Introduction to Computer Simulation of Alloys meeting 4 th May 2010

1. Preamble: phase diagrams of metal alloys

Contents
1. Preamble: phase diagrams of metal alloys
2. Gibbs free energy, equilibrium and chemical potential, Gibbs phase rule
3. Single component systems
1. dG(T)
2. Clausius-Clapeyron equation and the phase diagram of titanium
4. Binary (two component) systems
1. Ideal solutions
2. Regular solutions
3. Activity
4. Real solutions, ordered phases and Intermediate phases
5. Binary phase diagrams
1. Miscibility gap
2. Ordered alloys
3. Eutectics and peritectics
4. Additional useful relationships
5. Ternary diagrams
6. 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 (m 3 ), 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
Solid : Low atomic kinetic energy or E
 low T and small S
Liquid : Large E
 high T and large S metastable equilibrium
E.g. Metastable : Diamond
Equilibrium : Graphite
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
Assumption: Closed system
 ignore d

For purposes of most discussions : fix pressure (unless otherwise stated)
G
From thermodynamics: S liquid
> S solid
Phase transition occurs when:
G solid
= G liquid
For pressure dependence:
G solid
T
M
G liquid
T (K)
Similar arguments apply : V liquid
> V solid so increasing P implies liquid to solid transition

Clausius Clapeyron Equation more dense
(intermediate)
Less dense more dense
Less dense

4. Binary (two component) systems : Ideal solutions
Two species in the mixture: consider mole fractions X
A and X
B
X
A
+ X
B
= 1
Two contributions to G from mixing two components together:
1. G
1
– weighted molar average of the two components
2. Free Energy of mixing G
1
= X
A
G
A
+ X
B
G
B

G
MIX
=

H mix
T

S
MIX
Where

H mix is the heat absorbed or evolved during mixing or heat of solution

S
MIX is the entropy difference between the mixed and unmixed states

4. Binary (two component) systems : Ideal solutions
Simplest case : Ideal solution :

H
MIX
= 0
Some assumptions :
1. Free energy change is only due to entropy
2. Species A and B have the same crystal structure
(no volume change)
3. A and B mix to form substitutional solid solution
Boltzmann equation: S = k
B ln (

) S is the configurational entropy

- total number of microstates of system or total number of distinguishable ways of arranging the atoms
Using Stirling’s approximation and N a k
B
= R

G
MIX
= RT ( X
A ln X
A
+ X
B ln X
B
)
Mixing components lowers the free energy!

4. Binary (two component) systems : The chemical potential
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 =

S d T +

A
X
A
+

B
X
B
(+ V d P )
Simplified equations for an ideal liquid:

A
X
A
= G
A
+ RT ln X
A

B
X
B
= G
B
+ RT ln X
B
I.e.

A is the free energy of component A in the mixture

4. Binary (two component) systems : Regular solutions and atomic bonding
Generally:

H
MIX

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
Define:

H
MIX
= C
AB
 where C
AB is the number of A-B bonds and

=

AB
 ½( 
AA
+

BB
)

H
MIX
=

X
A
X
B
Where

= N a z

, z= bonds per atom
If

<0

A-B bonding preferred
If

>0

AA, BB bonding preferred

G
MIX
=

H
MIX
+ RT ( X
A ln X
A
+ X
B ln X
B
)
Point of note:

G
MIX always decreases on addition of solute

Free energy curves for various conditions:
Mixing always occurs at high
Temp. despite bonding


Mixing if A and
B atoms bond


A and B atoms repel
Phase separation in to 2 phases.

4. Binary (two component) systems : Activity, a of a component
Activity is simply related to chemical potential by:

A
= G
A
+ RT ln a
A
G
A

B
= G
B
+ RT ln a
B
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).
RT ln a
A

A
0
X
B

G
MIX
G
B
1

B
RT ln a
B i.e. For homogeneous mixing,

<0
 a
A
< X
A and a
B
< X
B
So the activity is the tendency of a component to leave solution

For low concentrations of B ( X
B
<<1)
And…
Henry’s Law (or everything dissolves)
Raoult’s Law

H
MIX
> 0

H
MIX
< 0
Homogeneous mixing

5. Binary phase diagrams : The Lever rule
Phase diagrams can be used to get quantitative information on the relative concentrations of phases using the Lever rule :
Temperature
Liquid, L
T l
 l
 i.e. ~25% solid and
~75% liquid at X
0
Solid, S
A X
0 B
At temperature, T and molar fraction X
0
, 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 : A and B are completely miscible and ideal solutions

5. Binary phase diagrams : The Miscibility gap
G
T
1 solid L
G
T
2 a b c d
S liquid

H
MIX
> 0
Common tangent
A B A B
A
G
L
T
3
S e
T
1
T
2 f
B
T
3
A e
X
B liquid
Single phase, mixed solid f
B
2 phase: (A+

B) and (B+

A)
Compositions e and f ;
“ The miscibility gap ”

Titanium-Vanadium revisited
(bcc)
(hcp)
What can we deduce?
1. Ti and V atoms bond weakly
2. There are no ordered phases
3. (Ti,V) phase : mixture of Ti and V in a fcc structure
4. Ti (hcp) phase does not dissolve V well
Blue : single phase
White : two phase
(bcc)

Equilibrium in heterogenous systems
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: X 0 <
 e
X 0 >
 e
  phase only
  phase only
Common tangent
If
 e
< X 0 >
 e then minimum free energy is G e
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:

4. Binary (two component) systems : Ordered phases
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 substitutional
Ordered structures can also tolerate deviations from stoichiometry. This gives the broad regions on the phase diagram

Random mixture
The Copper-Gold system
(fcc)
Single phases
(fcc)
Mixed phases
N.B. Always read the legend!!! (blue is not always ‘singe phase’)

An intermediate 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. Ni
3
Al
Narrow stability range broad stability range

5. Binary phase diagrams : Ordered phases
Peak in liquidus line : attraction between atoms

H
MIX
< 0 i.e. A and B attract
1 phase, solid
Weak attraction
Ordered phase

Strong attraction
Ordered
 phase extends to liquid phase

5. Binary phase diagrams : Simple Eutectic systems

H
MIX
 0 ; A and B have different crystal structures;
Eutectic point

Phase is A with

B dissolved (crystal structure A)

Phase is B with

A dissolved (crystal structure B)
Two phase
Single phase

Example : http://www.soton.ac.uk/~pasr1/index.htm
Eutectic systems and phase diagrams

5. Binary phase diagrams : 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 + K(

Na)
K(

Na) + KNa
2
(bcc)
L + KNa
2
L + Na(

K)
Peritectic line
(3 phase equil.)
(hcp)
KNa
2
+ Na(

K)
(bcc)

5. Binary phase diagrams : 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 X
V
(that gives a minimum free energy)

G
V
=

H
V

T

S
V
Where

H
V is the increase in enthalpy per mole of vacancies added and

S
V is the change in thermal entropy on adding the vacancies (changes in vibrational frequencies etc.).
X
V 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:

5. Binary phase diagrams : 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
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.

G a 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