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

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

2nd Revised Pages

404 • Chapter 11 / Phase Transformations

0

+

radius, r

4 r^2

 r



(^43) Gv
3
Free energy change,
G
(a) (b)
0

+

radius, r

r*

G*

Free energy change,

G



Figure 11.2 (a) Schematic curves for volume free energy and surface free energy

contributions to the total free energy change attending the formation of a spherical

embryo/nucleus during solidification. (b) Schematic plot of free energy versus

embryo/nucleus radius, on which is shown the critical free energy change (G∗) and the

critical nucleus radius (r∗).

to anactivation 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.

Sincer∗andG∗appear at the maximum on the free-energy-versus-radius curve

of Figure 11.2b, derivation of expressions for these two parameters is a simple matter.

Forr∗, we differentiate theGequation (Equation 11.1) with respect tor, set the

resulting expression equal to zero, and then solve forr(=r∗). That is,

d(G)

dr

=^43 πGv(3r^2 )+ 4 πγ(2r)= 0 (11.2)

which leads to the result

r∗=−

2 γ

Gv

(11.3)

For homogeneous

nucleation, critical

radius of a stable

solid particle nucleus

Now, substitution of this expression forr∗into Equation 11.1 yields the following

expression forG∗:

G∗=

16 πγ^3

3(Gv)^2

(11.4)

For homogeneous

nucleation, activation

free energy required

for the formation of a

stable nucleus

This volume free energy changeGvis the driving force for the solidification

transformation, and its magnitude is a function of temperature. At the equilibrium

solidification temperatureTm, the value ofGvis zero, and with diminishing tem-

perature its value becomes increasingly more negative.

It can be shown thatGvis a function of temperature as

Gv=

Hf(Tm−T)

Tm

(11.5)