Callister - An introduction - 8th edition

10.3 The Kinetics of Phase Transformations • 351
Liquid
Solid
Surface or interface
SL
SI
IL
Figure 10.5 Heterogeneous
nucleation of a solid from a liquid.
The solid–surface 1 SI 2, solid–liquid
1 SL 2, and liquid–surface 1 IL 2,
interfacial energies are
represented by vectors. The
wetting angle 1u2 is also shown.
For heterogeneous
nucleation of a solid
particle, relationship
among solid–surface,
solid–liquid, and
liquid–surface
interfacial energies
and the wetting angle
For heterogeneous
nucleation, critical
radius of a stable
solid particle nucleus
For heterogeneous
nucleation,
activation free
energy required for
the formation of a
stable nucleus
occur at surfaces and interfaces than at other sites. Again, this type of nucleation is
termed heterogeneous.
In order to understand this phenomenon, let us consider the nucleation, on a
flat surface, of a solid particle from a liquid phase. It is assumed that both the liquid
and solid phases “wet” this flat surface, that is, both of these phases spread out
and cover the surface; this configuration is depicted schematically in Figure 10.5.
Also noted in the figure are three interfacial energies (represented as vectors) that
exist at two-phase boundaries— SL , SI , and IL —as well as the wetting angle u (the
angle between the SI and SL vectors). Taking a surface tension force balance in
the plane of the flat surface leads to the following expression:
IL SI SL cos u
(10.12)
Now, using a somewhat involved procedure similar to the one presented above
for homogeneous nucleation (which we have chosen to omit), it is possible to derive
equations for r* and G*; these are as follows:
r* 2 SL
¢G y
¢G* a 16pg SL
3 b S1u2
3¢Gy
2
(10.13)
(10.14)
The S() term of this last equation is a function only of (i.e., the shape of the nucleus),
which will have a numerical value between zero and unity. 1
From Equation 10.13, it is important to note that the critical radius r* for heterogeneous
nucleation is the same as for homogeneous, inasmuch as SL is the same surface
energy as in Equation 10.3. It is also evident that the activation energy barrier
for heterogeneous nucleation (Equation 10.14) is smaller than the homogeneous barrier
(Equation 10.4) by an amount corresponding to the value of this S() function, or
¢G het * ¢G hom * S1u2
(10.15)
Figure 10.6, a schematic graph of G versus nucleus radius, plots curves for both
types of nucleation, and indicates the difference in the magnitudes of ¢G* het and
¢G* hom , in addition to the constancy of r*. This lower ¢G* for heterogeneous means
that a smaller energy must be overcome during the nucleation process (than for
1
For example, for angles of 30 and 90, values of S() are approximately 0.01 and 0.5,
respectively.