Order –disorder phenomena

3.3.4 Order –disorder phenomena

A substitutional solid solution can be one of two types, either ordered in which the A and B atoms are arranged in a regular pattern, or disordered in which the distribution of the A and B atoms is random. From the previous section it is clear that 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, A 3 B or

AB 3 . The following are common structures:

1. CuZn While the disordered solution is bcc with equal probabilities of having copper or zinc atoms

Figure 3.44 Examples of ordered structures, (a) CuZn, at each lattice point, the ordered lattice has copper

(b) Cu 3 Au, (c) CuAu, (d) Fe 3 Al .

80 Modern Physical Metallurgy and Materials Engineering simple cells are necessary to describe the complete

greater than Q are able to make the jump, where ordered arrangement. In this structure any individual

Q ˛!ˇ D H m H ˛ and Q ˇ!˛DH m H ˇ are the activation atom is surrounded by the maximum number of

enthalpies for heating and cooling, respectively. The unlike atoms and the aluminium atoms are arranged

probability of an atom having sufficient energy to jump tetrahedrally in the cell. Other examples of the D0 3 the barrier is given, from the Maxwell –Boltzmann dis-

include Fe 3 Si and Cu 3 Al.

5. Mg 3 Cd This ordered structure is based on the cph k is Boltzmann’s constant, T is the temperature and Q

is usually expressed as the energy per atom in electron Ti 3 Al, MgCd 3 and Ni 3 Sn.

lattice. Other examples of the D0 19 structure are

volts. 1

The rate of reaction is given by Another important structure which occurs in certain

intermetallics is the defect lattice. In the compound (3.8) NiAl, as the composition deviates from stoichiome-

try towards pure aluminium, the electron to atom ratio where A is a constant involving n and v , the frequency of vibration. To determine Q experimentally, the reac-

becomes greater than 3 2 , but to prevent the compound tion velocity is measured at different temperatures and, becoming unstable the lattice takes up a certain propor-

since

tion of vacancies to maintain the number of electrons per unit cell at a constant value of 3. Such defects obvi-

(3.9) ously increase the entropy of the alloy, but the fact that these phases are stable at low temperatures, where the

the slope of the In (rate) versus 1/T curve gives Q/k. entropy factor is unimportant, demonstrates that their

In deriving equation (3.8), usually called an stability is due to a lowering of internal energy. Such

Arrhenius equation after the Swedish chemist who defects produce an anomalous decrease in both the lat-

first studied reaction kinetics, no account is taken of tice parameter and the density above 50 at.% Al.

the entropy of activation, i.e. the change in entropy as a result of the transition. In considering a general reaction the probability expression should be written in terms of the free energy of activation per atom F

3.4 The mechanism of phase changes

or G rather than just the internal energy or enthalpy.

The rate equation then becomes Changes of phase in the solid state involve a redistri-

3.4.1 Kinetic considerations

bution of the atoms in that solid and the kinetics of

D A ⊲ 3.10⊳ the change necessarily depend upon the rate of atomic

migration. The transport of atoms through the crystal The slope of the ln (rate) versus 1/T curve then gives is more generally termed diffusion, and is dealt with

the temperature-dependence of the reaction rate, which in Section 6.4. This can occur more easily with the

is governed by the activation energy or enthalpy, and aid of vacancies, since the basic act of diffusion is the

the magnitude of the intercept on the ln (rate) axis movement of an atom to an empty adjacent atomic

depends on the temperature-independent terms and site.

include the frequency factor and the entropy term. Let us consider that during a phase change an

During the transformation it is not necessary for atom is moved from an ˛-phase lattice site to a more

the entire system to go from ˛ to ˇ at one jump favourable ˇ-phase lattice site. The energy of the atom

and, in fact, if this were necessary, phase changes should vary with distance as shown in Figure 3.45,

would practically never occur. Instead, most phase where the potential barrier which has to be overcome

changes occur by a process of nucleation and growth arises from the interatomic forces between the mov-

(cf. solidification, Section 3.1.1). Chance thermal fluc- ing atom and the group of atoms which adjoin it and

tuations provide a small number of atoms with suffi- the new site. Only those atoms (n) with an energy

cient activation energy to break away from the matrix (the old structure) and form a small nucleus of the new phase, which then grows at the expense of the matrix until the whole structure is transformed. By this mechanism, the amount of material in the intermedi- ate configuration of higher free energy is kept to a minimum, as it is localized into atomically thin lay- ers at the interface between the phases. Because of

1 Q may also be given as the energy in J mol 1 in which case the rate equation becomes

Figure 3.45 Energy barrier separating structural states . where R D kN is the gas constant, i.e. 8.314 J mol 1 K 1 .

Structural phases: their formation and transitions 81 this mechanism of transformation, the factors which

transformed, and G s , the extra free energy of the determine the rate of phase change are: (1) the rate

boundary atoms, becomes important due to the large of nucleation, N (i.e. the number of nuclei formed in

surface area to volume ratio of small nuclei. Therefore unit volume in unit time) and (2) the rate of growth,

before transformation can take place the negative term

G (i.e. the rate of increase in radius with time). Both G v must be greater than the positive term G s and, processes require activation energies, which in general

since G v is zero at the equilibrium freezing point, it are not equal, but the values are much smaller than

follows that undercooling must result. that needed to change the whole structure from ˛ to ˇ in one operation.

