1. Nucleation & Growth Driving Force
All phase transformations can be described as nucleation & growth processes.
In order to understand, predict and engineer phase transformations, we must understand the driving force, the kinetics and the mechanisms of transformations.
Driving Force
Example: solidification: as you cool a liquid below the liquidus, so the driving force for solidification increases. This driving force is often called undercooling or Supercooling.
Example: phase change: for a pure substance with more than one allotrope, as you cool it below the phase transformation temperature, so the driving force for the phase change increases.

2. Driving force for nucleation
It is important to realize the difference between the driving force for the reaction as a whole, which is given by the change in free energy between the supersaturated solid solution and the two-phase mixture,
Why a different free energy for nucleation?
Because the first nuclei of beta to appear do not significantly change the composition of the parent material. Thus the free energy change for nucleation is the rate of change of free energy for the new, product phase (beta).

3. Thermodynamic driving force for a phase transformation
Liquid
solidification
evaporation
sublimation
Solid
gas
melting
condensation
Solid state phase
transformation
Solid 2 1
Solid liquid interface
Liquid Gas interface
Solid Gas interface
Thermodynamic driving force for a phase transformation
Decrease in Gibbs free energy
Liquid-> solid
gS - gL= g = -ve

4. Homogeneous nucleation
Spherical ball of solid of radius R in the middle of the liquid at a temperature below Tm
Homogeneous nucleation r
gL = free energy of liquid per unit volume
gS = free energy of solid per unit volume
g = gS - gL

5. Heterogeneous nucleation
How does solidification begins?
Usually at the walls of the container
Heterogeneous nucleation.
Why?
To be discussed later.

6. Change in free energy of the system due to formation of the solid ball of radius r :p 2 4 r +
+ve: barrier to nucleation r r r*

7. G r f D = 3 4 p
Solid balls of radius r < r* cannot grow as it will lead to increase in the free energy of the system !!! D g p 2 4 r +
Solid balls of radii r > r* will grow G D r
r* is known as the CRITICAL RADIUS OF HOMOGENEOUS NUCLEATION r* G r D 3 4 p

8. g p 2 4 r +
Eqn. 9.4 r r*

9. g Tm gL gS T
g (T) T

10. g p 2 4 r + f
f1* g p 2 4 r +
f2* r
r2*
r1*
T2 < T1

11. The Nucleation Rate Critical particle
Formation of critical nucleus
Atoms surrounding the critical particle
Diffuse jump of a surrounding atom to the critical particle makes it a nucleation
Nt=total number of clusters of atoms per unit volume
N* = number of clusters of critical size per unit volume
By Maxwell-Boltzmann statistics

12. Growth
Increase in the size of a product particle after it has nucleated T U

13. Overall Transformation Kinetics
I : Nucleation rate T U
U : Growth rate
dX/dt
Overall transformation rate (fraction transformed per second) I
X=fraction of product phase

14. Fraction transformed as a function of timeX
Slow due to final impingement
Slow due to very few nuclei t ts tf