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
and others about a paramagnetic
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,
disordered phase
(FCC, cF4)
Low Temperature,
ordered phase
(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
L
with
temperature.
This behavior is called a “first order” phase
transition. At Tc the disordered and ordered
phases may coexist.
L
There is a latent
heat of
transformation
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
L=0
CsCl, B2
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:






 sit es : 000
111
 sit es :
222
p is the probability of finding A on 
A

p is the probability of finding B on 
B

pA is the probability of finding A on 
pB is the probability of finding B on 








p  p 1
A

B

p  p 1
A

B


Fhkl   f i e xp(2i( hu  kv  lw )
Structure factor
 ,
on the  sites : pAf A  pB f B
on the  sites : pAf A  pB f B
T hus Fhkl  pAf A  pB f B  ( pAf A  pB f B )(e xp(i( h  k  l))
Fhkl  p f  p f  (p f  p f )(exp(i(h  k  l))
A
 A
B
 B



B
 B
Specific Cases:



A
 A



p A  pA  X A
a) random
p B  pB  X B
Fhkl  (X A f A  X Bf B )[1  exp( i(h  k  l))]
T hisis theBCC 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
(44.35,100.0)
100
Diffraction Pattern of
A2 or BCC Structure
90
80
70
60
50
40
2,1,1
30
(81.64,22.7)
2,0,0
(64.52,13.3)
20
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B  1 p B  0
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
(44.35,100.0)
100
90
Diffraction Pattern of
B2 or CsCl Structure
80
Fhkl  f A  f B
70
60
Fhkl  f A  f B
50
40
1,0,0
30
2,1,1
(81.64,23.4)
(30.96,25.2)
20
1,1,1
(55.06,5.3)
10
2,0,0
(64.52,13.5)
2,1,0
(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
A2
70
60
50
40
2,1,1
30
(81.64,22.7)
2,0,0
(64.52,13.3)
20
10
2 q (°)
0
Intensity (%)
20
25
30
35
100
40 1,1,0
45
50
(44.35,100.0)
55
60
65
70
75
80
84
90
80
Superlattice peaks,
or reflections
70
60
50
B2
40
1,0,0
30
2,1,1
(81.64,23.4)
(30.96,25.2)
20
1,1,1
(55.06,5.3)
10
2,0,0
(64.52,13.5)
2,1,0
(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 (%)
Intensity (%)
Intensity (%)
1,1,0
(44.35,100.0)
100
1,1,0
(44.35,100.0)
100
90
90
80
80
70
70
60
60
50
50
40
40
2,1,1
(81.64,22.7)
30
2,0,0
(64.52,13.3)
20
90
80
70
60
50
40
2,1,1
(81.64,23.4)
30
20
10
2,0,0
(64.52,13.5)
1,0,0
(30.96,9.1)
10
2 q (°)
0
25
30
35
40
45
50
55
60
65
70
75
80
84
1,0,0
30
2,1,1
(81.64,23.4)
(30.96,25.2)
20
1,1,1
2,1,0
(55.06,1.9)
(73.27,1.9)
1,1,1
(55.06,5.3)
10
2,0,0
(64.52,13.5)
2,1,0
(73.27,5.3)
2 q (°)
0
20
1,1,0
(44.35,100.0)
100
2 q (°)
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
The Long Range Order parameter is a
macroscopic parameter, in that it is a measure
for the entire sample that is examined by the xrays 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 (%)
Intensity (%)
1,1,1
(43.32,100.0)
100
100
90
90
80
80
70
70
60
1,1,1
(43.75,100.0)
60
tetragonal
2,0,0
50
50
(50.45,45.0)
40
2,0,0
40
2,2,0
(74.13,22.0)
30
(50.45,31.6)
3,1,1
30
(89.94,23.2)
20
20
2,2,2
10
(95.15,6.7)
1,1,0
(35.08,15.5)
(90.24,18.5)
1,1,32,2,2
(51.99,14.3)
2,0,1
1,1,2
(74.13,8.3)2,2,1 3,1,0 (92.64,8.7)
(96.36,8.0)
(57.27,7.2)
(64.27,4.9)
(79.78,2.5)
(84.73,2.2)
10
2 q (°)
0
3,1,1
2,0,2
(75.37,15.8)
2,2,0
0,0,2
2 q (°)
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
l

