The mechanism of phase changes

2.4 The mechanism of phase changes

2.4.1 Kinetic considerations

Changes of phase in the solid state involve a redistribution of the atoms in that solid and the kinetics of the change necessarily depend upon the rate of atomic migration. The transport of atoms through the crystal is more generally termed diffusion, and is dealt with in Section 5.4. This can occur more easily with the aid of vacancies, since the basic act of diffusion is the movement of an atom to an empty adjacent atomic site.

Let us consider that during a phase change an atom is moved from an α-phase lattice site to a more favorable β-phase lattice site. The energy of the atom should vary with distance as shown in Figure 2.39, where the potential barrier which has to be overcome arises from the interatomic forces between the moving atom and the group of atoms which adjoin it and the new site. Only those atoms (n) with an energy greater than Q are able to make the jump, where Q α →β =H m −H α and Q β →α =H m −H β are the activation enthalpies for heating and cooling, respectively. The probability of an atom having sufficient energy to jump the barrier is given, from the Maxwell–Boltzmann distri- bution law, as proportional to exp [ −Q/kT ], where k is Boltzmann’s constant, T is the temperature and Q is usually expressed as the energy per atom in electron volts. 16

16 Q may also be given as the energy in J mol −1 , in which case the rate equation becomes Rate of reaction = A exp [−Q/RT ]

where R = kN is the gas constant, i.e. 8.314 J mol −1 K −1 and N is the Avogadro number 6.02 × 10 23 .

Phase equilibria and structure 87

Energy H b H

Atomic position

Figure 2.39 Energy barrier separating structural states.

The rate of reaction is given by Rate = A exp[−Q/kT ],

(2.8) where A is a constant involving n and v, the frequency of vibration. To determine Q experimentally,

the reaction velocity is measured at different temperatures and, since ln (Rate) = ln A − Q/kT ,

(2.9) the slope of the ln(Rate) versus 1/T curve gives Q/k.

In deriving equation (2.8), usually called an Arrhenius equation after the Swedish chemist who first studied reaction kinetics, no account is taken of 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 or G rather than just the internal energy or enthalpy. The rate equation then becomes

Rate = A exp[−F/kT ] (2.10) = A exp[S/k] exp[−E/kT ].

The slope of the ln(Rate) versus 1/T curve then gives the temperature dependence of the reaction rate,

i.e. the activation energy or enthalpy, and the magnitude of the intercept on the ln(Rate) axis depends on the temperature-independent terms, and includes the frequency factor and the entropy term. During the transformation it is not necessary for the entire system to go from α to β in one jump and, in fact, if this were necessary, phase changes would practically never occur. Instead, most phase changes occur by a process of nucleation and growth (cf. solidification, Section 2.1.1). Chance thermal fluctuations provide a small number of atoms with sufficient 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 intermediate configuration of higher free energy is kept to a minimum, as it is localized into atomically thin layers at the interface between the phases. Because of this mechanism of transformation, the factors which determine the rate of phase change are: (1) the rate of nucleation, N (i.e. the number of nuclei formed in unit volume in unit time) and (2) the rate of growth, G (i.e. the rate of increase in radius with time). Both processes require activation energies, which in general are not equal, but the values are much smaller than that needed to change the whole structure from α to β in one operation.

Even with such an economical process as nucleation and growth transformation, difficulties occur and it is common to find that the transformation temperature, even under the best experimental

88 Physical Metallurgy and Advanced Materials conditions, is slightly higher on heating than on cooling. This sluggishness of the transformation is

known as hysteresis, and is attributed to the difficulties of nucleation, since diffusion, which con- trols the growth process, is usually high at temperatures near the transformation temperature and is therefore not rate controlling. Perhaps the simplest phase change to indicate this is the solidification of a liquid metal.

