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• Publié par: Urku
• Nombre de lectures: 31
• Langue: English

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Chapter 5
Introduction to Order-Disorder Transitions
In the previous chapter, two new types of nanometer rod-shaped precipitates were
observed; they were named QP and QC and seemed to be precursors of the stable Q-
Al Cu Mg Si . Before a complete TEM and HREM study of these phases, subject of the next5 2 8 7
chapter, an introduction to ordering mechanisms is required. Order-disorder transitions will be
introduced in the global framework of phase transitions (solid-liquid-gas, ferro-para
magnetic, ferro-para electric, superfluids, polymers), without enlarging the presentation to
critical phenomena.
Most of the approach presented in this chapter is based on the simple following
thermodynamic concept: for a closed system in thermal equilibrium, the transition is a
consequence of a compromise: the energy tends to order and the entropy associated to the
temperature tends to break the order. Different classifications of phase transitions will be
presented in section 5.1. Phenomenological Landau’s approach by thermodynamics will be
treated in section 5.2. A more general approach by using statistical mechanics on an Ising
model, as well as Monte Carlo simulations, will be treated in section 5.3. It will help us to
introduce 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 most
appropriate observation means, i.e. TEM diffraction and HREM will be treated in section 5.5
to show their potential applications for the study of ordering mechanisms.
775. Introduction to Order-Disorder Transitions
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5.1 Classification of the Phase Transitions
5.1.1 Chemistry
A microscopic approach by crystal chemistry can provide a basis for the classification
of the phase transitions [118]. If a solid undergoes a phase transition at a critical temperature
T by absorbing thermal energy, the transformed phase possesses higher internal energy, thec
bonding between neighboring atoms or units are weaker than in the low-temperature
phase.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 the
specific heat is low (few J/g). The main characteristic is the group-subgroup relationship
between the phases. This permits for example to clearly define an order parameter used for the
thermodynamical description of the transition. These transitions can be of the first or second
order (these terms will be explained in the next section).
(2) Reconstructive transitions [120] proceed through the breaking of the primary or
secondary bonds. These transitions were firstly described by Buerger [121]. They imply large
atomic 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 main
characteristic is the absence of any group-subgroup relationship between the phases contrarily
to the case of Landau transitions (section 5.2). The transitions can even increase the symmetry
of the high temperature phase. This transition occurs in many materials such as ZnS, C, H O,2
Am, C, SiO , TiO . Bain transitions (BCC-FCC) and Buerger transitions (BCC-HCP) can be2 2
described as reconstructive.
(3) Order-disorder transitions proceed through substitution between atoms possibly
followed by small atomic displacements. They are commonly found in metals and alloys but
also in some ceramics. Some of them keep a group-subgroup relationship, as for the CuZn
transition (between BCC and simple cubic SC structure), others are also reconstructive as for
Am, Fe, Co, ZnS or SiC (FCC-HCP). These transitions can be described with the help of a
latent lattice common to the phases [120].
5.1.2 Thermodynamics
Let us consider a closed, isochore and diathermic system in thermal contact with a heat
bath. This system is characterized by its free energy F (minimum at equilibrium), given by
F = E - T.S (5.1)
E is the internal energy and depends on the bonding between the atoms. S is the entropy,
characteristic of the disorder by S = k .Log Ω (Ε) , where Ω is the complexion number, i.e.B
number of configurations of the system for a given energy E. At low temperatures, the
785.2. Landau s Phenomenological Approach
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entropic term is negligible and the system is driven by E (negative) which has its maximum
absolute value when the bonds of highest energy are formed (the system is ordered). At high
temperatures, 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 phase
transition must exist at a critical temperature T . In this first approach we have voluntarilyc
neglected the fact that the internal energy of the system can fluctuate. Actually the system must
be considered as a canonical ensemble (section 5.3.1).
Closed, expansible and diathermic systems are characterized by their free energy G
which remains continuous during the phase transition. However, thermodynamic quantities
like entropy S, volume V, heat capacity C , the volume thermal expansivity α or thep
compressibility β can undergo discontinuity. Ehrenfest classified the phase transitions in
function of the thermodynamic quantities that present a discontinuity. The order of the
transition is the same than the order of the derivation of G required to obtain a discontinuity:
∂G ∂G If V = or –S = has a discontinuity, the transition is of first order. ∂p ∂TT p
2 2 2
  ∂ G ∂V C∂ G ∂V ∂ G ∂S p If == –V β , or has a == V α ==– – ------ 2  2 ∂p T ∂Tp∂ ∂T ∂T T∂p p p ∂TT p
discontinuity, the transition is of second order. Higher order transitions would involve further
differential 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 continuous
function that can be developed in a Taylor series near the critical temperature T , depending onc
a parameter called the order parameter, and noted ξ. This parameter is characteristic of the
degree of order. It can be the magnetization for ferro-paramagnetic transition, the polarization
for ferro-paraelectric transition, or the percentage of atoms that are on their right sublattice for
an 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 operations
of the highest symmetric phase implied in the transition. The development of the free energy
keeps therefore only the even exponents of ξ
α()T 2 β()T 4 γ()T 6
FT(), ξ = F()T++------------ξ ----------- ξ+-----------ξ (5.2)0 2 4 6
Let us assume that β and γ do not depend on the temperature. Since F is an increasing
function with ξ at high temperatures (preponderance of the T.S component in F), we must have
γ > 0. If β > 0, the exponent 6 term can be ignored, if β ≤ 0, all the terms must be taken into
795. Introduction to Order-Disorder Transitions
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consideration. These two cases are the conditions of a second and a first order transition
respectively.
5.2.1 Second Order Transitions
Case β > 0. Since F is minimum for ξ = 0 when T ≥ T , and for ξ › 0 when T < T , thec c
sign of α must change at T . In first approximation α = α .(T − T ) with α > 0, and thec 0 c 0
expression of F is
α()TT– 2 β 40 cFT(), ξ = F()T++------------------------- ξ ---ξ0 (5.3)2 4
The stable states are given by
∂F 2 (5.4)==ξα()+βξ 0
∂ξ T
2
∂ F 2
= α+03βξ ≥ (5.5)
2∂ξ T
As shown in Fig. 5.1a, for T ≥ T , the system has one minimum at ξ = 0, and for T < T , twoc c
minima represented in Fig. 5.1b given by
α()TT–0 c (5.6)ξ = ± -------------------------
β
The specific heat L =T .∆S can be calculated by c (Tc)
∂F∂F α 2 0  (5.7)S==– – --- ξ – ∂T 2 ∂Tξ ξ
It can be noticed that S and ξ are continuous at T = T . This transition is a second orderc
transition 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 T and in firstc
approximation α = α .(T − T ). The stable states are given by 0 c
∂F 2 4==ξα()++βξ γξ 0 (5.8)∂ξ T
2
∂ F 2 4
= α++3βξ 5γξ >02 (5.9)∂ξ T
2
As shown in Fig. 5.1c, it can be noticed that for TT> = T + β ⁄()4γα , it exists only one
2 c 0
805.2. Landau s Phenomenological Approach
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phase, which corresponds to ξ = 0 (phase I). Just below T , it appears another metastable phase2
corresponding to ξ › 0 (phase II). This phase becomes stable as soon as F - F = 0 obtained for0
2
TT= = T + 3β ⁄()16γα . Bel