7
Phase Transformations of Nanoparticles
7.1
Thermodynamics of Nanoparticles
Although thermodynamics may be treated on a variety of levels of complexity and
precision, here an elementary introduction is presented and, therefore, in all cases,
the simplest possible description is used, neglecting any influential factors required
for an exact description of equilibria. The influence of the vapor phase is not
considered in any of the cases. However, because of the large surface of nanoparticulate materials, energy stored as surface energy must always be taken into
account when considering the thermodynamics of systems. It will be shown that, in
many cases, the amount of energy stored at the surface is in the same range as the
energy of phase transformations in the bulk. Accordingly, surface energy controls
the stability of multiphase systems and, therefore, the Gibb’s free enthalpy must be
written as:
G ¼ U À TS þ cA
ð7:1Þ
In this equation, G, U, S, and T have their usual meanings of free enthalpy,
enthalpy, S entropy, and temperature, respectively, with each parameter always being
related to 1 mole. Here, c is the surface energy and A is the surface of the system.
(Quantities per mole are written in capital letters, those related to one particle in
lower case letters.) It was shown in Section 3.2 that, in the case of small particles, the
energy connected to the surface is in the range of the energy of formation for oxides.
Therefore, a strong influence of particle size on phase transformations is expected.
This is shown graphically in Figure 7.1 for the melting of aluminum, where the
surface energy in the solid state G surfaceÀsolid ¼ A solid c solid and liquid state
G surfaceÀliquid ¼ A liquid c liquid , and their differences, are plotted. For comparison,
the enthalpy of melting is also shown in the graph. It should be noted that, for
particles smaller than 10 nm, the surface energy of the solid or the liquid state is
higher than of the enthalpy of melting; therefore, it is clear that particle size (in this
case aluminum particles) might have a major influence on melting. In more general
terms, particle size has a significant influence on phase transformations.
Nanomaterials: An Introduction to Synthesis, Properties and Applications, Second Edition. Dieter Vollath.
Ó 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
j135
Phase Transformations of Nanoparticles
7.1
Thermodynamics of Nanoparticles
Although thermodynamics may be treated on a variety of levels of complexity and
precision, here an elementary introduction is presented and, therefore, in all cases,
the simplest possible description is used, neglecting any influential factors required
for an exact description of equilibria. The influence of the vapor phase is not
considered in any of the cases. However, because of the large surface of nanoparticulate materials, energy stored as surface energy must always be taken into
account when considering the thermodynamics of systems. It will be shown that, in
many cases, the amount of energy stored at the surface is in the same range as the
energy of phase transformations in the bulk. Accordingly, surface energy controls
the stability of multiphase systems and, therefore, the Gibb’s free enthalpy must be
written as:
G ¼ U À TS þ cA
ð7:1Þ
In this equation, G, U, S, and T have their usual meanings of free enthalpy,
enthalpy, S entropy, and temperature, respectively, with each parameter always being
related to 1 mole. Here, c is the surface energy and A is the surface of the system.
(Quantities per mole are written in capital letters, those related to one particle in
lower case letters.) It was shown in Section 3.2 that, in the case of small particles, the
energy connected to the surface is in the range of the energy of formation for oxides.
Therefore, a strong influence of particle size on phase transformations is expected.
This is shown graphically in Figure 7.1 for the melting of aluminum, where the
surface energy in the solid state G surfaceÀsolid ¼ A solid c solid and liquid state
G surfaceÀliquid ¼ A liquid c liquid , and their differences, are plotted. For comparison,
the enthalpy of melting is also shown in the graph. It should be noted that, for
particles smaller than 10 nm, the surface energy of the solid or the liquid state is
higher than of the enthalpy of melting; therefore, it is clear that particle size (in this
case aluminum particles) might have a major influence on melting. In more general
terms, particle size has a significant influence on phase transformations.
Nanomaterials: An Introduction to Synthesis, Properties and Applications, Second Edition. Dieter Vollath.
Ó 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
j135