2.2 Elementary Consequences of Small Particle Size 17
of 5 nm diameter jump these simple calculations lead to a height of more than
one meter. Certainly, these games with number do not have physical reality;
however, they indicate that nanoparticles that are not fixed at a surface are moving
around. Doing electron microscopy, this dynamic becomes reality. Provided the
particles and the carbon film on the carrier mesh are clean, the particles of the
specimen move around like ants on the carbon film. This makes electron microscopy difficult.
The thermal instability presented here, demonstrates a simple consequence
of smallness; however, other physical properties may change significantly and
this may lead to new properties. The most important phenomenon of this group,
superparamagnetism, will be described in Chapter 8 on magnetic properties.
Fluctuations are also observed in connection with phase transformation, for
example, melting and crystallization of nanoparticles.
2.2.3
Diffusion Scaling Law
Diffusion is controlled by the two Fick’s laws. Solutions of these equations, important for nanotechnology, say that the squared mean diffusion path of the atoms
is proportional to the time. In other words: Assuming the particle diameter as
the diffusion path, doubling of the diameter leads to a four-fold time needed for
diffusion.
Box 2.3 Scaling Law of Diffusion
Mathematically, diffusion is described by the two laws of Fick, a set of two
partial differential equations. The solution, important for the considerations
connected to nanoparticles says:
x
Dt
2 ∝ .
(2.4)
The brackets 〈 〉 stand for the mean value of an ensemble; the quantity x stands
for the diffusion path; therefore, 〈x〉
2 is the mean square of the diffusion path,
D is the diffusion coefficient and t the time. Generally, the diffusion coefficient
depends exponentially on the temperature,
D
q
kT
∝
−
exp
,
(2.5)
which means that the diffusion will get faster with increasing temperature. The
quantity q is the activation energy.
This scaling law for diffusion has dramatic consequences when applied to
nanomaterials. As an example, the homogenization time necessary in the case of
conventional and nanomaterials are compared. Conventional materials usually
have grain sizes of around 10 μm. It is well known that at elevated temperatures,
of 5 nm diameter jump these simple calculations lead to a height of more than
one meter. Certainly, these games with number do not have physical reality;
however, they indicate that nanoparticles that are not fixed at a surface are moving
around. Doing electron microscopy, this dynamic becomes reality. Provided the
particles and the carbon film on the carrier mesh are clean, the particles of the
specimen move around like ants on the carbon film. This makes electron microscopy difficult.
The thermal instability presented here, demonstrates a simple consequence
of smallness; however, other physical properties may change significantly and
this may lead to new properties. The most important phenomenon of this group,
superparamagnetism, will be described in Chapter 8 on magnetic properties.
Fluctuations are also observed in connection with phase transformation, for
example, melting and crystallization of nanoparticles.
2.2.3
Diffusion Scaling Law
Diffusion is controlled by the two Fick’s laws. Solutions of these equations, important for nanotechnology, say that the squared mean diffusion path of the atoms
is proportional to the time. In other words: Assuming the particle diameter as
the diffusion path, doubling of the diameter leads to a four-fold time needed for
diffusion.
Box 2.3 Scaling Law of Diffusion
Mathematically, diffusion is described by the two laws of Fick, a set of two
partial differential equations. The solution, important for the considerations
connected to nanoparticles says:
x
Dt
2 ∝ .
(2.4)
The brackets 〈 〉 stand for the mean value of an ensemble; the quantity x stands
for the diffusion path; therefore, 〈x〉
2 is the mean square of the diffusion path,
D is the diffusion coefficient and t the time. Generally, the diffusion coefficient
depends exponentially on the temperature,
D
q
kT
∝
−
exp
,
(2.5)
which means that the diffusion will get faster with increasing temperature. The
quantity q is the activation energy.
This scaling law for diffusion has dramatic consequences when applied to
nanomaterials. As an example, the homogenization time necessary in the case of
conventional and nanomaterials are compared. Conventional materials usually
have grain sizes of around 10 μm. It is well known that at elevated temperatures,