MATE 664 Lecture 13

Phase Diagram and Phase Transformation

Dr. Tian Tian

2026-02-24

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:

Copyright: kintek.com

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

Artificial dendrite growth

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

Phase transformation scheme

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!