Presented by Dr. John Perepezko as part of the 2019 MRSEC Summer Seminar Series. MRSEC hosted this inaugural series of pedagogical seminars for the benefit of students and postdocs interested in a deeper dive into selected topics. Presentations are selected based on topics requested by students. This presentation covers the basic reaction pathways controlling the crystallization of amorphous materials. The topics include a survey of nucleation kinetics, phase section thermodynamics, growth kinetics and the representation of the overall transformation kinetics. Some of the ways that the popular analysis methods are used and abused are highlighted and the importance of incorporating a detailed microstructure evaluation in any kinetics analysis is pointed out.

1. Nucleation and Growth of Crystal Phases
from Amorphous Solids
MRSEC Lecture
John H. Perepezko
University of Wisconsin-Madison
Department of Materials Science and Engineering
1509 University Ave.
Madison, WI 53706
July 22, 2019

2. Outline
Introduction
•Background
•Thermodynamic Preliminaries
•Bulk Phase Equilibria
•Nanoscale Phase Equilibria
•Hierarchy of Equilibrium
•Driving Force Scaling
•Nucleation Reactions
•Nucleation Rate (Steady State)
•Transient Behavior
•Stochastic Behavior
•Kinetic Transitions
•Alloy Effects
•Glass Formation

3. Outline (continued)
•Nanocrystallization
• Reaction Classes
• Kinetic Rates
• Primary Crystallization
•Some Applications
• Nanocrystallization in Amorphous Al-Base
and Au-Base alloys
•Summary

4. Key Lessons
• Metastable vs. Unstable
• Stochastic (Probabilistic) vs. Deterministic Kinetics
• Spatial Heterogeneities
• Nanostructures are Nucleation-Controlled
(i.e. high nucleation rate and slow growth)
 Precursor Reactions for Synthesis

5. Synthesis of Amorphous/Nanocrystalline Alloys
Amorphous alloys
Crystalline
solids
mechanical mixing
Open system
(Driven system)
Aqueous
solution
electrodeposition
Closed system
Vapours
CVD/PVD
Closed system
Alloy melt
rapid solidification process
Closed system

6. Nucleation Control Signatures
• Development of Undercooling/ Supersaturation
• Reaction Hysteresis
• Metastability
• Statistical Behavior
• Strong Temperature Dependence of Product Phase Number Density
• Phase Selection Options
• Initial Stage of Intermediate Phase Formation- Interface Control

7. (a) large Ni- base single-crystal droplet after 20ºC/min cooling from a very pure melt; (b)/(c)
laser processed Al-26%Si, laser velocity 100 mm/s (b) and 500 mm/s (c), showing fine
equiaxed silicon crystals surrounded by a-Al cells and distributed in a fibrous eutectic matrix
(plane view); (d)/(e) AlY7Fe5 droplets, cooled in the DTA with two intermetallic particles
surrounded by a eutectic matrix (d) or water quenched with numerous intermetallic particles (e);
(f) AlY7Fe5 melt spun ribbon after an isothermal annealing treatment for 10 min at 275ºC,
density of nanocrystals is greater than 1022 m-3.
a) b) c)
d) e) f)

8. Mechanical Properties
A. L. Greer, “Metallic Glasses”, Science, 267 (1995) pp. 1947

9. HIERARCHY OF NON-EQUILIBRIUM
INCREASING
TRANSFORMATION
RATE
FULL DIFFUSIONAL
EQUILIBRIUM
LOCAL INTERFACIAL
EQUILIBRIUM
METASTABLE
LOCAL INTERFACIAL
EQUILIBRIUM
INTERFACIAL
NON-EQUILIBRIUM

10. Phase equilibria
G = H  TS
G = 0 at Equilibrium
For example in Liquid-Solid
GS  GL = (HS  HL)  Tm (SS  SL) = 0
Gf = Hf  Tm Sf = 0
Then, Sf = (Hf)/T Tm
At other temperatures
Gf = Hf  T Sf = Hf  T(Hf/ Tm)
Gf =
where T is the undercooling and a correction for heat capacity is
neglected
G = Gibbs Free Energy
H = Enthalpy
T = Temperature
S = Entropy
Hf (Tm  T)
Tm
=
Hf T
Tm

12. For alloy solutions
G = G(T, P, ni, nj, …)
dG = VdP  SdT +
where
also, i = i
0 + RT ln ai i = standard state, ai = activity = i Xi
i = activity coefficient, Xi = mole fraction
For an ideal solution, i = 1
For alloy formation (i.e. mixing) at constant T and P, for each phase
Gm = (1  XB) A + XB A
so: Gm = (1  XB) A + XB A
At equilibrium: A
L = A
S and B
L = B
S
G
ni
|T, P, nj
= i (chemical potential)
T, P, nj
G
ni
|T, P, nj
dni +
G
nj
|T, P, ni
dnj
T, P, nj T, P, ni

