The mechanism of phase changes

Gaseous 1 Gaseous 1

3.4 The mechanism of phase changes

3.4.1 Kinetic considerations

Changes of phase in the solid state involve a redistri- bution 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 6.4. 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 or-phase lattice site to a more favourable fl-phase lattice site. The energy of the atom should vary with distance as shown in Figure 3.45, where the potential barrier which has to be overcome arises from the interatomic forces between the mov- ing atom and the group of atoms which adjoin it and the new site. Only those atoms (n) with an energy

. . .

> .

_~_ . . .

A t o m i c p o s i t i o n

Figure 3.45 Energy barrier separating structural states.

greater than Q are able to make the jump, where Q ~ = Hm - Ho, and Qt~--,~=~/,,,-H~ 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 dis- tribution law, as proportional to exp [ - Q / k T ] where k is Boltzmann's constant, T is the temperature and Q is usually expressed as the energy per atom in electron volts. ~

The rate of reaction is given by

Rate = A exp [ - Q / k T ] (3.8)

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

In (Rate) = lnA - Q / k T (3.9) the slope of the In (rate) versus 1/T curve gives Q/k.

In deriving equation(3.8), 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 / k T ]

= A e x p [ S / k ] e x p [ - E / k T ] (3.10) The slope of the In (rate) versus I / T curve then gives the temperature-dependence of the reaction rate, which is governed by the activation energy or enthalpy, and the magnitude of the intercept on the In (rate) axis depends on the temperature-independent terms and include the frequency factor and the entropy term.

During the transformation it is not necessary for the entire system to go from c~ to 13 at 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 3.1.1). Chance thermal fluc- tuations 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 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

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

Rate of reaction = A exp I-Q/RT]

where R = kN is the gas constant, i.e. 8.314 J mol -I K -1.

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 o~ to/3 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 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 controls 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 transition, as given in a phase diagram, to occur when the bulk or chemical free energy change, A Gv, is infinitesimally small and negative, i.e. when a small but positive driv- ing 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 structure, 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 of interface. Con- sequently, the transformation will occur only when the sum AGo + AGs becomes negative, where AGs arises from the surface energy of solid-liquid interface. Nor- mally, for the bulk phase change, the number of atoms which form the interface is small and A G~ compared with A G,, can be ignored. However, during nucleation AGo is small, since it is proportional to the amount

Structural phases" their formation and transitions 81 transformed, and A Gs, 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 the negative term AGo must be greater than the positive term AGs and, since AGo is zero at the equilibrium freezing point, it follows that undercooling must result.

3.4.2 Homogeneous nucleation

Quantitatively, since A Gv depends on the volume of the nucleus and AG~ is proportional to its surface area, we can write for a spherical nucleus of radius r

A G = (4zrraAGo/3) + 4rcr2y (3.11) where AGo is the bulk free energy change involved in the formation of the nucleus of unit volume and 7, is the surface energy of unit area. When the nuclei are small the positive surface energy term 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 3.46a. This indicates that a critical nucleus size exists below which the free energy increases as the nucleus grows, and above which further growth can proceed with a lowering of free energy; A Gmax may be considered as the energy or work of nucleation W.

Both rc and W may be calculated since d A G / d r = 4rrr 2AGo + 8rrry = 0 when r = rc and thus rc =

- 2 y / A G o . Substituting for rc gives

W = 16rry3/3AGo 2 (3.12)

The surface energy factor y is not strongly dependent on temperature, but the greater the degree 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 AGo = A H - T AS, and at T = Te, AGo = 0, so that A H = TEAS. It therefore follows that

AGo = ( T e - T ) A S = A T A S and because AGo cx AT, then

W oc y 3 / A T 2 (3.13)

~ Gin8 I

t

Nucleus size ,---,,-

(a)

I

Nucleation d0fficult~es r~ this region

i . ~ ~ Oeffuson slow

Y

Degree of undercool~nq .---,..

(b)

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 Modem 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

Rate = A exp [ - Q / k T ] exp [ - A G m a x / k T

-- A exp [ - ( Q + AGmax)/kT] (3.14) The term exp [ - Q / k T ] 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, when A Gmax << Q, the rate of nucleation approaches exp [ - Q / k T ] and, because of the slowness of atomic 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 (105 K s - l ) 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 T T T curves, Chapter 8).

3.4.3 H e t e r o g e n e o u s nucleation

In practice, homogeneous nucleation rarely takes place and heterogeneous nucleation occurs either on the mould walls or on insoluble impurity particles. From equation (3.13) it is evident that a reduction in the interfacial energy V would facilitate nucleation at small values of AT. Figure 3.47 shows how this occurs at a mould wall or pre-existing solid particle, where the nucleus has the shape of a spherical cap to minimize the energy and the 'wetting' angle 0 is given by the balance of the interfacial tensions in the plane of the mould wall, i.e. cos0 = (YML -- YSM)/YSL.

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

A G = VAG~ + ASLYSL "~- ASMYSM -- ASMYML

= Jr/3(2 - 3 cos 0 + cos 30)r 3 AGv + 2rr(1 - cos O)r 2 YSL

+ zrr 2 sin E 0(FSM -- YLM) (3.15) Differentiation of this expression for the maximum, i.e.

d A G / d r = 0, gives rc = -2ysL/AG,, and W = (16zrv3/3AG~2)[(1 - c o s 0 f ( 2 + cos 0)/4]

(3.16) o r

W (heterogeneous) = W (homogeneous)[ S (0) ]

The shape factor S(O) _< 1 is dependent on the value of 0 and the work of nucleation is therefore less for

g,- 'sM

mould.

wall ?

, . , . . . . . . , . , = . . D

r COS 0

Figure 3.47 Schematic geometry of heterogeneous nucleation.

heterogeneous nucleation. When 0 = 180 ~ no wetting occurs and there is no reduction in W; when 0 --+ 0 ~ there is complete wetting and W--+ 0; and when 0 < 0 < 180 ~ there is some wetting and W is reduced.

3.4.4 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

A G = VAG,, + AV + V A G s (3.17) where A is the area of interface between the two phases and y the interfacial energy per unit area, and AGs is the misfit strain energy per unit volume of new phase.

For a spherical nucleus of the second phase

A G .__ 4 5yrr- (AGv - AGs) + 4zrr2y (3.18) and the misfit strain energy reduces the effective driv- ing force for the transformation. Differentiation of equation (3.18) gives

rc = - 2 y / ( A G , , - AGs), and W = 1 6 n ' g 3 / 3 ( A G v - AGs) 2

The value of V can vary widely from a few mJ/m 2 to several hundred mJ/m 2 depending on the coherency

(a) (b)

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 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 {000 1 } in cph. When the 'match' at the interface is not perfect it is still possible to maintain coherency by strain- ing one or both lattices, as shown in Figure 3.48a.

These coherency strains increase the energy and for large misfits it becomes energetically more favourable to form a semi-coherent interface in which the mis- match is periodically taken up by misfit dislocations. ~ 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 e, i.e. b/e. 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 4). The energy of a semi-coherent interface is 2 0 0 - 5 0 0 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 3.3) with its energy of 0.5 to 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 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 I A detailed treatment of dislocations and other defects is given in Chapter 4.

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 Widmanst~itten 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 bound- aries, 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 behaviour is considered again in Chapter 7.

Gaseous 1 Gaseous 1

3.4 The mechanism of phase changes

3.4.2 Homogeneous nucleation

t

I

Y

(b)

mould.

wall ?

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