The transformation temperature, as shown on the equilibrium diagram, represents the point at which the free energy of the solid phase is equal to that of the liquid phase. Thus, we may consider the v , is infinitesimally small and negative, i.e. when a small but positive driving force exists. However, such

a definition ignores the process whereby the bulk liquid is transformed to bulk solid, i.e. nucleation and growth. When the nucleus is formed the atoms which make up the interface between the new and old phase occupy positions of compromise between the old and new structures, and as a result these atoms have rather higher energies than the other atoms. Thus, there will always be a positive free energy term opposing the transformation as a result of the energy required to create the surface

s becomes s arises from the surface energy of solid/liquid interface. Normally, for the bulk

sv

v is small, since it is proportional to the amount s , the extra free energy of the boundary atoms, becomes important due to the large surface area-to-volume ratio of small nuclei. Therefore, before transformation can take place

v is zero at the equilibrium freezing point, it follows that undercooling must result.

Worked example

C. How long does it take at the lower temperature of 700 ◦

A thermally activated transformation is complete after 180 seconds at 900 ◦

C if the activation energy for the process is 167.5 kJ mol −1 ?

Solution

Rate of reaction ∝ 1/time = 1/t. Rate

Ae −Q/RT 2 2 2 1 1 Q

1/t

Rate = 1/t = 1 1 Ae −Q/RT 1 = exp − − T + 1 T 2 R

1/180 =e −3.4 or t 2 = 180 e = 2220 s = 37 min.

2.4.2 Homogeneous nucleation

s is proportional to its surface area, we can write for a spherical nucleus of radius r:

(2.11) v is the bulk free energy change involved in the formation of the nucleus of unit volume

G = (4πr 3 G v / 3)

+ 4πr 2 γ ,

and γ is the surface energy of unit area. When the nuclei are small the positive surface energy term

Phase equilibria and structure 89

⌬ G max

Nucleation difficulties in this region

Diffusion slow ⌬ G

in this region

Rate of precipitation

0 Degree of undercooling Nucleus size

(a) (b) Figure 2.40 (a) Effect of nucleus size on the free energy of nucleus formation. (b) Effect of

undercooling on the rate of precipitation.

predominates, while when they are large the negative volume term predominates, so that the change in free energy as a function of nucleus size is as shown in Figure 2.40a. This indicates that a critical nucleus size exists below which the free energy increases as the nucleus grows, and above which

max may be considered as the energy or work of nucleation W . Both r c = 4πr 2 G v + 8πrγ = 0

when r =r c and thus r c v . Substituting for r c gives

2 v . (2.12) The surface energy factor γ is not strongly dependent on temperature, but the greater the degree

W = 16πγ 3 /

of undercooling or supersaturation, the greater is the release of chemical free energy and the smaller the critical nucleus size and energy of nucleation. This can be shown analytically, since

e S. It therefore follows that

G v = (T e

3 ∝γ 2 T . (2.13) Consequently, since nuclei are formed by thermal fluctuations, the probability of forming a smaller

nucleus is greatly improved, and the rate of nucleation increases according to Rate

max /kT ]

max )/kT ].

The term exp [ −Q/kT ] is introduced to allow for 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,

max ≪ Q, the rate of nucleation approaches exp[−Q/kT ] and, because of the slowness of atomic mobility, this becomes small at low temperature (Figure 2.40b). 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

90 Physical Metallurgy and Advanced Materials

Mold wall

Figure 2.41 Schematic geometry of heterogeneous nucleation. rapidly (10 5 Ks −1 ) and an amorphous solid may be produced.) Nevertheless, the principles are of

importance in metallurgy since in the isothermal transformation of eutectoid steel, for example, the rate of transformation initially increases and then decreases with lowering of the transformation temperature (see TTT curves, Chapter 7).

2.4.3 Heterogeneous nucleation

In practice, homogeneous nucleation rarely takes place and heterogeneous nucleation occurs either on the mold walls or on insoluble impurity particles. From equation (2.13) it is evident that a reduction

this occurs at a mold wall or pre-existing solid particle, where the nucleus has the shape of a spherical cap to minimize the energy and the ‘wetting’ angle θ is given by the balance of the interfacial tensions in the plane of the mold wall, i.e. cos θ = (γ ML −γ SM )/γ SL .

The formation of the nucleus is associated with an excess free energy given by

G v +A SL γ SL +A SM γ SM −A SM γ ML

= π/3(2 − 3 cos θ + cos 3 θ )r 3 G v

+ 2π(1 − cos θ)r 2 γ SL (2.15)

+ πr 2 sin 2 θ (γ SM −γ LM ).

