atomic ordering in alloys david e. laughlin alcoa professor of physical metallurgy
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Atomic Ordering in AlloysDavid E. Laughlin
ALCOA Professor of Physical Metallurgy
Materials Science and Engineering Department
Electrical and Computer Engineering Department
Data Storage Systems Center
Carnegie Mellon University
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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
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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!
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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
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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
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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.
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A Phase Diagram Which Includes a Typical Disorder to Order Transformation
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High Temperature, disordered phase
(FCC, cF4)
Low Temperature, ordered phase
(L10, tP4)
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Order ParameterWhen 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, , is one which has an equilibrium value, so that we can always write:
0G
P,T
since G, the Gibbs free energy is a minimum at equilibrium
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Order Parameter as a Function of T
There are two distinct ways that L may vary with temperature.
L
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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 in this type of transformation.
L
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This behavior is called a “higher order” phase transition. At Tc the disordered and ordered phases do not coexist.
There is no latent heat of transformation in this type of transformation.
L
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The Order Parameter in Ferromagnetic Transitions is the Magnetization, M
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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, B2L = 0 1 L 0
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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
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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
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2
1
2
1
2
1 :sites
000 :sites
on B finding ofy probabilit theis p
on A finding ofy probabilit theis p
on B finding ofy probabilit theis p
on A finding ofy probabilit theis p
B
A
B
A
Using the following terms we can quantify the ordering:
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1pp
1ppBA
BA
))()(exp((
)(exp(,
lkhiffffThus
ffon
ffon
lwkvhui2fF
BABA
BA
BA
ihkl
BABAhkl
BA
BA
pppp F
pp :sites the
pp :sites the
Structure factor
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Specific Cases:
a) random
5.0XX if )ff(F
)fXfX(2F evenlkh
0F odd lkh
:case BCC theis This
))]lkh(iexp(1)[fXfX(F
Xpp
Xpp
BABAhkl
BBAAhkl
hkl
BBAAhkl
BBB
AAA
))lkh(i)(exp(fpfp(fpfpF BB
AA
BB
AA
hkl
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Intensity (%)
2 q (° )
20 25 30 35 40 45 50 55 60 65 70 75 80 84
0
10
20
30
40
50
60
70
80
90
100(44.35,100.0)
1,1,0
(64.52,13.3)
2,0,0(81.64,22.7)
2,1,1
Diffraction Pattern of A2 or BCC Structure
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Specific cases:
b) complete order
BAhkl
BAhkl
BAhkl
BB
AA
ffFeven is lkh if
ffF odd is lkh if
))lkh(iexp(ffF
0p 1p
0p 1p
))lkh(i)(exp(fpfp(fpfpF BB
AA
BB
AA
hkl
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Intensity (%)
2 q (° )
20 25 30 35 40 45 50 55 60 65 70 75 80 84
0
10
20
30
40
50
60
70
80
90
100
(30.96,25.2)
1,0,0
(44.35,100.0)
1,1,0
(55.06,5.3)
1,1,1 (64.52,13.5)
2,0,0
(73.27,5.3)
2,1,0
(81.64,23.4)
2,1,1
Diffraction Pattern of B2 or CsCl Structure
BAhkl ffF
BAhkl ffF
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Intensity (%)
2 q (° )
20 25 30 35 40 45 50 55 60 65 70 75 80 84
0
10
20
30
40
50
60
70
80
90
100(44.35,100.0)
1,1,0
(64.52,13.3)
2,0,0(81.64,22.7)
2,1,1
Intensity (%)
2 q (° )
20 25 30 35 40 45 50 55 60 65 70 75 80 84
0
10
20
30
40
50
60
70
80
90
100
(30.96,25.2)
1,0,0
(44.35,100.0)
1,1,0
(55.06,5.3)
1,1,1 (64.52,13.5)
2,0,0
(73.27,5.3)
2,1,0
(81.64,23.4)
2,1,1
Superlattice peaks, or reflections
A2
B2
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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
Intensity (%)
2 q (° )
20 25 30 35 40 45 50 55 60 65 70 75 80 84
0
10
20
30
40
50
60
70
80
90
100(44.35,100.0)
1,1,0
(64.52,13.3)
2,0,0(81.64,22.7)
2,1,1
Intensity (%)
2 q (° )
20 25 30 35 40 45 50 55 60 65 70 75 80 84
0
10
20
30
40
50
60
70
80
90
100
(30.96,25.2)
1,0,0
(44.35,100.0)
1,1,0
(55.06,5.3)
1,1,1 (64.52,13.5)
2,0,0
(73.27,5.3)
2,1,0
(81.64,23.4)
2,1,1
Intensity (%)
2 q (° )
20 25 30 35 40 45 50 55 60 65 70 75 80 84
0
10
20
30
