132 7 Thermodynamics of Nanoparticles and Phase Transformations
The graph in Figure 7.9 shows that, as theoretically expected, solid-state transformations also follow the simple considerations, which led to Eq. (7.7). However,
this very general statement needs a caveat: The transformation tetragonal–
monoclinic is, in the case of zirconia, a martensitic transformation, which means
that the whole particle transforms within an extremely short time interval. The
experimental proof would be significantly more difficult in the case of a diffusioncontrolled transformation, which takes time.
7.3
Thermal Instabilities Connected to Phase Transformations
In the previous section, phase transformations of nanoparticles were treated in
a particular way, as they would be, except for the surface energy, more or less
identical to phase transformation in the world of macroscopic objects. The validity
of this approach is, however, limited. It does not take note that nanosized objects
are small. The thermal energy u of an object at a temperature T, independent of
its size, is according to Boltzmann
u kT
= .
(7.12)
The quantity k in Eq. (7.12) is the Boltzmann constant. (The denomination Boltzmann constant was introduced by Max Plank. Interestingly, Ludwig Boltzmann
himself thought that it would be more or less impossible to determine the exact
numerical value.) As the free enthalpy of phase transformation for one particle
g is proportional to its mass, the energy needed, respectively, released at the
phase transformation of one particle gets smaller when the mass of the particle
is reduced. There it is possible that the thermal energy of a particle gets larger
than the energy necessary for phase transformation. This leads to the limiting
condition
∆g kT
≤ ,
(7.13)
where thermal instabilities are expected. Thermal instabilities lead to fluctuations,
which are, in this context defined as: “spontaneous transitions from an equilibrium
phase to a nonequilibrium phase, followed by a back-transformation” [5]. As the first
question may be: Is this pure theory or is there experimental evidence supporting
these considerations? Therefore, first some experimental findings, which are not
explicable without the assumption of thermal fluctuations, are presented.
Iijima and Ichihashi [6] made a series of high-resolution electron micrographs of gold particles with 2 nm diameter. One of these series is presented in
Figure 7.10.
The series of high-resolution electron micrographs presented in Figure 7.10
shows pictures of one individual particle, which were taken in time intervals of
1
60
s at a temperature around 370 K. The short time that was available for one
picture, explains the blurring by noise. (Normally, taking electron micrographs
The graph in Figure 7.9 shows that, as theoretically expected, solid-state transformations also follow the simple considerations, which led to Eq. (7.7). However,
this very general statement needs a caveat: The transformation tetragonal–
monoclinic is, in the case of zirconia, a martensitic transformation, which means
that the whole particle transforms within an extremely short time interval. The
experimental proof would be significantly more difficult in the case of a diffusioncontrolled transformation, which takes time.
7.3
Thermal Instabilities Connected to Phase Transformations
In the previous section, phase transformations of nanoparticles were treated in
a particular way, as they would be, except for the surface energy, more or less
identical to phase transformation in the world of macroscopic objects. The validity
of this approach is, however, limited. It does not take note that nanosized objects
are small. The thermal energy u of an object at a temperature T, independent of
its size, is according to Boltzmann
u kT
= .
(7.12)
The quantity k in Eq. (7.12) is the Boltzmann constant. (The denomination Boltzmann constant was introduced by Max Plank. Interestingly, Ludwig Boltzmann
himself thought that it would be more or less impossible to determine the exact
numerical value.) As the free enthalpy of phase transformation for one particle
g is proportional to its mass, the energy needed, respectively, released at the
phase transformation of one particle gets smaller when the mass of the particle
is reduced. There it is possible that the thermal energy of a particle gets larger
than the energy necessary for phase transformation. This leads to the limiting
condition
∆g kT
≤ ,
(7.13)
where thermal instabilities are expected. Thermal instabilities lead to fluctuations,
which are, in this context defined as: “spontaneous transitions from an equilibrium
phase to a nonequilibrium phase, followed by a back-transformation” [5]. As the first
question may be: Is this pure theory or is there experimental evidence supporting
these considerations? Therefore, first some experimental findings, which are not
explicable without the assumption of thermal fluctuations, are presented.
Iijima and Ichihashi [6] made a series of high-resolution electron micrographs of gold particles with 2 nm diameter. One of these series is presented in
Figure 7.10.
The series of high-resolution electron micrographs presented in Figure 7.10
shows pictures of one individual particle, which were taken in time intervals of
1
60
s at a temperature around 370 K. The short time that was available for one
picture, explains the blurring by noise. (Normally, taking electron micrographs