= 0, gives r c = −2γ SL G v and W

(2.16) or W (heterogeneous) =W (homogeneous) [S(θ)].

2 v )[(1

− cos θ) 2 (2 + cos θ)/4]

Phase equilibria and structure 91 The shape factor S(θ) ≤ 1 is dependent on the value of θ and the work of nucleation is therefore less

for heterogeneous nucleation. When θ = 180 ◦ , no wetting occurs and there is no reduction in W ; when θ →0 ◦ there is complete wetting and W → 0; and when 0 < θ < 180 ◦ there is some wetting and W

is reduced.

Worked example

Heterogeneous nucleation of a β-phase in an α-phase occurs on a grain boundary. If γ αβ is 250 mJ m −2 and the grain boundary energy is 1 2 Jm −2 determine how the nucleation process is aided relative to the homogeneous process.

Solution

Assuming the nucleation is of a spherical nucleus,

G = −Aγ gb 2 + 4πr 4 γ αβ − π r 3 G V ,

where A is the area of the grain boundary removed by the nucleus,

G = −πr 2 γ gb + 4πr 2 γ

αβ − π r 3 G V

3 = −2πrγ 2 gb + 8πrγ αβ − 4πr G V =0

4γ αβ r

−γ gb

For homogeneous nucleation, r c is given by the above formula with γ gb = 0 (see also Section 2.4.2), so r c (heterogeneous)

1 r (homogeneous) =

(4γ αβ −γ gb V 4γ αβ −γ gb 1000 − 500

So, r c (heterogeneous) 1 = 2 r c (homogeneous).

2.4.4 Nucleation in solids

When the transformation takes place in the solid state, i.e. between two solid phases, a second factor giving rise to hysteresis operates. The new phase usually has a different parameter and crystal structure from the old so that the transformation is accompanied by dimensional changes. However, the changes in volume and shape cannot occur freely because of the rigidity of the surrounding matrix, and elastic strains are induced. The strain energy and surface energy created by the nuclei of the new phase are positive contributions to the free energy and so tend to oppose the transition.

The total free energy change is

92 Physical Metallurgy and Advanced Materials

(b) Figure 2.42 Schematic representation of interface structures. (a) A coherent boundary with misfit

(a)

strain. (b) A semi-coherent boundary with misfit dislocations.

where A is the area of interface between the two phases and γ 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:

(2.18) and the misfit strain energy reduces the effective driving force for the transformation. Differentiation

3 + 4πr γ

of equation (2.18) gives

s ), and

W = 16πγ 3 /

The value of γ can vary widely from a few mJ m −2 to several hundred mJ m −2 depending on the coherency of the interface. A coherent interface is formed when the two crystals have a good ‘match’ and the two lattices are continuous across the interface. This happens when the interfacial plane has the same atomic configuration in both phases, e.g. {1 1 1} in fcc and {0 0 0 1} in cph. When the ‘match’ at the interface is not perfect it is still possible to maintain coherency by straining one or both lattices, as shown in Figure 2.42a. These coherency strains increase the energy and for large misfits it becomes energetically more favorable to form a semi-coherent interface (Figure 2.42b) in which

the mismatch is periodically taken up by misfit dislocations. 17 The coherency strains can then be relieved by a cross-grid of dislocations in the interface plane, the spacing 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 contribution as for fully coherent interfaces, plus the energy of the dislocations (see Chapter 3). The energy of a semi-coherent interface is 200–500 mJ m −2 and increases with decreasing dislocation spacing until the dislocation strain fields overlap. When this occurs, the discrete nature of the dislocations is lost and the interface becomes incoherent. The incoherent interface is somewhat similar to a high-angle grain boundary (see Figure 2.3) with its energy of 0.5–1 J m −2 relatively independent of the orientation.

