area of 1 mole of particles). Finally, one obtains the surface energy of particles with
diameter d per mole:
U surface ¼
6M
r
c
1
d
ð3:2Þ
Equation (3.2) states that the surface energy per mole increases with 1/d and in
some cases, especially those related to very small particles, this may have dramatic
consequences.
The same considerations are valid for polycrystalline materials, where the volume
related to the grain boundaries increases as the grain size decreases. In contrast to
the well-ordered crystalline areas, the atoms or ions in the grain boundaries are, in a
first approximation, arranged randomly. The famous picture of Gleiter [1] representing the arrangement of grains and grain boundaries is shown in Figure 3.2.
Figure 3.2 Grain boundaries in polycrystalline material with grains in the nanometer range. A
large portion of the material is associated with the surface [1] (Reproduced by permission of
Elsevier).
0
5
10
15
20
particle diameter [nm]
0
0.2
0.4
0.6
0.8
1
ratio
(surface
layer
/
volume)
Thickness of surface layer
0.5 nm
1.0 nm
Figure 3.1 Ratio of the surface layer over the total particle volume. The thickness of the surface
layer was assumed to be 0.5 or 1.0 nm.
24j 3 Surfaces in Nanomaterials