# The thermodynamics of phase transitions

Mechanics

Oct 27, 2013 (4 years and 6 months ago)

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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

1.
Preamble: phase diagrams of metal alloys

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.

5.
Ternary diagrams

6.
Kinetics of Phase transformations

Contents

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

⡊(浯l)

PV

small

ignore

Equilibrium

: the most stable state defined by lowest possible
G

dG = 0

equilibrium

metastable

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

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

Gibbs phase rule for equilibrium phase :

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
.

Number of degrees of freedom F = C

K +2

C, number of components

K, number of phases in equilibrium

3. Single component systems

For purposes of most discussions :

fix pressure (unless otherwise stated)

For pressure dependence:

Similar arguments apply :
V
liquid

>
V
solid

so increasing
P

implies liquid to solid transition

From thermodynamics:

S
liquid

>

S
solid

G

T (K)

T
M

G
solid

G
liquid

Phase transition occurs when:
G
solid
=
G
liquid

Assumption: Closed system

ignore
d

Clausius Clapeyron Equation

Less

dense

more

dense

Less

dense

more

dense

(intermediate)

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

G
1

=

X
A
G
A

+
X
B
G
B

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
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

-

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

Boltzmann equation:

S

=
k
B

ln (

)

S

is the configurational entropy

G
MIX

=
RT
(
X
A
ln
X
A

+
X
B
ln
X
B
)

Mixing components lowers the free energy!

Molar 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

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.

Free energy curves for various conditions:

4.
Binary (two component) systems :

Activity, a

of a component

-
RT
ln
a
A

0

1

X
B

-
RT
ln
a
B

B

A

G
MIX

A

=
G
A

+
RT
ln
a
A

G
A

G
B

Activity is simply related to chemical potential by:

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).

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)

Henry’s Law (or everything dissolves)

Raoult’s Law

And…

Homogeneous mixing

H
MIX

> 0

H
MIX

< 0

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
:

At temperature, T and molar fraction
X
0
, the solid and liquid phase will coexist in
equilibrium according the ratio:

Temperature

A

B

T

Solid, S

Liquid, L

X
0

l

l

n

l

=
n

l

i.e. ~25% solid and
~75% liquid at
X
0

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

A

B

T
1

G

liquid

solid

L

Common tangent

A

B

G

S

a b c d

T
2

S

A

B

T
3

G

L

e

f

H
MIX

> 0

A

B

X
B

liquid

T
1

T
2

T
3

e

f

Single phase, mixed solid

2 phase: (A+

B) and (B+

A)

Compositions
e

and
f

;

The miscibility gap

Titanium
-

(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

phase only

X
0
>

e

phase only

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:

Common tangent

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.

Ordered substitutional

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

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

The Copper
-
Gold system

Random mixture

Single phases

Mixed phases

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

(fcc)

(fcc)

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

5.
Binary phase diagrams

: Ordered phases

H
MIX

< 0

i.e. A and B attract

Weak attraction

Strong attraction

Ordered

phase extends to liquid phase

1 phase, solid

Ordered phase

Peak in liquidus line : attraction between atoms

5.
Binary phase diagrams

: Simple Eutectic systems

H
MIX

0 ; A and B have different crystal structures;

Phase is A with

B dissolved (crystal structure A)

Phase is B with

A dissolved (crystal structure B)

Single phase

Two phase

Eutectic point

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.

Solid solution K(+

Na)

Solid solution Na(+

K)

(bcc)

(hcp)

(bcc)

Peritectic line

(3 phase equil.)

L + KNa
2

L + Na(

K)

L + K(

Na)

K(

Na) + KNa
2

KNa
2
+ Na(

K)

5.
Binary phase diagrams

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