# Order Disorder Phase Transformation

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Order Disorder Phase Transformation document sample

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```							Atomic Ordering in Alloys
David E. Laughlin
ALCOA Professor of Physical Metallurgy
Materials Science and Engineering Department
Electrical and Computer Engineering Department
Data Storage Systems Center

Carnegie Mellon University
The phrase disorder to order or order / disorder
in alloys is an ambiguous term. Depending on
your background it may mean different things.
For example if I say “disordered alloy”
some people think about an amorphous material
as opposed to a crystalline one
others about a random distribution of atoms on a
crystal lattice as opposed to an ordered distribution

alloy or paraelectric alloy!
Today’s talk will focus on the ordering of two
(or more) types of atoms on an underlying
“lattice”. There will be some application to
magnetic ordering as well!
Topics of today’s talk include:
order parameter and its measurement
microstructure of the transformation
crystallography and domains
thermodynamics / kinetics
Applications
An atomic disorder to order transformation is a
change of phase. It entails a change in the
crystallographic symmetry of the high temperature,
disordered phase, usually to a less symmetric low
temperature atomically ordered phase.
This can be understood from a basic equation of
phase equilibria in the solid state, namely the
definition of the Gibbs Free Energy:
G = H - TS
where     G is the Gibbs free energy
H is the enthalpy
S is the entropy of the material
G = H - TS
At constant T and P the system in equilibrium will
be the one with the lowest Gibbs Free Energy

At high temperatures the TS term dominates the
phase equilibria and the equilibrium phase is more
“disordered” (higher entropy) than the low
temperature equilibrium phase.
Examples: Liquid to Solid
Disorder to Order
In both cases the high temperature equilibrium phase
is more “disordered” than the low temperature
“ordered” phase.
A Phase Diagram Which Includes a Typical
Disorder to Order Transformation
High Temperature,   Low Temperature,
disordered phase     ordered phase
(FCC, cF4)          (L10, tP4)
Order Parameter
When an disorder to order transformation occurs
there is usually a thermodynamic parameter, called
the order parameter, which can be used as a measure
of the extent of the transformation.
This order parameter, h, is one which has an
equilibrium value, so that we can always write:
 G 
     0
 h 
    T ,P
since G, the Gibbs free energy is a minimum at equilibrium
Order Parameter as a Function of T

There are two
distinct ways
that L may
vary with       L
temperature.
This behavior is called a “first order” phase
transition. At Tc the disordered and ordered
phases may coexist.

There is a latent
heat of
transformation
L
in this type of
transformation.
This behavior is called a “higher order” phase
transition. At Tc the disordered and ordered
phases do not coexist.

L

There is no
latent heat of
transformation
in this type of
transformation.
The Order Parameter in Ferromagnetic
Transitions is the Magnetization, M
How Do We Measure the Atomic Order
Parameter?
We will do this for the easiest case or disorder to
order, namely the BCC to CsCl transition

BCC, A2                 CsCl, B2
L = 0                   1  L  0
In the disordered case (BCC) the probability of
an A atom being at the 000 site is the same as
being at the ½½½ site.

There are two
equivalent sites per unit
cell (of volume a3) in this
structure
In the ordered case (B2) the probabilities are not
equal: there is a tendency for A atoms to occupy
one site and B atoms to occupy the other site.
In the fully ordered case, all the A atoms are on
one type of site (e.g. 000) and all the B atoms are
on the other type (e.g. ½ ½ ½ )

There is only one
equivalent site per unit
cell (of volume a3) in this
structure. This is a loss
in translational
symmetry
              Using the following terms we
can quantify the ordering:
            
                   sites : 000
             
111
                           sites :
222
p  is the probabilit y of finding A on 
A

p is the probabilit y of finding B on 
B


p is the probabilit y of finding A on 
A


p is the probabilit y of finding B on 
B

                
                                   p  p 1
A

B

                       p  p 1
A

B

                
                

Fhkl   f i exp(2i( hu  kv  lw )          Structure factor
 ,

on the  sites : p f A  p f B
A        B

on the  sites : p f A  p f B
A        B

Thus Fhkl  p f A  p f B  ( p f A  p f B )(exp(i( h  k  l))
A        B          A        B
Fhkl  p f  p f  (p f  p f )(exp(i(h  k  l))
A
 A
B
 B
A
 A
B
 B

