SIZE DEPENDENT INTERFACE ENERGY - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 216

Similar to the simplification of γ sl ( D ), Eq. (3.4) can also be simplified as

γ ss ( D )/γ ss0 = 1- D ′ 0 /(4 D ). (3.5)

Figure 1 shows an agreement between the model predictions of Eq. (3.4) and the
computer simulation results for Cu [112]. It is interesting that the use of a negative interface
stress for γ ss ( D ) in Eq. (3.4) leads to a full agreement with the computer simulation results,
which implies that f < 0.
If we compare Eq. (2.22) and Eq. (3.5), it can be found that γ sl ( D )/γ sl0 < γ ss ( D )/γ ss0 which
implies that the stiffer surrounding of particles brings out less decrease of the interface energy
as D decreases. For the grain boundaries, even if when D → D ′ 0 , γ ss ( D )/γ ss0 ≈ 75% while
γ sl ( D )/γ sl0 = 0. Since when D ≈ 2 D ′ 0 , the grains are no more stable and will transform to
amorphous solids in terms of the computer simulation results [112], the smallest value of
γ ss ( D ′ 0 )/γ ss0 could be about 85%.

Solid-Vapor Interface Energy or Surface Energy

The Bulk Surface Energy γ sv0

The surface energy γ sv0 usually is defined as the difference between the free energy of the
surface and that of the bulk or simply as the energy needed to split a solid in two along a
plane, which is one of the basic quantities to understand the surface structure and phenomena
[32,77,113]. Despite of its importance, γ sv0 value is difficult to determine experimentally. The
most of these experiments are performed at high temperatures where surface tension of liquid
is measured, which are extrapolated to zero Kelvin. This kind of experiments contains
uncertainties of unknown magnitude [114-115] and corresponds to only γ sv0 value of an
isotropic crystal [116]. Note that many published data determined by the contact angle of
metal droplets or from peel tests disagree each other, which can be induced by the presence of
impurities or by mechanical contributions, such as dislocation slip or the transfer of material
across the boundary [117]. In addition, there are hardly the experimental data on the more
open surfaces except for the classic measurements on Au, Pb and In [118] to our knowledge.
Therefore, a theoretical determination of γ sv0 values especially for open surface is of vital
importance.
During the last years there have been several attempts to calculate γ sv0 values of metals
using either ab initio techniques [119-121] with tight-binding (TB) parameterizations [122]
or semi-empirical methods [123]. γ sv0 values, work functions and relaxation for the whole
series of bcc (A2) and fcc (A1) 4 d transition metals have firstly been studied [119] using
the full-potential (FP) linear muffin-tin orbital (LMTO) method in conjunction with the
local-spin density approximation to the exchange-correlation potential [124-126]. In the
same spirit, γ sv0 values and the work functions of the most elemental metals including the
light actinides have been carried out by the Green′s function with LMTO method [120-
121,127-128]. Recently, the full-charge density (FCD) Green′s function LMTO technique
in the atomic-sphere approximation (ASA) with the generalized gradient approximation
(GGA) has been utilized to construct a large database that contains γ sv0 values of low-index