3.4.2 Homogeneous nucleation

Even with such an economical process as nucleation and growth transformation, difficulties occur and it is

Quantitatively, since G v depends on the volume of common to find that the transformation temperature,

the nucleus and G s is proportional to its surface area, even under the best experimental conditions, is slightly

we can write for a spherical nucleus of radius r higher on heating than on cooling. This sluggishness

2 (3.11) of the transformation is known as hysteresis, and is

G D ⊲

3 G v /

attributed to the difficulties of nucleation, since dif- where G v is the bulk free energy change involved fusion, which controls the growth process, is usually

in the formation of the nucleus of unit volume and high at temperatures near the transformation tempera-

is the surface energy of unit area. When the ture and is, therefore, not rate-controlling. Perhaps the

nuclei are small the positive surface energy term simplest phase change to indicate this is the solidifica-

predominates, while when they are large the negative tion of a liquid metal.

volume term predominates, so that the change in free The transformation temperature, as shown on the

energy as a function of nucleus size is as shown in equilibrium diagram, represents the point at which the

Figure 3.46a. This indicates that a critical nucleus size free energy of the solid phase is equal to that of the

exists below which the free energy increases as the liquid phase. Thus, we may consider the transition, as

nucleus grows, and above which further growth can given in a phase diagram, to occur when the bulk or

proceed with a lowering of free energy; G max may chemical free energy change, G v , is infinitesimally

be considered as the energy or work of nucleation W. small and negative, i.e. when a small but positive driv-

Both r c and W may be calculated since dG/dr D ing force exists. However, such a definition ignores the

c and thus r c D process whereby the bulk liquid is transformed to bulk

2 G v

2 /G v . Substituting for r c gives solid, i.e. nucleation and growth. When the nucleus is

3 / 3G v formed the atoms which make up the interface between 2 (3.12) the new and old phase occupy positions of compromise

WD

The surface energy factor is not strongly dependent between the old and new structure, and as a result

on temperature, but the greater the degree of under- these atoms have rather higher energies than the other

cooling or supersaturation, the greater is the release atoms. Thus, there will always be a positive free energy

of chemical free energy and the smaller the critical term opposing the transformation as a result of the

nucleus size and energy of nucleation. This can be energy required to create the surface of interface. Con-

shown analytically since G v D , and at sequently, the transformation will occur only when the

TDT e , G v D 0, so that H D T e S . It therefore sum G v

C G s becomes negative, where G s arises

follows that

from the surface energy of solid–liquid interface. Nor- mally, for the bulk phase change, the number of atoms

G v D ⊲T e T⊳S D TS which form the interface is small and G s compared

and because G v / T , then with G v can be ignored. However, during nucleation

G v is small, since it is proportional to the amount W/ 3 /T 2 (3.13)

Figure 3.46 (a) Effect of nucleus size on the free energy of nucleus formation. (b) Effect of undercooling on the rate of precipitation .

82 Modern Physical Metallurgy and Materials Engineering Consequently, since nuclei are formed by thermal fluc-

tuations, the probability of forming a smaller nucleus is greatly improved, and the rate of nucleation increases according to

max /kT

D A max ⊳/kT ]

the fact that rate of nucleus formation is in the limit controlled by the rate of atomic migration. Clearly, with very extensive degrees of undercooling, when G max − Q , the rate of nucleation approaches exp

mobility, this becomes small at low temperature (Figure 3.46b). While this range of conditions can

be reached for liquid glasses the nucleation of liquid metals normally occurs at temperatures before this condition is reached. (By splat cooling, small droplets

of the metal are cooled very rapidly ⊲10 5 Ks 1 ⊳ and

an amorphous solid may be produced.) Nevertheless, the principles are of importance in metallurgy since

Figure 3.47 Schematic geometry of heterogeneous in the isothermal transformation of eutectoid steel, for

nucleation .

example, the rate of transformation initially increases and then decreases with lowering of the transformation

° , no wetting temperature (see TTT curves, Chapter 8).

there is complete wetting and W ! 0; and when

° there is some wetting and W is reduced. In practice, homogeneous nucleation rarely takes place

3.4.3 Heterogeneous nucleation

and heterogeneous nucleation occurs either on the

3.4.4 Nucleation in solids

mould walls or on insoluble impurity particles. From When the transformation takes place in the solid state, equation (3.13) it is evident that a reduction in the

i.e. between two solid phases, a second factor giving interfacial energy would facilitate nucleation at small

rise to hysteresis operates. The new phase usually values of T. Figure 3.47 shows how this occurs at

has a different parameter and crystal structure from

a mould wall or pre-existing solid particle, where the the old so that the transformation is accompanied by nucleus has the shape of a spherical cap to minimize

dimensional changes. However, the changes in volume and shape cannot occur freely because of the rigidity of

balance of the interfacial tensions in the plane of the the surrounding matrix, and elastic strains are induced.