                   Specific Cases:
             
a) random
              p   p  X A
A    A

             
p   p  X B
B    B

             
Fhkl  (X A f A  X Bf B )[1  exp(i(h  k  l))]
This is the BCC case:
h  k  l  odd Fhkl  0
h  k  l  even Fhkl  2(X A f A  X Bf B )
 Fhkl  (f A  f B ) if X A  X B  0.5
Intensity (%)                  1,1,0
Diffraction Pattern of
100
(44.35,100.0)
A2 or BCC Structure
90

80

70

60

50

40

30                                                                            2,1,1
(81.64,22.7)
2,0,0
20
(64.52,13.3)
10
2 q (°)
0
20    25   30   35   40    45    50   55   60   65        70   75     80    84
Fhkl  p f  p f  (p f  p f )(exp(i(h  k  l))
A
 A
B
 B
A
 A
B
 B

Specific cases:
b) complete order
p 1 p  0
A

A


p  1 p   0
B       B

Fhkl  f A  f B exp(i(h  k  l))
if h  k  l is odd Fhkl  f A  f B
if h  k  l is even Fhkl  f A  f B
Intensity (%)                        1,1,0

100
(44.35,100.0)         Diffraction Pattern of
90                                                     B2 or CsCl Structure
80

70

60
Fhkl  f A  f B

Fhkl  f A  f B
50

40
1,0,0
2,1,1
30               (30.96,25.2)                                                             (81.64,23.4)
2,0,0
20
1,1,1      (64.52,13.5)      2,1,0
10                                                (55.06,5.3)                  (73.27,5.3)
2 q (°)
0
20    25      30      35   40    45    50       55    60       65        70    75      80   84
Intensity (%)                        1,1,0
(44.35,100.0)
100

90

80

70

A2   60

50

40

30                                                                                           2,1,1
(81.64,22.7)
2,0,0
20
(64.52,13.3)
10
2 q (°)
0
Intensity (%)
20     25      30      35   40 1,1,0
45    50       55    60       65        70    75      80   84
(44.35,100.0)
100

90

80

70
Superlattice peaks,
60
or reflections
50

B2   40

30
1,0,0
2,1,1
(30.96,25.2)                                                             (81.64,23.4)
2,0,0
20
1,1,1      (64.52,13.5)      2,1,0
10                                                (55.06,5.3)                  (73.27,5.3)
2 q (°)
0
20    25      30      35   40    45    50       55    60       65        70    75      80   84
It can be shown that the intensity of a
superlattice reflection is I = L2 F2
Thus the order parameter can be obtained
from the relative intensities of the superlattice
reflections
L=0        L = 0.6                L=1                                                                                                                                                                Intensity (%)                        1,1,0
Intensity (%)                  1,1,0                                                             Intensity (%)                      1,1,0                                                                                                        (44.35,100.0)
(44.35,100.0)                                                                                        (44.35,100.0)                                                                   100
100                                                                                              100
90
90                                                                                               90
80
80                                                                                               80
70
70                                                                                               70

60                                                                                                                                                                                                              60
60

50                                                                                               50                                                                                                             50

40                                                                                               40                                                                                                             40
1,0,0
2,1,1                                                                                                          2,1,1                                                                                                            2,1,1
30                                                                                               30                                                                                                             30               (30.96,25.2)
(81.64,22.7)                                                                                                   (81.64,23.4)                                                                                                     (81.64,23.4)
2,0,0                                                                                                         2,0,0                                                                                                            2,0,0
20                                                                                               20                                                                                                             20
(64.52,13.3)                                                        1,0,0                                     (64.52,13.5)                                                                                                     (64.52,13.5)
1,1,1                        2,1,0
10                                                                                                                (30.96,9.1)                       1,1,1                        2,1,0
10                                                                                                             10                                                (55.06,5.3)                  (73.27,5.3)
2 q (°)                                                   (55.06,1.9)                  (73.27,1.9)             2 q (°)                                                                                                          2 q (°)
0                                                                                                0                                                                                                              0
20    25   30   35   40    45    50   55   60   65        70   75     80    84                   20    25     30     35   40    45    50       55    60       65        70    75      80   84                   20    25      30      35   40    45    50       55    60       65        70    75      80   84
The Long Range Order parameter is a
macroscopic parameter, in that it is a measure
for the entire sample that is examined by the x-
rays or electrons. It may or may not be
homogeneous within the sample. We will now
look at this is some detail.
Broadly speaking there are two kinds of
transformations that occur in materials:

Homogeneous
Heterogeneous
In a homogeneous transformation the entire
system (sample) transforms at the same time.
All regions of the sample are transforming

