Order Disorder Transition

77

Chapter 5

Introduction to Order-Disorder Transitions

In the previous chapter, two new types of nanometer rod-shaped precipitates wereobserved; they were named QP and QC and seemed to be precursors of the stable Q-Al5Cu2Mg8Si7. Before a complete TEM and HREM study of these phases, subject of the nextchapter, an introduction to ordering mechanisms is required. Order-disorder transitions will beintroduced in the global framework of phase transitions (solid-liquid-gas, ferro-paramagnetic, ferro-para electric, superfluids, polymers), without enlarging the presentation tocritical phenomena.

Most of the approach presented in this chapter is based on the simple followingthermodynamic concept: for a closed system in thermal equilibrium, the transition is aconsequence of a compromise: the energy tends to order and the entropy associated to thetemperature tends to break the order. Different classifications of phase transitions will bepresented in section 5.1. Phenomenological Landau’s approach by thermodynamics will betreated in section 5.2. A more general approach by using statistical mechanics on an Isingmodel, as well as Monte Carlo simulations, will be treated in section 5.3. It will help us tointroduce the order parameters and approximate methods such as the Bragg-Williams method.Since we are interested in disordered nano-precipitates present in a matrix, the mostappropriate observation means, i.e. TEM diffraction and HREM will be treated in section 5.5to show their potential applications for the study of ordering mechanisms.

5. Introduction to Order-Disorder Transitions______________________________________________________________________

78

5.1 Classification of the Phase Transitions

5.1.1 Chemistry

A microscopic approach by crystal chemistry can provide a basis for the classificationof the phase transitions [118]. If a solid undergoes a phase transition at a critical temperatureTc by absorbing thermal energy, the transformed phase possesses higher internal energy, thebonding between neighboring atoms or units are weaker than in the low-temperaturephase.This results in a change in the nature of the first and second-nearest neighbor bonds.Phase transition in solids may be classified into three categories:

(1) Displacive transitions [119] proceed through a small distortion of the bonds(dilatational or rotational). The atomic displacements are reduced to 0.01-0.1Å and thespecific heat is low (few J/g). The main characteristic is the group-subgroup relationshipbetween the phases. This permits for example to clearly define an order parameter used for thethermodynamical description of the transition. These transitions can be of the first or secondorder (these terms will be explained in the next section).

(2) Reconstructive transitions [120] proceed through the breaking of the primary orsecondary bonds. These transitions were firstly described by Buerger [121]. They imply largeatomic displacements with 10-20% of distortion of the lattice, the specific heat is important(~kJ/g). These transformations are sluggish since the barrier of energy is high. The maincharacteristic is the absence of any group-subgroup relationship between the phases contrarilyto the case of Landau transitions (section 5.2). The transitions can even increase the symmetryof the high temperature phase. This transition occurs in many materials such as ZnS, C, H2O,Am, C, SiO2, TiO2. Bain transitions (BCC-FCC) and Buerger transitions (BCC-HCP) can bedescribed as reconstructive.

(3) Order-disorder transitions proceed through substitution between atoms possiblyfollowed by small atomic displacements. They are commonly found in metals and alloys butalso in some ceramics. Some of them keep a group-subgroup relationship, as for the CuZntransition (between BCC and simple cubic SC structure), others are also reconstructive as forAm, Fe, Co, ZnS or SiC (FCC-HCP). These transitions can be described with the help of alatent lattice common to the phases [120].

5.1.2 Thermodynamics

Let us consider a closed, isochore and diathermic system in thermal contact with a heatbath. This system is characterized by its free energy F (minimum at equilibrium), given by

E is the internal energy and depends on the bonding between the atoms. S is the entropy,characteristic of the disorder by S = kB.Log Ω (Ε), where Ω is the complexion number, i.e.number of configurations of the system for a given energy E. At low temperatures, the

F = E - T.S (5.1)

5.2. Landau s Phenomenological Approach______________________________________________________________________

79

entropic term is negligible and the system is driven by E (negative) which has its maximumabsolute value when the bonds of highest energy are formed (the system is ordered). At hightemperatures, the system is driven by T.S which is maximum for a disordered system.Therefore, it appears that T is the balancing coefficient between order and disorder: a phasetransition must exist at a critical temperature Tc. In this first approach we have voluntarilyneglected the fact that the internal energy of the system can fluctuate. Actually the system mustbe considered as a canonical ensemble (section 5.3.1).

Closed, expansible and diathermic systems are characterized by their free energy Gwhich remains continuous during the phase transition. However, thermodynamic quantitieslike entropy S, volume V, heat capacity Cp, the volume thermal expansivity α or thecompressibility β can undergo discontinuity. Ehrenfest classified the phase transitions infunction of the thermodynamic quantities that present a discontinuity. The order of thetransition is the same than the order of the derivation of G required to obtain a discontinuity:

If or has a discontinuity, the transition is of first order.

If , or has a

discontinuity, the transition is of second order. Higher order transitions would involve furtherdifferential quantities.

