Page 389 - Materials Science and Engineering An Introduction

10.3 The Kinetics of Phase Transformations • 361

Figure 10.3 Schematic free
r* T < T energy-versus-embryo/nucleus-
+ 1 2 1
radius curves for two different
r* temperatures. The critical free
2 G * 1 at T 1 energy change ( G*) and critical
G *
G 2 nucleus radius (r*) are indicated
0
radius, r for each temperature.

at T 2
–

expressions are relatively insensitive to temperature changes.) Figure 10.3, a schematic
G-versus-r plot that shows curves for two different temperatures, illustrates these re-
lationships. Physically, this means that with a lowering of temperature at temperatures
below the equilibrium solidification temperature (T m ), nucleation occurs more readily.
Furthermore, the number of stable nuclei n* (having radii greater than r*) is a function
of temperature as
G*
n* = K 1 expa - b (10.8)
kT
where the constant K 1 is related to the total number of nuclei of the solid phase.
For the exponential term of this expression, changes in temperature have a greater
effect on the magnitude of the G* term in the numerator than the T term in the
denominator. Consequently, as the temperature is lowered below T m , the exponen-
tial term in Equation 10.8 also decreases, so that the magnitude of n* increases.
This temperature dependence (n* versus T) is represented in the schematic plot of
Figure 10.4a.
Another important temperature-dependent step is involved in and also influences
nucleation: the clustering of atoms by short-range diffusion during the formation of
nuclei. The influence of temperature on the rate of diffusion (i.e., magnitude of the
diffusion coefficient, D) is given in Equation 5.8. Furthermore, this diffusion effect
is related to the frequency at which atoms from the liquid attach themselves to the

Figure 10.4 For
solidification, T m T m T m
schematic plots T
of (a) number of
stable nuclei versus d
temperature,
(b) frequency of Temperature Temperature exp – Q d Temperature .
atomic attachment exp – G* kT N
kT
versus temperature,
and (c) nucleation n*
rate versus tem-
perature (the dashed
curves are repro- .
duced from parts Number of stable nuclei, n* Frequency of attachment, d n*, , N
d
a and b). (a) (b) (c)