In a heterogeneous transformation there are
regions which have transformed and regions
which have not transformed
Heterogeneous Ordering in an FePd Alloy

From Klemmer
Homogeneous Ordering
Transformation of a Particle
L = 0 < L              <     L     <      L     < L
< L =1

time

The colors represent the degree of order in the grains.
Note that the way the order is represented is homogeneous.
Homogeneous Ordering
Transformation of a Particle

FePt L10 Particle
Heterogeneous Ordering
Transformation of a Particle

FePt L10 Particle
Heterogeneous and Homogeneous
Ordering in Polycrystalline Sample

L = 0.5

L =
0.5
The FCC to L1o Disorder to Order
Transformation
Intensity (%)         1,1,1                                                                                  Intensity (%)        1,1,1
(43.32,100.0)                                                                                                (43.75,100.0)
100                                                                                                          100

90                                                                                                           90

80                                                                                                           80

70                                                                                                           70

60                                                                                                           60

50
2,0,0
(50.45,45.0)                                                                   50                                                     tetragonal
40                                                                                                           40                            2,0,0
3,1,1                                                (50.45,31.6)
30                                                          2,2,0                                            30
(74.13,22.0)        (89.94,23.2)                                                                                                    3,1,1
1,1,0                 0,0,2                        2,0,2
20                                                                                                           20                                                                          (90.24,18.5)
(35.08,15.5)           (51.99,14.3)                (75.37,15.8)
2,2,0                  1,1,32,2,2
2,2,2                                                      2,0,1
1,1,2
10                                                                                  (95.15,6.7)                                                  (57.27,7.2)                    2,2,1 3,1,0 (92.64,8.7)
(74.13,8.3)                (96.36,8.0)
10                                           (64.27,4.9)
2 q (°)                                                                   (79.78,2.5)
(84.73,2.2)
2 q (°)
0                                                                                                            0
30   35    40      45       50     55   60   65   70   75        80   85    90      95     100               30   35   40     45      50       55   60   65    70    75    80    85    90     95   100

There are superlattice reflections from the
ordering as well as split reflections due to the new
tetragonal structure
Since the lattice parameters and symmetry
change during the transformation there will be
changes in the diffraction pattern.
1 h k 2
l2   2
For the
2
   2
 2            tetragonal phase
d     a    c
The 111FCC reflection does not split, but the
200FCC reflection as well as others such as the
311FCC do split due to the tetragonality of the
new phase.
That is the 311L1o does not have the same d
spacing as the 113L1o
Intensity (%)

100
If the transformation is
90

80
FCC                                       discontinuous or
70

60
heterogeneous, there
50

40
will be a time during
3,1,1
30

20
(89.94,27.1)

2,2,2
which both the FCC
10

0
3,1,0
(84.73,2.1)
(95.15,7.9)
2 q (°)   phase and L1o
80             85            90                          95              98

Intensity (%)                                                                                 tetragonal phase is
100

90
L1o                                       present
80

70

60

50

40
Note the
30

20
3,1,1
(90.24,18.5)
1,1,3                2,2,2
splitting in the
3,1,0                      (92.64,8.7)        (96.36,8.0)

311
10
(84.73,2.2)
2 q (°)
0
80             85            90                          95              98
The 311L10 increases in
Note the two phase
equilibria at 6 and 8 hr.        intensity and the 311FCC
decreases. However the
peak position does not
change much showing
pretty much the
equilibrium composition
and hence order
parameter

DISCONTINUOUS or Heterogeneous

K1 and K2 observed because of the large 2q angle
Here, the 311L10
increases in intensity and
the 311FCC decreases.
However the peak
position changes
continuously showing
that the initial L1o was
very similar to the FCC
phase.
No obvious two phase
equilibrium

CONTINUOUS or Homogeneous
The Crystallography of the L10 Formation
Ordering Temp. < 825oC
FCC  (CoPt)                                       L10 CoPt
Easy
Axis

c                                c

b                                 b
a                               a

Co or Pt        Pt     Co

There are changes in the translational symmetry and in
the point group symmetry
FCC para

FCC para to L1o para
L1o-para          48/16 = 3 structural domains
4 to 2 eq. Sites = 2 orientation
domains per structural domain
L1o-ferro
6 DOMAINS in TOTAL due to
FCC to L10

Let’s first look at the
Co Pt              translational domains
Anti-phase translation