5.2 Landau’s Phenomenological Approach

A phenomenological treatment of phase transitions has been given by Landau in 1937[122]. The theory is based on the assumption that the free energy of the system is a continuousfunction that can be developed in a Taylor series near the critical temperature Tc, depending ona parameter called the order parameter, and noted ξ. This parameter is characteristic of thedegree of order. It can be the magnetization for ferro-paramagnetic transition, the polarizationfor ferro-paraelectric transition, or the percentage of atoms that are on their right sublattice foran order-disorder transition (for this type of transition, details will be given in section 5.4.2).

The main property of the free energy is to remain unchanged by the symmetry operationsof the highest symmetric phase implied in the transition. The development of the free energykeeps therefore only the even exponents of ξ

Let us assume that β and γ do not depend on the temperature. Since F is an increasingfunction with ξ at high temperatures (preponderance of the T.S component in F), we must haveγ > 0. If β > 0, the exponent 6 term can be ignored, if β ≤ 0, all the terms must be taken into

(5.2)

Vp∂

∂G

T= S–

T∂∂G

p=

p2

2

∂∂ G

Tp∂

∂V

TV β–= =

T p∂

2

∂∂ G

T∂∂V

pV α= =

T2

2

∂∂ G

pT∂

∂S –

p

Cp

T------–= =

F T ξ,( ) F0 T( ) α T( )2

------------ξ2 β T( )4

-----------ξ4 γ T( )6

-----------ξ6+ + +=


80

consideration. These two cases are the conditions of a second and a first order transitionrespectively.

5.2.1 Second Order Transitions

Case β > 0. Since F is minimum for ξ = 0 when T ≥ Tc, and for ξ › 0 when T < Tc, thesign of α must change at Tc. In first approximation α = α0.(T − Tc) with α0 > 0, and theexpression of F is

The stable states are given by

As shown in Fig. 5.1a, for T ≥ Tc, the system has one minimum at ξ = 0, and for T < Tc, twominima represented in Fig. 5.1b given by

The specific heat L =Tc.∆S (Tc) can be calculated by

It can be noticed that S and ξ are continuous at T = Tc. This transition is a second ordertransition in the Ehrenfest classification.

5.2.2 First Order Transitions

Case γ > 0 and β < 0. Similarly to the precedent case, the sign of α changes at Tc and in firstapproximation α = α0.(T − Tc). The stable states are given by

As shown in Fig. 5.1c, it can be noticed that for , it exists only one

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

F T ξ,( ) F0 T( )α0 T Tc–( )

2-------------------------ξ2 β

4---ξ4+ +=

ξ∂∂F

Tξ α βξ 2+( ) 0= =

ξ2

2

∂∂ F

T

α 3βξ2+ 0≥=

ξα0 T Tc–( )

β-------------------------±=

ST∂

∂F

ξ–

α2---ξ2–

T∂∂F0

ξ–= =

ξ∂∂F

Tξ α βξ 2 γξ4+ +( ) 0= =

ξ2

2

∂∂ F

T

α 3βξ2 5γξ4+ + 0>=

T T2> Tc β24γα0( )⁄+=

5.2. Landau s Phenomenological Approach______________________________________________________________________

81

phase, which corresponds to ξ = 0 (phase I). Just below T2, it appears another metastable phasecorresponding to ξ › 0 (phase II). This phase becomes stable as soon as F - F0 = 0 obtained for

. Below T1, the phase I becomes metastable until Tc is reached forT < Tc, only phase II exists. The order parameter corresponding to the metastability (T1 < T <T2) or to the stability (T < T1) of phase II is represented in Fig. 5.1d

It can be noticed that a temperature range ∆T = T2-Tc exists where the two phases cancoexist. Each of them are successively stable, metastable and unstable. This situation isgenerally observed with a thermal hysteresis. The order parameter ξ, as well as the specificheat L (equation (5.7)) are discontinuous at T1. This transition is a first order transition in theEhrenfest classification.

(5.10)

T T1= Tc 3β216γα0( )⁄+=

ξβ– β2 4α– 0 T Tc–( )( )

1 2⁄+

2γ------------------------------------------------------------------±=

0

1

ξ

T T2 T1 Tc

0

1

ξ

T Tc

T < Tc

T = Tc

T > TcF(ξ)-F0

ξ

T < Tc

T > T2

T = T1

T < T2

F(ξ)-F0

ξ

Fig. 5.1 Landau’s treatment of phase transitions: (a,b) and (c,d) second and first ordertransitions respectively: (a,c) the free energy curves in function of the order parameter and(b,d) the order parameter curves in function of temperature.

(a) (b)

(d)(c)

2d order transitions

1st order transitions


82

5.3 Statistical Mechanics Approach

5.3.1 Canonical Ensembles

A closed isochore (N, V = cst) and diathermic system in thermal contact with a heat bathis in equilibrium when its free energy F is minimum. Its internal energy can fluctuate andactually the system must be considered as a set of all the microstates, defined as a canonicalensemble [125]. The probability Px that the system has the energy Ex (and is in a configurationx) is given by the Boltzmann distribution law

The constant of proportionality Z is called the canonical partition function, it does not dependon the specific state of the system and is determined by the normalization requirement

This partition function is characteristic of the thermodynamics of the system since the freeenergy and the average energy can be deduced from it by

where β = 1/(kB.T). The <> denote the thermal average, i.e. the average on all theconfigurations with their respective probability.

