Basics of Phases and Phase Transformations W. Püschl University of Vienna

Content 1. Historical context 2. Classification of phase transformations 3. Graphical thermodynamics – Phase diagrams 4. Miscibility Gap – Precipitation nucleation vs. spinodal decomposition 5. Order 6. Ising model: atomic and magnetic spin configuration 7. Martensitic transformations

Early technological application of poly-phase systems: Damascus Steel

Aloys v. Widmannstätten 1808 Iron meteorite cut, polished, and etched: Intricate pattern appears

Oldest age hardening curve: Wilms Al-Cu(Mg,Mn,Fe, Si) alloy Retarded precipitation of a disperse phase.

A scientific understanding of phases and phase transformation begins to develop end 19th / beginning 20th centuries  physical metallurgy Experimental: Gustav Tammann (Göttingen) Theoretical: Josiah Willard Gibbs

What is a phase? Region where intrinsic parameters have (more or less) the same value lattice structure, composition x, degree of order , density ,… Need not be simply (singly) connected. Expreme example: disperse phase and matrix phase where it is embedded (like Swiss cheese) When is a phase thermodynamically stable? How can we determine wihich phase is stable at a certain composition, temperature (and pressure, magnetic field…) What happens if this is not the case  metastability or phase transition How can a phase transition take place?

Ehrenfest (1933) 1st oder phase transition2nd order (generally: higher order)

Free energy vs. order parameter according to Landau Higher-order phase transition 1st oder phase transition

Chemical potentials g i of the components

Gibbs phase rule f =  (n - 1) – n (  - 1) + 2 = n -  + 2

Liquid-solid transition of a two-component System (Ge-Si)

Excess enthalpy and miscibility gap

Excess enthalpy and miscibility gap

Precipitation: alternative mechanisms

Heterophase fluctuation corresponds to nucleation Homophase fluctuation corresponds to spinodal decomposition

Free energy of a spherical precipitate particle

Ni 36 Cu 9 Al 55 Precipitation by nucleation and growth: N V particle number,  c supersaturation, mean particle radius

Spinodal Decomposition

Excess enthalpy Positive: like atoms preferred: Phase separation Negative: unlike atoms preferred: ordering Short range order: there is (local) pair correlation Cowley- Warren SRO parameter Decay with distance from reference atom If they do not decay  long range order

disordered state (bcc) Long range order: out of bcc structure the B2 (CsCl) structure arises ordered state (B2) disordered state (bcc)

Long range order: B2 (CsCl) representatives

D0 3 stoichiometry 3:1 Out of the same bcc structure:

L1 2 Long range order out of the fcc structure:

L1 2 ordered state L1 2 disordered state (fcc)

L1 2 representatives

fcc  L1 0 stoichiometry 1:1

representatives fcc  L1 0 stoichiometry 1:1

Different long range ordered structures in the Cu-Au phase diagram L1 2 L1 0 L1 2 L1 0

L1 2 L1 0 CuAu II (long period.) L1 2 L1 0 Different long range ordered structures in the Cu-Au phase diagram

Fcc  L1 1 stoichiometry 1:1

Statistical physics of ordered alloys Partition function Possibly different vibration spectrum for every atom configuration Does it really matter?

FePd: Density of phonon states g( ) L1 0 - ordered fcc disordered Mehaddene et al. 2004

Bragg – Williams model: only nn pair interactions, disregard pair correlations R long range order parameter tanh R/   <1  >1 Simplifying almost everything:

Different levels of approximation in calculating internal interaction energy Bragg-Williams Experiment Quasi-chemical quasi-chemical Experiment Bragg-Williams

Ising model (Lenz + Ising 1925) Can be brought to Ising form by identifying (for nn interaction) Hamiltonian for alloy (pair interaction model) p i n atom occupation function

Idea of mean field model: treat a few local interactions explicitly, environment of similar cells is averaged and exerts a mean field of interaction

Local interaction only 1 atom  Bragg- Williams – model Correspondences: Phase-separating ferromagnetic Long range ordering antiferromagnetic ferromagnetic

Structure on polished surface after martensitic transformation: roof-like, but no steps. A scratched line remains continuous

Martensite morphologies

Homogeneous distorsion by a martensitic transformation

First step :Transformation into a new lattice type: Bain transformation

Second step: Misfit is accomodated by a complementary transformation: twinning or dislocation glide

Thermoelastic Martensites: Four symmetric variants per glide plane: Can be transformed into one another by twinning

Shape Memory effect

Final remarks: As the number of components grows and interaction mechanisma are added, phase transformations can gain considerable complexity For instance: Phase separation and ordering (opposites in simple systems) may happen at the same time. I have completely omitted many interesting topics, for instance Gas-to-liquid or liquid-to-liquid transformations The role of quantum phenomena at low-temperature phases Dynamical phase transformations, self-organized phases far from equilibrium