15. Nanostructure Considerations
The nanoscale is often reported as a linear dimension but the
important interfacial effects should be considered in terms of the
interfacial area per unit volume (A/V). For example, for a sphere
A/V=3/r=3x107 m-1 for r = 100 nm
This is significant!
Interfacial effects can be included as
The increment in free energy due to interfaces is represented by the
Gibbs-Thomson relation as:
G= = ( r1
-1 +r2
-1) = 2/r
Depending on the relative magnitude of γ for each phase the phase
stability can be modified.
  idAiidniSdTVdPdG 

16. G
T  T
n

T
n

n

α stable at macroscale
γα < γβ
Here the stability of  is enhanced at the nanoscale

17. G
T T
n

T
n

n

α stable at macroscale
γα > γβ
Here there is a reversal of phase stability at the nanoscale

18. G
G
1/r
T
L
 




Tm

Tm

Tm(r) = Tm() - 2slVm/(rSm)

20. Driving Free Energies
Reaction Process Free Energy Typical value
(J/mol)
Remarks
Crystallization Hf T/Tm 3x103
Mixing or
Interdiffusion
RT(xAlnxA +
xBlnxB)
5x103
Ideal solution behavior
Oxidation G0
= RTlnK 5x104
– 5x106
G0
formation of oxide
Sublimation/
Deposition
Hv T/Ts 104
– 105
Hv – sublimation enthalpy,
Ts - sublimation temperature
Grain growth 2/r 20 for r = 1m
2x103
for r = 10 nm
 = 1 J/m2
Precipitation RTln(C/C0) 104
C/C0 ~ 10
Cold work (stored
energy)
Gb2
102
– 103
G – shear modulus,
b – Burgers vector,
 – dislocation density
T = 1000 K

23. Rapid Solidification Process: Melt-Spinning (MS)
Cooling rate:
V ~ 50m/s
 mKa
VaT
/102.1
;
4


Typically:
V  33~55m/s
i.e. 3 105
~ 7105(K/s)

25.   vGrrrG  3
3
424 
 
 



 

kT
nG
CnC l exp
 ** nCSJ SL
0
G(r)
1 2 3
r[-Gv]
-1
4r
2

4r
2
r
3
Gv/3
4r
3
Gv/3
TH
mVmT
vG
r
f






2
2*
223
22316
23
316
TfH
mVmT
vG
*
G










 













kT
G
C
a
r
a
D
J l
L *
exp
*4
2
2
2


26. • Homogeneous Nucleation
• Nucleation rate
• Heterogeneous
Nucleation
 θfaa
nn sv







 







kTΔG
a
exp
kT
ΔG
νexpJ 2
v
3
slA
v

nv

30. t
D
V
2



32. Kinetics of Nanocrystallization

































RT
vG
RT
DQ
VV exp1exp0













RT
DQ
VV exp0
2
4


IDV 
t
D
V
2









3
exp1
43
JtV
X

kT
ctD
rr

 3
0
3
 







kT
θfΔG
expΩJ
*
ii

33. Annealing of Melt-Spun Al88Y7Fe5
Crystallization of -Al
nanocrystals occurs at
temperatures below
273ºC and the growth of
the nanocrystals is
impeded when the
diffusion field
impingement occurs
Isotherm at 245ºC 10 min 30 min 100 min
200 250 300 350 400
-8
-6
-4
-2
0
2
245
o
C
X
Primary Crystallization
Onset: 273
o
C
T(
o
C)
Temperature (
o
C)

34. • Critical cooling rate – kinetics analysis
• Existing models
• Formulated in terms of reduced variables
• Based upon steady-state kinetics
• Treat polymorphic crystallization
• Basic relations:
 0exp /( )B T T  
0 0
exp
( )(1 )
r r
r
r I I
B T
T T T

 
  
  
3
2
16
exp
3( )
r
r
r r r
T
I
G T
 

 
  
 
/mS R  
/r mT T T
m
 03 a
kT
D 

35. 4 6
( , ) 1 exp ( )/ 10r r r rX T R Y T R 
     
 (1 exp / )r
r r r
r
T
U G T

   
3
1
4
( ) ( ) ( `) `
3 r r
x
r r r
T T
Y T I x dx U x dx
  
  
  
 

36. 0.6 0.8
1
Tf
-16

37. Melt-spun sample
Annealed:150ºC, 10 min

39. XSm
Gf
(kJ/g-atom)
Al-fcc LiquidAl Sm11 3
Al-fcc
GL

Al Sm
GL
 11 3
Free energy curves
Common tangent line
Parallel tangent lines
Tangent line on liquid

