Atomistic Approximation of Solid Surface Energy and Its Anisotropy
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he surface energy of solids is an important physical parameter in controlling a wide range of phenomena, e.g., the stress for brittle fracture, the rate of sintering, and the growth rate during particle coarsening.[1] However, experimental determination of the surface energy of solids is both difficult and subject to numerous errors.[1,2] The electron theory[3] is capable of giving quantitatively accurate descriptions of the solid interior,[4–6] but for solid surfaces, great additional difficulties are produced by the rapid decrease of electron density and by the loss of translational symmetry.[3] Consequently, various approximations are needed and present electron-based calculations on the surface energy of solids can only be carried out at 0 K.[7–9] The discrepancies among available electron-based data usually go up to around 30 pct and, as will shown below, regarding the dependence of solid surface energy on planar packing fraction, these data may seem unrealistic in some cases. Atomistic approximations toward the solid surface energy, and in particular to its anisotropy, are regrettably still missing. Here we show that, interestingly, by comparing the estimated new surface generated during
LIANWEN WANG is with the School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, P.R. China. Contact e-mail: [email protected] Manuscript Submitted April 15, 2020.
METALLURGICAL AND MATERIALS TRANSACTIONS A
vacancy formation with the reported vacancy formation energy, the anisotropic solid surface energy can be reasonably approximated. In the present work, complex computer simulation is not invoked and the data are obtained by solving two simple equations. When compared with literature data, the present results are found more reasonable. A significant finding here is that atomistic approximation also works on the issue of solid surface energy. On this basis, in a general broader view, atomistic approximations are suggested to be a workable choice at hand for important materials science issues such as the free energy of grain boundaries and solid–liquid interfaces, on which some preliminary efforts are carrying out by the author. It has been for a long time agreed[10] that a vacancy is formed when an atom leaves its lattice site and occupies free positions on the surface. However, the variation in the surface area during vacancy formation in relation with surface energy and vacancy formation energy has not been considered quantitatively. See Figure 1, when a vacancy is formed by moving an atom of radius r to the crystal surface, an area S1 = pr2 originally occupied by this atom disappears. For the moved atom, a fraction of its surface is exposed inwards to the solid which is the area of the lower-part spherical crown S2 = 2pr2(1 cosh). Consequently, after formation of the vacancy, the area of this crystal surface is increased by DS ¼ 4pr2 S1 S2 ¼ 4pr2 pr2 2pr2 ð1 cos hÞ ¼ pr2 ð1 þ 2 cos hÞ: ½1 For a particular crystal facet (hkl), the surface increment DShkl accompanying vacancy formation can be e
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