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. Widmannstätten1808Iron meteorite cut, polished,and etched:Intricate pattern appears

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

A scientific understanding of phases and phase transformationbegins to develop end 19th / beginning 20th centuriesphysical metallurgy

Experimental:Gustav Tammann (Göttingen)

Theoretical:Josiah Willard Gibbs

What is a phase?

Region where intrinsic parameters have (more or less) the same valuelattice structure, composition x, degree of order , density ,…

Need not be simply (singly) connected.Expreme example: disperse phaseand 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?

g

T

g

T

Phase 1

Phase 2

Tu

Phase 1

Phase 1

Phase 2

Phase 2

Phase 2

Tu

a) b)

Abb. 1-3

Ehrenfest (1933)

1st oder phase transition 2nd order (generally: higher order)

Free energy vs. order parameter according to Landau

Higher-order phase transition 1st oder phase transition

Chemical potentials gi of the components

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

Liquid-solid transitionof a two-componentSystem (Ge-Si)

Excess enthalpyand miscibility gap

Excess enthalpyand miscibility gap

Precipitation:alternative mechanisms

Heterophase fluctuationcorresponds to nucleation

Homophase fluctuationcorresponds tospinodal decomposition

Free energy of a spherical precipitate particle

Ni36Cu9Al55

Precipitation by nucleation and growth:NV particle number, c supersaturation, mean particle radiusR

Spinodal Decomposition

Excess enthalpy

Positive: like atoms preferred: Phase separation

Negative: unlike atoms preferred: ordering

Short range order: there is (local) pair correlation

jm

ijnm

jm

in

jm

inij

nmp

P

pp

pp 11

Cowley- Warren SRO parameter

Decay with distance from reference atom

If they do not decay long range order

L12

Long range order out of the fcc structure:

Long range order out of the fcc structure:

L12

ordered state L12 disordered state (fcc)

fcc L10

stoichiometry 1:1

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

L12

L12

L10

L12

L10

L12

L12

L10

CuAu II (long period.)L12

L10

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

Statistical physics of ordered alloys

Partition function

kT

RWRZRZ r

r

rV

)(exp)()(

N

jrj

rrV

kT

hkTZ

3

1

0

exp1

1exp

Possibly different vibration spectrum for every atom configuration

Does it really matter?

FePd: Density of phonon states g()

L10 - ordered fcc disordered

Mehaddene et al. 2004

Bragg – Williams model:

only nn pair interactions, disregard pair correlationsR long range order parameter

tanh R/

<1>1

20 4

)1ln()1()1ln()1(2ln22

RNzV

WRRRRNkT

FC

0R

FC

Simplifying almost everything:

Different levels of approximation in calculatinginternal interaction energy

Bragg-Williams

Experiment

Quasi-chemical

quasi-chemical

Experiment

Bragg-Williams

Ising model (Lenz + Ising 1925)

mn n

nmnnmn hJH,2

1

i n

in

i

mn

jm

in

ijnm

i n

in

iin

in pppVppHpH

,2

1~

nnp 12

1 ABBBAA VVVV 2VJ nm 4

1 BBAABA VVh

4

1

2

1

Can be brought to Ising form by identifying (for nn interaction)

Hamiltonian for alloy (pair interaction model)

pin atom

occupation function

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

Local interaction only 1 atom Bragg- Williams – model

Correspondences:

Phase-separating ----- ferromagnetic

Long range ordering ----- antiferromagnetic

ferromagnetic

Structure on polished surfaceafter 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 acomplementary transformation: twinning or dislocation glide

Thermoelastic Martensites:

Four symmetric variants per glide plane:

Can be transformed intoone another by twinning

Shape Memory effect

Final remarks:

As the number of components growsand 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 transformationsThe role of quantum phenomena at low-temperature phasesDynamical phase transformations, self-organized phasesfar from equilibrium