40
50
60
70
80
90
100
(30.96,9.1)
1,0,0
(44.35,100.0)
1,1,0
(55.06,1.9)
1,1,1
(64.52,13.5)
2,0,0
(73.27,1.9)
2,1,0
(81.64,23.4)
2,1,1
L = 0 L = 0.6 L = 1
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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
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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
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Massive ordering
From Klemmer
untransfo
rmed
untransfo
rmed
Heterogeneous Ordering in an FePd Alloy
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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
L = 0 < L < L < L < L < L =1
time
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Homogeneous Ordering Transformation of a Particle
FePt L10 Particle
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Heterogeneous Ordering Transformation of a Particle
FePt L10 Particle
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L = 0.5
L = 0.5
Heterogeneous and Homogeneous Ordering in Polycrystalline Sample
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Intensity (%)
2 q (° )
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
10
20
30
40
50
60
70
80
90
100(43.32,100.0)
1,1,1
(50.45,45.0)
2,0,0
(74.13,22.0)
2,2,0(89.94,23.2)
3,1,1
(95.15,6.7)
2,2,2
Intensity (%)
2 q (° )
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
10
20
30
40
50
60
70
80
90
100
(35.08,15.5)
1,1,0
(43.75,100.0)
1,1,1
(50.45,31.6)
2,0,0
(51.99,14.3)
0,0,2
(57.27,7.2)
2,0,1
(64.27,4.9)
1,1,2(74.13,8.3)
2,2,0(75.37,15.8)
2,0,2
(79.78,2.5)
2,2,1
(84.73,2.2)
3,1,0
(90.24,18.5)
3,1,1
(92.64,8.7)
1,1,3
(96.36,8.0)
2,2,2
There are superlattice reflections from the ordering as well as split reflections due to the new tetragonal structure
The FCC to L1o Disorder to Order Transformation
tetragonal
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Since the lattice parameters and symmetry change during the transformation there will be changes in the diffraction pattern.
2
2
2
22
2 c
l
a
kh
d
1
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
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Intensity (%)
2 q (° )
80 85 90 95 98
0
10
20
30
40
50
60
70
80
90
100
(84.73,2.2)
3,1,0
(90.24,18.5)
3,1,1
(92.64,8.7)
1,1,3
(96.36,8.0)
2,2,2
Intensity (%)
2 q (° )
80 85 90 95 98
0
10
20
30
40
50
60
70
80
90
100
(84.73,2.1)
3,1,0
(89.94,27.1)
3,1,1
(95.15,7.9)
2,2,2
FCC
L1o
Note the splitting in the 311
If the transformation is discontinuous or heterogeneous, there will be a time during which both the FCC phase and L1o tetragonal phase is present
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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
K1 and K2 observed because of the large 2q angle
Note the two phase equilibria at 6 and 8 hr.
DISCONTINUOUS or Heterogeneous
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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
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Co or Pt Pt Co
L10 CoPtFCC (CoPt)Ordering Temp. < 825oC
3.75 Å
3.75
Å
3.75 Å
ba
c
3.79 Å
3.69
Å3.79 Å
ba
c
EasyAxis
The Crystallography of the L10 Formation
There are changes in the translational symmetry and in the point group symmetry
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FCC para
L1o-para
L1o-ferro
FCC para to L1o para
48/16 = 3 structural domains
4 to 2 eq. Sites = 2 orientation domains per structural domain
6 DOMAINS in TOTAL due to FCC to L10
Let’s first look at the translational domainsCo Pt
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Anti-Phase Boundary
C axis
Translation vector is 1/2 back and 1/2 up 1/2[101]
Anti-phase translation
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Translational Domains (Anti-phase)
FePd, after Zhang and Soffa
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The Three Structural Domains (Variants) of L1o
Structural Domains
Changes in the point group symmetry:
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Structural Domains (Variants) Translational Domains (Anti-phase)
FePd, after Zhang and Soffa
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Bo Bian
FePt particle
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Phase diagram of FePd alloy
Fe or Pd
Fe Pd
c-axis
3.7
23
Å
3.852Å
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Structure of L10 materials
Fe Pd
C3 axis
C2 axis
C1 axisTwin boundary
Structural variants are formed due to symmetry breaking down. FCC-> L10
Magnetic domains are formed when paramagnetic L10 phase transforms into Ferromagnetic phase.
Magnetic properties depends on the coupling between these two type of domains.
Fe or Pd
Twin boundary =Magnetic domain wall
M// c axis
M
Magnetic domain wall
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Polytwinned microstructureStructural variants are formed due to symmetry breaking down. FCC-> L10
C3 axis
C2 axis
C1 axis(101)
<111>
(011)Three variants can form polytwinned structure to minimize the strain energy.
