At the nanoscale, the melting temperature T _m and melting enthalpy ΔH _m , for free-standing nanostructures can be expressed as function of their bulk corresponding property, the size of the structure and one shape parameter 9 .

where the shape parameter, α_shape, is defined as α _shape = AD(γ _s -γ _l )/(VΔH _{m, ∞}); D being the size of the structure (i.e. for a sphere, D is the diameter), A (meter squared) and V (cubic meter) are the surface area and volume of the nanostructure, respectively. ΔH _m,∞ is the bulk melting enthalpy (Joule per cubic meter), whereas γ _l and γ _s are the surface energy in the liquid and solid phases (Joule per square meter), respectively. γ _l and γ _s are considered size independent. This is justified by the fact that the size effect on the surface energies is less than 4% for sizes higher than 4 nm 10 11 . Indeed, below this size, edges, and corners of the structures begin to play a significant role in the surface energy 12 .

The size-dependent melting temperatures of platinum and palladium are plotted in Figures 1 and 2 respectively. The materials properties of the considered materials are indicated in Table 1. The melting properties for the sphere have been calculated using for the solid surface energy the mean value of experimental data 13 . For the other polyhedra shapes, we have considered the fcc crystal structure of the metals and the respective solid surface energy for each face 14 . Tables 2 and 3 indicate the parameters used for the calculation of the melting properties. Experimentally, the melting of agglomerated Pt nanocrystals (tetrahedrons and cubes) with an average size around approximately 8 nm starts at approximately 900 K 15 in relative good agreement with our theoretical predictions. Molecular dynamics simulations 16 have calculated the size effect on the melting temperature of Pd and found α_sphere = 0.95 nm while our theory predicts 1.68 nm.

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 23 24 :

where x _solidus (x _liquidus) is the composition in the solid (liquid) phase at a given T, respectively. 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 3. 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 .

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:

where z ₁ is the first nearest neighbor atoms; z _1ν 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 z ₁ = 12, z _1ν = 4 for (100) faces and three for (111) faces. ΔH _vap is the difference between the bulk vaporization enthalpies of the two pure elements, . ΔH _sub 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, ΔH _sub = 0 and ΔH _vap = 0, there is no segregation and we retrieve Equation 3. x _solidus and x _liquidus 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 4 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 28 29 and other theoretical calculations 29 30 31 . 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 .

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 34 35 . 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, E _ca 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 38 39 . 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 38 39 .

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, E _ca seems to evolve quadratically with the composition when the segregation is not considered; which is not the case when the segregation is considered (Figure 5). For the segregated Pt-Pd nanoparticle, a maximum in the catalytic activation energy is reached around 16% of Pt composition.