5.3.2 The Ising Model

The Ising model is probably the simplest statistical model whose solution is not trivial.It was introduced by Lenz and Ising in 1925 [123]. Let us consider a simple lattice andsuppose that there is a magnetic moment at each lattice site which can only have twoorientations along a given direction, up and down, further noted 1 and respectively. Thesimplest coupling is then introduced by considering that the nearest neighbor spins interact: apair of parallel spins has an energy -J and a pair of antiparallel spins has an energy J. Ofcourse it is quite easy to calculate the energy of each configuration for a finite system, but thehigh number of configuration impedes the calculations of an explicit expression of thepartition function when the dimension of the system is d ≥ 3. However this model is veryinteresting because it brings most of the important and basic ideas about phase transitions.

For d = 1, one can easily show that there is no phase transition in the absence of anyexternal field. Indeed, let us imagine a long chain of N ordered spins. The free energy requiredto create a simple antiphase boundary, i.e. the free energy difference between two possible

(5.11)

(5.12)

(5.13)

(5.14)

Px1Z---

Ex

kBT---------–

exp=

Z T N V, ,( )Ex

kBT---------–

expx∑=

F kBT Zlog–=

E⟨ ⟩ PxExx∑ β∂

∂ Zln

N V,–= =

1

5.3. Statistical Mechanics Approach______________________________________________________________________

83

kinds of configurations: (1,1,...,1,1,...,1,1) and (1,1,...,1, ,...., , ) is J. Since, there are Nsimple states in antiphase boundary configuration, the free energy change is

which is always negative for any reasonable value of N and T. This implies that the disorderingoccurs spontaneously in 1-D system.

For the case d = 2, the energy required to create an antiphase boundary is of the order of.J and the entropy of the order of ln( . ). A phase transition is therefore possible at

kBTc of the order of J [118]. The exact rigorous solution is far more difficult to obtain and wasgiven only in 1944 by Onsager: kB.Tc = 2.2692 |J| [124].

The usual Ising model can be generalized by considering the nth nearest interaction. Letus denote H(σn) the energy of a configuration characterized by the spin numbers σn in thepresence of a magnetic field h

where Jnm is the pair energy of the (n - m) nearest neighbors.

5.3.3 Monte Carlo Simulations

There are two general classes of simulation. One is called the molecular dynamicsmethod. Here, one considers a classical dynamical model for atoms and molecules, and thetrajectory is formed by integrating Newton’s equations of motion. The procedure providesdynamical information as well as equilibrium statistical properties. The other, subject of thissection, is called the Monte Carlo method1. This procedure is more generally applicable thanmolecular dynamics in that it can be used to study quantum systems and lattice models as wellas classical assemblies of molecules. For more simplicity, the Monte Carlo method will bediscussed on the base of the magnetic Ising model [125, 126].

A trajectory is a chronological sequence of configurations for a system. A configurationof a lattice Ising magnet is the list of spin variables σ1, σ2, σ3..., σN. Let us call x a point in theN-dimensional configurational space (also called phases space) obtained along the trajectory attime t: x(t) = (σ1, σ2, σ3..., σN), for example, x(t) = (1, , ,1...1,1). The aim of the Monte Carlomethod is to simulate trajectories representative of the thermal equilibrium state of the system,so that the thermal average value of a property P follows

This means that trajectories should be ergodic and constructed in such a way that theBoltzmann distribution law is in agreement with the relative frequencies with which the

∆F = J - kBT lnN (5.15)

(5.16)

1. The origin of the name comes from the city in south of France well known for its roulette and otherhazard games.

. (5.17)

1 1 1

N 3 N N

H σn ( ) 12--- Jnmσnσm

n m,∑– h σn

n∑–=

1 1

P⟨ ⟩ 1T---

Tlim Px t( )

t 1=

T

∑=


84

different configurations are visited. Let us consider a trajectory going through two states x =(...1, ,1,1...) and x’ = (...1, , ,1...), produced by flipping a spin in the x state. The two statesx and x’ have a probability of existence, respectively px and px’, given by the Boltzmann’s law(5.11). The energy associated to a possible change of state ∆Exx’ = Ex’ - Ex governs therelative probability of this change through the Boltzmann distribution law. Lets call wxx’ thisprobability of change per unit time. The evolution of the states probabilities follows theMaster equation

At equilibrium in the canonical ensemble, , which associated to (5.11) results in

Provided a trajectory obeys this condition, the statistics acquired from this trajectory willcoincide with those obtained from a canonical ensemble. In the Metropolis algorithm [127],the following particular values of wxx’ have been chosen

That is if ∆Exx’ ≤ 0, the move is accepted and if ∆Exx’ ≥ 0, the move is accepted only with anexponential probability which depends on both temperature and difference of energy.