41. Medium range order sites
• Fluctuation Electron Microscopy (FEM)
• Nanoscale coherent beam diffraction
• Identification of defective planar
arrangements
• Planar spacing indicates pure Al composition
• Solute in remaining volume
• Y and Fe are rarely coordinated
• Y has large Al CN
• Fe tightly bound to Al
• FEM results seem to indicate solute
containing portions are closer to DRP
AlY
Fe

42. Median Range
Order (MRO)
Seeded Nucleation
Cryst
al
MR
O
MR
O
42

43. Nucleation Kinetics
*
exp r
SS
B
G
J Z
k T

 
  
 
2
*2 4
0 3
4
8
V a
B
G v
r DC a Z
k T
 
 
 
 
*2 *
0
4 3
4 2
exp
8
V a r
SS
B
B
r DC G v G
J
k Ta k Ts
 

  
  
 
expt SSJ J
t
 
  
 

44. Nucleation characteristics
• Initial increase
• Balance between steady state
rate and transient effect
• Steady state region
• Subject to energetics and the
available site density
• Saturation
• Sites subject to nucleation
• Sites incorporated during
growth

45. steady state nucleation rate
Nanocrystal number density vs.
annealing time at
237°C, 240°C, 245°C, 247°C.
τ

46. Steady State Nucleation Model
 Volume-dependent
nucleation rate:
 Accounts for the observed
 Nucleation rate, but on the
wrong side of the maximum.
*2 3
0
24 3
4 2 16
exp
38
V a
SS
B VB
r DC G v
J
k T Ga k Ts
  

  
  
 
exp
r
SS
B
G
J Z
k T

 
  
 
8
4
3
2
4
0
2*


Tk
vG
ZaDCr
B
aV
 
----Al88Y6Fe5Cu1
----Al88Y7Fe5
----Al87Y7Fe5Cu1

47. Impact of Transient Kinetics
• Initially low rates
• Higher temperatures can be
reached during constant
heating
• Increases toward JSS
• Saturation limits nucleation
expt SSJ J
t
 
  
 
3 4
2 4 2
0
161 B
v a
k T a
Z D G C v
 


 

10 K/min
 = 0 exp[B/(TT0)]
r
kT
D
6


48. r
 *
rΔG
 rG
kT

Nucleation Process: Uphill Struggle Enabled by Fluctuations
r* =-2γ/ΔGv = -2γTm/ΔHv ΔT

49. Transient Nucleation
During isothermal annealing of metallic glasses, a finite period
at the very beginning of annealing is expected during which
the steady state distribution of clusters assumed in classical
nucleation theory are established.

50. *
expSS
G
J Z
kT

 
  
 
2
1/ 2Z 
*
2
expSS
G
J
Z kT



 
  
 
*
2 1
ln( ) ln( ) ( )SS
G
J
Z k T



 
  
 
*
expSS
G
J Z
kT

 
  
 
intercept slope
Theory 60.93 -1
Expt Fit 60.03±1.81 -0.80±0.24

51. Size Distribution
Plot size histogram of the
size distribution
Scale Conversion
According to the time length
of annealing, convert the
size scale to the time scale
Image analysis
Collect size distribution data
from TEM negatives on
isothermally annealed
samples
Expression for N(t)
Fit N(t) as a polynomial or
exponential function
N(t) vs. t
Calculate: N(t) = SNi and
plot N(t) vs t
Expression for I(t)
The nucleation rate is the
time derivative of the
number of nuclei:
I(t) = dN(t)/dt
Parabolic Growth
u = dr/dt = 0.5kt-1/2
Use the radius of the largest
particle formed during
different annealing time to
calculate k.

52. Heterogeneous Catalysis
Impurity Particles
1023 x 102 = 1025 m-3 0.1%
Precursor Liquid Phase Separation
Site Density
Thermodynamics
Homophase/Heterophase Catalysis
Local Structure
Impurity Cores

53. Key Lessons
• Metastable vs. Unstable
• Closed vs. Open (Driven) Systems
• Stochastic (Probabilistic) vs. Deterministic Kinetics
• Dynamic vs. Static Conditions
• Nanostructures are Nucleation-Controlled
(i.e. high nucleation rate and slow growth)
 Precursor Reactions for Synthesis

54. Summary
• Nucleation limited kinetics allows for
deep undercoolings to Tg in bulk
volumes- a nanostructure precursor
• Primary Nanocrystalline Reactions
• Transient heterogeneous
nucleation kinetics
• Growth limited kinetics
control
• Microstructure information
is essential for modeling
• Kinetic control offers flexibility in
structure selection BUT also a
challenge to processing modeling-
(Multicomponent Alloys, Solute
Partitioning