1. Description of Phase Transformation
(i) Equilibrium phase transformations – occurring at the transition
temperature ( e. g. freezing at Tm)
(ii) Non-equilibrium phase transformations
(freezing below Tm)
Equilibrium phase transitions:
Recall the Ehrenfest (“classical”) classification scheme.
An nth order transition is defined by the lowest- order derivative
which becomes discontinuous at the transition temperature.

2. Modern Scheme of classification:
Classifies transitions as either 1st or 2nd order
1st order transition has a latent heat
2nd order transition has no latent heat
For example consider two order – disorder transitions
-brass (2nd order)
Cu3Au (1st order) G T Tc G T Tc

3. Long range order parameterH T
Latent heat H T Tc Tc Cp T  Cp T
“” transition W T
1.0
Discontinuous change in W at Tc W T
1.0
Continuous change
in W to Tc
Long range order parameter

4. Non – equilibrium Transformations:
Tid bits
• 1st order transition has infinite Cp at Tc
• Cp has a lambda point for 2nd order transition
• 2nd order (continuous) transitions are studied as the “physics of
critical phenomena”
Non – equilibrium Transformations:
Many important phase transformations do not occur at equilibrium.
(e.g. undercooling during solidification)
Since these transitions do not occur at Tc, we can’t use the previous classification schemes.

5. Scheme for classification of non-equilibrium phase transformations:
There are two general mechanisms of non-equilibrium transformation.
(1) Nucleation and growth -  
 nucleates & grows
* motion of a well- defined  /  interface
* Criteria for 1st order transition
This process is large in degree and small in extent.

6. (2) Continuous transformation (e.g., spinodal decomposition)
Phase separation occurs by gradual amplification of composition variations.
Small in degree and large in extent. C r
* No movement of a sharp interface – 2nd order non-equilibrium phase transformation

7. Classical Theory of Nucleation
Phase transition requires formation of fluctuation or nucleus.   ’ ’ 
Homogeneous Nucleation
Barrier - interface
critical nucleus size  /
Foreign substances involved which serve as nucleation sites.
Heterogeneous Nucleation
Heterogeneous Nucleation

8. Thermodynamics of Nucleation :
(i) Homogeneous Nucleation – (1925 Volmer-Weber)
Series of bimolecular reactions I
(A)
Here, i* is the critically sized nucleus. Other methods of cluster formation such as simultaneous collisions are less probable.
Volmer’s kinetic analysis considered only the forward rate of reaction (A). The back reacting rate was considered small and was ignored

9. Thermodynamic balance of reactions up to (A)
Mixing of ni clusters of size i increases the entropy of the system  some cluster populations are always present.
Dilute solution theory for mixture of n1 ……. ni clusters:
The chemical potential of an i atom cluster :
 # per unit volume
in equilibrium :
(*chemical potential per mole the same in all clusters  likely an incorrect assumption!)

10. Approximation :
(i.e., # of clusters of i  1 is small)
where

11. The nucleation of ’ creates an interface which costs energy
The nucleation of ’ creates an interface which costs energy. The free energy of nucleus formation, Gi, is generally composed of 2 terms :
(a) A volume or bulk term describing the free energy change driving the
transformation
(b) an interfacial energy term opposing the transformation.
volume of α’
assumed independent of i and positive
area
negative
These assumptions fit a central force nn bond model
each atom has Z bonds of energy 
 bulk energy of solid = -1/2 Z per atom

12. for spherical clusters :
critical nucleus : r
4r2 r* G
G*
Gv 4/3r3

13. The equilibrium conc. of critical nuclei
What to use for Gv?
Pure materials
Gas pressure
Gas  Liquid
Gas  Solid
Equil. vap
pressure
(2) Condensed phases
Entropy of transformation
Recall: liq  sol
; Te = equil. transition temp.

14. Thermodynamic Description of Phase Transitions
1. Component Solidification G
Gsolid
@ T= Tm
Gl =Gs
dGl = dGs ; G = 0
Gliquid T T* Tm
Where L is the enthalpy change of the transition or the heat of fusion (latent heat).

15. Alloys: a’ nucleus can have a composition different from a. e.g.,
binary system, x is atom fraction of one element.

16. (ii) Heterogeneous Nucleation –

-phase  
aS
S
S-substrate
 contact angle
s = bS - aS

17. Superhydrophobic surface – Lotus leaf

18.  = 0
Perfect wetting
  90
  90
 = 180
No wetting
For  >  : No wetting  foreign substrate does not help nucleation

19. Shape : assume spherical cap of radius r, contact angle 
If surface deforms
  
S  θ
, 
S
Determination of free energy of formation (GS) of a nucleus on a substrate:
Shape : assume spherical cap of radius r, contact angle 
Volume of cap:   r

20. Contact circle area
Cap volume
Cap area
bS - aS

21. (no undercooling)
(no help from foreign substrate)
Concentration of nuclei
# of single atom sites on substrate