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

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

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

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

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

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.

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.

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, B2 L = 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

Using the following terms we can quantify the ordering: b

a b Structure factor

Specific Cases: a) random a b

Diffraction Pattern of A2 or BCC Structure

Specific cases: b) complete order

Diffraction Pattern of B2 or CsCl Structure

A2 Superlattice peaks, or reflections 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 L = 0 L = 0.6 L = 1

Transformation Microstructure

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

Heterogeneous Ordering in an FePd Alloy untransformed Massive ordering untransformed 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

The L1o Transformation

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

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

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

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

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

Crystallographic Domains

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

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

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]

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

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

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

Bo Bian FePt particle

FePd Alloys Microstructures Domain Structures

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

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

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.

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]

Differential Scanning Calorimetry of Ordered Magnetic Alloys

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

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

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

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)

Kinetics

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

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

Application: Patterning by IONs

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

Ordered Alloys with a Magnetic/Non-Magnetic Transition

Application FePt Thin Films

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 50nm 55nm 530 C deposition Average grain size ~10-15nm In-plane XRD Glass MgO 8nm FePt ~ 9 nm 110 200

Application Self Organized Media

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

Extra Stuff

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

We move onto FCC Derivative Structures

Thermodynamics Stat. Mech

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

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:

X  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

Specific cases: c) incomplete order

L Fhkl =L(fA - fB)

Kinetics

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)

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 Ln Rate 1/T

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

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

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

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

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

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 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!

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!