Order parameter

The phenomenon of ordering is discussed first, in the context of thermodynamics, before introducing the crystallographic aspects of typical ordering reactions in iron alloys.

The classical Bragg-Williams model contains many of the essential features for ordering. The derivation presented here is due to Christian [101] and is for an equiatomic binary alloy of components A and В when ordering changes a cubic-I lattice into one which is primitive cubic. The most popular example of this kind of an ordering reaction is CuZn, but FeAl and FeCo are relevant examples in the context of iron. The iron aluminides in particular may become important engineering materials in their own right.

The extent of order is described in terms of a long-range order parameter L which is unity for the fully ordered crystal and zero when the distribution of atoms is random in the context of a long range:

where гд is the probability of an A site being rightly occupied by an A atom, equal in our example to the probability of a В site being occupied by a В atom, so that with л-д = дв = 0.5,

When r= 1, the alloy is fully ordered with L = 1 and when r = L = 0 with the alloy being fully disordered.

The probability of A-A pairs is the chance of finding an A site that is occupied correctly (r) multiplied by that of finding a В site which is incorrectly occupied (1 — r). A similar rationale for the B-B and A-В pairs gives:

since only j sites can be occupied correctly by either species for the cubic-I to cubic-P ordering transition, z is a coordination number and N is the number of atoms. In deducing the number of unlike bonds, r2 is the probability of finding two adjacent sites that are occupied correctly and the additional term (1 - r)2 accounts for the probability of finding two adjacent sites which are incorrectly occupied. Both of these circumstances lead to A-В bonds.

The configurational internal energy U is then given by the sum of all bond strengths.

The factor of two drops out because of the way in which the binding energy is defined (Figure 2.32). The configurational Gibbs free energy is given by assuming that the enthalpy of ordering is equal to the internal energy, and adding the contribution from the change in configurational entropy. The ordered lattice can be imagined to consist of a series of sublattices to enable the entropy of any distribution of atoms on these sublattices to be estimated; for a binary alloy without vacancies, this is given by22

where n is the number of sublattices and Дд the fraction of A atoms on the sublattice For the present example, N = 2 so:

When r = 1, the configurational entropy is zero. To find the equilibrium value of r, dG/dr is set to 0, to obtain

The variation of the long range order parameter as a function of the reduced temperature T/Тс is illustrated in Figure 2.45. 7c is the temperature where L = 0. The increase in disorder is at first small as the temperature rises towards 7c, but this resistance to disordering decreases once the process gets hold.

The equilibrium value of the long-range order parameter as a function of the reduced temperature, using the Bragg-Williams model

Figure 2.45 The equilibrium value of the long-range order parameter as a function of the reduced temperature, using the Bragg-Williams model.

Short-range order

This Bragg-Williams model for the order-disorder transition is analogous to the Weiss model for ferromagnetism by spin orientation ordering. They both assume that the atoms (or moments) are distributed at random on each sublattice of the superlattice structure and hence predict completely random mixing at all temperatures above Tq. Short-range order cannot therefore be treated with these models.2' As with magnetism, the probability of finding A-В pairs remains larger than random when the long-range order parameter becomes zero. An excess heat capacity (i.e., relative to an ideal solution model) persists beyond 7c; important implications of this phenomena are discussed in Chapter 3 which deals with diffusion.

It is well established that models based on near neighbour interactions alone cannot explain the stability of many of the superlattices found in practice. Second, sometimes third near neighbour interactions have to be taken into account. This will become obvious when the detailed crystal structures are discussed in the next section. In some cases it becomes necessary to consider clusters of atoms rather than just pairwise interactions in order to predict the correct equilibrium state. The historical development of the theory can be described in terms of quasichemical approximations, where the Bragg-Williams model is the zeroth, and the Beth model the first order quasichemical approximation. Cluster variation methods are higher approximations of the quasichemical method. These are not described here partly because they have been discussed thoroughly by Christian [101 j but also because in the context of iron, there are few data against which they can be tested. One difficulty is that many important ordering reactions occur at temperatures where it takes a long time to reach equilibrium.

A pragmatic approach is to apply Inden’s heat capacity equations (Section 2.19.4) to the problem of chemical ordering. As with magnetism, all that is required is a knowledge of the critical temperature and the factor fa which gives the ratio of the enthalpy due to short-range order to the total amount of enthalpy due to ordering. Inden [128] has suggested that in the absence of good experimental data, f0 can be taken to be identical to /ц, i.e., 0.4 for bcc and 0.28 for fee systems.

Superlattices

Ordered crystals

Ordering leads to an increase in the volume of the primitive cell, hence the term superlattice is used to describe crystal structures in which the different atomic species are ordered. Some of the technologically important superlattices of iron-based intermetallic compounds are listed in Table 2.11. All of these structures lead to an increase in the number of unlike neighbours when compared with the case where the atoms are disordered, randomly dispersed.

