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### The thermodynamics of phase transformations

Robin Perry

School of Physics and Astronomy,

Edinburgh

Introduction to Computer Simulation of Alloys meeting 4th May 2010

- Preamble: phase diagrams of metal alloys
- Gibbs free energy, equilibrium and chemical potential, Gibbs phase rule
- Single component systems
- dG(T)
- Clausius-Clapeyron equation and the phase diagram of titanium

- Binary (two component) systems
- Ideal solutions
- Regular solutions
- Activity
- Real solutions, ordered phases and Intermediate phases

- Binary phase diagrams
- Miscibility gap
- Ordered alloys
- Eutectics and peritectics
- Additional useful relationships
- Ternary diagrams

- Kinetics of Phase transformations

2. Gibbs free energy, equilibrium and chemical potential, Gibbs phase rule

Definition of terms :

Phase, K: portion of the system with homogeneous properties and composition. Physically distinct.

Components, C : chemical compounds that make up a system

Gibbs free energy, G (J/mol): measure of relative stability of a phase at constant temperature and pressure

G = E + PV TS + N

Intensive variables : Temperature, T (K); Pressure, P (Pa);

Extensive variables : Internal energy E (J/mol); Volume, V (m3), Entropy (J/K mol) particle number, N;Chemical potential (J/mol)

Solids/liquid transitions in metals: PV small ignore

2. Gibbs free energy, equilibrium and chemical potential, Gibbs phase rule

Equilibrium : the most stable state defined by lowest possible G

dG = 0

metastable

equilibrium

E.g. Metastable : Diamond

Equilibrium : Graphite

Solid : Low atomic kinetic energy or E

low T and small S

Liquid : Large E

high T and large S

Chemical potential or partial molar free energy governs how the free energy changes with respect to the addition/subtraction of atoms.

This is particularly important in alloy or binary systems.(particle numbers will change)

2. Gibbs free energy, equilibrium and chemical potential, Gibbs phase rule

Gibbs phase rule for equilibrium phase :

Number of degrees of freedom F = C – K +2

Examples :

Single component system C=1 and F = 3 K

If 1 phases in equilibrium (e.g. solid) 2 degrees of freedom i.e. can change T and P without changing the phase

If 2 phases in equilibrium (e.g. solid and liquid) 1 degree of freedom i.e. T is dependent on P (or vice-versa)

If 3 phases in equilibrium (e.g. solid, liquid and ) 0 degrees of freedom. 3 phases exist only at one fixed T and P.

C, number of components

K, number of phases in equilibrium

3. Single component systems Gibbs phase rule

For purposes of most discussions :

fix pressure (unless otherwise stated)

G

From thermodynamics:Sliquid >Ssolid

Gsolid

Phase transition occurs when: Gsolid=Gliquid

Gliquid

TM

T (K)

For pressure dependence:

Similar arguments apply : Vliquid > Vsolid so increasing P implies liquid to solid transition

Assumption: Closed system ignore d

Clausius Clapeyron Equation Gibbs phase rule

more

dense

(intermediate)

more

dense

Less

dense

Less

dense

- Binary (two component) systems : Gibbs phase rule Ideal solutions

G1 = XAGA + XBGB

Two species in the mixture: consider mole fractions XA and XB

XA + XB = 1

- Two contributions to G from mixing two components together:
- G1 – weighted molar average of the two components
- Free Energy of mixing

GMIX = Hmix - T SMIX

Where Hmix is the heat absorbed or evolved during mixing or heat of solution

SMIX is the entropy difference between the mixed and unmixed states

- Binary (two component) systems : Gibbs phase rule Ideal solutions

Boltzmann equation:

S = kB ln ( )

S is the configurational entropy

- total number of microstates of system or total number of distinguishable ways of arranging the atoms

Mixing components lowers the free energy!

