1. 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

2. 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

3. 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!

4. 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

5. where G is the Gibbs free energy
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

6. 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.

7. A Phase Diagram Which Includes a Typical Disorder to Order Transformation

8. High Temperature, disordered phase (FCC, cF4)
Low Temperature, ordered phase (L10, tP4)

9. Order Parameter

10. 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:
since G, the Gibbs free energy is a minimum at equilibrium

11. Order Parameter as a Function of TL
There are two distinct ways that L may vary with temperature.

12. This behavior is called a “first order” phase transition
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.

13. This behavior is called a “higher order” phase transition
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.

14. The Order Parameter in Ferromagnetic Transitions is the Magnetization, M

15. 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

16. 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

17. 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

18. Using the following terms we can quantify the ordering:b

19. a b
Structure factor

20. Specific Cases:
a) random a b

21. Diffraction Pattern of A2 or BCC Structure

22. Specific cases:
b) complete order

23. Diffraction Pattern of B2 or CsCl Structure

24. A2
Superlattice peaks, or reflections B2

25. 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 = L = 1

26. Transformation
Microstructure

27. 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

28. 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

29. Heterogeneous Ordering in an FePd Alloy
untransformed
Massive ordering
untransformed
From Klemmer

30. 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.

31. Homogeneous Ordering Transformation of a Particle
FePt L10 Particle

32. Heterogeneous Ordering Transformation of a Particle
FePt L10 Particle

33. Heterogeneous and Homogeneous Ordering in Polycrystalline Sample

34. The L1o
Transformation

35. The FCC to L1o Disorder to Order Transformation
tetragonal
There are superlattice reflections from the ordering as well as split reflections due to the new tetragonal structure

36. Since the lattice parameters and symmetry change during the transformation there will be changes in the diffraction pattern.
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

37. If the transformation is discontinuous or heterogeneous, there will be a time during which both the FCC phase and L1o tetragonal phase is present
FCC
L1o
Note the splitting in the 311

38. DISCONTINUOUS or Heterogeneous
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
Note the two phase equilibria at 6 and 8 hr.
DISCONTINUOUS or Heterogeneous
Ka1 and Ka2 observed because of the large 2q angle

39. CONTINUOUS or Homogeneous
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

40. Crystallographic
Domains

41. The Crystallography of the L10 Formation
Co or Pt Pt Co
L10 CoPt
FCC a (CoPt)
Ordering Temp. < 825oC
3.75 Å b a c
3.79 Å
3.69 Å
Easy
Axis
There are changes in the translational symmetry and in the point group symmetry

42. Let’s first look at the translational domains
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 domains
Co Pt

43. Translation vector is 1/2 back and 1/2 up 1/2[101]
Anti-phase translation
C axis
Anti-Phase Boundary
Translation vector is 1/2 back and 1/2 up 1/2[101]

44. FePd, after Zhang and Soffa
Translational Domains (Anti-phase)
FePd, after Zhang and Soffa

45. The Three Structural Domains (Variants) of L1o
Changes in the point group symmetry:
Structural Domains
The Three Structural Domains (Variants) of L1o

46. FePd, after Zhang and Soffa
Translational Domains (Anti-phase)
Structural Domains (Variants)
FePd, after Zhang and Soffa

47. Bo Bian
FePt particle

48. FePd Alloys
Microstructures
Domain Structures

49. Phase diagram of FePd alloy
Fe or Pd Fe Pd
c-axis
3.723Å
3.852Å

50. Structure of L10 materials
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 Fe Pd
C3 axis
C2 axis
C1 axis
Twin boundary

51. Polytwinned microstructure
Structural 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 variant
C1 variant
(111)
(110)
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.

