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Nanomaterials 2020, 10, 767 4 of 15 of nanoclusters/nanoparticle is ill-defined, we below discuss three different ways of assigning the (molar) volume to the studied nanoclusters/nanoparticles. The radius r is then set equal to the radius of a sphere, which has the volume equal to the product of the number of particles in the nanocluster/nanoparticle and the (specifically assigned) volume (per atom). The surface area (to be multiplied by σsur) is then put equal to the surface of a sphere with the radius equal to the above discussed radius r. The fact, that the shape of studied nanoclusters/nanoparticle is non-spherical, decahedral, is taken into account by a shape factor C is introduced [25,43,45] into the Equation (3): Gsur=3·C·Vm·σs, C=Ashape/Asph, (4) r where the shape factor C is defined as the ratio of the surface area of the calculated nanoparticle Ashape to the surface area of a spherical particle with the same volume. For the decahedron we use Adec = a2 ·5·√3·sin36◦ ·1−0.75·(sin36◦)2 and Vdec = a3 ·(5/4)·sin36◦ ·cos36◦ (5) with the length parameter a defined in Figure 1 (also equal to the height of the decahedron, a pentagonal dipyramid). The values for a few commonly occurring shapes of (nano-)particles are listed in Table 1. Importantly, following the procedure described above we do not need a Tolman length to define the surface of particle [46] as it is defined by molar volume and radius of a spherical particle. Table 1. Shape factors for different shapes of (nano-)particles as collected from selected literature sources. By a liquid spherical shape we mean an ideal sphere (without any atomic structure manifesting itself) while solid spherical shape represents a spherical nanoparticle with its atomic structure which is making its surface not ideally spherical. Shape of Particle Shapefactor spherical - liquid 1.00 spherical - solid 1.05 regular icosahedron 1.06 regular dodecahedron 1.10 regular octahedron 1.18 cube 1.24 decahedron 1.28 regular tetrahedron 1.49 [12,43] Analogous to the Equations (2) and (4) is the total Gibbs energy of cluster, gtot equal to: gtot = Gtot ·N = gref +gsur where gref = N·gφ (6) where N is the number of atoms in a studied nanocluster/nanoparticle, NA is the Avogadro constant and the gφ is the atomic Gibbs energy of pure constituent Ag. As far as the Gibbs energy gφ is Ag concerned, we use the Gibbs energy of the bulk fcc Ag per atom Ebulk, calculated by DFT. Ag The equation of surface contribution for one nanocluster/nanoparticle is then changed to: gsur = N·3·C· Vat ·σsur = N·3·C· Vm ·σsur. (7) where Vat corresponds to the volume of one atom. One of the consequences of a high surface Gibbs energy is a surface strain [17]. It is caused by the minimization of the surface energy of the studied (nano-)particles and it leads to the reduction of the mean molar volume of the nanoparticle. As an extreme case, a particle without any surface energy exhibits zero surface strain and its volume is equal to that of the bulk. r r · NA References [12,22,47,48] [22,47,48] [12,49] [49] [12,49] [12,49] this work NA AgPDF Image | Quantum-Mechanical of the Energetics of Silver Decahedron Nanoparticles
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