16 2 Nanoparticles – Nanocomposites
Schematically, this situation is depicted in Figure 2.11.
Looking at thermal instabilities (fluctuations), one can design a simple example.
One can ask for the size of a zirconia particle (ρ = 5.6 × 10
3 kg m
−3 ) that could be
lifted at room temperature to a height equal to its diameter. The answer is somewhat surprising, the diameter of 1100 nm. If one asks, how high could a particle
Box 2.2 Thermal Instability
Thermal energy u th of an isolated particle is given by
u
kT
th = ,
with k the Boltzmann constant and T the temperature.
Assuming an energy that depends on the volume of the particle u(v). The
system is no longer stable, it fluctuates, if the condition
u v kT
( ) ≤
(2.3)
is fulfilled.
A simple example: The energy necessary to lift a particle with the density ρ
the elevation x.
u v
vx
( ) = ρ .
This particle moves around thermally and jumps up to a height x, if the
condition
ρ
ρ
vx kT
T
v
k
x
≤
≥
or
is fulfilled.
Figure 2.11 Schematic visualization of thermal fluctuation. In this case, the thermal energy is
larger than the energy difference between level 1 and level 2.
Level 1
Level 2
kT
u Level 2 − u Level 1 < kT
two neighboring states. Now the system is no longer stable, and the system
fluctuates.