Page 105 - Science at the nanoscale

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June 9, 2009
surface of the solid material:
δG
5.1. Surface Energy 95 ch05
γ = (5.1)
δA T,P
G = Gibbs free energy
A = Surface area
Apparently, some bonds have been broken in creating this new
surface. Extensive model calculations using such “broken-bond”
models have been performed for a variety of crystal planes and
structures 1
We have learnt in Section 4.3 that atoms are arranged differently
on different crystal planes. In the following, we use a simplified
version of the “broken-bond model” to illustrate that different
crystal planes have different surface energies. By ignoring the
interactions of higher order neighbours, we estimate γ as half of ε,
the bond strength, multiplying with the number of broken bonds
(N ):
b
1
γ = N ερ a (5.2)
b
2
ρ a = number of atoms per unit area on the new surface
This model is obviously oversimplified since it assumes that the
bond strength is the same throughout the whole crystal structure,
and is similar for both surface and bulk atoms. The former may be
true in the case of elemental metallic crystals composed of just one
type of atoms. In a simple picture, let us consider cutting away
one unit cell along the {100} facet of a fcc metallic solid (Fig. 5.2).
Since each atom on the “new” {100} surface is left with 8 coor-
dinating neighbours, it is clear that four bonds have been broken
2
and γ = (4ε)/a according to Eq. 5.2.
A similar schematic can be drawn for the {111} surface to show
√
2
that three bonds are broken and γ = (2ε 3)/a . Thus, this sim-
plified model allows us to see that different crystal facets possess
different surface energies. Broken-bond model calculations that
take into account second and higher order interactions predict
that γ < γ < γ for fcc metals. 1 Generally, it was
(111) (100) (110)
found that crystal surfaces with lower Miller indices have a lower
surface energy than those with higher Miller indices. Commonly

1 S. G. Wang, E. K. Tian and C. W. Lung, J. Phys. and Chem. Solids, 61, 1295 (2000).