The surface and strain energy effects discussed above play an important role in phase separation. When there is coherence in the atomic structure across the interface between precipitate and matrix

17 A detailed treatment of dislocations and other defects is given in Chapter 3.

Phase equilibria and structure 93 the surface energy term is small, and it is the strain energy factor which controls the shape of the

particle. A plate-shaped particle is associated with the least strain energy, while a spherical-shaped particle is associated with maximum strain energy but the minimum surface energy. On the other hand, surface energy determines the crystallographic plane of the matrix on which a plate-like precipitate forms. Thus, the habit plane is the one which allows the planes at the interface to fit together with the minimum of disregistry; the frequent occurrence of the Widmanstätten structures may be explained on this basis. It is also observed that precipitation occurs most readily in regions of the structure which are somewhat disarranged, e.g. at grain boundaries, inclusions, dislocations or other positions of high residual stress caused by plastic deformation. Such regions have an unusually high free energy and necessarily are the first areas to become unstable during the transformation. Also, new phases can form there with a minimum increase in surface energy. This behavior is considered again in Chapter 6.

Problems

2.1 Change the following alloy compositions (i) Cu–40 wt% Zn, (ii) Cu–10 wt% Al and (iii) Cu–20 wt% In into (a) atomic percent and (b) electron–atom ratio, e/a. What is the significance of the e/a ratio? Why is Cu–In different from Cu–Zn and Cu–Al?

2.2 A hypothetical alloy transforms from a simple cubic structure to a close packed hexa- gonal structure. Assuming a hard sphere model, calculate the volume change during the transformation.

2.3 A pressure–temperature diagram for a typical metal is shown in Figure 2.7. How is this diagram modified for the metals Bi and Ga, and why?

2.4 With reference to Figure 2.11, prove the ‘Lever Rule’ as described in Section 2.2.4.1.

2.5 From the Cu–Ni phase diagram (Figure 2.11) determine for a Cu–40% Ni alloy the composition and amount of each phase at 1250 ◦ C.

2.6 From the Pb–Sn phase diagram (Figure 2.15) determine for the eutectic alloy the amount and composition of each phase, just below the eutectic temperature at 182 ◦ C.

2.7 (a) The Cu–Ni alloy system is an ideal solid solution system in which entire solubility is observed, while the Cu–Ag system is a typical eutectic system in which limited solubility of Cu into Ag and vice versa is observed. Which one of the the Hume-Rothery alloying rules best accounts for this difference? (b) Which one of the Hume-Rothery rules best explains the fact that in the Cu–Zn alloy system, Cu can dissolve up to ∼40 wt% of Zn but Zn can only dissolve only about 2 wt% of Cu?

2.8 Compare the size of the interstitial sites 1 2 , 1 4 ,0 and

0, 0, 1 2 for a bcc structure containing

iron, and explain why the smaller site is occupied by C in low-carbon steel.

2.9 For the Fe–C system, calculate (a) the amount of ferrite and cementite in pearlite formed at 722 ◦

C, and (b) the amount of Fe 3 C and pearlite for a 0.2% C steel at room temperature.

Further reading

Beeley, P. R. (1972). Foundry Technology. Butterworths, London. Brandes, E. A. and Brook, G. B. (1998). Smithells Metals Reference Book. Butterworth-Heinemann,

Oxford. Campbell, J. (1991). Castings. Butterworth-Heinemann, London. Chadwick, G. A. (1972). Metallography of Phase Transformations. Butterworths, London. Davies, G. J. (1973). Solidification and Casting. Applied Science, London. Driver, D. (1985). Aero engine alloy development, Inst. of Metals Conf., Birmingham, ‘Materials at their

Limits’ (25 September 1985). Flemings, M. C. (1974). Solidification Processing. McGraw-Hill, New York.

94 Physical Metallurgy and Advanced Materials Hume-Rothery, W., Smallman, R. E. and Haworth, C. (1969). Structure of Metals and Alloys, 5th edn.

Institute of Metals, London. Kingery, W. D., Bowen, H. K. and Uhlmann, D. R. (1976). Introduction to Ceramics, 2nd edn. Wiley- Interscience, New York. Massalski, T. B., Okamoto, H. eds (1996). Binary Alloy Phase Diagrams, ASM International. Quets, J. M. and Dresher, W. H. (1969). Thermo-chemistry of hot corrosion of superalloys. Journal of

Materials, ASTM, JMSLA, 4, (3), 583–599. Rhines, F. N. (1956). Phase Diagrams in Metallurgy: their development and application. McGraw-Hill, New York. West, D. R. F. (1982). Ternary Equilibrium Diagrams, 2nd edn. Macmillan, London.

Chapter 3

Crystal defects