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

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 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

Order Parameter as a Function of T

There are two distinct ways that L may vary with temperature.

L

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

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

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, B2L = 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

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:

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

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

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

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

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

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

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

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

Massive ordering

From Klemmer

untransfo

rmed

untransfo

rmed

Heterogeneous Ordering in an FePd Alloy

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

Homogeneous Ordering Transformation of a Particle

FePt L10 Particle

Heterogeneous Ordering Transformation of a Particle

FePt L10 Particle

L = 0.5

L = 0.5

Heterogeneous and Homogeneous Ordering in Polycrystalline Sample

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

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

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

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

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

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

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

Anti-Phase Boundary

C axis

Translation vector is 1/2 back and 1/2 up 1/2[101]

Anti-phase translation

Translational Domains (Anti-phase)

FePd, after Zhang and Soffa

The Three Structural Domains (Variants) of L1o

Structural Domains

Changes in the point group symmetry:

Structural Domains (Variants) Translational Domains (Anti-phase)

FePd, after Zhang and Soffa

Bo Bian

FePt particle

Phase diagram of FePd alloy

Fe or Pd

Fe Pd

c-axis

3.7

23

Å

3.852Å

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

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.

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

EXOTHERMIC

0 STH Thus

0SSS

STH

0G when :ST-HG

Order toDisorder

L1 toFCC

disorderorder

0

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

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]

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

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

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

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

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

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

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

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

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

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

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

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

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

l

Temperature

Critical temperature

This represents a higher order transition. Just like the para to ferromagnetic transition

))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

))()(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)

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)

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

1/T

Ln Rate

The slope is -Q/k

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

Time-Temperature-Transformation

T

Time

No transformationTransformation 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 3 / 3 gv2

and gv goes to zero at the transition temperature

Time-Temperature-Transformation

T

Time

No transformationTransformation nearly complete

Importance of quench rate

Knee of the curve, etc

))kt(exp(1X n

This equation is sometimes called the Johnson/Mehl/ Avrami equation

)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.

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!

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!