SIZE DEPENDENT INTERFACE ENERGY - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 228

Table 12. Necessary parameters in Eq. (4.12). Note that γ sv0 values are cited from Ref.
[116] while that of Al is cited from Ref. [118] since this value in Ref. [116] is doubtful
where γ sv0 of Al (110) is smaller than that of Al (111), which is physically unacceptable

h
(nm)

D
(nm)

E
(kJ/mol)

T b
(K)

S b
(J/mol-K)

γ svo
(J/m 2 )

Element Surface

1.83 Be (0001) 1.90 0.222 292.4 2745 106.5

Mg (0001) 2.80 0.320 127.4 1363 93.5 0.79

Na (110) 3.45 0.372 97.0 1156 83.9 0.26

4.29

1.30 Al (110) 0.286 293.4 2793 105.0

5.72

1.28 (111)

3.80 0.288 334.4 3130 106.8

1.63 (100)

Au

1.70 (110)

Ref 163-166 42 42 42

It is known that the surface energy ratio between different facets is a more important
parameter in determining the crystalline shapes. Eq. (4.12) indicates that [162],

1 1

γ ( D ) γ

(4.15)

sv

=

sv 0

γ

2

( D ) γ

2

sv sv 0

where the superscripts 1 and 2 denote different facets. Eq. (4.15) implies that although the
surface energy is size-dependent, the surface energy ratio between different facets is size-
independent and is equal to the corresponding bulk ratio. Eq. (4.15) can also be compared
with the theoretical results for Au where for example,

( 100 ) ( 111 )

and

γ / γ ≈ 1 24

sv sv

( 110 ) ( 111 )

sv γγ when D = 3.8 nm [166], which correspond well to the corresponding bulk
ratios of 1.27 and 1.32 [116].
Noted that the structures of Be and Mg, Na, Al and Au belong to hcp, bcc and fcc
structures, respectively. Owing to the above agreements shown in figures 5 and 6, the model
should be applicable for all crystalline structures with different facets. Thus, Eq. (4.12) not
only supplies a simple way to determine γ sv ( D ) values of different facets without any free
parameter but also has an evident thermodynamic characteristic.

/ ≈ 1 28

sv

Liquid-Vapor Interface Energy or Surface Tension

The Bulk Surface Tension γ lv0 and its Temperature Coefficient γ′ lv0

The γ lv0 function and its temperature dependence are fundamental and important in the
theory and practice of materials processing (e.g. crystal growth, welding and sintering), and
its temperature coefficient γ′ lv0 ( T ) = dγ lv0 ( T )/d T governs the well-known Marangoni
convection on the surface of melt. There are several characteristics for the liquid surface. The
first, the configuration of a liquid surface produced either by cleavage or by stretching is the
same because the mobility of liquid molecules is high and the surface takes the equilibrium