Unfortunately, a certain degree of experimentation is always required to indicatewhether the statistical results are reliable. And while the indications can be convincing, theycan never be definitive. Indeed, some problems may arise when the system is sluggish such asin substitutional transitions, because the trajectory can be blocked in a local minimumsurrounded by large energy barriers.

5.3.4 Phase Diagrams

Let us chose the generalized multi-body Ising model. Usually the variation of energyproduced by a spin flipping or by an exchange of atoms is calculated with the multipleinteraction energies between the spins with equation (5.16) or between the atoms (as detailedin the next section) with equation (5.26). Depending on the resulted values, Monte Carlomethod makes possible to predict the different kinds of thermodynamically stable structuresin function of temperature.

At T = 0K, the problem is reduced to the minimization of the internal energy (groundstate) and can be treated analytically. Details are given by Ducastelle [128]. Let us define agiven cluster αi = n1,..., nri a given set of lattice sites (the index i to specify the type of the

(5.18)

(5.19)

(5.20)

1 1 1

tdd px wxx'px– wx'xpx'+[ ]

x'∑=

tdd px 0=

wxx'

wx'x

--------px

px'

-----∆Exx'–

kBT---------------

exp= =

wxx'

1 ∆Exx' 0≤,

∆Exx'–

kBT---------------

∆Exx' 0>,exp

=

5.3. Statistical Mechanics Approach______________________________________________________________________

85

cluster: pair, triangle, ...). Let us define its occupation number σα = σn1...σnri. The Hamiltonian(5.16) can be written in a generalized way by

where νi only depends on the bonding energies Vnmi limited to the size of the type-i cluster.Noting xi the correlation function xi = < >, the energy of the system is

where Xi = ri.vi, where ri is the number of type-i clusters in the system. Some relationshipsexist between the xi. These ones were given by Kanamori [129]. Let us note pn the numberequalling 1 if σn = 1 and 0 if σn = 1. pn and σn are linked by equation (5.28). The relationshipsbetween the xi can be obtained by expressing the fact that <pnpm>, <pn(1-pm)>, <(1-pn)pm>,<(1-pn)(1-pm)>, and <pnpmpl>, <pnpm(1-pl)>, etc.., being probabilities, should be positive andlower than 1. A general expression of the relationships was given by Ducastelle [128] whoexprimed the probability of finding a type-i cluster ρi by

where |α | is the number of sites in the cluster α. Equation (5.23) looks complicated but itsapplication is easy and direct. For example, for triangular clusters, it takes the form

We must minimize the energy given by equation (5.22) respecting the linear inequalities givenby 0 ≤ ρi ≤ 1 with ρi given in equation (5.23). The problem can be solved by linearprogramming. A simple geometrical solution was given by Kudo and Katsura in 1976 [130]:the inequalities given by (5.23) define a polyhedra in the xi space, and the minimization of theenergy, which is a linear form of the xi, is obtained for one of the vertices of the polyhedra. Thesolutions xi0 give a configuration of the clusters that constitute the phases, and depend on theXi values (linked to the bonding energies Vnmi). It will be seen in section 6.4.2 that animportant and interesting problem which occurs for some lattices is the construction ofperiodic structures with some of the cluster solution xi0. These lattices are called frustrated.The construction may be impossible and may imply a degeneration of the solution in infiniteapproximate solutions.

(5.21)

(5.22)

(5.23)

(5.24)

H σn ( ) να iσα i

i∑=

σα i

E⟨ ⟩ H⟨ ⟩ Xixii

∑= =

ρi1

2 α------- 1 xj σβjβj α i⊂∑

j ∅≠∑+

=

ρ n m p, , 18--- 1 x1 σn σm σp+ +( ) x2 σnσm σmσp σnσp+ + )( x3σnσmσp+++ ][=


86

5.4 Ordering in Binary Alloys

5.4.1 Equivalence with the Ising Model

For a AB binary alloy which features an order-disorder phase transition, the Ising latticeis the Bravais lattice of the highest symmetry phase. Let us define the parameter pn

The expression of the Hamiltonian is (apart from an irrelevant constant) [128]

where Vnm = V(n-m) represents the energy of the creation of an A-B pair separated by (n - m)

where Vnm ij = V ij (n-m) is the energy of the (n - m) nearest neighbors occupied by i and j

atoms, µ = µA - µB + V BB - V

AB. µ i is the chemical potential of the element i, and V ij=

. The two expressions (5.16) and (5.26) are formally equivalent, with

It can be noticed that an order-disorder transition in a binary AB alloy is equivalent to amagnetic transition where h › 0 (because the number of atom A and B must remain constant,which is not the case for the number of up or down spins). An A-B change corresponds to aspin flip. This equivalence can be generalized to many physical phenomena called criticalphenomena1.

