Environmental Engineering Reference
In-Depth Information
where γ
lv0
is the corresponding bulk value of γ
lv
(
D
), δ
denotes a vertical distance from the
surface of tension to the dividing surface where the superficial density of fluid vanishes [113].
As a first order approximation, although there is no direct experimental evidence to support
Eq. (4.9), Eq. (4.9) should also be applicable to predicting γ
sv
(
D
) since the structural
difference between solid and liquid is very small in comparison with that between solid and
gas or between liquid and gas. In addition, it is unknown whether
D
in Eq. (4.9) can be
extended from micron size to nanometer size. Hence, a theoretical determination of γ
sv
(
D
) is
meaningful.
Although both the expressions of Eqs. (4.1) to (4.5) and the corresponding results are
different, all of them indicate that,
γ
sv
=
kE
b
/(
N
a
A
S
)
(4.10)
where
k
< 1 is a function of
CN
.
If the nanocrystals have the same structure of the corresponding bulk,
k
is size-
independent. Thus, Eq. (4.10) may be extended to nanometer size as [162],
γ
sv
(
D
) =
kE
(
D
)/(
N
a
A
S
).
(4.11)
Combining Eq. (4.11) with Eq. (2.30), there is [162],
1
2
S
1
⎡
⎤
⎛
⎞
.
(4.12)
γ
(
D
)
/
γ
=
1
−
exp
−
b
⎝
⎠
⎣
⎦
sv
sv
2
D
/
h
−
1
3
R
2
D
/
h
−
1
In terms of Eq. (4.12), comparisons of γ
sv
(
D
) of Be, Mg, Na, Al thin films and Au
particles with different facets between model predictions and experimental and other
theoretical results [163-166] are shown in figures 5 and 6 where the related parameters in Eq.
(4.12) are listed in table 12. It is evident that our predictions are in agreement with the
experimental values of Be and Mg (0001), and with other theoretical results for Na (110) and
for three low-index surfaces of Au. The deviations in all comparisons are smaller than 5%
except that for Al (110) with a deviation of about 10%.
As shown in the figures 5 and 6, γ
sv
(
D
) decreases with a decrease in size. This trend is
expected since
E
(
D
) of the nanocrystals increases as the size decreases [162]. In other words,
γ
sv
(
D
) as an energetic difference between surface atoms and interior atoms decreases as
energetic state of interior atoms increases.
Considering the mathematical relation of exp (-
x
) ≈ 1-
x
when
x
is small enough, Eq.
(4.12) can be rewritten as,
γ
sv
(
D
)/γ
sv
≈ 1-
S
b
h
/(3
RD
)
(4.13)
Eq. (4.13) is in agreement with the general consideration that the decrease of the any
size-dependent thermodynamic quantity is proportional to 1/
D
[27]. If γ
sv
(
D
) function of Eq.
(4.13) and γ
lv
(
D
) function of Eq. (4.9) have the same size dependence, δ =
S
b
h
/(12
R
) ≈
h
when
S
b
≈ 12
R
as seen in table 12. Namely, the transition zone separating a solid phase and a
vapor phase is only one atomic layer, which is an understandable result. This determined δ
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