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The thermodynamics of phase transformations. Robin Perry School of Physics and Astronomy, Edinburgh. Introduction to Computer Simulation of Alloys meeting 4 th May 2010. Preamble: phase diagrams of metal alloys. Contents. Preamble: phase diagrams of metal alloys

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The thermodynamics of phase transformations

The thermodynamics of phase transformations

Robin Perry

School of Physics and Astronomy,


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



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)


From thermodynamics:Sliquid >Ssolid


Phase transition occurs when: Gsolid=Gliquid



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











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

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


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

Two component system need to consider partial molarA and B.

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

Simplified equations for an ideal liquid:



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

Generally: HMIX0 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


Where =Naz, z=bonds per atom

If <0  A-B bonding preferred

If  >0  AA, BB bonding preferred


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.


-RTln aB

-RTln aA






Activity is simply related to chemical potential by:


A = GA + RT ln aA


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)


Raoult’s Law

HMIX > 0

HMIX < 0

Homogeneous mixing


Liquid, L




Solid, S




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:

nl = nl

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







a b c d


Common tangent








Single phase, mixed solid






2 phase: (A+B) and (B+A)

Compositions e and f ;

“The miscibility gap”











HMIX > 0

Titanium-Vanadium revisited Gibbs phase rule

Blue : single phase


White : two phase



  • 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:

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



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

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

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 rule

Eutectic systems and phase diagrams

  • 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 ruleK)

Solid solution Na(+K)

Solid solution K(+Na)

L + K(Na)

L + KNa2

Peritectic line

(3 phase equil.)

K(Na) + KNa2


KNa2 + Na(K)



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  TSV

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:

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