5.4.2 Order Parameters

During an order-disorder transition in an alloy, one of the symmetry or translational

1 if the site n is occupied by the atom A0 if the site n is occupied by the atom B

(5.25)

(5.26)

Vnm = Vnm AA + Vnm

BB - 2Vnm

AB (5.27)

(5.28)

(5.29)

(5.30)

1. Critical phenomena is certainly one of the most interesting branches of modern physics. It gives thesame fundamental base of a priori many different phenomena, such as superconductivity, transitions inpolymers, ferroelectrics, superfluids. It can even be applied to the quarks bond in protons and neutrons[131, 132]. It plays a central role in the unification quest (grand unified theory, superstring theory): ouractual universe with 4 forces would come from a hot and condensed universe governed by one force,which would have been subject to a symmetry breaking during its expansion and cooling after the BigBang [133].

pn = pnA =

H pn ( ) 12--- Vnmpnpm

n m,∑ µ pn

n∑–=

Vij n( )n∑

σn 2pn 1–=

h12---µ 1

4--- V n( )

n∑–=

Jnm14---Vnm–=

5.4. Ordering in Binary Alloys______________________________________________________________________

87

elements of the high temperature phase is broken and the Bravais lattice is separated into twosublattices. Let us illustrate the concepts of short and long order parameters with the exampleof an AB alloy (50%, 50%) with the same number of sites in each sublattice, such as the CuZntransition illustrated in Fig. 5.2. At T = 0K, all the A atoms are on one of the sublattices, thatwill be noted α. For 0 < T ≤ Tc, most of the A atoms remain on their right α sublattice, andother A atoms can change the sublattice to go on the wrong sublattice (noted β). Let us callcA and cB the fractional portions of A and B atoms (cA = cB = 1/2). For the magnetic Isingmodel, the order parameter is the normalized magnetization M/N = . For the order-disorder transition, a similar parameter is not relevant because h › 0. Indeed, it can be deducedfrom equation (5.28) that its value would be always cA-cB = 0. Therefore, let us introduce thenew parameters

where rA denotes the number of A atoms on the right α sublattice divided by the total numberof A atoms NA, and wA the number of A atoms on the wrong β sublattice divided by NA. Bydefinition, rA + wA = 1. At T = 0K, rA = 1, at T = , rA = wA. Moreover, cA = cB = 1/2, andthe number of sites of each sublattice are equal (Nα = Nβ = N/2); it follows that rA = rB + cB -Nα/N = rB = r and wA = wB = w. Therefore, these two parameters can be used to create an orderparameter similar to the magnetization

ξ is called the long range order LRO parameter, it is a measure of the fraction of atoms A

(5.31)

(5.32)

(5.33)

Cu 50%Zn 50%

Cu Zn

Fig . 5 .2 Disordered (h ightempera ture ) and orderedstructure (low temperature) ofCuZn. The broken symmetrycor re sponds t o a changebetween the BCC and the SCstructures.

1N---- σn∑⟨ ⟩

rA1

NA------ pn

A

n α∈∑=

wA1

NA------ pn

A

n β∈∑=

∞

ξ r w–r w+------------ 2r 1–= =


88

sitting on their right sublattice site. Its value is ξ = 1 at T = 0 K, 0 < ξ < 1 for 0 < T ≤ Tc, and0 for T > Tc. Similarly to equation (5.31), let us introduce the parameter

which corresponds to the number of right nth nearest neighbors divided by 2Nzn the number ofpairs formed by the nearest neighbors of the system (independently whether A is in the rightor wrong position). Here again rn + wn = 1. Similarly to equation (5.33), let us define

ξn is a short range order SRO parameter. It corresponds to the pair correlation coefficientbetween the nth nearest neighbors. Since the atoms always tend to have for nearest neighbors fthe right ones, we have ξ1 ≥ ξ2 ≥ ...≥ ξn. Moreover, for very large distances between sites

, the probability of finding a right pair rn becomes independent of n and can beexpressed in function of the probability of finding a pair of atoms both on their right site orboth on their wrong site. Equation (5.34) therefore becomes

which, with equation (5.33), becomes

The long range order parameter corresponds to the square of the limit of the short range orderparameter series.

5.4.3 Approximate Methods

There exist many methods for treating the phase transitions on an Ising lattice. The firstone was introduced by Bragg and Williams in 1934 for the order-disorder transition [134], andis usually referred to as the B-W method. This is similar to the Weiss’s method for thetreatment of ferro-paramagnetic transitions [135]. Let us illustrate this method with the CuZnphase transition.The model is based on the simplification provided by the assumption that ξ1 =ξ2 =... = ξ . It means that all the sites of a sublattice α or β have the same probability to beoccupied by A or B independently of their neighbors.There is absolutely no correlation length(in magnetic terms it is the maximum distance over which the flip of a spin has no moreinfluence on the other spin). This method neglects all the fluctuations of the system. The meannumber of the 1st nearest neighbor A-B bonds is by definition NAB = (N/2) z r1, where z is the

(5.34)

(5.35)

, (5.36)

(5.37)

rn1

2Nzn------------ pj

Apj n+B

j α∈( ) j n+ β∈( )j β∈( ) j n+ α∈( )

∑=

ξn

rn wn–rn wn+----------------- 2rn 1–= =

n ∞→

r∞ r2 w2+=

ξ∞ 2r∞ 1– ξ2= =

5.4. Ordering in Binary Alloys______________________________________________________________________

89

number of nearest neighbors (its value is 8 for the CuZn phases). Under these conditions

