Modern Physical Metallurgy and Materials Engineering Part 4 pot

These take the form of: ž Point defects, such as vacant atomic sites or simply vacancies and interstitial atoms or simply intersti-tials where an atom sits in an interstice rather than a

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 ˛ to ˇ

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

dif-fusion, which controls the growth process, is usually

high at temperatures near the transformation

tempera-ture and is, therefore, not rate-controlling Perhaps the

simplest phase change to indicate this is the

solidifica-tion 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, 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 GvCGsbecomes negative, where Gsarises

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 Gs compared

with Gv can be ignored However, during nucleation

Gv is small, since it is proportional to the amount

transformed, and Gs, the extra free energy of theboundary atoms, becomes important due to the largesurface area to volume ratio of small nuclei Thereforebefore transformation can take place the negative term

Gvmust be greater than the positive term Gsand,since Gvis zero at the equilibrium freezing point, itfollows that undercooling must result

in the formation of the nucleus of unit volume and

is the surface energy of unit area When thenuclei are small the positive surface energy termpredominates, while when they are large the negativevolume term predominates, so that the change in freeenergy as a function of nucleus size is as shown inFigure 3.46a This indicates that a critical nucleus sizeexists below which the free energy increases as thenucleus grows, and above which further growth canproceed with a lowering of free energy; Gmax may

be considered as the energy or work of nucleation W.Both rc and W may be calculated since dG/dr D

4 r2GvC8 r D 0 when r D rc and thus rcD

2 /Gv Substituting for rcgives

The surface energy factor is not strongly dependent

on temperature, but the greater the degree of cooling or supersaturation, the greater is the release

under-of chemical free energy and the smaller the criticalnucleus size and energy of nucleation This can beshown analytically since GvDH  TS, and at

T D Te, GvD0, so that H D TeS It thereforefollows that

GvDTeTS D TSand because Gv/T, then

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

precipitation.

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 D A exp [Q/kT] exp [Gmax/kT

DA exp [Q C Gmax/kT] 3.14

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, when

Gmax−Q, the rate of nucleation approaches exp

[Q/kT] 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 105K s 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 8).

3.4.3 Heterogeneous 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 would facilitate nucleation at small

values of T 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  is given by the

balance of the interfacial tensions in the plane of the

mould wall, i.e cos  D  ML SM/ SL

The formation of the nucleus is associated with an

excess free energy given by

WheterogeneousDWhomogeneous[S]

The shape factor S  1 is dependent on the value

of  and the work of nucleation is therefore less for

Figure 3.47 Schematic geometry of heterogeneous

nucleation.

heterogeneous nucleation When  D 180°, no wetting

occurs and there is no reduction in W; when  ! 0°

there is complete wetting and W ! 0; and when

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 givingrise to hysteresis operates The new phase usuallyhas a different parameter and crystal structure fromthe old so that the transformation is accompanied bydimensional changes However, the changes in volumeand shape cannot occur freely because of the rigidity ofthe surrounding matrix, and elastic strains are induced.The strain energy and surface energy created by thenuclei of the new phase are positive contributions tothe free energy and so tend to oppose the transition.The total free energy change is

where A is the area of interface between the two phasesand the interfacial energy per unit area, and Gsisthe misfit strain energy per unit volume of new phase.For a spherical nucleus of the second phase

G D 43 r3GvGs C 4 r2 ... atoms

of the dislocation, as shown in Figure 4. 4a and 4. 4b

This causes the dislocation to climb, as discussed

in Section 4. 3 .4 The process whereby vacancies

are annihilated... screw dislocation (seeSection 4. 3 .4) and also of particular importance inmaterials that have been subjected to irradiation byhigh-energy particles

4. 2.2 Point defects in non-metallic... dislocation can be resolved into edge andscrew components The atomic structure of a simpleedge and screw dislocation is shown in Figure 4. 1 3and 4. 14

4. 3.3 The Burgers vector