nanocrystalline materials is based on model calculations, not least because experiments are often impaired by secondary influences such as porosity or grain growth
at elevated temperatures.
11.2
Bulk Metallic and Ceramic Materials
11.2.1
Influence of Porosity
When considering mechanical properties, it is essential to study deformation
mechanisms as a function of materials’ structure. Important structural features
in this context are grain size and porosity. As bulk nanomaterials are not in
thermodynamic equilibrium, grain growth at elevated temperature or during
deformation must be taken into account.
The influence of porosity on mechanical properties has been the subject of many
studies and in this context the porosity p is defined as:
p ¼ 1 À
r specimen
r theor
ð11:2Þ
where r specimen is the actual, experimentally determined density of the specimen
and r theor is the theoretical density of the full, dense material. Most important is the
influence on elastic properties and one of many formulae describing this was
provided by MacKenzie [3], who applied a Taylor series development for Young’s
modulus E to estimate the influence of the porosity p:
E ¼ E 0 ð1 þ a 1 p þ a 2 p
2 þ Á Á ÁÞ
ð11:3Þ
where E 0 is Young’s modulus of the full dense material, p ¼ 0, and a 1 and a 2 are
fitting parameters. For small values of porosity the linear term is sufficient, but for
higher porosities an increasing number of series elements are necessary. In addition
to this simple series development, a huge number of theoretically developed
equations are described. However, theoretically well-based formulae also require
fitting parameters, and Figure 11.4 depicts Young’s modulus for nanocrystalline
copper and palladium as a function of the porosity. This is a good example of small
porosity, where the linear term is sufficient. Additionally, it demonstrates the
unavoidable scattering of experimental values that, most likely, is due to the different
grain sizes for specimens with different porosities. When examining these heavily
scattered experimental data, it is clear that a fit using more than the linear
approximation is not justified.
Evaluation of the experimental data depicted in Figure 11.4 led to the following
fitting parameter for Young’s moduli (in GPa):
E Cu ¼ 121 1 À 3:8p
ð
Þ
E Pd ¼ 132 1 À 2:5p
ð
Þ
ð11:4Þ
302j 11 Mechanical Properties of Nanoparticles
at elevated temperatures.
11.2
Bulk Metallic and Ceramic Materials
11.2.1
Influence of Porosity
When considering mechanical properties, it is essential to study deformation
mechanisms as a function of materials’ structure. Important structural features
in this context are grain size and porosity. As bulk nanomaterials are not in
thermodynamic equilibrium, grain growth at elevated temperature or during
deformation must be taken into account.
The influence of porosity on mechanical properties has been the subject of many
studies and in this context the porosity p is defined as:
p ¼ 1 À
r specimen
r theor
ð11:2Þ
where r specimen is the actual, experimentally determined density of the specimen
and r theor is the theoretical density of the full, dense material. Most important is the
influence on elastic properties and one of many formulae describing this was
provided by MacKenzie [3], who applied a Taylor series development for Young’s
modulus E to estimate the influence of the porosity p:
E ¼ E 0 ð1 þ a 1 p þ a 2 p
2 þ Á Á ÁÞ
ð11:3Þ
where E 0 is Young’s modulus of the full dense material, p ¼ 0, and a 1 and a 2 are
fitting parameters. For small values of porosity the linear term is sufficient, but for
higher porosities an increasing number of series elements are necessary. In addition
to this simple series development, a huge number of theoretically developed
equations are described. However, theoretically well-based formulae also require
fitting parameters, and Figure 11.4 depicts Young’s modulus for nanocrystalline
copper and palladium as a function of the porosity. This is a good example of small
porosity, where the linear term is sufficient. Additionally, it demonstrates the
unavoidable scattering of experimental values that, most likely, is due to the different
grain sizes for specimens with different porosities. When examining these heavily
scattered experimental data, it is clear that a fit using more than the linear
approximation is not justified.
Evaluation of the experimental data depicted in Figure 11.4 led to the following
fitting parameter for Young’s moduli (in GPa):
E Cu ¼ 121 1 À 3:8p
ð
Þ
E Pd ¼ 132 1 À 2:5p
ð
Þ
ð11:4Þ
302j 11 Mechanical Properties of Nanoparticles