Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-11 GTBL042-Callister-v3 October 4, 2007 11:59

2nd Revised Pages

406 • Chapter 11 / Phase Transformations

Number of stable nuclei, n*

exp –G*

Temperature kT

Tm

Frequency of attachment, d

exp –

Qd

kT

Temperature

Tm

n*, d, N

d

.

.

n*

N

T

Temperature

Tm

(a) (b) (c)

Figure 11.4 For

solidification,

schematic plots of

(a) number of stable

nuclei versus

temperature,

(b) frequency of

atomic attachment

versus temperature,

and (c) nucleation

rate versus

temperature (also

shown are curves for

partsaandb).

same as for the diffusion coefficient—namely,

νd=K 2 exp

(

−

Qd

kT

)

(11.9)

whereQd is a temperature-independent parameter—the activation energy for

diffusion—andK 2 is a temperature-independent constant. Thus, from Equation 11.9

a diminishment of temperature results in a reduction inνd. This effect, represented

by the curve shown in Figure 11.4b, is just the reverse of that forn∗as discussed

above.

The principles and concepts just developed are now extended to a discussion of

another important nucleation parameter, the nucleation rateN ̇(which has units of

nuclei per unit volume per second). This rate is simply proportional to the product

ofn∗(Equation 11.8) andνd(Equation 11.9); that is,

N ̇=K 3 n∗νd=K 1 K 2 K 3

[

exp

(

−

G∗

kT

)

exp

(

−

Qd

kT

)]

(11.10)

Nucleation rate

expression for

homogeneous

nucleation

HereK 3 is the number of atoms on a nucleus surface. Figure 11.4cschematically plots

nucleation rate as a function of temperature and, in addition, the curves of Figures

11.4aand 11.4bfrom which theN ̇curve is derived. Note (Figure 11.4c) that, with a

lowering of temperature from belowTm, the nucleation rate first increases, achieves

a maximum, and subsequently diminishes.

The shape of thisN ̇curve is explained as follows: for the upper region of the

curve (a sudden and dramatic increase inN ̇with decreasingT),G∗is greater than

Qd, which means that the exp(–G∗/kT) term of Equation 11.10 is much smaller than

exp(–Qd/kT). In other words, the nucleation rate is suppressed at high temperatures

due to a small activation driving force. With continued diminishment of temper-

ature, there comes a point at whichG∗becomes smaller than the temperature-

independentQdwith the result that exp(–Qd/kT)<exp(–G∗/kT), or that, at lower

temperatures, a low atomic mobility suppresses the nucleation rate. This accounts for

the shape of the lower curve segment (a precipitous reduction ofN ̇with a continued

diminishment of temperature). Furthermore, theN ̇curve of Figure 11.4cnecessarily

passes through a maximum over the intermediate temperature range where values

forG∗andQdare of approximately the same magnitude.