Environmental Engineering Reference
In-Depth Information
terms of Eq. (2.22) with
E
(
D
)/
E
function in terms of Eq. (2.30) where
S
b
≈ 12
R
is assumed
except for Sb and Bi [42], a similar size dependence is found (see figure 4). This agreement
indicates that the size dependence of γ
sl
(
D
) is originated from the size dependence of bond
energy. In terms of Eq. (2.28) or Eq. (2.29), γ
sl
(
D
n
,
T
n
)/γ
sl0
(
T
m
) is within 10% of the value of
E
(
D
n
)/
E
b
, which also confirms the validity of Eq. (2.28) or Eq. (2.29).
Another linear relationship between γ
sl
(
D
n
,
T
n
) or γ′
sl0
and
T
m
with large scatter among
different groups of elements [11] has been rejected as the basis for an empirical rule in favor
of the correlation with
H
m
[98]. However, direct calculations for γ′
sl0
values of transition fcc
metals with hard-sphere systems have shown a strong correlation with
T
m
[88,98], which can
also be extended to γ
sl0
(
T
m
) for all fcc metals. Eq. (2.28-b) can confirm this correlation since
H
m
=
T
m
S
m
and
c
4
S
m
= 1.59 ± 7% nm J/mol K for all fcc metals listed in table 5. For other
elements,
c
4
S
m
values show large scatter in a range of 1.04 to 5.44 nm J/mol K due to the
scatter of
S
m
values. This indicates a disagreement for a linear relationship between
T
m
and
H
m
. Thus,
H
m
values, but not
T
m
values, in general characterize γ
sl0
(
T
m
) value better.
Recently, it is thought [99] that Skapsky′s assumption, e.g. γ
sl0
(
T
m
) = γ
sv0
(
T
m
)-γ
lv0
(
T
m
)
[100], has been used in the derivation of Eq. (2.13) where subscript v denotes vapor phase.
Because there is no nucleation barrier for surface melting, solids can be hardly superheated
above
T
m
, which leads to γ
sl0
(
T
m
) ≤ γ
sv0
(
T
m
)-γ
lv0
(
T
m
). Thus, γ
sl0
(
T
m
) in terms of Eq. (2.13) is
the upper limit of γ
sl
. However, it is not the case. As a monotone function of
T
and
D
in terms
of Eq. (2.26), γ
sl
(
D
,
T
) takes its minimum at
D
n
and
T
n
(
D
n
and
T
n
are the smallest values
possibly to be taken) and its maximum at
D
→∞ and
T
→
T
m
. Combining the values of
D
n
/
h
and θ listed in table 5, Eq. (2.26) gives the both limits of γ
sl
for elements as,
0.62 ≤ 3
V
m
R
γ
sl
(
D
,
T
)/(2
hS
vib
H
m
) ≤ 1.
(2.31)
Eq. (2.31) shows the reason why γ
sl0
(
T
m
) in terms of Eq. (2.13) is the upper limit of γ
sl
and overestimates γ
sl
(
D
n
,
T
n
) 50% to 100% [46].
γ
sl
(D
n
,T
n
) for Alkali Halides
In terms of Eqs. (2.26-b) and (2.27-b), the γ
sl
(
D
n
,
T
n
) values for alkali halides (ionic
crystals) are also predicted and shown in table 6. For comparison, the corresponding γ
CNT
values [101] are also listed.
As shown in table 6, the difference between γ
sl
(
D
n
,
T
n
) and γ
CNT
is smaller than 20%
except for CsF where the difference reaches to 33%. Considering that both γ
sl
(
D
n
,
T
n
) and γ
CNT
values are very smaller, tiny difference in absolute values will make the deviation larger.
Thus, it can also claim that an agreement between γ
sl
(
D
n
,
T
n
) and γ
CNT
is found.
Moreover, for alkali halides, most of the
D
n
/
h
values are smaller while the corresponding
θ
n
values are larger than those of metallic and semiconductors elements, which leads to that
γ
sl
(
D
n
,
T
n
)/γ
sl0
values of alkali halides will be smaller than those of elements in terms of Eq.
(2.26).
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