C axis

Anti-Phase Boundary

Translation vector is 1/2 back and 1/2 up 1/2[101]
Translational Domains (Anti-phase)

FePd, after Zhang and Soffa
Changes in the point group symmetry:

Structural Domains

The Three Structural Domains (Variants) of L1o
Structural Domains (Variants)    Translational Domains (Anti-
phase)

FePd, after Zhang and Soffa
Bo Bian

FePt particle
Phase diagram of FePd alloy

Fe or Pd

c-axis        3.852Å

3.723Å
Fe       Pd
Structure of L10 materials
Structural variants are formed due to symmetry breaking down. FCC-> L10
C3 axis
C1 axis
Twin boundary

Fe       Pd
Fe or Pd

Magnetic domains are formed when paramagnetic L10 phase transforms into
Ferromagnetic phase.
M
M// c axis

Magnetic domain wall

Magnetic properties depends on the coupling between these two type of domains.

Twin boundary =Magnetic domain wall
Polytwinned microstructure
Structural variants are formed due to symmetry breaking down. FCC-> L10

C3 axis
(011)
Three variants can form polytwinned
C1 axis     structure to minimize the strain
(101)
energy.
C2 axis

<111>
(110)
(111)
C3 and C2 variants intersect at
(011) twin boundary.
C1 and C3 variants intersect at                                      C3 variant
(101) twin boundary.                                        (101)
(011)
C1 and C2 variants intersect at
(110) twin boundary.
C1 variant
C2 variant
Micro-Magnetics in polytwinned
microstructure
 Trace analysis can be used to determine the surface orientation of the
polytwinned microstructure and the c axis orientation of the twin variants.

[130]          [120]

Fresnel under-focus           Fresnel in-focus                          Fresnel over-focus

p[010] p[001]
Surface normal [1, 7, 19]
DW1
[010]
87.3o   70.4o
[100]
DW2
63.65o
45.0o
D(101)                               p[100]
25.4o                                            [001]
A(01 1 )
C (10 1 )   B(110)                                  19.8o

Schematic diagram of          C axis orientation projection                    Surface orientation
magnetization directions      In the plane of observation
FCC to L10
Disorder to Order

G  H - TS : whenG  0
H  TS
S  Sorder  Sdisorder  0
Thus H  TS  0
EXOTHERMIC
DSC Traces and the Kissinger Plot for FePt
(Barmak, Kim, Svedberg, Howard)
20                                                        -12.4
          Tpeak                                              Q = 1.7 ± 0.1 eV
-12.8
16         (oC/min)      (oC)

ln(/Tp ) [1/Ks]
Exotherm Down (mW)

-13.2
12            20         395

2
-13.6
40         410
8                                                        -14.0
80         426
4                                                        -14.4
16.6    16.8 17.0 17.2       17.4
1/ (kBTp) [1/eV]
0
-4
Fe0.50Pt0.50 1000 nm
-8
o
20 C/min
-12               o
40 C/min
o
-16             80 C/min

0      100       200       300                      400            500     600          700
*  : Constant Heating Rate                                                                         o
Temperature ( C)
DSC Traces and the Kissinger Plot for CoPt (Barmak,
Kim, Svedberg, Howard)
-12.8
          Tpeak                                           Q = 2.8 ± 0.2 eV
8                                                      -13.2
(oC/min)      (oC)

ln(/Tp ) [1/Ks]
Exotherm Down (mW)

-13.6
20         517

2
-14.0
4            40         531
-14.4
80         544
-14.8
14.2      14.4        14.6
1/ (kBTp) [1/eV]
0

Co0.45Pt0.55 1000 nm
-4                o
20 C/min
o
40 C/min
o
80 C/min
-8
0     100        200     300                  400            500       600         700
o
Temperature ( C)
DSC measurement of Curie
Temperature FePd FCC and L10                                                    455oC
DSC scan of FePd with different composition
0.55
Fe50Pd_FCC
0.5
Fe50Pd_L10
Heat capacity (arbitrary unit)

0.45      Fe55Pd_FCC                           450oC

0.4      Fe55Pd_L10
419oC
Fe60Pd_FCC
0.35
Fe60Pd_L10
0.3                                 399oC

0.25
340oC
0.2
0.15           320oC

0.1

0.05
0
200   250     300      350      400      450      500   550   600
Temperature (oC)
M-T measurement of Tc for FePd
FCC and L10
Fe-50, 55, 60 at%Pd M-T
1.2