 2
2
2
d
a
c
2
2
2
For the
tetragonal phase
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
FCC
90
80
70
60
50
40
3,1,1
30
(89.94,27.1)
20
2,2,2
3,1,0
(84.73,2.1)
10
(95.15,7.9)
2 q (°)
0
80
85
90
95
98
Intensity (%)
100
L1o
90
80
If the transformation is
discontinuous or
heterogeneous, there
will be a time during
which both the FCC
phase and L1o
tetragonal phase is
present
70
60
50
40
30
3,1,1
(90.24,18.5)
20
1,1,3
(92.64,8.7)
3,1,0
(84.73,2.2)
10
2,2,2
(96.36,8.0)
2 q (°)
0
80
85
90
95
98
Note the
splitting in the
311
Note the two phase
equilibria at 6 and 8 hr.
The 311L10 increases in
intensity and the 311FCC
decreases. However the
peak position does not
change much showing
that the initial L1o had
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
FCC  (CoPt)
Ordering Temp. < 825oC
L10 CoPt
Easy
Axis
c
c
a
b
b
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
Co Pt
6 DOMAINS in TOTAL due to
FCC to L10
Let’s first look at the
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 (Antiphase)
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
Twin boundary
C1 axis
Fe
Fe or Pd
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
(011)
C3 axis
(101)
C1 axis
Three variants can form polytwinned
structure to minimize the strain
energy.
C2 axis
<111>
C3 and C2 variants intersect at
(011) twin boundary.
C1 and C3 variants intersect at
(101) twin boundary.
C1 and C2 variants intersect at
(110) twin boundary.
(110)
(111)
(011)
C2 variant
(101)
C3 variant
C1 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 over-focus
Fresnel in-focus
p[010] p[001]
Surface normal [1, 7, 19]
DW1
87.3o
[010]
45.0o
p[100]
D(101)
25.4o
C(101)
Schematic diagram of
magnetization directions
[100]
DW2
63.65o
70.4o
A(011)
B(110)
C axis orientation projection
In the plane of observation
[001]
19.8o
Surface orientation
FCC t o L10
Disorder t oOrder
G  H - TS : when G  0
H  TS
S  Sorder  Sdisorder  0
T hus H  TS  0
EXOT HERMIC
DSC Traces and the Kissinger Plot for FePt
(Barmak, Kim, Svedberg, Howard)
-12.4
Tpeak
(oC)
12
20
395
40
410
80
426
8
Q = 1.7 ± 0.1 eV
-12.8
-13.2
2
16

(oC/min)
ln(/Tp ) [1/Ks]
Exotherm Down (mW)
20
4
-13.6
-14.0
-14.4
16.6
16.8 17.0 17.2
1/ (kBTp) [1/eV]
0
17.4
-4
Fe0.50Pt0.50 1000 nm
-8
o
20 C/min
o
40 C/min
o
80 C/min
-12
-16
0
*  : Constant Heating Rate
100
200
300
400
500
o
Temperature ( C)
600
700
DSC Traces and the Kissinger Plot for CoPt (Barmak,
Kim, Svedberg, Howard)
-12.8
20
517
40
531
80
544
Q = 2.8 ± 0.2 eV
-13.2
2
Exotherm Down (mW)
Tpeak
(oC)
ln(/Tp ) [1/Ks]

