Surface energy of metals

The calculation of the surface energy of metals has been along two rather different lines. The first has been that of Skapski, outlined in Section III-IB. In its simplest form, the procedure involves simply prorating the surface energy to the energy of vaporization on the basis of the ratio of the number of nearest neighbors for a surface atom to that for an interior atom. The effect is to bypass the theoretical question of the exact calculation of the cohesional forces of a metal and, of course, to ignore the matter of surface distortion. 

The approximate calculation of the surface energies of metals as a function of crystal structure described earlier uses the enthalpy of sublimation, s, and the co-ordination number to calculate the energy as a function of the atomic concentration on the surface. The atomic areas of the principal configurations are as follows ... 

Figure 6.7. Experimental values of the work of adhesion for non-reactive pure metal/monocrystalline AI2O3 systems versus the liquid surface energies of metals. From (Eustathopoulos and Drevet 1998) [17].

Although it is very difficult to measure surface energies, they may today relatively easily and reliably be calculated from first-principles [26,27], even in the cases of quite open surfaces [28]. In Fig. 2 we show the surface energies of metals in the 4d-series of the Periodic Table obtained by first-principles calculations [27]. 

Carr, A.K. (1997), Increase ofthe Surface Energy of Metal and Polymeric Surfaces Using the One Atmosphere Uniform Flow Discharge Plasma, MS Thesis, University of Tetmessee. 

To get a rough estimate of the surface energy of metals, we calculate the overall change in bond energy as an atom moves from the bulk to the surface is calculated. 

The PKZB meta-GGA achieves very accurate atomization energies of molecules, surface energies of metals, and lattice constants of solids [108]. These properties are greatly improved over GGA. On the other hand, meta-GGA s that are heavily fitted to molecular properties tend to give surface energies and lattice constants that are less accurate than those of non-empirical GGA s or even LSD [108]. 

Studies on jeUium surfaces provide a partial explanation for why LDA outperforms GGA for the calculation of the surface energy of metals. Specifically, they show that the exchange-correlation contribution to the surface energy is calculated more reliably with the LDA because of a favorable cancellation of errors LDA overestimates the exchange contribution to the surface energy, but underestimates the correlation contribution, whereas PBE underestimates both quantities [138-140], Since these results on jellium surfaces are apparently transferable to real metals, it is reasonable to anticipate that functionals that are superior to LDA for jellium surfaces will also be superior to LDA for real metal surfaces. In this regard, it appears that the meta-GGA TPSS, PBE-WC, and a functional from Mattsson and coworkers based on a subsystem functional approach offer some promise [28,140, 141]. However, whether these functionals live up to expectations for real metal surfaces and solid surfaces in general remains to be seen. 

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