Bond-Energy and Surface-Energy Calculations in Metals

Abstract

One of the major goals of materials science is to relate the properties of materials to the potential functions that characterize the atomic and molecular interactions that hold the material together. Sharing this ambitious and ongoing quantum-mechanical program with students is usually beyond the scope of an introductory course in materials science. However, there are situations where an elementary approach to connecting atomic interactions to material properties is possible. One of these areas is the use of the lattice energies and bond energies of metals to predict the metal surface energies. These calculations are presented here at a level that is suitable for an introductory materials science course.Many phenomena of importance in solids depend on the properties of the surfaces or interfaces of the materials in question, rather than properties of the bulk material. Surface and interfacial tensions are key properties in understanding phenomena such as crystal faceting, grain structure, crack propagation, capillaritydriven mass transport, vacancy formation, wetting, and adhesion (1). The bonding concepts considered here are similar to those applied to the covalent bond energy that is a central idea in general, organic, inorganic, and physical chemistry courses (2). Once determined, the metallic bond strengths are seen to be somewhat smaller than the energies of the more familiar covalent bonds. For body-centered cubic (bcc), face-centered cubic (fcc), and hexagonalclosest-packed (hcp) metals, the bond energy and the lattice constant can then be used to make reasonably good estimates the surface energies of solid metals. Bond Energies for the Bcc and Fcc MetalsThe calculation of the bond energy in a metallic lattice is similar to that in gaseous molecules except that students find it a little more challenging to determine the number of bonds broken during atomization. The atomization of a crystalline metal, M(c), can be represented by the processwhere the molar energy of atomization of the metal lattice is Δ a U and the gaseous metal is assumed to be entirely monatomic. The atomization energy is also called the energy of sublimation, the cohesive energy, or the lattice energy. The calculation of bond energies from Δ a U will initially be limited to bcc and fcc metals. Each metallic crystal structure has its own coordination number, Z, which is the number of nearest neighbors surrounding a central atom. The crystal lattice is assumed to be composed entirely of "hard spheres" of identical radius so that all of the nearest neighbors "touch" the central atom they surround. The unit cell geometries for the bcc and fcc metals are shown in Figures 1A and 2A. The coordination numbers for these two crystal structures areZðfccÞ ¼ 12Because the removal of one atom from a metallic lattice causes the breaking of Z bonds, it is tempting, but not correct, to assume that Δ a U = Zε, where ε is the molar bond energy. It is true that the Z bonds surrounding this atom are broken during atomization; however, this process also brea...

Amorphous/Crystalline Hetero‐Phase Pd Nanosheets: One‐Pot Synthesis and Highly Selective Hydrogenation Reaction

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Size-, Shape-, and Composition-Dependent Model for Metal Nanoparticle Stability Prediction

“…Contrary to the rest of the metals, we note that the CEs of Zr MNPs are under-predicted, likely because Zr is an hcp metal. In hcp metals, the (0001) plane intralayer bond lengths are not equivalent with interplane bond lengths, meaning that a bulk hcp atom more accurately has six nearest neighbors and another six near–nearest neighbors . Note that the values of CB as used in the SRB model only depend on the bulk behavior of the given metal, although the Zr MNPs in this work are built from fcc-like initial structures.…”

“…In hcp metals, the (0001) plane intralayer bond lengths are not equivalent with interplane bond lengths, meaning that a bulk hcp atom more accurately has six nearest neighbors and another six near−nearest neighbors. 54 Note that the values of CB as used in the SRB model only depend on the bulk behavior of the given metal, although the Zr MNPs in this work are built from fcc-like initial structures. Thus, a CB value is likely appropriately assigned as less than 12, shifting all the SRB CE values (eq 1) higher for an hcp metal such as Zr.…”

Correlation between the electron work function of metals and their bulk moduli, thermal expansion and heat capacity via the Lennard-Jones potential

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