At the nanoscale, the shape which exhibits the highest melting temperature is the one which minimizes the most the Gibbs' free energy (G
= H - TS
); and is then the favored one. From Figures and , the four most-stable shapes among the ones considered are the dodecahedron, truncated octahedron, icosahedron, and the cuboctahedron. Experimentally, truncated octahedron and cuboctahedron are observed for platinum nanoparticles [8
] whereas icosahedron, decahedron, truncated octahedron and cuboctahedron are observed for palladium nanoparticles [8
]. Therefore, our predictions are in relative good agreement with the observations for palladium and platinum except that dodecahedron and icosahedron are not observed for platinum. Other theoretical calculations confirmed that the dodecahedron is a stable shape for palladium [17
]. More generally, according to Yacaman et al.
], the most often observed shapes at the nanoscale are the cuboctahedron, icosahedron, and the decahedron.
Furthermore, care has to be taken when we compare theoretical results with experimental ones due those materials properties depend on the synthesis process [18
]. And then predicted properties from thermodynamics may differ from the experimentally observed if the synthesis process is not running under thermodynamical equilibrium. Moreover, thermal fluctuations are often observed in nanoparticles [20
] meaning that the shape stability is much more complicated than just a minimisation of the A/V
ratio with faces exhibiting the lowest surface energy.
Nano-phase diagram of Pt-Pd
According to the Hume-Rothery's rules, platinum and palladium forms an ideal solution [21
]. In this case, considering no surface segregation, the liquidus and solidus curves of bulk and nanostructures are calculated from the following equations [22
) is the composition in the solid (liquid) phase at a given T
is the size-dependent melting temperature of the element i
is the size-dependent melting enthalpy of the element i
The phase diagram of the Pt-Pd alloy is plotted in Figure . We note that the lens shape of the phase diagram is conserved at the nanoscale; however, the lens width increases for the shapes characterized by a small melting enthalpy and melting temperature, i.e.
, exhibiting a strong shape effect. Moreover, the melting temperature increases with the concentration of Pt in agreement with Ref. [25
Phase diagram of the Pt-Pd system for different shapes. Different shapes at a size equal to 4 nm and at the bulk scale. The solid lines indicate the liquidus curves whereas the dashed lines indicate the solidus ones.
In order to predict nanomaterials properties more accurately, we are considering a possible surface segregation which is known as the surface enrichment of one component of a binary alloy. At the nanoscale, surface segregation leads to a new atomic species repartition between the core and the surface. According to Williams and Nason [26
], the surface composition of the liquid and solid phase are given by:
is the first nearest neighbor atoms; z1ν
is the number of first nearest atoms above the same plane (vertical direction). In the case of face-centered cubic (fcc) crystal structure of Pt and Pd materials, we have z1
= 12, z1ν
= 4 for (100) faces and three for (111) faces. ΔHvap
is the difference between the bulk vaporization enthalpies of the two pure elements,
is the difference between the bulk sublimation enthalpies of the two pure elements,
. Element A
is chosen to be the one with the highest sublimation and vaporization enthalpies. If the two components are identical, ΔHsub
= 0 and ΔHvap
= 0, there is no segregation and we retrieve Equation 3. xsolidus
are obtained from solving Equation 3. Assuming an ideal solution, only the first surface layer will be different from the core composition.
Considering the surface segregation in the Pt-Pd system, we can see in Figure that the lens shape of the surface liquidus/solidus curves is deformed compared to the core. At a given temperature, the liquidus and solidus curves of the surface are enriched in Pd compared to the core; meaning that the surface is depleted of Pt (the higher bond energy element) which is in agreement with experimental observations[27
] and other theoretical calculations[29
]. This is due to the fact that Pd has a lower solid surface energy, a lower cohesive energy compared to Pt and also because diffusion is enhanced at the nanoscale [32
Phase diagram of the Pt-Pd system considering the surface segregation effect. Surface segregation effect at a size equal to 4 nm for a spherical nanoparticle.
Size-dependent catalytic activation energy of Pt-Pd
The catalytic activation energy is the energy quantity that must be overcome in order for a chemical reaction to occur in presence of a catalyst. The low the catalytic activation energy is, the most active the catalyst is. It is thus an important kinetic parameter linked to the chemical activity. Indeed, the catalytic activation energy is a linear function of the work function [33
]. For pure materials, the catalytic activity depends on the fraction of surface atoms on corners and edges while for binary compounds it depends also on the surface segregation. Recently, it has been showed by Lu and Meng in Ref. [36
] that the size-dependent catalytic activation energy, Eca
could be obtained from the following relation:
Therefore, it means that the size-dependent catalytic activation energy decreases with size.
To compare with experimental results, the ratio of the catalytic activation energies between tetrahedral (D
= 4.8 nm) and spherical (D
= 4.9 nm) pure platinum nanoparticles has been determined around 0.66 in excellent agreement with the experimental value of 0.62 ± 0.06 announced by Narayanan and El-Sayed [37
]. Moreover, the ratio of the catalytic activation energies between cubic (D
= 7.1 nm) and spherical (D
= 4.9 nm) pure platinum nanoparticles is around 1.01 in relative good agreement with the experimental value of 1.17 ± 0.12 [37
From the size-dependent Pt-Pd phase diagram, the melting temperature of the alloy can be deduced. Equation 6 describes the melting temperature of the bulk Pt-Pd while Equations 7 and 8 describe the nanoscaled melting temperature of a non-segregated and segregated spherical nanoparticle (with a diameter equal to 4 nm), respectively.
where x represents the alloy composition. For a spherical Pt-Pd nanoparticle with a diameter equal to 4 nm, by combining Equations 5-8, Eca seems to evolve quadratically with the composition when the segregation is not considered; which is not the case when the segregation is considered (Figure ). For the segregated Pt-Pd nanoparticle, a maximum in the catalytic activation energy is reached around 16% of Pt composition.
Composition dependency of the catalytic activation energy for a spherical nanoparticle of Pt-Pd. Nanoparticle of Pt-Pd with a size equal to 4 nm.