52. 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.
[100]
19.8o
[010]
[001]
Surface normal [1, 7, 19]
70.4o
87.3o
25.4o
45.0o
63.65o
DW1
DW2
Fresnel under-focus
Fresnel over-focus
Surface orientation
Fresnel in-focus
C axis orientation projection
In the plane of observation
Schematic diagram of
magnetization directions
[130]
[120]

53. Differential Scanning Calorimetry of Ordered Magnetic Alloys

55. DSC Traces and the Kissinger Plot for FePt (Barmak, Kim, Svedberg, Howard)
(oC/min)
Tpeak
(oC) 20
395 40
410 80
426
* F : Constant Heating Rate

56. DSC Traces and the Kissinger Plot for CoPt (Barmak, Kim, Svedberg, Howard)
(oC/min)
Tpeak
(oC) 20
517 40
531 80
544

57. DSC measurement of Curie Temperature FePd FCC and L10
455oC
450oC
419oC
399oC
340oC
320oC

58. M-T measurement of Tc for FePd FCC 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)

59. Kinetics

60. Phase Diagram of FePd FCC  L10 on cooling
Curie temperature (Tc) of Ordered FePd alloy (L10).
Phase diagram, ASM International
FCC  L10 on cooling

61. C-Curve Kinetics of FePd
Driving Force ~ DHvDT/Tc
Temperature Tc
Long time because of small DT
Long time because of small amount of diffusion
after Guschin, 1987
time
After Klemmer

62. Application:
Patterning by IONs

63. CrPt3 – Example of Order/Disorder Magnetic/NM
Magnetic (Ordered)
Non-Magnetic (Disordered) Cr
Random
3/4 Pt
1/4 Cr Pt
Order Parameter vs Ion Dose
Order Parameter vs Ion Dose
Magnetic Properties vs Ion Dose

64. Ordered Alloys with a Magnetic/Non-Magnetic Transition

65. Application
FePt Thin Films

66. 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

67. Plan view TEM <001> 50nm 50nm 55nm 530 C deposition
Average grain size ~10-15nm
In-plane XRD
Glass
MgO 8nm
FePt ~ 9 nm
110
200

68. Application
Self Organized Media

69. Ordered FePt particles
Questions: will very small size particles order? Can ordering occur without sintering?…etc. etc.

70. 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

72. Extra
Stuff

73. Now we will look at cases with V1 < 0
We start with BCC derivative structures

75. We move onto FCC Derivative Structures

76. Thermodynamics
Stat. Mech

77. 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:
Where P is the partition function, the sum is over all possible energy levels and b = 1/kT

78. 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)

79. The Excess Configurational Gibbs free energy of a partially ordered solid solution can be shown to be:

80. X 
The equilibrium order parameter l is determined by noting where the curve and the line intersect.

81. l
Critical temperature
Temperature 
This represents a higher order transition. Just like the para to ferromagnetic transition

83. Specific cases:
c) incomplete order

84. L
Fhkl =L(fA - fB)

85. Kinetics

86. How fast does a phase form
Kinetics
How fast does a phase form
This is often more important than what phase is the equilibrium one!
I = K exp( -DG*/kT)
I is the rate of nucleation
DG* is barrier to nucleation
(all precipitation reactions have a barrier to their initiation)

87. 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

88. Typical plots are as shown below
The slope is -Q/k
Ln Rate
1/T

89. Another important equation that has this form is the one for the temperature dependence of the diffusion coefficient
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

90. Time-Temperature-Transformation
No transformation
Transformation
nearly complete
The lower region follows Arrhenius’ law. Why not the upper?

91. Look at the nucleation rate equation
I = K exp( -DG*/kT)
As the temperature approaches the transition temperature, Dg* gets larger and larger because it is equal to
DG* = 16 p s3 / 3 Dgv2
and Dgv goes to zero at the transition temperature

92. Time-Temperature-Transformation
No transformation
Transformation
nearly complete
Importance of quench rate
Knee of the curve, etc

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

94. 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.

95. Back to the Nucleation rate equation
DG* = 16 p s3 / 3 Dgv2
Note the importance of the surface energy term, s
and the driving force term, DGv
Let us look at Dgv
How do we obtain this value?
From the Free Energy Curves!

96. we must look at the surface energy term!
Note that the value of Dgv 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, Dg*
is smaller!