reasons for the existence of nanoparticles with shapes far from spherical. Apart from
the highly specific methods of synthesis that result in such nanostructures, three
major reasons can be proposed for the existence of stable nanotubes and nanorods;
these are discussed in the following subsections.
5.1.1
Conditions for the Formation of Rods and Plates
The first point for discussion is the influence of surface energy. For nonspherical
nanostructures, this is especially important in the case of anisotropic (noncubic)
structures. For reasons of simplicity, and without any loss of generality, tetragonal
bodies with the sides a and c, and surface energies c a and c c , are assumed. The
surface energy u surf of such a prism is:
u surf ¼ 4c a ac þ 2c c a
2
ð5:1Þ
By assuming a constant volume v:
v ¼ a
2 c ) c ¼
v
a 2 ) u surf ¼ 4c a
v
a
þ 2c c a
2
a minimum of surface energy is found by the condition:
@u surf
@a
¼ À4c a
v
a 2 þ 4c c a ¼ À4c a c þ 4c c a ¼ 0
This leads to the important relationship:
c a
c c
¼
a
c
ð5:2Þ
Equation (5.2) states that the ratio of the sides of a tetragonal prism is equal to
the ratio of the surface energies. Lastly, this is the thermodynamic basis for the
formation of nanorods or nanoplates. In the case of c a ¼ c c , which is to be
expected in cubic structures, one obtains a ¼ c (i.e., a cube). The derivation is
practically identical for hexagonal structures. However, it must be noted that, by
attaching surface-active compounds, the surface energy of lattice planes may be
modified in such a way as to influence the habitus of nanoparticles in a
significant manner; this is used widely in the synthesis of nonspherical
nanoparticles.
The next question to be answered in this context is the geometry of the minimum
surface energy of agglomerates. Again, using the example of a prism, the question to
be asked is: “Which of the configurations depicted in Figure 5.6 has the least surface
energy?”
The configuration according to Figure 5.6a has a surface energy of:
u a ¼ 8c a ac þ 2c c a
2
ð5:3Þ
Analogously, for the arrangement depicted in Figure 5.6b:
u b ¼ 6c a ac þ 4c c a
2
ð5:4Þ
5.1 General Considerations j93
the highly specific methods of synthesis that result in such nanostructures, three
major reasons can be proposed for the existence of stable nanotubes and nanorods;
these are discussed in the following subsections.
5.1.1
Conditions for the Formation of Rods and Plates
The first point for discussion is the influence of surface energy. For nonspherical
nanostructures, this is especially important in the case of anisotropic (noncubic)
structures. For reasons of simplicity, and without any loss of generality, tetragonal
bodies with the sides a and c, and surface energies c a and c c , are assumed. The
surface energy u surf of such a prism is:
u surf ¼ 4c a ac þ 2c c a
2
ð5:1Þ
By assuming a constant volume v:
v ¼ a
2 c ) c ¼
v
a 2 ) u surf ¼ 4c a
v
a
þ 2c c a
2
a minimum of surface energy is found by the condition:
@u surf
@a
¼ À4c a
v
a 2 þ 4c c a ¼ À4c a c þ 4c c a ¼ 0
This leads to the important relationship:
c a
c c
¼
a
c
ð5:2Þ
Equation (5.2) states that the ratio of the sides of a tetragonal prism is equal to
the ratio of the surface energies. Lastly, this is the thermodynamic basis for the
formation of nanorods or nanoplates. In the case of c a ¼ c c , which is to be
expected in cubic structures, one obtains a ¼ c (i.e., a cube). The derivation is
practically identical for hexagonal structures. However, it must be noted that, by
attaching surface-active compounds, the surface energy of lattice planes may be
modified in such a way as to influence the habitus of nanoparticles in a
significant manner; this is used widely in the synthesis of nonspherical
nanoparticles.
The next question to be answered in this context is the geometry of the minimum
surface energy of agglomerates. Again, using the example of a prism, the question to
be asked is: “Which of the configurations depicted in Figure 5.6 has the least surface
energy?”
The configuration according to Figure 5.6a has a surface energy of:
u a ¼ 8c a ac þ 2c c a
2
ð5:3Þ
Analogously, for the arrangement depicted in Figure 5.6b:
u b ¼ 6c a ac þ 4c c a
2
ð5:4Þ
5.1 General Considerations j93