Provided that the thermal energy kT is greater than the energies E 1 and E 2 , the
system fluctuates between both energetically possible states 1 and 2. Certainly, it
does not make any difference to these considerations if E 1 and E 2 are equal or more
than two different states are accessible with thermal energy at temperature T.
The second example describes the temperature increase by the absorption of light
quanta. Again, a zirconia particle with density r ¼ 5.6 Â 10
3 kg m
À3 , a heat capacity
C p ¼ 56.2 J mol
À1 K
À1 equivalent to c p ¼ 457 J kg
À1 K
À1 , and, in this case, a particle
diameter of 3 nm is assumed. After the absorption of one photon with a wavelength,
l, of 300 nm, a photon, which is typical for the UV-range, the temperature increase
DT is calculated from c p rnDT ¼ hn ¼ g c=l
ð Þ (c is the velocity of light and h is
Planck’s constant) to 18 K. Being an astonishingly large value, this temperature
increase must be considered when interpreting optical spectra of nanomaterials
with poor quantum efficiency or composites with highly UV-absorbing kernels.
2.2.3
Diffusion Scaling Law
Diffusion is controlled by the two laws defined by Fick. The solutions of these equations,
which are important for nanotechnology, imply that the mean square diffusion path of
the atoms hxi
2 is proportional to Dt, where D is the diffusion coefficient and t is the
time. The following expression will be used in further considerations:
x
h i
2 / Dt
ð2:4Þ
Equation (2.4) has major consequences, but in order to simplify any further
discussion it is assumed that hxi
2 is proportional to the squared particle size. The
angular brackets h i denote the mean value. Conventional materials usually have
grain sizes of around 10 mm and it is well known that at elevated temperatures these
materials require homogenization times of the order of many hours. When
energy
state 2
state 1
E
1
E
2
Figure 2.11 Graphical representation of the energy barrier, showing the energy necessary to
jump from state 1 to state 2 and vice versa.
14j 2 Nanomaterials and Nanocomposites
system fluctuates between both energetically possible states 1 and 2. Certainly, it
does not make any difference to these considerations if E 1 and E 2 are equal or more
than two different states are accessible with thermal energy at temperature T.
The second example describes the temperature increase by the absorption of light
quanta. Again, a zirconia particle with density r ¼ 5.6 Â 10
3 kg m
À3 , a heat capacity
C p ¼ 56.2 J mol
À1 K
À1 equivalent to c p ¼ 457 J kg
À1 K
À1 , and, in this case, a particle
diameter of 3 nm is assumed. After the absorption of one photon with a wavelength,
l, of 300 nm, a photon, which is typical for the UV-range, the temperature increase
DT is calculated from c p rnDT ¼ hn ¼ g c=l
ð Þ (c is the velocity of light and h is
Planck’s constant) to 18 K. Being an astonishingly large value, this temperature
increase must be considered when interpreting optical spectra of nanomaterials
with poor quantum efficiency or composites with highly UV-absorbing kernels.
2.2.3
Diffusion Scaling Law
Diffusion is controlled by the two laws defined by Fick. The solutions of these equations,
which are important for nanotechnology, imply that the mean square diffusion path of
the atoms hxi
2 is proportional to Dt, where D is the diffusion coefficient and t is the
time. The following expression will be used in further considerations:
x
h i
2 / Dt
ð2:4Þ
Equation (2.4) has major consequences, but in order to simplify any further
discussion it is assumed that hxi
2 is proportional to the squared particle size. The
angular brackets h i denote the mean value. Conventional materials usually have
grain sizes of around 10 mm and it is well known that at elevated temperatures these
materials require homogenization times of the order of many hours. When
energy
state 2
state 1
E
1
E
2
Figure 2.11 Graphical representation of the energy barrier, showing the energy necessary to
jump from state 1 to state 2 and vice versa.
14j 2 Nanomaterials and Nanocomposites