187
As shown in Figure 6.18, the overall trend remains for the case of
the cube, but the significant variation in surface-to-volume ratio is
observed at larger critical dimensions compared with the sphere
and cylinder cases.
After stressing the importance of the increase in surface area in
nanomaterials relative to traditional larger-scale materials, let’s put
this information into context. With the help of a few simple calculations, we can determine how much of an increase in surface
area will result—for example, from a spherical particle of 10 µm
to be reduced to a group of particles with 10 nanometers, assuming that the volume remains constant. To do this, first we calculate
the volume of a sphere with 10 microns. Following Equation 6.2
gives V (10 µm) = 5.23 × 10
11 nm
3
. We then calculate the volume
of a sphere with 10 nm. Again, with the help of Equation 6.2, we
get V (10 nm) = 523 nm
3
. Because the mass of the 10 micron particle is converted to a group of nanosized particles, the total volume
remains the same. Therefore, to calculate the number of nanosized
particles generated by the 10 micron particle, we simply need to
divide V (10 µm) by V (10 nm)in the form:
N
V
m
V
nm
=
(
)
(
)
=
×
= ×
10
10
5 23 10
523
1 10
11
9
µ
.
particles
(6.6)
Hence, so far we can conclude that one single particle with 10
microns can generate 1 billion nanosized particles with a diameter
of 10 nm, whereas the total volume remains the same.
We are thus left with the task of finding the increase in surface area
in going from one particle to 1 billion particles. This can be done by
first calculating the surface area of the 10 micron particle. Following
Equation 6.1 gives A (10 µm) = 3.14 × 10
8 nm
2
. On the other hand,
for the case of the 10 nm particle A (10 nm) = 314 nm
2 . However,
since we have 1 billion 10 nm particles, the surface area of all these
particles amounts to 3.14 × 10
11 nm
2 . This means an increase in
surface area by a factor of 1000.
magic numbers
As discussed, for a decrease in particle radius, the surface-to-volume
ratio increases. Therefore the fraction of surface atoms increases as
the particle size goes down. In general, for a sphere, we can relate the
number of surface and bulk atoms according to the expressions
V
r n
A
=
4
3
3
π
(6.7)
Size Effects
As shown in Figure 6.18, the overall trend remains for the case of
the cube, but the significant variation in surface-to-volume ratio is
observed at larger critical dimensions compared with the sphere
and cylinder cases.
After stressing the importance of the increase in surface area in
nanomaterials relative to traditional larger-scale materials, let’s put
this information into context. With the help of a few simple calculations, we can determine how much of an increase in surface
area will result—for example, from a spherical particle of 10 µm
to be reduced to a group of particles with 10 nanometers, assuming that the volume remains constant. To do this, first we calculate
the volume of a sphere with 10 microns. Following Equation 6.2
gives V (10 µm) = 5.23 × 10
11 nm
3
. We then calculate the volume
of a sphere with 10 nm. Again, with the help of Equation 6.2, we
get V (10 nm) = 523 nm
3
. Because the mass of the 10 micron particle is converted to a group of nanosized particles, the total volume
remains the same. Therefore, to calculate the number of nanosized
particles generated by the 10 micron particle, we simply need to
divide V (10 µm) by V (10 nm)in the form:
N
V
m
V
nm
=
(
)
(
)
=
×
= ×
10
10
5 23 10
523
1 10
11
9
µ
.
particles
(6.6)
Hence, so far we can conclude that one single particle with 10
microns can generate 1 billion nanosized particles with a diameter
of 10 nm, whereas the total volume remains the same.
We are thus left with the task of finding the increase in surface area
in going from one particle to 1 billion particles. This can be done by
first calculating the surface area of the 10 micron particle. Following
Equation 6.1 gives A (10 µm) = 3.14 × 10
8 nm
2
. On the other hand,
for the case of the 10 nm particle A (10 nm) = 314 nm
2 . However,
since we have 1 billion 10 nm particles, the surface area of all these
particles amounts to 3.14 × 10
11 nm
2 . This means an increase in
surface area by a factor of 1000.
magic numbers
As discussed, for a decrease in particle radius, the surface-to-volume
ratio increases. Therefore the fraction of surface atoms increases as
the particle size goes down. In general, for a sphere, we can relate the
number of surface and bulk atoms according to the expressions
V
r n
A
=
4
3
3
π
(6.7)
Size Effects