C3 variant
C2 variantC1 variant
(111)(110)
(011)(101)
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.
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Micro-Magnetics in polytwinned microstructure
[130] [120]
Trace analysis can be used to determine the surface orientation of the polytwinned microstructure and the c axis orientation of the twin variants.
[100]
19.8o
[010]
[001]
Surface normal [1, 7, 19]
70.4o87.3o
)101(A)110(C
]010[p
)110(B
)101(D
25.4o
45.0o63.65o
]100[p
]001[pDW1
DW2
Fresnel under-focus Fresnel over-focus
Surface orientation
Fresnel in-focus
C axis orientation projectionIn the plane of observation
Schematic diagram of magnetization directions
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EXOTHERMIC
0 STH Thus
0SSS
STH
0G when :ST-HG
Order toDisorder
L1 toFCC
disorderorder
0
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DSC Traces and the Kissinger Plot for FePt (Barmak, Kim, Svedberg, Howard)
0 100 200 300 400 500 600 700
-16
-12
-8
-4
0
4
8
12
16
20
Fe0.50
Pt0.50
1000 nm
Exo
the
rm D
ow
n (
mW
)
Temperature (oC)
20 oC/min
40 oC/min
80 oC/min
(oC/min)
Tpeak
(oC)
20 395
40 410
80 426
16.6 16.8 17.0 17.2 17.4-14.4
-14.0
-13.6
-13.2
-12.8
-12.4
Q = 1.7 ± 0.1 eV
ln(
/Tp2 )
[1/K
s]
1/ (kBT
p) [1/eV]
* : Constant Heating Rate
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0 100 200 300 400 500 600 700-8
-4
0
4
8
Co0.45
Pt0.55
1000 nm
Exo
the
rm D
ow
n (
mW
)
Temperature (oC)
20 oC/min
40 oC/min
80 oC/min
DSC Traces and the Kissinger Plot for CoPt (Barmak, Kim, Svedberg, Howard)
(oC/min)
Tpeak
(oC)
20 517
40 531
80 544
14.2 14.4 14.6-14.8
-14.4
-14.0
-13.6
-13.2
-12.8
Q = 2.8 ± 0.2 eV
ln(
/Tp2 )
[1/K
s]
1/ (kBT
p) [1/eV]
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DSC measurement of Curie Temperature FePd FCC and L10
DSC scan of FePd with different composition
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
200 250 300 350 400 450 500 550 600
Temperature (oC)
Heat
capaci
ty (
arb
itra
ry u
nit
)
Fe50Pd_ FCC
Fe50Pd_ L10
Fe55Pd_ FCC
Fe55Pd_ L10
Fe60Pd_ FCC
Fe60Pd_ L10
455oC
450oC
419oC
399oC
340oC
320oC
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M-T measurement of Tc for FePdFCC and L10
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)
Fe-50, 55, 60 at% Pd M-T
0
0.2
0.4
0.6
0.8
1
1.2
200 250 300 350 400 450 500 550Temperature (oC)
Rel
etiv
e m
om
ent
50_FCC_1
50_FCC_2
50_FCC_3
50_L10
55_FCC_1
55_FCC_2
55_L10
60_FCC_1
60_FCC_2
60_FCC_3
60_L10
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Phase Diagram of FePd
Curie temperature (Tc) of Ordered FePd alloy (L10).Phase diagram, ASM International
FCC L10 on cooling
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C-Curve Kinetics of FePd
time
Tem
per
atu
re
Tc
Driving Force ~ HvT/Tc
after Guschin, 1987
Long time because of small T
Long time because of small amount of diffusion
After Klemmer
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CrPt3 – Example of Order/Disorder Magnetic/NM
a
b
c
xyz
Magnetic (Ordered)a
b
c
xyz
Non-Magnetic (Disordered)Cr
Pt
Random3/4 Pt1/4 Cr
Order Parameter vs Ion DoseOrder Parameter vs Ion Dose Magnetic Properties vs Ion Dose
150
100
50
0
Mr,M
s (e
mu/
cc)
1011
1012
1013
1014
1015
1016
Ion Dose Density (1/cm2)
7000
6000
5000
4000
3000
2000
1000
0
Hc
(Oe)
Ms Mr Hc
1.0
0.8
0.6
0.4
0.2
0.0
Lon
g-R
ange
Ord
er P
aram
eter
, S
1011
1012
1013
1014
1015
1016
Ion Dose (B+/cm
2)
No Implant
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Ordered Alloys with a Magnetic/Non-Magnetic Transition
Alloy Atomic Ordering Disordered Ordered Disordered Ordered Disordered OrderedTemp. (deg C) Structure Structure Magnetic Magnetic Tc (deg C) Tc (deg C)
O Mag. -> D NonO Non -> D Mag.High -> Low Ms
Tc < Room TempVanadium Alloys
VPt3 1015 fcc L12 / D022 P F/F n/a -30 / -60
Chromium AlloysCrPt3 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 AlloysMnPt3 1000 fcc L12 P F n/a 100
MnxAl1-xCy, tau 850 fcc L10 P F n/a
Iron AlloysFePt3 1352 fcc L12 F A -100FeAl 1310 bcc B2 F P
Nickel AlloysNiPt 645 fcc L10 F A -158
Ni3Mn 510 fcc L12 F, low Ms F, high Ms
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L10 High Anisotropy Media Toward Ultra High Density of 1 terabits/inch2
Substrate
C-axes
001 fiber texture
Magnetic HysteresisPerpendicular
Anisotropy
Si or Glass
underlayer
FePt 001
Soft Magnetic Layer will be inserted
Grains
Small Grainmagnetic isolation
Minimizing FCC phaseLowering ordering Temperature
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20 30 40 50 60 70
INT
EN
SIT
Y (
a.u
.)