Calling V = V(1)/2 the mean energy to form an A-B bond, with V(1) defined in (5.27), themixing energy becomes

Additionally, the complexion number in the B-W approximation is

and therefore, using the Stirling approximation and the fact that NA = NB = N/2, the entropy ofconfiguration can be simplified

With equation (5.33) and the fact that rA = rB = r, it becomes

Therefore, the free energy of mixing is given by

To find the order parameter for the equilibrium state, we are looking for the minimum of theF(ξ) obtained for

Equation (5.44) does not have any analytical solution but can be solved numerically. As shownin Fig. 5.3, it has always the solution ξ = 0, and when T < Tc, it has also a second distinctsolution. For T < Tc the solution ξ = 0 is unstable and the other solution is stable: the alloy isordered, even if not completely ordered. For T ≥ Tc, the solution ξ = 0 is stable: the alloy isdisordered. There is no discontinuity of the order parameter when T goes through Tc: it is asecond order transition.

The B-W model is also called the mean-field approximation, generally used for themagnetic Ising model. Indeed, the B-W approximation is equivalent to consider that only V(1)› 0, and the other V(n) = 0, which, by considering equation (5.29), is the same as to replace afluctuating field by a mean field. Apart from a constant term corresponding to the mean field of

NAB = (N/2) z r = N z (ξ2 + 1) /4 (5.38)

E = - N V z (ξ2 + 1) /4 (5.39)

(5.40)

(5.41)

(5.42)

(5.43)

(5.44)

where (5.45)

ΩNA!

NArA!NA 1 rA–( )!------------------------------------------

NB!NBrB!NB 1 rB–( )!------------------------------------------⋅=

S kN2

-------– rA rAln 1 rA–( ) 1 rA–( )ln rB rBln 1 rB–( ) 1 rB–( )ln+ + + ][⋅=

SkBN

2---------–

1 ξ+( ) 1 ξ+( )ln 1 ξ–( ) 1 ξ–( )ln 2 2ln–+ ][⋅=

FN---- Vz

4------–

ξ2 1+( )kBT

2---------+ 1 ξ+( ) 1 ξ+( )ln 1 ξ–( ) 1 ξ–( )ln 2 2ln–+ ][⋅=

ξ∂∂F

T0=

1 ξ+1 ξ–------------

ln⇔ 2=TTc-----ξ ξ

TcξT

-------- tanh=⇔

TcVz2kB--------=


90

the completely disordered state, the expression of the equivalent mean field is

This can also be obtained by the differentiation of (5.39). Considering that the proportion r/wobeys the Boltzmann law, the result (5.44) can be found again

We have introduced the B-W in the example of AB (50%, 50%) alloy with a transitionbetween CC and SC structures. Actually, the model can be generalized to any AB (cA, cB)alloy. The B-W model can also be applied to transitions between FCC and SC structures. Inthis case, the model is more complex and must take into account the 1st-antiliberals the α-αand β-β bonds in addition to the α-β bonds. Its results in a prevision of a first order transition1.The B-W model is generally in good agreement with the observations of order-disorder phasetransition. For instance, the CuZn BCC-SC transition is of second order whereas the AuCu3FCC-SC transition is of first order. No metastable CuZn structure phase exists at room

(5.46)

(5.47)

1. The first order character of the transition of FCC alloy can also be understood by the the fact that theFCC lattice is a very constained or frustrated lattice, i.e. not easily ordered [128].

-2 -1 1 2

-3

-2

-1

1

2

3

Tanh ξ

y = 0.5 ξ

y = 1.5 ξ

y = ξ

ξ1

-0.2

0.1

0

F(ξ) - F(0) N kBT

0.6

0.8

0.911.2 1.1

0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1

TTc

ξ

Fig. 5.3 Bragg-Williams treatment of order-disorder phase transitions: (a) free energycurve in function of the order parameter, (b)graphical resolution of equation (5.44),leading to the drawing of (c) the orderparameter curve in function of temperature.

(a) (b)

(c)

h0zVξ

2---------=

rw----

2h0

kBT---------

expzVξkBT---------

exp 1 ξ+1 ξ–------------= = =

5.5. Scattering and HREM Images of Disordered Particles______________________________________________________________________

91

temperature after quenching, in contrary to the AuCu3 system. Nevertheless, there are manyinadequacies in the B-W approximation. The critical exponents (representative of the variationwith T near Tc of parameters such as the order parameters, the correlation length or the heatcapacity) are not correct. A better approximate model has been given by Bethe which considersnot only the point correlation function (as the B-W method), but also the pair correlationfunctions [136]. This method has been generalized by considering the polyhedra interactions,this has been done mainly by Kikuchi (1951) with its Cluster Variation method (CV) [137,138]. It gives very good approximations for the critical exponents.

Another very performing method is based on the renormalization group theorydeveloped by Wilson in 1972 [139] (he was award the 1982 Nobel Prize in Physics for hiscontribution). Wilson’method is very general and has wide applicability extending wellbeyond the field of phase transitions. The method is based on finding a link between thecoupling parameters (such as J in the simple Ising model) so that the problem remainsinvariant, i.e. the partition function must be unchanged during the scaling steps (for examplewhen a group of 4 spins is replaced by one spin according to a certain averaging law). Themethod can predict with a very high degree of accuracy the critical exponent (behavior close toTc).

