21
Surfaces in Nanomaterials
3
3.1
General Considerations
The surface forms a sharp interface between a particle and the surrounding atmosphere or between a precipitated phase and the parent phase. In mathematics, the
surface of a body, for example, a sphere or a polyhedron, is clearly defined. As
already mentioned in Chapter 2, in a sphere, the ratio surface over volume is
indirectly proportional to the diameter. This is different in the case of a real, physically existing material. In this case, one has to distinguish between free surfaces
in the case of particulate materials and grain boundaries in bulk material. As
nanoparticles are small, they have large surfaces. However, what is a surface of a
real solid? The answer must not be restricted to the geometrical surface. Looking
at a solid and its behavior, one has to take note of the surface-influenced volume.
A simplified model assumes a layer with a thickness δ at the surface. Depending
on the property in question, this thickness is found experimentally in the range
between 0.5 and 1 nm.
Box 3.1 Physical Surface of Particles
Assuming a sphere with the diameter d and a layer with the thickness δ, which
is influenced by the surface. In this case the volume of this shell is
v
d
d
d
d
shell =
−
−
(
) =
− −
(
)
π
π
δ
π
δ
6
6
6
2
3
3
3
3 ,
(3.1)
Now a dimensionless volume ratio R
* is defined as:
R
v
v
d
d
d
d
d
*
.
=
=
− −
(
)
= −
−
shell
sphere
π
δ
π
δ
6
2
6
1
2
3
3
3
3
(3.2)
This ratio approaches one if d ≈ 2δ.
Nanoparticles – Nanocomposites – Nanomaterials: An Introduction for Beginners, First Edition. Dieter Vollath.
© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
Surfaces in Nanomaterials
3
3.1
General Considerations
The surface forms a sharp interface between a particle and the surrounding atmosphere or between a precipitated phase and the parent phase. In mathematics, the
surface of a body, for example, a sphere or a polyhedron, is clearly defined. As
already mentioned in Chapter 2, in a sphere, the ratio surface over volume is
indirectly proportional to the diameter. This is different in the case of a real, physically existing material. In this case, one has to distinguish between free surfaces
in the case of particulate materials and grain boundaries in bulk material. As
nanoparticles are small, they have large surfaces. However, what is a surface of a
real solid? The answer must not be restricted to the geometrical surface. Looking
at a solid and its behavior, one has to take note of the surface-influenced volume.
A simplified model assumes a layer with a thickness δ at the surface. Depending
on the property in question, this thickness is found experimentally in the range
between 0.5 and 1 nm.
Box 3.1 Physical Surface of Particles
Assuming a sphere with the diameter d and a layer with the thickness δ, which
is influenced by the surface. In this case the volume of this shell is
v
d
d
d
d
shell =
−
−
(
) =
− −
(
)
π
π
δ
π
δ
6
6
6
2
3
3
3
3 ,
(3.1)
Now a dimensionless volume ratio R
* is defined as:
R
v
v
d
d
d
d
d
*
.
=
=
− −
(
)
= −
−
shell
sphere
π
δ
π
δ
6
2
6
1
2
3
3
3
3
(3.2)
This ratio approaches one if d ≈ 2δ.
Nanoparticles – Nanocomposites – Nanomaterials: An Introduction for Beginners, First Edition. Dieter Vollath.
© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.