3
Surfaces in Nanomaterials
3.1
General Considerations
In nanomaterials, the surface forms a sharp interface between a particle and its
surrounding atmosphere or between a precipitated phase and the parent phase.
These are free surfaces in the case of particulate materials or grain boundaries in
bulk material. Nanomaterials have large surfaces, a fact that can be demonstrated
by using spherical particles as examples. As mentioned Chapter 2, nanoparticles
demonstrate a large ratio R
0 of surface area a to volume v. Assuming a mathematical surface, the surface area/volume ratio, R
0
¼ a/v ¼ 6/d, is inversely proportional to the particle diameter d. Realistically, however, the surface has a certain
thickness, influencing partly the volume. Based on many physical properties, it is
known that the region of a particle that is influenced by the surface has a thickness
d between 0.5 and 1.5 nm. Therefore, a modified, dimensionless ratio R
à must be
defined as:
R
à ¼
d
3 À d À 2d
ð
Þ
3
d
3
¼ 1 À
d À 2d
d
3
ð3:1Þ
This ratio, for an assumed surface thickness of 0.5 and 1.0 nm, is shown
graphically in Figure 3.1. On examining Figure 3.1, it is clear that in the case of
a 5-nm particle, 49% or 78%, respectively, of the volume belongs to the surface or,
more precisely, to the surface-influenced volume.
As surface is related to energy, the amount of surface energy per particle u surface
is equal to ca, where c is the specific surface energy and a is the surface area of
one particle. In this context, one considers the geometric surface area of
the particle, which is calculated from a ¼ pd
2 (physical values related to one
particle are denoted by lower-case letters, while those related to molar quantities
are denoted by upper-case letters.) For thermodynamic considerations, the surface
energy per mole of material is the essential quantity. Hence, if N is the number of
particles per mole, one obtains Nca ¼ ðM=rvÞca ¼ cA (where r is the density of
the material, M is the molar weight, d is the particle diameter, and A is the surface
Nanomaterials: An Introduction to Synthesis, Properties and Applications, Second Edition. Dieter Vollath.
Ó 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
j23
Surfaces in Nanomaterials
3.1
General Considerations
In nanomaterials, the surface forms a sharp interface between a particle and its
surrounding atmosphere or between a precipitated phase and the parent phase.
These are free surfaces in the case of particulate materials or grain boundaries in
bulk material. Nanomaterials have large surfaces, a fact that can be demonstrated
by using spherical particles as examples. As mentioned Chapter 2, nanoparticles
demonstrate a large ratio R
0 of surface area a to volume v. Assuming a mathematical surface, the surface area/volume ratio, R
0
¼ a/v ¼ 6/d, is inversely proportional to the particle diameter d. Realistically, however, the surface has a certain
thickness, influencing partly the volume. Based on many physical properties, it is
known that the region of a particle that is influenced by the surface has a thickness
d between 0.5 and 1.5 nm. Therefore, a modified, dimensionless ratio R
à must be
defined as:
R
à ¼
d
3 À d À 2d
ð
Þ
3
d
3
¼ 1 À
d À 2d
d
3
ð3:1Þ
This ratio, for an assumed surface thickness of 0.5 and 1.0 nm, is shown
graphically in Figure 3.1. On examining Figure 3.1, it is clear that in the case of
a 5-nm particle, 49% or 78%, respectively, of the volume belongs to the surface or,
more precisely, to the surface-influenced volume.
As surface is related to energy, the amount of surface energy per particle u surface
is equal to ca, where c is the specific surface energy and a is the surface area of
one particle. In this context, one considers the geometric surface area of
the particle, which is calculated from a ¼ pd
2 (physical values related to one
particle are denoted by lower-case letters, while those related to molar quantities
are denoted by upper-case letters.) For thermodynamic considerations, the surface
energy per mole of material is the essential quantity. Hence, if N is the number of
particles per mole, one obtains Nca ¼ ðM=rvÞca ¼ cA (where r is the density of
the material, M is the molar weight, d is the particle diameter, and A is the surface
Nanomaterials: An Introduction to Synthesis, Properties and Applications, Second Edition. Dieter Vollath.
Ó 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
j23