Plan view TEM
a
b
c
xyz
<001>
55nm55nm
530 C depositionAverage grain size ~10-15nm
55nm
In-plane XRD
50nm 50nm
GlassMgO 8nm
FePt ~ 9 nm 110200
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Ordered FePt particles
Questions: will very small size particles order? Can ordering occur without sintering?…etc. etc.
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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
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Now we will look at cases with V1 < 0
We start with BCC derivative structures
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We move onto FCC Derivative Structures
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Statistical Models for Solid Solutions
From statistical thermodynamics (for example Guggenheim’s text on Mixtures) we know that we can write:
kukk )Eexp()g(E P where
PlnkTFG
Where P is the partition function, the sum is over all possible energy levels and = 1/kT
After Lupis, Chemical Thermodynamics of Materials
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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)
g(Ek) is the degeneracy factor if the kth state, which is the number of states that have the same energy
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The Excess Configurational Gibbs free energy of a partially ordered solid solution can be shown to be:
( ) ABBBAA
2C
CCCCC
EEE2
1E where
)]1ln()1()1ln()1(2ln2[2
RT)1(E
2
ZnG
STESTHG
)N 2n (here, kT
ZE
)-(1
)(1ln
:obtain wealgebra someafter Thus
0G
that know wemequilibriuAt
0
C
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ykT2
ZE of valuesh variousorigin wit the
through lines and kT2
ZE versustanhplot we
yZE
kT2
ykT2
ZEtanh thus
tanh2)-(1
)(1ln
1
1
1
The equilibrium order parameter l is determined by noting where the curve and the line intersect.
X
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l
Temperature
Critical temperature
This represents a higher order transition. Just like the para to ferromagnetic transition
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))lkh(i)(exp(fpfp(fpfpF BB
AA
BB
AA
hkl
Specific cases:
c) incomplete order
)fXfX(2F
X2pp and X2pp but
f)pp(f)pp(F
)fpfp(fpfpF
evenlkhFor
BBAAhkl
BBB
AAA
BBB
AAA
hkl
BB
AA
BB
AA
hkl
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))()(exp(( lkhiffff BABA BABA
hkl ppppF
)pp(
)ff)(pp(F
toreduceswhich
)fpfp(fpfpF
oddlkhFor
AA
BAAA
hkl
BB
AA
BB
AA
hkl
LFhkl =L(fA - fB)
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KineticsHow 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)
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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
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Typical plots are as shown below
1/T
Ln Rate
The slope is -Q/k
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Another important equation that has this form is the one for the temperature dependence of the
diffusion coefficient
)RT
Qexp(DD D
O
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
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Time-Temperature-Transformation
T
Time
No transformationTransformation nearly complete
The lower region follows Arrhenius’ law. Why not the upper?
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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 3 / 3 gv2
and gv goes to zero at the transition temperature
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Time-Temperature-Transformation
T
Time
No transformationTransformation nearly complete
Importance of quench rate
Knee of the curve, etc
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))kt(exp(1X n
This equation is sometimes called the Johnson/Mehl/ Avrami equation
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)X1(tnkdt
dXThus
))kt(exp(1X
1nn
n
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.
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Back to the Nucleation rate equation
G* = 16 3 / 3 gv2
Note the importance of the surface energy term,
and the driving force term, Gv
Let us look at gv
How do we obtain this value?
From the Free Energy Curves!
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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!