Nucleation and Growth

• Goal
Understand the basic thermodynamics behind the nucleation
and growth processes

• References
Handout
Ch. 8, Intro. to Ceramics by Kingery, Bowen and Uhlmann
Most books on glasses and glass-ceramics

• Homework
None

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 Phase Transformations
• Considered as a transformation of a homogeneous solution to a mixture
of two phases

• For a stable solution, ∆Gmix is less than zero. In other words, the solution
is more stable than the individual components

• ∆Gmix is composed of entropic (-T∆Smix) and enthalpic (∆Hmix) parts

• Consider
1. ∆Hmix less than zero: stable solution
2. ∆Hmix = zero (ideal solution), stable solution due to entropic
3. ∆Hmix slightly greater than zero: stable solution entropy dominates
4. ∆Hmix >> 0: enthalpy dominates, phase separation occurs

• Note: in all cases as T increases, entropy becomes more important, so at

very high temperatures, solutions are usually favored

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 Phase separation
• If ∆Hmix is greater than zero, the overall ∆Gmix can be greater than
zero meaning that phase separation is favored

• As T increases, homogeneous solution is favored

• Tc, the consulate temperature is the point above which solution is

favored

• Behavior described by a series of G vs. composition curves at

different temperatures
Inflection points and minima plotted on T vs. comp. Diagram

• Spinodals from inflection points

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 Spinodal Decomposition
• A continuous phase transformation
Initially, small composition changes that are wide-ranging
Give interpenetrating microstructure (2 continuous phases)

• No thermodynamic barrier to phase separation

One phase separates into two
Infinitesimal composition changes lower the system free energy

• Very important in glass and liquids

Vycor
Liquid-liquid phase separation

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 Nucleation and Growth
• Important for:
Phase transitions, precipitation, crystallization of glasses
Many other phenomena
• Nucleation has thermodynamic barrier Vβ π
= I v u3t 4
• Initially, large compositional change V 3
Small in size Nucleation and growth
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• Volume transformations

Volume Fraction Transformed

α to β phase transformation 0.8

Avrami equation 0.6

Vβ is the volume of second phase

0.4
V is system volume
Iv is the nucleation rate 0.2

u is the growth rate

t is time 0
0 0.2 0.4 0.6 0.8 1

Sigmoidal transformation curves Normalized Time of Reaction

• Infinitesimal changes raise system free energy

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 Volume Energy
• ∆Gv is ∆Grxn (energy/volume) times the new phase volume

• Spherical clusters have the minimum surface area/volume ratio

• So: the volume term can be:

( volume ) ∆G v or
4 πr 3 ∆G
3 v

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 Surface Energy
• The LaPlace equation shows the importance of surface energy
2γ
∆P =
r
Where: ∆P is the pressure drop across a curved surface
γ is the surface energy LaPlace Equation/Kelvin Effect
15000
r is particle radius

• Surface energy is important for small particles

10000

∆P (atm)
• Nuclei are on the order of 100 molecules

• More generally, surface energy is given by: 5000

 ∂G 
γ = 
 ∂A  T, P, composition 0
0.001 0.01 0.1 1 10
Radius (µm)

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 Nucleation
• Consider the nucleation of a new phase at a temperature T
The transition temperature (T) is below that predicted by
thermodynamics when surface or volume are not considered

• We can estimate the free energy change

as a function of the radius of the nuclei Nucleation

from the volume and surface terms 4 10-13 2

Surface Term (~x )
3
Volume Term (~x )
Sum of Surface and Volume

• When r is small, surface dominates 2 10-13

∆ G*

∆G (J)
0 100
• When r is large, volume dominates r*

-2 10-13
• r* is the inflection point
T To -4 10-13

0 2 10- 8 4 10- 8 6 10- 8 8 10- 8 1 10- 7

∆T = T0 − T β phase stable α phase stable
Radius (m)
∆T

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Increasing Temperature

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 Nucleation
• r* is the critical size nucleus and inflection point on the curve
∂ ( ∆G r )
At r*: =0
∂r

• We can use this to calculate r* and ∆Gr*

2γ 16πγ 3
r* = − ∆G* =
∆G v 3( ∆G v ) 2

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 Critical Nuclei
• The number of molecules in the critical nucleus, n*, can be
calculated by equating the volume of the critical nucleus, 4/3π(r*)3,
with the volume of each molecule, V, times the number of
molecules per nucleus
4
π( r*) 3 = n * V
3

• Substituting the previous equations and solving gives

32πγ 3
n* = −
3V( ∆G v ) 3

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 Nucleus Formation
• The number of nuclei can be calculated using statistical entropy
 Nr   Nr   N   N  
∆G n = Nr ∆G r + kT  ln  +  ln 
 N + Nr   N + Nr   N + Nr   N + Nr  

Where: ∆Gn is the free energy for cluster formation

Nr is the number of clusters of radius r per unit volume
N is the number of molecules per unit volume

• At equilibrium, Nr <<N so the previous equation simplifies to:

 ∆G * 
Nr* = N exp − 
 kT 

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 Nucleation Rate
• The nucleation rate, I, is then the product of a thermodynamic
barrier described by Nr* and a kinetic barrier given by the rate of
atomic attachment
 ∆G*  kT  ∆G m 
I = N exp −  NS h exp − kT 
 kT 

• As the degree of undercooling increases, the thermodynamic

driving force increases, but atomic mobility decreases
Nucleation Rate

Thermo Kinetic
Driving Limitation
Force

∆T
To ∆T increasing
T increasing

kT  ∆G m   16πγ 3To2 
I = Ns exp −  N exp − 
WS2002 h  kT  2 2
 3( ∆T) ( ∆H rxn ) kT 12

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 Heterogeneous Nucleation
• In many cases (some argue all cases), nucleation occurs at a
surface, interface, impurity, or other heterogeneities in the system

• The energy required for nucleation is reduced by a factor related to

the contact angle of the nucleus on the foreign surface

∆G*het = ∆G*hom o f ( θ )
( 2 + cos θ )( 1 − cos θ ) 2
f (θ) =
4

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 Growth
• Compared to nucleation, growth is relatively simple
Assume that stable nuclei exist prior to growth
Add molecules to a stable cluster
Driven by free energy decrease of phase change
Kinetically limited
  ∆G m  
u = νa o 1 − exp − 
  kT  

Where: u = growth rate per unit area of interface

ao = distance across the α-β interface (~ 1 atomic dia.)
∆Gm = activation energy for mobility or diffusion
ν = frequency factor
kT
ν=
3πa 3o η

Where: η is atomic mobility of viscosity

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 Summary
• The thermodynamic driving force for both nucleation and growth increases
as undercooling increases, but both become limited by atomic mobility
Growth Nucleation

I and u Rate
I II III
IV

To ∆T increasing ∆T
T increasing
• As we cool from the reaction temperature To we find 4 regions:
Region I, α is metastable, no β grows since no nuclei have formed
Region II, mixed nucleation and growth
Region III, nucleation only
Region IV, no nucleation or growth due to atomic mobility

• Implications for tailoring microstructure

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