Atomic Ordering in Alloys David E. Laughlin ALCOA Professor of Physical Metallurgy

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

12. 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.

13. 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.

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

15. CsCl, B2 BCC, A2 L = 0 1 L 0 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

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

19. b a a a a a a a a Structure factor

20. b a a a a a a a a Specific Cases: a) random

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. L = 0 L = 0.6 L = 1 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

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. L = 0 < L < L < L < L < L =1 time Homogeneous Ordering Transformation of a Particle 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 L = 0.5 L = 0.5

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. 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 FCC L1o

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

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

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

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

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

47. Bo Bian FePt particle

48. FePd Alloys Microstructures Domain Structures

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

50. Structural variants are formed due to symmetry breaking down. FCC-> L10 C3 axis C1 axis Twin boundary C2 axis Fe or Pd Magnetic domains are formed when paramagnetic L10 phase transforms into Ferromagnetic phase. Fe Pd Magnetic properties depends on the coupling between these two type of domains. M M// c axis Magnetic domain wall Twin boundary =Magnetic domain wall Structure of L10 materials