Simplest case : Ideal solution : HMIX = 0

- Some assumptions :
- Free energy change is only due to entropy
- Species A and B have the same crystal structure (no volume change)
- A and B mix to form substitutional solid solution

Using Stirling’s approximation and NakB=R

Molar Free Energy

GMIX = RT(XAlnXA + XBlnXB)

- Binary (two component) systems : Gibbs phase rule The chemical potential

Chemical potential : governs the response of the system to adding component

Two component system need to consider partial molarA and B.

Total molar Gibbs free energy = SdT + AXA + BXB (+VdP)

Simplified equations for an ideal liquid:

AXA = GA +RTlnXA

BXB = GB +RTlnXB

I.e. Ais the free energy of component A in the mixture

- Binary (two component) systems : Gibbs phase rule Regular solutions and atomic bonding

Generally: HMIX0 i.e. internal energy of the system must be considered

In a binary, 3 types of bonds: A-A, B-B, A-B of energies AA, BB, AB

= AB ½(AA +BB)

Define: HMIX= CAB where CAB is the number of A-B bonds and

HMIX= XAXB

Where =Naz, z=bonds per atom

If <0 A-B bonding preferred

If >0 AA, BB bonding preferred

GMIX = HMIX + RT(XAlnXA + XBlnXB)

Point of note:

GMIX always decreases on addition of solute

Free energy curves for various conditions: Gibbs phase rule

Mixing if A and B atoms bond

Mixing always occurs at high Temp. despite bonding

A and B atoms repel

Phase separation in to 2 phases.

- Binary (two component) systems : Gibbs phase ruleActivity, a of a component

GMIX

-RTln aB

-RTln aA

B

A

0

1

XB

Activity is simply related to chemical potential by:

GB

A = GA + RT ln aA

GA

B = GB + RT ln aB

It is another means of describing the state of the system. Low activity means that the atoms are reluctant to leave the solution (which implies, for example, a low vapour pressure).

i.e. For homogeneous mixing, <0 aA<XA and aB<XB

So the activity is the tendency of a component to leave solution

For low concentrations of B ( Gibbs phase ruleXB<<1)

Henry’s Law (or everything dissolves)

And…

Raoult’s Law

HMIX > 0

HMIX < 0

Homogeneous mixing

- Binary phase diagrams Gibbs phase rule : The Lever rule

Temperature

Liquid, L

T

l

l

Solid, S

X0

A

B

Phase diagrams can be used to get quantitative information on the relative concentrations of phases using the Lever rule :

i.e. ~25% solid and ~75% liquid at X0

At temperature, T and molar fraction X0, the solid and liquid phase will coexist in equilibrium according the ratio:

nl = nl

Where n/n is ratio of liquid to solid

Solid to liquid phase diagram in a two component system Gibbs phase rule : A and B are completely miscible and ideal solutions

- Binary phase diagrams Gibbs phase rule : The Miscibility gap

G

G

T2

T1

solid

L

a b c d

S

Common tangent

liquid

A

B

A

B

G

liquid

Single phase, mixed solid

T3

T1

L

T2

S

2 phase: (A+B) and (B+A)

Compositions e and f ;

“The miscibility gap”

f

e

e

f

T3

A

B

A

XB

B

HMIX > 0

Titanium-Vanadium revisited Gibbs phase rule

Blue : single phase

(bcc)

White : two phase

(hcp)

(bcc)

- What can we deduce?
- Ti and V atoms bond weakly
- There are no ordered phases
- (Ti,V) phase : mixture of Ti and V in a fcc structure
- Ti (hcp) phase does not dissolve V well

Equilibrium in heterogenous systems Gibbs phase rule

For systems with phase separation ( and ) of two stable structures (e.g. fcc and bcc), we must draw free energy curves.

G is the curve for A and B in fcc structure ( phase)

G is the curve for A and B in bcc structure ( phase)

For: X0<e phase only

X0>e phase only

Common tangent

If e<X0>e then minimum free energy is Ge

And two phases are present (ratio given by the Lever rule – see later)

When two phases exist in equilibrium, the activities of the components must be equal in the two phases:

- Binary (two component) systems : Gibbs phase ruleOrdered phases

Ordered substitutional

Previous model gross oversimplification : need to consider size difference between A and B (strain effects) and type/strength of chemical bonding between A and B.