5.5 Scattering and HREM Images of Disordered Particles

This section is dedicated to the diffraction, HREM observations and simulations ofdisordered particles. Other methods usually more effective for ordering characterization [140],like X-ray and neutron diffraction will no be presented because they are not appropriate to ourstudy of the nanosized precipitates present with many other phases in a matrix. Manydisordered structures can be generated along a trajectory during Monte Carlo simulations of anordering mechanism. Their representations are not easily comparable in the direct space, but iflarge enough, they exhibit the same diffraction patterns. Nevertheless, as it will be shown, thisdiffraction pattern is not enough to completely characterize the ordering mechanisms.

5.5.1 Diffuse Scattering and Disorder

As introduced in section 3.4.2, the diffuse scattering comes from a potential disorder in acrystal: the finite size of a particle breaks the translational and infinite order and makes thespots broader, the thermal motions (Debye-Waller) or small displacements of atoms aroundtheir mean positions lower the spot intensities for high G vectors. The diffraction generallycomes from the order, which, for a binary substitutional alloy, depends on the correlations thatexist between the substitutions (as shown in section 5.4.2). Cowley [141] has expressed thetotal diffracted intensity by

(5.48)I u( ) N fAcA fBcB+( )2 2πiu Rn•[ ]expn∑ N fA

2 fB2–( ) s0sn⟨ ⟩ 2πiu Rn•[ ]exp

n∑+=


92

where N is the number of atoms, Rn the vectors between the lattice points n, fA and fB theiratomic diffusion factors, and <s0sn> the correlation between the site 0 and n. sn represents thedifference between the real occupation of the site n and the mean occupation

with pnΑ defined in (5.25). Here the brackets <> represent the spatial average over all sites1.

<snsm> is linked to the probability <PnAPm

B> noted PnmAB to find the A and B atoms

separated by a vector m - n given by

<snsm> are other SRO parameters. They are more often used than the ξn SRO parametersdefined in 5.4.2 because their one-point and two-point correlations can directly be obtained byinversion of the second term of equation (5.48) noted Id(u)

The parameter αnm =<snsm>/cAcB is called the Warren-Cowley parameter. It is also oftenused. Similarly to ξn, it tends to the square of the long range order ξ when the modulus of m-ntends to , with m and n respectively on α and β.

The other multiple-point correlations <snsmsl...> cannot be obtained by diffractionexperiments. However, they are essential to unambiguously define the complete arrangementof the atoms, and thus to completely determine the order. Indeed, Welberry [142] has shownthat substantial differences in the multi-point lattice averages can exist without affecting thetwo-point correlations, and thus proving that very different kinds of ordering clearlydistinguishable in direct space can exhibit the same diffraction patterns with the sameintensities. Therefore, equation (5.48) cannot be used to prove the unicity of a model, but canonly be used to simulate diffractions when strong physical and chemical arguments govern themodel and only when few parameters -in general the interaction constants of the Monte-Carloprocess- are fitted by the program; for example for 1,3-dibromo-2,5-diethyl-4,6-dimethylbenzene and for mullite [142].

In equation (5.48), the first term gives the Bragg peaks and the second term gives thediffuse scattering. It can be noticed that this diffuse intensity part produces sharpsuperstructure Bragg spots when the perfect superstructure is obtained (ξ = 1), diffuse peaksor streaks if the structure just exhibits a ˙tendency towards ordering¨ (0 < ξ < 1), and auniform background for disordered structure (ξ = 0). The superstructure peaks for 0 < ξ < 1are diffuse due to the small size of the ordered clusters (linked to SRO parameter). Theirintensities decrease rapidly with u because the order is only partially obtained inside of eachcluster (similar to a Debye-Waller effect). This description is a simplified presentation, more

sn = pn Α - cA = cB - pn

B (5.49)

1. In a large system in thermal equilibrium, this spatial averaging is equivalent to a temporal averagingor to an averaging on all the configurations with their respective probability.

PnmAB = <pn

Α pmB> = cAcB + <snsm>. (5.50)

(5.51)s0sn⟨ ⟩ 1

N fA fB–( )2-----------------------------ℑ 1– Id u( )[ ]=

∞


93

details are given in [141, 142, 143, 144].

