This ratio is inversely proportional to the particle size and, as a consequence, the
surface increases with decreasing particle size. The same is valid for the surface per
mole A, a quantity that is of extreme importance in thermodynamic considerations:
A ¼ Na ¼
M
r
p
6
d
3
pD
2 ¼
6M
rd
ð2:2Þ
where N is the number of particles per mole, M is the molecular weight, and r is the
density of the material. Similar to the surface/volume ratio, the area per mole
increases inversely in proportion to the particle diameter; hence, huge values of area
are achieved for particles that are only a few nanometers in diameter.
It should be noted that as the surface is such an important topic for nanoparticles,
Chapter 3 of this book has been devoted to surface and surface-related problems.
2.2.2
Thermal Phenomena
Each isolated object – in this case a nanoparticle – has a thermal energy of kT (k is the
Boltzmann constant and T is the temperature). First, let us assume a property of the
particle that depends for example on the volume v of the particle; the energy of this
property may be u v
ð Þ. Then, provided that the volume is sufficiently small such that
the condition:
u v
ð Þ < kT
ð2:3Þ
is fulfilled, one may expect thermal instability. As an example, one may ask for the
particle size where thermal energy is large enough to lift the particle. In the simplest
case, one estimates the energy necessary to lift a particle of density r over the
elevation x: u v
ð Þ ¼ rvx ¼ kT. Assuming a zirconia particle with a density of
5.6 Â 10
3 kg m
À3 , at room temperature the thermal energy would lift a particle of
diameter 1100 nm to a height equal to the particle diameter d. If one asks how high
might a particle of 5 nm diameter jump, these simple calculations indicate a value of
more than 1 m. Clearly, although these games with numbers do not have physical
reality, they do show that nanoparticles are not fixed, but rather are moving about on
the surface. By performing electron microscopy, this dynamic becomes reality and,
provided that the particles and carbon film on the carrier mesh are clean, the
specimen particles can be seen to move around on the carbon film. On occasion,
however, this effect may cause major problems during electron microscopy studies.
Although the thermal instability shown here demonstrates only one of the
consequences of smallness, when examining the other physical properties then
an important change in the behavior can be realized. Details of the most important
phenomenon within this group – superparamagnetism – are provided in Chapter 8.
In the case of superparamagnetism, the vector of magnetization fluctuates between
different “easy” directions of magnetization and these fluctuations may also be
observed in connection with the crystallization of nanoparticles. In a more generalized manner, thermal instabilities leading to fluctuations may be characterized
graphically, as shown in Figure 2.11.
2.2 Elementary Consequences of Small Particle Size j13
surface increases with decreasing particle size. The same is valid for the surface per
mole A, a quantity that is of extreme importance in thermodynamic considerations:
A ¼ Na ¼
M
r
p
6
d
3
pD
2 ¼
6M
rd
ð2:2Þ
where N is the number of particles per mole, M is the molecular weight, and r is the
density of the material. Similar to the surface/volume ratio, the area per mole
increases inversely in proportion to the particle diameter; hence, huge values of area
are achieved for particles that are only a few nanometers in diameter.
It should be noted that as the surface is such an important topic for nanoparticles,
Chapter 3 of this book has been devoted to surface and surface-related problems.
2.2.2
Thermal Phenomena
Each isolated object – in this case a nanoparticle – has a thermal energy of kT (k is the
Boltzmann constant and T is the temperature). First, let us assume a property of the
particle that depends for example on the volume v of the particle; the energy of this
property may be u v
ð Þ. Then, provided that the volume is sufficiently small such that
the condition:
u v
ð Þ < kT
ð2:3Þ
is fulfilled, one may expect thermal instability. As an example, one may ask for the
particle size where thermal energy is large enough to lift the particle. In the simplest
case, one estimates the energy necessary to lift a particle of density r over the
elevation x: u v
ð Þ ¼ rvx ¼ kT. Assuming a zirconia particle with a density of
5.6 Â 10
3 kg m
À3 , at room temperature the thermal energy would lift a particle of
diameter 1100 nm to a height equal to the particle diameter d. If one asks how high
might a particle of 5 nm diameter jump, these simple calculations indicate a value of
more than 1 m. Clearly, although these games with numbers do not have physical
reality, they do show that nanoparticles are not fixed, but rather are moving about on
the surface. By performing electron microscopy, this dynamic becomes reality and,
provided that the particles and carbon film on the carrier mesh are clean, the
specimen particles can be seen to move around on the carbon film. On occasion,
however, this effect may cause major problems during electron microscopy studies.
Although the thermal instability shown here demonstrates only one of the
consequences of smallness, when examining the other physical properties then
an important change in the behavior can be realized. Details of the most important
phenomenon within this group – superparamagnetism – are provided in Chapter 8.
In the case of superparamagnetism, the vector of magnetization fluctuates between
different “easy” directions of magnetization and these fluctuations may also be
observed in connection with the crystallization of nanoparticles. In a more generalized manner, thermal instabilities leading to fluctuations may be characterized
graphically, as shown in Figure 2.11.
2.2 Elementary Consequences of Small Particle Size j13