Systems with strong A-B bonds can form Ordered and/or intermediate phases

Ordered phases occur for (close to) integer ratios.

i.e. 1:1 or 3:1 mixtures. But entropy of mixing is very small so increasing temperature can disorder the phase. At some critical temperature, long range order will disappear.

Ordered structures can also tolerate deviations from stoichiometry. This gives the broad regions on the phase diagram

The Copper-Gold system Gibbs phase rule

Random mixture

(fcc)

(fcc)

Single phases

Mixed phases

N.B. Always read the legend!!! (blue is not always ‘singe phase’)

An Gibbs phase ruleintermediate phase is a mixture that has different structure to that of either component

Range of stability depends on structure and type of bonding (Ionic, metallic, covalent…)

Intermetallic phases are intermediate phase of integer stoichiometry e.g. Ni3Al

Narrow stability range

broad stability range

- Binary phase diagrams Gibbs phase rule : Ordered phases

HMIX < 0

Peak in liquidus line : attraction between atoms

i.e. A and B attract

1 phase, solid

Weak attraction

Strong attraction

Ordered phase

Ordered phase extends to liquid phase

- Binary phase diagrams Gibbs phase rule : Simple Eutectic systems

Single phase

Eutectic point

HMIX 0 ; A and B have different crystal structures;

Two phase

Phase is A with B dissolved (crystal structure A)

Phase is B with A dissolved (crystal structure B)

Example : Gibbs phase rulehttp://www.soton.ac.uk/~pasr1/index.htm

Eutectic systems and phase diagrams

- Binary phase diagrams Gibbs phase rule : Peritectics and incongruent melting

- Sometimes ordered phases are not stable as a liquid. These compounds have peritectic phase diagrams and display incongruent melting.
- Incongruent melting is when a compound melts and decomposes into its components and does not form a liquid phase.
- These systems present a particular challenge to material scientists to make in a single phase. Techniques like hot pouring must be used.

L + Na( Gibbs phase ruleK)

Solid solution Na(+K)

Solid solution K(+Na)

L + K(Na)

L + KNa2

Peritectic line

(3 phase equil.)

K(Na) + KNa2

(hcp)

KNa2 + Na(K)

(bcc)

(bcc)

- Binary phase diagrams Gibbs phase rule : Additional equations

A. Equilibrium vacancy concentration

So far we have assumed that every atomic site in the lattice is occupied. But this is not always so. Vacancies can exist in the lattice.

Removing atoms: increase internal energy (broken bonds) and increases configuration entropy (randomness).

Define an equilibrium concentration of vacancies XV (that gives a minimum free energy)

GV=HV TSV

Where HV is the increase in enthalpy per mole of vacancies added and SV is the change in thermal entropy on adding the vacancies (changes in vibrational frequencies etc.).

XV is typically 10-4-10-3 at the melting point of the solid.

B. Gibbs-Duhem relationship

This relates the change in chemical potential that results from a change in alloy composition:

- Binary phase diagrams Gibbs phase rule : Ternary phase diagrams

These are complicated.

- 3 elements so triangles are at fixed temperature
- Vertical sections as a function of T and P are often given.

Blue – single phase

White – two phase

Yellow – three phase

6. Kinetics of phase transformations Gibbs phase rule

So far we have only discussed systems in equilibrium. But we have said nothing of rate of a phase transformation. This is the science of Kinetics.

G is the driving force of the transformation.

Ga is the activation free energy barrier.

Atoms must obtain enough thermal energy to overcome this barrier.

General equation for the rate of the transformation is the Arrhenius rate equation:

i.e. high temperature implies faster rate

N.B. some rates are very long e.g. diamond graphite

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