5.5.2 Diffuse Scattering Simulations

In chapter 6, diffuse scattering simulations will be performed to confirm the validity ofan ordering model developed for the precipitation in the Al-Cu-Mg-Si alloys. We will study anorder-disorder transition inside a precipitate family containing four elements (Al, Mg, Si, andCu), with substitutions and atomic displacements. Since this transition is more complicatedthan the substitution ordering with two species, it will not be possible to apply equation (5.48).To simulate diffuse scattering, we have preferred to build an image of the total projectedpotential of the disordered structure to perform its FFT. Indeed, from equation (3.7), noting nthe zone axis, one can notice that the intensity of the diffraction pattern is proportional to thesquare of the modulus of the FFT of the projected potential

The projected potentials are calculated for the atomic species using the method described inannex A and integrated along the observation direction (which can be different from a zoneaxis). Such potential images are represented in Fig. 5.4. Diffraction and HREM simulationsusually require periodic conditions to perform Discrete Fourier Transforms. These conditionsare obvious for translational periodic structures (perfect crystals), but are arbitrary and requirelarge supercell sizes for quasicrystals, structures containing defects and disordered structures.The advantage of the method presented here is that no periodic conditions are required due tothe use of Continuous Fourier Transform to calculate the atomic potential of a single atom.Moreover, projected potential simulations can be performed for comparisons with theexperimental HREM images (if the weak phase object approximation is obtained), permittingto distinguish two potential disordering processes producing the same point and paircorrelation parameters.

5.5.3 Filtered HREM Images

In the weak phase object approximation at Scherzer defocus a HREM image is nearly therepresentation of the projected potential of the object. If the observed area of the crystal islarge enough, it is possible to calculate from the HREM images the multiple point correlationsand thus completely define the order. Even if the weak phase object approximation is notobtained, for simple A-B substitutional transformations, with a large difference between the Aand B diffusion factors, a direct interpretation of the atomic position remains possible [145].

However, at least for the precipitates studied in this work, the thin parts of the samples

(5.52)

I u( ) A u( ) 2= K2 V r( ) 2πiu r•[ ]exp vdΩ∫

2=

K2 V

r n||∫

r ⊥ n∫ r( ) 2πiu r•[ ]exp vd

2K2 Vp r( ) 2πiu r•[ ]exp sd

S∫

2==


94

are too amorphous to be observed. Thus, the HREM images were acquired in thicker partswhich impeded the weak phase object approximation. Moreover, the transformation is morecomplex than a simple A-B substitutional transformation since it involves four elements (Al,Mg, Si and Cu) with substitutions and atomic displacements (see chapter 6). In addition, noclear pattern can be identified in the observed precipitates due to the slight disorder betweeneach atomic plane normal to the electron beam. Nevertheless, as usual in ordering phenomena,the atoms are disordered around the lattice of the final ordered phase. Applying a mask filterin the reciprocal space of the HREM images allows to show an image of the mean atomicpositions, i.e. the latent lattice. Indeed, the filtering corresponds to apply to the FFT of theimage (noted I(u)) a Gaussian mask (size noted 1/m [nm-1]) with the periodicity (noted 1/a[nm-1]). The filtered image If (u) can be written in one dimension by

Equation (5.53) shows that applying a mask filter in the reciprocal space corresponds toaverage each pattern separated by the periodicity a of the direct space image with theneighboring other patterns on a distance m. Thus, this makes appear a pattern (of size a) onlyif mean positions (with random disorder around these positions) exist inside this pattern. Thesize of the mask (1/m) must be chosen to contain the diffuse intensity, which means that the

(5.53)

-5 0 5 10 15 20

5

10

15

20

25

30

Fig. 5.4 Disordered structure represented in (a) illustrative model: Si, Mg, Cu, and (b) projected potential calculated with annex A.

(a) (b)

ℑ

If x( ) ℑ 1– ℑ I u( ) δ u n a⁄–( ) u2– m⁄[ ]exp⊗n ∞–=

∞

∑•=

I x( ) ℑ 1– δ u n a⁄–( ) u2– m⁄[ ]exp⊗n ∞–=

∞

∑⊗=

I x( ) πma δ x ha–( ) mπ2x2–[ ]exp•h ∞–=

∞

∑⊗=


95

average must be performed on the correlation length distance. Therefore, this method iseffective only for slightly disordered structures (largely below the critical transitiontemperature). This method was used for comparisons between experimental and simulatedHREM images based on a refined model of the crystal structure (section 6.3.5).

5.5.4 Dark Field Superstructure Images

Some slightly disordered structures can be mixed, i.e. constituted with two or morephases corresponding to superstructures. Dark field superstructure images consist in acquiringdark field images with two diffracted spots (single plane stacking information) or, if it ispossible, with 3 non-colinear spots (atomic column arrangement information) of eachsuperstructure phase. If the information limit is sufficient to transfer the spatial frequencies, thelattices of each structure appear by interference. Therefore, these images can be used toobserve the ordered parts of a disordered precipitate, and possibly to determine the correlationlength. To obtain such images, the disordering must occur two—dimensionally so that thecrystal remains perfectly periodic in the third direction (direction of the electron beam) and sothat superposition effects are eliminated. This method can be applied to thick regions with arelatively low resolution microscope. For example, a conventional electron microscope notnecessarily dedicated to HREM is sufficient for periodicities higher than 0.4 nm. This type ofimages is widely used for ordered-disordered structures [146, 147] such as Cu-Zn, Cu-Al orNi-Mo[148], Au4Mg, AuCd, AuMn [149, 150]. In this work, DF superstructure images will beused to show different superordered parts in the AlCuMgSi rod-shaped precipitates (section6.1.3).


96

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