1
50_FCC_1
50_FCC_2
50_FCC_3
Reletive moment

0.8
50_L10
55_FCC_1
0.6                                                            55_FCC_2
55_L10
60_FCC_1
0.4
60_FCC_2
60_FCC_3
60_L10
0.2

0
200     250   300      350     400       450   500     550
o
Temperature ( C)

Fe-50at.%Pd            Fe-55at.%Pd             Fe-60at.%Pd
FCC                        748 K (475oC)          698 K (425oC)           618 K (345oC)
L10                        723 K (450oC)          668 K (395oC)           593 K (320oC)
Phase Diagram of FePd

Curie temperature
(Tc) of Ordered
FePd alloy (L10).
Phase diagram, ASM International

FCC  L10
on cooling
C-Curve Kinetics of FePd
Driving Force ~ HvT/Tc
Tc

Long time because of small T
Temperature

Long time because of small
amount of diffusion
after Guschin, 1987

time

After Klemmer
CrPt3 – Example of Order/Disorder Magnetic/NM
Cr Magnetic (Ordered)
a
Random Non-Magnetic (Disordered)
a
Pt                                           b
3/4 Pt
b
1/4 Cr
y x                                                                                                y x
z                                                                                                  z
c                                                                                                     c

Order Parameter vs Ion Dose                                                                                         Magnetic Properties vs Ion Dose
7000
Long-Range Order Parameter, S

1.0
6000
No Implant                                                                               150

Mr,Ms (emu/cc)
0.8
5000

Hc (Oe)
0.6                                                                                                                                                                              4000
100

3000
0.4

50
Ms                                                     2000

0.2
Mr
Hc                                                     1000

0.0                                                                                                    0                                                                         0
11          12        13            14        15            16
11            12            13          14            15        16                              10          10        10            10        10            10
10            10            10          10            10        10
2
Ion Dose (B /cm )
+         2                                                            Ion Dose Density (1/cm )
Ordered Alloys with a Magnetic/Non-Magnetic Transition
Alloy        Atomic Ordering Disordered Ordered Disordered      Ordered Disordered Ordered
Temp. (deg C)   Structure Structure Magnetic      Magnetic T c (deg C) T c (deg C)
O Mag. -> D Non
O Non -> D Mag.
High -> Low Ms
Tc < Room Temp
VPt3             1015          fcc    L12 / D022      P           F/F        n/a      -30 / -60

Chromium Alloys
CrPt3             1130          fcc       L12          P            I         n/a       ~ 200
CrPd              570           fcc       L10          P            F         n/a        350
Cr2Pd3             505           fcc       L12          P            F         n/a        350
(CrxMn1-x)Pt3                       fcc       L12          P            F         n/a

Manganese Alloys
MnPt3               1000          fcc       L12          P            F         n/a        100
MnxAl1-xCy, tau        850           fcc       L10          P            F         n/a

Iron Alloys
FePt3             1352         fcc        L12          F            A                    -100
FeAl              1310         bcc        B2           F            P

Nickel Alloys
NiPt              645           fcc       L10            F           A       -158
Ni3Mn              510           fcc       L12       F, low Ms   F, high Ms
L10 High Anisotropy Media
Toward Ultra High Density of 1 terabits/inch2
C-axes
FePt 001
001 fiber
underlayer
texture
Si or Glass
Grains

Soft Magnetic Layer
will be inserted

Substrate

Magnetic Hysteresis
Perpendicular
Anisotropy                  Small Grain
magnetic isolation
Minimizing FCC phase
Lowering ordering Temperature
Plan view TEM
<001>
c

50nm                    50nm
55nm
z
yx

b
a

55nm             55nm
530 C deposition
Average grain size ~10-15nm                                      In-plane XRD

INTENSITY (a.u.)
110
FePt ~ 9 nm                                                     200
MgO 8nm
Glass
20    30   40     50    60            70
Ordered FePt particles

Questions: will very small size particles order? Can
ordering occur without sintering?…etc. etc.
Summary
We have looked at several of the aspects of the
atomic disorder to order phase change in alloys:
Thermodynamics
Phase Diagrams
Transformations
Kinetics
Crystallography
Diffraction
Applications
Now we will look at cases with V1 < 0

We move onto FCC Derivative Structures
Statistical Models for Solid Solutions
After Lupis, Chemical Thermodynamics of Materials

From statistical thermodynamics (for example
Guggenheim’s text on Mixtures) we know that we can
write:

