SIZE DEPENDENT INTERFACE ENERGY - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 215

Table 7. Comparison of γ ss0 (T m ) between γ ss0 by Eq. (3.1) and available theoretical or
experimental results γ′ ss0 for metals [39,105-111] and organic crystals [48-49]

γ ss0 γ′ ss0 γ′ sl0

System

(mJ/m 2 )

Cu 584 601

Ag 398 392,375

Au 430 400

Al 302 300~380

Ni 844 866,757

Co 706 650

Nb 804 760

Ta 944 900

Sn 158 160,164

Pb 96 111.5±15.6

Bi 176 140.5±14.1

Pivalic acid 4.8

5.2±0.4 2.7±0.2

Succinonitrile 14.6

15.0±2.0 7.9±0.8

On the contrary, when γ CNT values are used, e.g. 132, 126, 177, 255 and 54.4 mJ/m 2 [11]
for Au, Ag, Cu, Ni and Bi, it is obvious that 2γ CNT values are about 20~40% smaller than the
corresponding theoretical or experimental values 400, 392, 601, 757 and 140.5±14.1 mJ/m 2
[105-107,109,111], which is unreasonable [11]. Moreover, this comparison supports again the
claim that γ′ sl0 ≈ γ sl ( r c , T n ) and γ′ sl0 ≠ γ sl0 ( T m ).

The Size Dependence of Solid-solid Interface Energy γ ss ( D )

In the deduction of γ ss ( D ), the way is similar to the deduction of γ sl ( D ) while Eq. (2.15)
and Eq. (2.16) must be modified due to different interface conditions [77]. We assume that f
as a first order approximation keeps constant for both solid-liquid and solid-solid interface at
T m , which leads to the same strain on both sides of the interface when the grains are isotropic.
However, elastic modulus of grain boundaries should be larger than that of the solid-liquid
interface with less strain under the same stress. This is introduced by the fact that A = 3 V /( D )
for solid-solid interface because two solid-liquid interfaces of particles combine to form one
grain boundary with

P = 2 fA /(3 V ) = 2 f / D (3.2)

and

ε = -2κ f /(3 D ). (3.3)

Both equations indicate that the strain on the grain boundary is only a half of that on the
solid-liquid interface and thus Δ A / A = 2ε = -4κ f /(3 D ). Now Δγ ss = γ ss ( D )-γ ss0 = (Δ A / A )( f- γ ss )= -
4κ f /(3 D )( f -γ ss ), or

γ ss ( D )/γ ss0 = [1- D ′ 0 /(4 D )]/[1-γ ss0 D ′ 0 /(4 fD )]. (3.4)