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Nanomaterials 2020, 10, 767 9 of 15 In order to determine the molar volumes from the Murnaghan equation of state and the definition of the bulk modulus (shown in Figure 5a) we used the hydrostatic pressure equal to the surface stress p = Psur which was found from the calculations of the surface energy of Ag for T = 0 K for the {111} and {100} terminations of the bulk fcc Ag. The obtained value of the surface energy for the {111} facet is equal to 0.80 Jm−2 and for the {100} surface orientation to 1.14 Jm−2. Our values agree quite well with the experimental mean surface energy of 1.1–1.3 Jm−2 (see Table 2), reported for much higher temperature of 1073 K in Ref. [58], or with the theoretical values obtained using the LDA approximation in Ref. [59]. Table 2. Our computed surface energies for Ag surfaces with different crystallographic orientations in comparison with available experimental data [58]. (111) (100) (110) exp. [58] (1073 K) eV/atom 0.409 0.646 0.801 J/m2 0.881 1.206 1.057 1.1–1.3 Our calculations also reproduce fairly well the lattice constant and the bulk modulus of the bulk fcc Ag. Our theoretical lattice constant of fcc Ag is equal to 4.1555 Å in an acceptable agreement with the experimental value of 4.0853 Å. Our computed bulk modulus of 90 GPa lies between the experimental values of 84 GPa and 118 GPa [60]. In a similar way we analyze also the ratio of the coordination number of surface atoms of the studied nanoparticles, the coordination number of an fcc bulk lattice is 12. The coordination number of the surface atoms is lower. For an infinite surface of a bulk fcc (see the schematics in Figure 3) it is equal to 9 and so the ratio of the coordination numbers of surface atoms of the bulk with respect to the coordination number of atoms in the fcc bulk is 0.75 (see this value as the horizontal green dashed line Figure 5b). The coordination numbers of surface atoms at the {111} facets of the studied nanoclusters/nanoparticles apparently converge to the coordination number of surface atoms at the {111} surface of the bulk only very slowly as a function of the number of atoms in the nanoparticle. Using our computational approaches it is now possible to evaluate an energy contribution related to the fact that the studied systems are nanoclusters/nanoparticles (with respect to the energy of the bulk). As this energy has a character of an excess energy Eex (the total energy of nanoclusters/nanoparticles without the cohesion energy of the bulk): Eex = Etot − N · Ebulk (13) N we show it (per atom) in Figure 5c as a function of the number of atoms in the studied nanoclusters/nanoparticles. The excess energy per atom decreases with increasing radius of nanoparticles. Let us note that this excess energy is different from the excess Gibbs energy GE employed in Equation 1, similarly as in other papers dealing with nanoparticles, e.g., Ref. [61]. While Figure 5c clearly shows that the absolute values of the excess energies (per atom) as determined using (i) our phenomenological thermodynamic approach based on bulk-related properties (obtained by DFT calculations) very well match (ii) those from direct DFT calculations of actual nanoclusters/nanoparticles EDFT, it is important to evaluate the differences more precisely. Therefore, ex we analyze the excess energy differences as relative values: E −EDFT D = ex ex · 100%. (14) EDFT exPDF Image | Quantum-Mechanical of the Energetics of Silver Decahedron Nanoparticles
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