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Nanomaterials 2020, 10, 767 8 of 15 3. Results and Discussion The molar volumes determined from direct quantum-mechanical calculations using the procedure of averaging the interatomic distances (see Figure 4) are compared with those determined from the Murhaghan equation and the definition of the bulk modulus within our thermormodynamic approach in Figure 5a. The volumes obtained from the direct calculations of the electronic structure of nanoclusters/nanoparticles are represented by the DFT black data points, and the molar volumes from our phenomenological thermodynamic approach are continuous blue and red lines for the volumes based on the Murnaghan equation and the definition of the bulk modulus, respectively. It is evident that the volumes from direct DFT calculations of nanoclusters/nanoparticles agree very well with those based on the definition of the bulk modulus (Equation (10)). a) 11.0 10.5 10.0 9.5 9.0 8.5 b) 0.8 0.6 0.4 0.2 8.00 c) 50 Number of atoms N 0.00 50 100 150 200 Number of atoms N 100 150 200 Murnaghan bulk modulus DFT bulk experimental d) Ratio of coordination numbers bulk 2.5 2.0 1.5 1.0 0.5 0 −10 −20 −30 −40 050100150200 Number of atoms N Murnaghan bulk modulus bulk DFT 0 50 100 150 200 Number of atoms N Murnaghan bulk modulus bulk DFT Figure 5. The dependencies of (a) the molar volume and (b) the ratio of coordination numbers of nanocluster surface atoms with respect to that of {111} surface of the bulk (which is equal to 9) as functions of the number of atoms in the studied nanoclusters/nanoparticles. The green dashed lines in parts (a,b) are the values corresponding to the bulk (or its {111} surface), blue and red lines represent volumes assigned to nanoparticles using Murnaghan equation and the definition of the bulk modulus, respectively. The full black circles represent results of direct DFT calculations of nanoparticles. The part (c) shows absolute (and part (d) also relative) differences of excess energies per atom determined by our phenomenological thermodynamic approach w.r.t. to the excess energies of the direct DFT calculations of nanoparticles when the volume of nanoparticles in the phenomenological thermodynamic modeling is determined from the Murnaghan equation (blue curves and blue data points in part (d)), from the definition of the bulk modulus (red curves and red data points) and from determining the volume of the nanoparticles from the volume of bulk fcc Ag (green lines and green data points). Also added is the experimental molar volume (the horizontal brown dashed line in (a)). Excess energy of decahedron per atom [eV/atom] Mean Vm · 10−6 [m3mol−1] Difference of calculated excess Ratio of coordination numbers energies D [%] of surface atomsPDF Image | Quantum-Mechanical of the Energetics of Silver Decahedron Nanoparticles
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