FeAl and FeCo intermetallic compounds have a cubic-I structure in the disordered state and cubic-P when ordered. The tendency for ordering (i.e., enthalpy change on ordering) is much larger for FeAl than FeCo so that the former is often described as a strongly ordered compound, a description that is not a measure of the order parameter.

Ni.iFe is a classic ordered compound which in the disordered state is cubic-F that on ordering becomes cubic-P with Ni atoms at the face centres and Fe atoms at the cube corners. Fe.^Al and Fe.iSi have a cubic-I crystal structure when disordered.

Table 2.11

Some of the ordered compounds that occur in iron alloys.

Compound

Crystal Structure Ordered Disordered

7ш/К

7с/К

Density /kg m 3

B2, Cubic-P

Cubic-I

1523-1673

1523-1673

5560

LI2, Cubic-P

Cubic-F

LI2, Cubic-P

Cubic -F

1673

1223

7800

LI2, Cubic-P

Cubic -F

1673

953

7600

DO3, Cubic-F

Cubic-I

1813

813

6750

DO3, Cubic-F

Cubic-I

1543

1543

7250

These and other ordered crystals can, for the purposes of thermodynamic analysis (Section 2.19.1), be represented conveniently by subdividing the bcc and fee arrangements of atoms into four and eight sublattices as illustrated in Figure 2.46.

Table 2.12

Location of atoms in the sublattices of the perfectly ordered phases. The sites 1, 2 ... 8 are identified in Figure 2.46. The first two compounds are based on the four sublattices of the bcc arrangement, and the last on the eight sublattices of the fee arrangement.

Compound

I

II

III

IV

V

VI

VII

VIII

AB, B2

A

A

В

В

-

-

-

-

A3B, DO3

В

A

A

A

-

-

-

-

A3B, LI2

В

В

A

A

A

A

A

A

Four and eight sublattices of the bcc and fee arrangements respectively, as an aid to illustrating the ordering of atoms (adapted from [128])

Figure 2.46 Four and eight sublattices of the bcc and fee arrangements respectively, as an aid to illustrating the ordering of atoms (adapted from [128]).

Thermodynamics of irreversible processes

Thermodynamics as a subject is limited to the equilibrium state. Properties such as entropy and free energy are, on an appropriate scale, static and time-invariant during equilibrium. There is an extension of the subject to systems that are close to equilibrium so that they can be divided into subsystems where the rules of equilibrium can be applied locally [139]. Parameters not relevant to the discussion of equilibrium, such as thermal conductivity, diffusivity and viscosity, then enter the picture because they can describe a second kind of time independence, that of the steady state. For example, the concentration profile does not change during steady-state diffusion, even though energy is being dissipated during diffusion.

The thermodynamics of irreversible processes deals with systems that are not at equilibrium but are nevertheless stationary. The theory in effect uses thermodynamics to deal with kinetic phenomena. There is, nevertheless, a distinction between the thermodynamics of irreversible processes and kinetics. The former applies strictly to the steady state, whereas there is no such restriction on kinetic theory.

Reversibility

A process, the direction of which can be changed by an infinitesimal alteration in the external conditions is called reversible, because an exact reversal leads to no net dissipation of energy. Figure 2.47 shows the response of an ideal gas contained at uniform pressure within a cylinder, any change being achieved by the motion of the piston. For any starting point on the pressure-volume curve, the application of an infinitesimal force may cause the piston to move to an adjacent position still on the curve, while the removal of the infinitesimal force restores the system to its original state. This process is reversible because there is no net dissipation in displacing and recovering the frictionless piston.

If the motion of the piston in the cylinder entails friction, then deviations occur from the P/V curve as illustrated by the cycle in Figure 2.47. An infinitesimal force cannot move the piston because energy must be dissipated to overcome the friction; this energy is the area enclosed by the cycle on the P/V plot. A process such as this, which involves the dissipation of energy, is classified as irreversible with respect to an infinitesimal change in the external conditions. More generally,

The curve represents the variation in pressure within the cylinder as the volume of the ideal gas is altered by the frictionless positioning the piston

Figure 2.47 The curve represents the variation in pressure within the cylinder as the volume of the ideal gas is altered by the frictionless positioning the piston. The cycle represents the dissipation of energy when the motion of the piston causes friction.

reversibility means that it is possible to pass from one state to another without appreciable deviation from equilibrium. Real processs are not reversible so equilibrium thermodynamics can only be used approximately, though the same principles define whether or not a process can occur spontaneously without ambiguity.

For irreversible processes the equations of classical thermodynamics become inequalities. For example, at the equilibrium melting temperature, the free energies of the liquid and solid are identical (Giiquid = GSOiid) but not so below that temperature (Gaquid > Gsoijd). Such inequalities are much more difficult to deal with though they indicate the natural direction of change. For steady-state processes however, the thermodynamic framework for irreversible processes as developed by Onsager [140] is particularly useful in obtaining relationships even though the system may not be at equilibrium.

 
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