Cu–Ni alloy (fcc) ![](./public/complete-miscible-phase-diag.png) ## Phase diagram example 2: miscibility gap - Solid state $\Delta H_{\text{mix}} > 0$ - Will be our example for spinodal decomposition ![](./public/miscibility-gap-phase-diag.png) ## Phase diagram example 3: eutectic alloy (same lattice) - Eutetic formation due to dominant solid-phase ![](./public/eutetic-phase-diagram.png) ## Additional effects for phase stability - Phase diagram does not tell everything! - Stability of a phase depends on - Molar free energy (what phase diagram tell) - Interfacial energy (extra work to stablize interface) - Kinetic stability (metastable & unstable regions) - Actual important issue for kinetic course - Interface influence: **nucleation theory** - Kinetic stability: **spinodal decomposition** ## Preview: influence of interfacial energy ![](./public/interfacial-influence.png) ## Preview: second-order stability in spinodal decomposition ![](./public/spinodal.png) ## Summary In today's lecture, we overviewed the origin of phase transformation and phase diagram. - Phase transformation can be described by free energy as function of an ordered parameter - Equilibrium phase diagram is constructed by the lowest free energy phase - Free energy of mixing --> chemical potential --> constructing phase diagram - Basic literacy in phase diagram reading!