size, appears to be general and was even observed for high-temperature superconductors [16]. Whenever materials exhibiting phase transformations are analyzed,
the same observations are made: with decreasing particle size, a transition to the
high-entropy (usually high-temperature) phase is observed. Additionally, it is
remarkable that in many phase transitions, first-order transitions in the case of
coarse materials, the lattice changes, looking at nanoparticles, are continuous and
not, as has been observed for conventional materials, abrupt in nature. (According to
Ehrenfest, a phase transition is of the nth order if the (n – 1)th derivative of the free
enthalpy G with respect to temperature, volume, or pressure is continuous, whereas
the nth derivative is discontinuous.) This behavior is quite strange, although in this
context it must not be forgotten that the determination of phases and particle sizes is
usually made from an evaluation of X-ray diffraction line profiles. There are,
however, two points that cause these evaluations to be problematic: (i) the tetragonality of the structures is very small and (ii) in the case of small particles the
diffraction lines, which are broadened due to the small particle size, are not split.
This makes it difficult to decide whether a diffraction line profile consists of a split
line of the tetragonal phase or whether it is a superposition of the diffraction lines of
the tetragonal and cubic phases. In the latter case, the transition from one phase to
the next would not be quasicontinuous but rather abrupt, possibly superimposed by
fluctuation processes between the two phases.
7.6
A Closer Look at Nanoparticle Melting
When discussing the melting of nanoparticles, the point was mentioned that in the
case of larger particles the melting process starts from a thin surface layer. In case of
lead, a value of approximately 3 nm was found for the thickness of the surface layer.
Hence, the question arises of how this behavior, which is so fundamentally different
compared to that of particles with conventional grain sizes, can be explained. Chang
and Johnson [18] showed, on the basis of theoretical considerations using Landau’s
parameter of ordering M, that small nanoparticles do not have the degree of ordering
that is observed in bulk materials. In that case, the ordering parameter is defined in a
way that the value is 1 for ideal crystals and 0 for the melted phase. Chang and
Johnson [18] also provided a formula to calculate this order parameter M (e.g., tin) as
a function of the radius and particle sizes. In Figure 7.18, M is shown as a function of
the radius for particles of different sizes. In the case of large particles (e.g., radius
10 nm), a perfect ordering is achieved in the interior of the particle and a reduced
ordering close to the surface. The thickness of this layer with reduced ordering is
approximately 3 nm. When considering very small particles (e.g., those with a radius
of 1 nm), a perfect ordering is not attained even in the center of the particle. In such a
case the maximum ordering would be less than 0.3, which means that small-sized
particles would act more like a melted material than a crystallized one. By applying
this theory, Chang and Johnson were able to fit the data for size-dependent melting
point of tin quite well.
7.6 A Closer Look at Nanoparticle Melting j153
the same observations are made: with decreasing particle size, a transition to the
high-entropy (usually high-temperature) phase is observed. Additionally, it is
remarkable that in many phase transitions, first-order transitions in the case of
coarse materials, the lattice changes, looking at nanoparticles, are continuous and
not, as has been observed for conventional materials, abrupt in nature. (According to
Ehrenfest, a phase transition is of the nth order if the (n – 1)th derivative of the free
enthalpy G with respect to temperature, volume, or pressure is continuous, whereas
the nth derivative is discontinuous.) This behavior is quite strange, although in this
context it must not be forgotten that the determination of phases and particle sizes is
usually made from an evaluation of X-ray diffraction line profiles. There are,
however, two points that cause these evaluations to be problematic: (i) the tetragonality of the structures is very small and (ii) in the case of small particles the
diffraction lines, which are broadened due to the small particle size, are not split.
This makes it difficult to decide whether a diffraction line profile consists of a split
line of the tetragonal phase or whether it is a superposition of the diffraction lines of
the tetragonal and cubic phases. In the latter case, the transition from one phase to
the next would not be quasicontinuous but rather abrupt, possibly superimposed by
fluctuation processes between the two phases.
7.6
A Closer Look at Nanoparticle Melting
When discussing the melting of nanoparticles, the point was mentioned that in the
case of larger particles the melting process starts from a thin surface layer. In case of
lead, a value of approximately 3 nm was found for the thickness of the surface layer.
Hence, the question arises of how this behavior, which is so fundamentally different
compared to that of particles with conventional grain sizes, can be explained. Chang
and Johnson [18] showed, on the basis of theoretical considerations using Landau’s
parameter of ordering M, that small nanoparticles do not have the degree of ordering
that is observed in bulk materials. In that case, the ordering parameter is defined in a
way that the value is 1 for ideal crystals and 0 for the melted phase. Chang and
Johnson [18] also provided a formula to calculate this order parameter M (e.g., tin) as
a function of the radius and particle sizes. In Figure 7.18, M is shown as a function of
the radius for particles of different sizes. In the case of large particles (e.g., radius
10 nm), a perfect ordering is achieved in the interior of the particle and a reduced
ordering close to the surface. The thickness of this layer with reduced ordering is
approximately 3 nm. When considering very small particles (e.g., those with a radius
of 1 nm), a perfect ordering is not attained even in the center of the particle. In such a
case the maximum ordering would be less than 0.3, which means that small-sized
particles would act more like a melted material than a crystallized one. By applying
this theory, Chang and Johnson were able to fit the data for size-dependent melting
point of tin quite well.
7.6 A Closer Look at Nanoparticle Melting j153