Page 388 - Materials Science and Engineering An Introduction

360 • Chapter 10 / Phase Transformations

sum of both terms (Figure 10.2b) first increases, passes through a maximum, and finally
decreases. In a physical sense, this means that as a solid particle begins to form as atoms
in the liquid cluster together, its free energy first increases. If this cluster reaches a size
corresponding to the critical radius r*, then growth will continue with the accompani-
ment of a decrease in free energy. However, a cluster of radius less than the critical
value will shrink and redissolve. This subcritical particle is an embryo, and the particle
of radius greater than r* is termed a nucleus. A critical free energy, G*, occurs at the
critical radius and, consequently, at the maximum of the curve in Figure 10.2b. This
G* corresponds to an activation free energy, which is the free energy required for the
formation of a stable nucleus. Equivalently, it may be considered an energy barrier to
the nucleation process.
Because r* and G* appear at the maximum on the free energy-versus-radius curve
of Figure 10.2b, derivation of expressions for these two parameters is a simple matter.
For r*, we differentiate the G equation (Equation 10.1) with respect to r, set the result-
ing expression equal to zero, and then solve for r ( r*). That is,

d( G)
= p G y (3r ) + 4pg(2r) = 0 (10.2)
4
2
dr 3
which leads to the result
For homogeneous 2g
nucleation, critical r*= - (10.3)
radius of a stable G y
solid particle nucleus
Now, substitution of this expression for r* into Equation 10.1 yields the following
expression for G*:

For homogeneous 16pg 3 (10.4)
nucleation, activation G*= 3( G y ) 2
free energy required
for the formation of
a stable nucleus This volume free energy change G y is the driving force for the solidification trans-
formation, and its magnitude is a function of temperature. At the equilibrium solidifica-
tion temperature T m , the value of G y is zero, and with decreasing temperature its value
becomes increasingly more negative.
It can be shown that G y is a function of temperature as

H f (T m - T)
G y = (10.5)
T m
where H f is the latent heat of fusion (i.e., the heat given up during solidification), and
T m and the temperature T are in Kelvin. Substitution of this expression for G y into
Equations 10.3 and 10.4 yields
Dependence of
critical radius on
surface free energy, r*= a - 2gT m b a 1 b (10.6)
latent heat of fusion, H f T m - T
melting temperature,
and transformation and
temperature
3 2
Activation free G*= a 16pg T m b 1 (10.7)
energy expression 3 H f 2 (T m - T) 2

Thus, from these two equations, both the critical radius r* and the activation free
energy G* decrease as temperature T decreases. (The g and H f parameters in these