Environmental Engineering Reference
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semiconductors are directly applied to the eutectic system, it may be reflect some evident
differences in γ
sl0
(
T
) between solid metal-liquid and solid semiconductor-liquid systems.
Combining our previous work [92], the model predictions for both
T
n
and γ
sl
(
D
n
,
T
n
)
correspond well to Turnbull′s experimental results although both γ
sl
(
D
,
T
) and
g
m
(
T
) functions
differ from those in the CNT. The possible reason may be the mutual compensation between
γ
sl
(
D
,
T
) and
g
m
(
T
) functions. However, there is about 30% difference in the value of
D
n
between CNT (
D
n
≈ 8
h
) and this model (
D
n
≈ 11
h
), which may result from neglecting of
derivative of γ
sl
(
D
,
T
)
with respect to
D
in the CNT. Although we cannot confirm the above
difference from experiments due to the experimental difficulties, we believe that a little larger
D
n
is more reasonable.
According to Eq. (2.27), the size of
D
n
is decided by
S
vib
and θ due to the introduction of
γ
sl
(
D
,
T
) function, and it increases with an increase in
S
vib
or a decrease in θ. Since
S
vib
values,
which differ from the cases of
S
m
, are similar for different bond natures of elements while θ
determined from experiments for distinct elements are also similar,
D
c
is in fact independent
on the elemental types.
Let γ
sl
(
D
,
T
) of Eq. (2.26) is also expressed as γ
sl
(
D
,
T
) =
c
2
hH
m
/
V
m
, there are
2
S
3
h
7
T
2
S
3
h
T
a
2
and
c
2
. The mean values of
S
vib
/
R
,
D
n
/
h
c
=
vib
(
−
)(
)
c
=
vib
(
−
)(
)
2
2
3
R
D
T
+
6
T
3
R
D
T
m
m
and θ for concerned metals and semiconductors listed in table 5 are 0.94, 11.2 and 0.187, and
0.98, 11.0 and 0.167, respectively, which leads to
c
2
a
= 0.43 and
c
2
c
= 0.33 at
T
n
and
D
n
.
These values correspond well to the initial findings of
c
1
′
a
= 0.45 and
c
1
′
b
= 0.32 in Eq. (2.3)
[11]. This correspondence confirms again that γ
CNT
≈ γ
sl
(
D
n
,
T
n
).
If the temperature dependence of
H
m
(
T
) is neglected, both γ
sl0
and γ
sl
(
D
,
T
) increase in
terms of Eqs. (2.25) and (2.26), which leads to
c
2
a
= 0.46 and
c
2
c
= 0.48. These are similar to
c
2
= 0.49±0.08 for 22 metals and 4 semiconductors where
H
m
(
T
) ≈
H
m
(
T
m
) is used [93].
However, this treatment will result in errors in analyzing the nucleation process due to the
unreasonable assumption of
H
m
(
T
) ≈
H
m
(
T
m
).
It seems from Eq. (2.3) that the size of
h
also affects γ′
sl0
value. To accurately estimate
the influence of
h
, all
h
values of elements should be unified to
h
′ values where the elements
with different structures have the same
CN
of 12. According to the Goldschmidt premise for
lattice contraction [71],
h
contracts 3%, 4%, and 12% if
CN
of the atom reduces from 12 to 8,
6, and 4, respectively. The additional correlation between
h
and
h
′ for the elements, which
CN
is not 4, 6 or 8, can be found in Ref. [94]. Replacing
h
in Eq. (2.3) with
h
′, Eq. (2.3) is
simplified as the following form [95],
γ′
sl0
=
c
3
H
m
/
V
(2.28-a)
where
c
3
≈ 0.11 ± 10% nm is the slope except semi-metals Pb, Sn and Ga shown in figure 2
while Eq. (2.13) can also be simplified as the similar form [95],
γ
sl0
(
T
m
) =
c
4
H
m
/
V
(2.28-b)
where
c
4
= 2
S
vib
h
′/(3
R
) ≈ 0.18 ± 15% nm is the slope except Fe, Al and Ga. Eq. (2.28) does
not need to distinguish the bond natures of the elements with a unique
c
3
or
c
4
value rather
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