Order- Disorder Transformation

Presented By:

Zaahir Salam

M.Tech NS&T

Order and Disorder designate the presence or absence of

some symmetry or correlation in a many-particle system.

In physics systems typically are ordered at low

temperatures; upon heating, they undergo one or several

phase transitions into less ordered states.

Examples for such an order-disorder transition are:

The melting of ice: solid-liquid transition, loss of crystalline order

The demagnetization of iron by heating above the Curie

temperature: ferromagnetic-paramagnetic transition, loss of

magnetic order.

What Does Order and Disorder Mean?

The degree of freedom that is ordered or disordered can be translational (crystalline

ordering), rotational (ferroelectric ordering), or a spin state (magnetic ordering).

Depending on how the correlations decay with distance, one speaks of long-range

order or short-range order.

If a disordered state is not in thermodynamic equilibrium, it is quenched disorder.

For instance, a glass is obtained by quenching (supercooling) a liquid.

Some other quenched states are called spin glass, orientational glass and the opposite

of quenched disorder is annealed disorder.

Short-range order (SRO):

Tendency for A-(B-) atoms to be

preferencially surrounded by B- or A-atoms

(correlation functions).

Long-range order (LRO):

Differentiation „” of probalilities of

particular sublattice sites being occupied

by particular atoms.

Ordered Phases

If the atoms in a substitutional solid solution, are randomly arranged each

atom position is equivalent and the probability that any given site in the

lattice will contain an A atom will be equal to the fraction of A atoms in the

solution XA , similarly XB for the B atoms.

In such solutions PAB, the number of A-B bonds,

PAB = Na z XAXB (bonds mol-1)

where Na Avogadro's number, and z is the number of bonds per atom.

PROBABILISTIC

OCCUPATION

Positionally ordered

Probabilistically ordered

A B

Probability of

occupation:

A 50%

B 50%

is the difference between A-B bond energy and the average of A-A and B-B .

= 0 , ∆Hmix =0 - Ideal Solution.

< 0 the atoms in the solution will prefer to be surrounded by atoms of

the opposite type and this will increase PAB.

Whereas > 0, PAB Will tend to be less than in a random solution.

∆Hmix= Ω Xa Xb

Ω= Na z

Ω <0 the no of A-B bonds is greater than this, the solution is said to contain

short range order (SRO)

s = PAB – PAB(random)

PAB(max) –PAB(random)

PAB(max)- max no. of bonds possible

PAB(random)- no. of bonds in random solution.

Cu-Au alloys

Cu and Au – FCC structured and totally miscible.

At Higher Temp: Cu or Au can occupy any site and the lattice can be

considered as fcc with a “random” atom at each lattice( Fig 1.20 a).

At Lower Temp: XCu=XAu =0.5 i.e 50/50 Cu/ Au mixture. (ordered

structure) (Fig 1.20 b)

CuAu Superlattice.

Cu3 Au Superlatice.

Cu3Au

CuAu

The necessary condition for the formation of a superlattice, i.e. an ordered solid solution, is

that dissimilar atoms must attract each other more than similar atoms. In addition, the alloy

must exist at or near a composition which can be expressed by a simple formula such as AB,

A3B or AB3. The following are common structures:

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

Disorder to Order Transformation

Order Parameter

Disorder Order

Thermodynamic parameter, called the order parameter, ƞ which can be used as a measure

of the extent of the transformation.

This order parameter ƞ, is one which has an equilibrium value, so that we can always

write:

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

0,

PT

G

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.

G H T S

↓T Ordered

↑T Disordered

A Phase Diagram Which Includes a Typical Disorder to Order Transformation

High Temperature,

disordered phase (FCC, cF4)Low Temperature, ordered

phase (L10, tP4)

Ordering Transformations:

What Happens when Ordered Single Crystal( CuZn or Cu3Au) is Heated from low temperatures to above the disordering temperature.

Here We quantify the degree of order in crystal by defining Long Range Order parameter L,

L=1 : Fully Ordered Alloy (all Atoms occupy their correct sites).

L=0 : Completely Random Distribution

1

A A

A

r XL

X

rA → probability that A sublattice is occupied with the right atom

XA → mole fraction of A in the alloy

L → Long Range Order

TC

Long Range Order

1

0

Short Range Order

CuZn Type

T

Second Order

~ First Order

At 0K : Min Free Energy by choosing Highly order Arrangement (L=1) :

Lowest internal energy. Configurational Entropy =0;

At higher temp: Min free energy state will contain some disorder. (diffusion)

Entropy effects become increasingly important with rising temperature so L

continuously decreases until above some critical ( Tc) L=0.Results for CuZn and Cu3Au Superlattices

Above Tc it is impossible to distinguish separate sublattices extending over long distances and L=0;

Solid solutions which have a negative enthalpy of mixing (Hmix < 0) prefer unlike nearest

neighbours → show tendency for ordering phases at low temperature.

No Sudden change in order at Tc : U and H

will be continous

Substantial change in order at Tc :

(U ,H )disorder state> (U,H) Ordered State

Two Mechanism to create disordered superlattice ordered superlattice

MechanismNucleation and Growth

Continuous increase in SRO

1. Continuous increase in SRO

Local rearrangements occurring homogenously throughout the crystal

which finally leads to LRO.

2. Nucleation and Growth

Due to an energy barrier to the formation of ordered domain,

transformation takes place with the help of nucleation and growth.

The two are equivalent to spinodal decomposition and precipitation

as mechanisms for formation of coherent zones in alloys with

∆Hmix >0 .

1st mechanism: Second order transformations or at very high

supercooling below Tc.

2nd mechanism: more common.

Atoms can order on either the A or B sublattice , the independently nucleated domains will be “out of phase”

Domains Subsiquintly grow together a boundary will form (antiphase domain boundary (APB)) across which

atoms will have wrong neighbours.

Examples of common ordered structures (superlattices)

L12: Cu3AuL10: CuAu (I)DO3: Fe3Al

Structure Examples

L20 CuZn, FeCo, NiAl, FeAl, AgMg

L12 Cu3Au, Au3Cu, Ni3Mn, Ni3Fe, Ni3Al, Pt3Fe

L10 CuAu, CoPt, FePt

DO3 Fe3Al, Fe3Si, Fe3Be, Cu3Al

DO19 Mg3Cd, Cd3Mg, Ti3Al, Ni3Sn

CuAu

Lattice parameter(s)a = 3.96Å, c =

3.67Å

Space Group P4/mmm (123)

Strukturbericht notation L10

Pearson symbol tP4

Other examples with this

structureTiAl

CuAu

Cu

Au

Cu

Au

Wyck

off

positio

n

x y z

Au1 1a 0 0 0

Au2 1c 0.5 0.5 0

Cu 2e 0 0.5 0.5

Cu3AuLattice parameter(s) a = 3.75 Å

Space Group Pm-3m (221)

Strukturbericht notation L12

Pearson symbol cP4

Other examples with this structure Ni3Al, TiPt3

Cu3Au

Cu

Au

Cu

Au

Thank You