Environmental Engineering Reference
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2
⎛
⎞
2
hH
S
T
γ
c
sl
(
T
)
=
m
vib
⎜
⎝
⎟
⎠
.
(2.25-c)
0
3
RV
T
m
γ
sl0
(
T
) in terms of Eq. (2.25) decreases as
T
drops. As
T
→
T
m
, γ
a
sl0
≈ γ
b
sl0
≈ γ
c
sl0
due to the
decreased effect of Δ
C
p
on
g
m
(
T
). Although negative temperature dependence for γ
sl0
has been
considered [90], it differs from the usual understanding that differences of structure and
surface state between crystal and liquid decrease with
T
[72-74].
Substituting Eq. (2.25) into Eq. (2.21), the integrated size- and temperature-dependent
interface energy can be read as,
2
hH
S
3
h
7
T
a
sl
2
(2.26-a)
γ
(
D
,
T
)
=
m
vib
(
−
)(
)
3
RV
D
T
+
6
T
m
2
hH
S
3
h
2
T
b
sl
2
(2.26-b)
γ
(
D
,
T
)
=
m
vib
(
−
)(
)
3
RV
D
T
+
T
m
2
hH
S
3
h
T
γ
c
sl
(
D
,
T
)
=
m
vib
(
−
)(
)
2
.
(2.26-c)
3
RV
D
T
m
Substituting Eq. (2.23) into Eq. (2.1) and γ′
sl0
is replaced by γ
sl
(
D
,
T
) in terms of Eq.
(2.26), the critical size of nuclei
D
n
can be determined by letting ∂Δ
G
(
D
,
T
)/∂
r
= 0,
D
n
2
/
h
=
2
A
+
A
−
3
A
θ /
/
2
)
θ
(2.27-a)
D
n
2
/
h
=
2
4
B
+
16
B
−
18
B
)
/(
3
θ
)
,
(2.28-b)
D
n
2
/
h
=
2
C
+
C
−
3
C
/
2
)
(2.27-c)
T
−
T
14
S
1
−
θ
6
S
1
−
θ
θ
m
T
A
=
vib
B
=
vib
where
is the degree of undercooling.
,
and
3
R
7
−
θ
R
2
−
θ
m
2
S
1
−
θ
C
=
vib
.
3
R
θ
Substituting Eq. (2.27) into Eq. (2.26) with experimentally determined θ values, the
interface energy γ
sl
(
D
n
,
T
n
) of the nucleus-liquid can be determined.
γ
sl
(D
n
,T
n
) for Metallic and Semiconductors Elements
Since metallic and semiconductors elements were firstly dealt with by Turnbull [11], the
comparison between the predicted γ
sl
(
D
n
,
T
n
) values in terms of Eq. (2.26) and experimentally
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