The strain energy and surface energy created by the The formation of the nucleus is associated with an

ML

SM ⊳/ SL .

nuclei of the new phase are positive contributions to excess free energy given by

the free energy and so tend to oppose the transition. G D VG v C A SL SL C A SM SM A SM The total free energy change is ML

3 3 G D VG v C A C VG s (3.17)

D G v

2 C where A is the area of interface between the two phases SL and the interfacial energy per unit area, and G s is

the misfit strain energy per unit volume of new phase.

For a spherical nucleus of the second phase Differentiation of this expression for the maximum, i.e.

dG/dr D 0, gives r c 2 SL /G v and

and the misfit strain energy reduces the effective driv- WD⊲

3 / 3G v 2 ⊳

ing force for the transformation. Differentiation of

equation (3.18) gives

or r c 2 /⊲G v G s ⊳, and W ⊲ heterogeneous⊳ D W ⊲ homogeneous⊳

3 / 3⊲G v G s ⊳ 2 The value of can vary widely from a few mJ/m 2

WD

to several hundred mJ/m 2 depending on the coherency

Structural phases: their formation and transitions 83

Figure 3.48 Schematic representation of interface structures. (a) A coherent boundary with misfit strain and (b) a semi-coherent boundary with misfit dislocations .

of the interface. A coherent interface is formed when plate-like precipitate forms. Thus, the habit plane is the two crystals have a good ‘match’ and the two lat-

the one which allows the planes at the interface to fit tices are continuous across the interface. This happens

together with the minimum of disregistry; the frequent when the interfacial plane has the same atomic config-

occurrence of the Widmanst¨atten structures may be uration in both phases, e.g. f1 1 1g in fcc and f0 0 0 1g

explained on this basis. It is also observed that precip- in cph. When the ‘match’ at the interface is not perfect

itation occurs most readily in regions of the structure it is still possible to maintain coherency by strain-

which are somewhat disarranged, e.g. at grain bound- ing one or both lattices, as shown in Figure 3.48a.

aries, inclusions, dislocations or other positions of high These coherency strains increase the energy and for

residual stress caused by plastic deformation. Such large misfits it becomes energetically more favourable

regions have an unusually high free energy and neces- to form a semi-coherent interface in which the mis-

sarily are the first areas to become unstable during the match is periodically taken up by misfit dislocations. 1 transformation. Also, new phases can form there with

The coherency strains can then be relieved by a cross-

a minimum increase in surface energy. This behaviour grid of dislocations in the interface plane, the spac-

is considered again in Chapter 7. ing of which depends on the Burgers vector b of the

dislocation and the misfit ε, i.e. b/ε. The interfacial energy for semi-coherent interfaces arises from the change in composition across the interface or chemical

Further reading

contribution as for fully-coherent interfaces, plus the energy of the dislocations (see Chapter 4). The energy Beeley, P. R. (1972). Foundry Technology. Butterworths,

2 of a semi-coherent interface is 200–500 mJ/m London. and Campbell, J. (1991). Castings. Butterworth-Heinemann, Lon- increases with decreasing dislocation spacing until the

don.

dislocation strain fields overlap. When this occurs, the Chadwick, G. A. (1972). Metallography of Phase Transfor- discrete nature of the dislocations is lost and the inter-

mations . Butterworths, London. face becomes incoherent. The incoherent interface is

Davies, G. J. (1973). Solidification and Casting. Applied Sci- somewhat similar to a high-angle grain boundary (see

ence, London.

Driver, D. (1985). Aero engine alloy development, Inst. of independent of the orientation.

Figure 3.3) with its energy of 0.5 to 1 J/m 2 relatively

Metals Conf., Birmingham. ‘Materials at their Limits’ (25 The surface and strain energy effects discussed September 1985). Flemings, M. C. (1974). Solidification Processing. McGraw-

above play an important role in phase separation.

Hill, New York.

When there is coherence in the atomic structure across Hume-Rothery, W., Smallman, R. E. and Haworth, C. the interface between precipitate and matrix the sur-

(1969). Structure of Metals and Alloys, 5th edn. Institute face energy term is small, and it is the strain energy

of Metals, London.

factor which controls the shape of the particle. A Kingery, W. D., Bowen, H. K. and Uhlmann, D. R. (1976). plate-shaped particle is associated with the least strain

Introduction to Ceramics , 2nd edn. Wiley-Interscience, energy, while a spherical-shaped particle is associated

New York.

with maximum strain energy but the minimum surface Rhines, F. N. (1956). Phase Diagrams in Metallurgy: their energy. On the other hand, surface energy determines

development and application . McGraw-Hill, New York. the crystallographic plane of the matrix on which a

Quets, J. M. and Dresher, W. H. (1969). Thermo-chemistry of hot corrosion of superalloys. Journal of Materials, ASTM, JMSLA, 4, 3, 583–599.

1 A detailed treatment of dislocations and other defects is West, D. R. F. (1982). Ternary Equilibrium Diagrams, 2nd given in Chapter 4.

edn. Macmillan, London.