(oC/min)
8
4
-13.6
-14.0
-14.4
-14.8
14.2
0
14.4
14.6
1/ (kBTp) [1/eV]
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
o
Temperature ( C)
600
700
DSC measurement of Curie
Temperature FePd FCC and L10
455oC
DSC scan of FePd with different composition
Heat capacity (arbitrary unit)
0.55
0.5
Fe50Pd_FCC
Fe50Pd_L10
0.45
Fe55Pd_FCC
0.4
Fe55Pd_L10
0.35
Fe60Pd_FCC
0.3
450oC
419oC
Fe60Pd_L10
0.25
399oC
340oC
0.2
320oC
0.15
0.1
0.05
0
200
250
300
350
400
450
Temperature (oC)
500
550
600
M-T measurement of Tc for FePd
FCC and L10
Fe-50, 55, 60 at%Pd M-T
1.2
1
50_FCC_1
Reletive moment
50_FCC_2
50_FCC_3
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
o
450
500
550
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
Temperature
Long time because of small T
after Guschin, 1987
time
After Klemmer
Long time because of small
amount of diffusion
CrPt3 – Example of Order/Disorder Magnetic/NM
Cr Magnetic (Ordered)
a
Pt
Random Non-Magnetic (Disordered)
a
3/4 Pt
b
1/4 Cr
b
y x
y x
z
c
Order Parameter vs Ion Dose
Magnetic Properties vs Ion Dose
7000
1.0
6000
150
0.8
0.6
0.4
5000
4000
100
3000
Ms
Mr
Hc
50
0.2
2000
1000
0
0.0
10
11
10
12
10
13
10
+
14
10
2
Ion Dose (B /cm )
15
10
16
0
10
11
10
12
10
13
10
14
10
15
10
2
Ion Dose Density (1/cm )
16
Hc (Oe)
No Implant
Mr,Ms (emu/cc)
Long-Range Order Parameter, S
z
c
Ordered Alloys with a Magnetic/Non-Magnetic Transition
Alloy
O Mag. -> D Non
O Non -> D Mag.
High -> Low Ms
Tc < Room Temp
Vanadium Alloys
VPt3
Atomic Ordering Disordered Ordered Disordered
Temp. (deg C)
Structure Structure Magnetic
Ordered Disordered Ordered
Magnetic T c (deg C) T c (deg C)
1015
fcc
L12 / D022
P
F/F
n/a
-30 / -60
1130
570
fcc
fcc
L12
L10
P
P
I
F
n/a
n/a
~ 200
350
505
fcc
fcc
L12
L12
P
P
F
F
n/a
n/a
350
Manganese Alloys
MnPt3
MnxAl1-xCy, tau
1000
850
fcc
fcc
L12
L10
P
P
F
F
n/a
n/a
100
Iron Alloys
FePt3
FeAl
1352
1310
fcc
bcc
L12
B2
F
F
A
P
Nickel Alloys
NiPt
Ni3Mn
645
510
fcc
fcc
L10
F
F, low Ms
A
F, high Ms
Chromium Alloys
CrPt3
CrPd
Cr2Pd3
(CrxMn1-x)Pt3
L12
-100
-158
L10 High Anisotropy Media
Toward Ultra High Density of 1 terabits/inch2
C-axes
FePt 001
001 fiber
texture
underlayer
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>
50nm
c
50nm
55nm
z
yx
b
a
55nm
530 C deposition
Average grain size ~10-15nm
FePt ~ 9 nm
MgO 8nm
Glass
In-plane XRD
INTENSITY (a.u.)
55nm
110
20
30
200
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 start with BCC derivative structures
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
1
where E  (E AA  E BB )  E AB
2
G C
At equilibrium we know that
0

T hus aftersome algebra we obtain:
(1 ) ZE
ln

(1- )
kT
(here, 2n  N 0 )
 (1  ) 
  2 tanh1 
ln
 (1- ) 
ZE
1
thus tanh  
y
2kT
2kT

y
ZE
X
ZE
we plot tanh  versus
and lines through
2kT
ZE
theorigin with variousvaluesof
y
2kT
1
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

Fhkl  2(X A f A  X B f B )
B

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
No transformation
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
No transformation
Importance of quench rate
Knee of the curve, etc
Transformation
nearly complete
X  1  exp((kt) )
n
This equation is sometimes called the Johnson/Mehl/
Avrami equation
X  1  exp((kt) n )
dX
T hus
 nkn 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!