G  F  kT ln P
where P   g(Ek ) exp(E k )
uk

Where P is the partition function, the sum is over all
possible energy levels and  = 1/kT
g(Ek) is the degeneracy factor if the kth state, which is
the number of states that have the same energy

Thus in order to obtain expressions for the
thermodynamic functions we need to know the energy
levels and how the system is distributed over the
energy levels, viz we need to know the:
Hamiltonian (ENERGY)
Distribution function (ENTROPY)
The Excess Configurational Gibbs free energy of a
partially ordered solid solution can be shown to be:

G C  H C  TSC  E C  TSC
Zn             RT
G 
C
E(1   ) 
2
[2 ln 2  (1  ) ln(1  )  (1  ) ln(1  )]
2              2
where E  (E AA  E BB )  E AB
1
2
G C
At equilibriu m we know that        0

Thus after some algebra we obtain :
(1  ) ZE
ln                     (here, 2n  N 0 )
(1- )   kT
 (1  ) 
ln 
 (1- )   2 tanh 1 
         
1     ZE
thus tanh               y
2kT
2kT
         y
ZE
X

1        ZE
we plot tanh  versus          and lines through
2kT
ZE
the origin with various values of       y
2kT
The equilibrium order parameter l is determined by noting where
the curve and the line intersect.
l                           Critical
temperature

Temperature 

This represents a higher order transition. Just
like the para to ferromagnetic transition
Fhkl  p f  p f  (p f  p f )(exp(i(h  k  l))
A
 A
B
 B
A
 A
B
 B

Specific cases:
c) incomplete order
For h  k  l  even
Fhkl  p f  p f  (p f  p f )
A
 A
B
 B
A
 A
B
 B

Fhkl  (p  p )f A  (p  p )f B
A

A

B

B


but p  p  2X A and p  p  2X B
A

A

B

B


Fhkl  2(X A f A  X Bf B )
Fhkl  p f  p f  ( p f  p f )(exp(i(h  k  l))
A
 A
B
 B
A
 A
B
 B

For h  k  l  odd
Fhkl  p f  p f  (p f  p f )
A
 A
B
 B
A
 A
B
 B

which reduces to
Fhkl  (p  p )(f A  f B )
A

A


L  (p  p )
A       A
       
Fhkl =L(fA - fB)
Kinetics
How fast does a phase form
This is often more important than what phase is the
equilibrium one!
I = K exp( -G*/kT)

I is the rate of nucleation
G* is barrier to nucleation
(all precipitation reactions have a barrier to their
initiation)
Let us look at the form of this
equation
rate = K exp( -Q/kT)
as T increases, the rate increases
or
as Q decreases, the rate increases
Q is called activation energy
The equation is Arrhenius’ law
Typical plots are as shown below

The slope is -Q/k

1/T
Another important equation that has this form
is the one for the temperature dependence of the
diffusion coefficient
 QD
D  D O exp(      )
RT
Here, QD is the activation energy for
diffusion which in substitutional solid
solutions is usually the sum of the activation
energies of the formation of vacancies and
the motion of vacancies
Time-Temperature-Transformation
Time

T

Transformation
No transformation                   nearly complete

The lower region follows Arrhenius’ law. Why not
the upper?
Look at the nucleation rate equation

I = K exp( -G*/kT)
As the temperature approaches the transition
temperature, g* gets larger and larger
because it is equal to
G* = 16  s3 / 3 gv2
and gv goes to zero at the transition
temperature
Time-Temperature-Transformation
Time

T

Transformation
No transformation              nearly complete

Importance of quench rate
Knee of the curve, etc
X  1  exp( (kt ) )                   n

This equation is sometimes called the Johnson/Mehl/
Avrami equation
X  1  exp((kt ) n )

dX
Thus     nk n t n 1 (1  X)
dt

Note that for t = 0, the rate is zero and for
large t, the rate goes to zero as well.

A maximum exists with respect to time.
Back to the Nucleation rate equation

G* = 16  s3 / 3 gv2

Note the importance of the surface
energy term, s

and the driving force term, Gv

Let us look at gv

How do we obtain this value?

From the Free Energy Curves!
Note that the value of gv is largest for the more stable
phase. At first sight it looks like this means that the
barrier to nucleation is smallest for the stable phase.

BUT

we must look at the surface energy term!

This term comes in as a cubic. This is the secret to why less
stable phases form faster than stable ones! It is almost
always because the surface energy term of the less stable is
smaller than that of the stable phase. Hence the value of
the barrier to nucleation, g*
is smaller!

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