Environmental Engineering Reference
In-Depth Information
Substituting Eq. (2.27) into Eq. (2.26) and plotting Eq. (2.26), the curve is linearly
regressed as a function of θ, which leads to [95],
γ
sl
(
D
n
′,
T
n
′) ≈ (1.78-3.83θ)
H
m
/
V
.
(2.29)
where
T
n
′ is any nucleation temperature and
D
n
′ is the corresponding radius of a nucleus. The
standard deviations for 1.78 and 3.83 are 0.01 and 0.16, respectively. Eq. (2.29) indicates that
Eq. (2.28) denotes two extreme cases where θ = θ
n
(the maximum undercooling, which is
nearly a constant of 0.18±0.02 for the most elements [43,96]) and θ = 0, respectively. Thus, at
any θ value, the relationship of γ
sl
(
D
n
′,
T
n
′) ∝
H
m
/
V
exists always. As an example, figure 3
shows such a relationship at θ = 0.1 where the slope is the linear function, which equals to
1.40 as indicated in Eq. (2.29).
1.2
1.2
0.9
0.9
0.6
0.6
0.3
0.3
0.0
0.0
0
20
40
60
80
100
D
/
D
′
0
Figure 4. Comparison between γ
sl
(
D
,
T
)/γ
sl0
(
T
) shown as solid line in terms of Eq. (2.22) and
E
(
D
)/
E
shown as dot in terms of Eq. (2.30) where
D
′
0
= 2
h
.
This linear relationship of Eq. (2.29) between γ
sl
and
H
m
could be considered as that
H
m
is
related to bond energy of crystalline atoms while γ
sl
denotes the bond energy difference
between surface atoms and interior atoms of a crystal. The behavior displayed by γ
sl
has been
found to be fruitfully compared to that of cohesive energy
E
b
[96]. It is known that cohesive
energy determines the size of bond energy while
E
is also size dependent, which has been
determined by [97],
1
⎛
2
S
1
⎞
⎡
⎤
(2.30)
E
(
D
)
/
E
=
1
−
exp
⎜
⎝
−
b
⎟
⎠
⎣
⎦
b
2
D
/
h
−
1
3
R
2
D
h
−
1
where
S
b
=
E
b
/
T
b
is the solid-vapor transition entropy of crystals with
E
b
being the bulk
cohesive energy and
T
b
being the bulk boiling temperature. Comparing γ
sl
(
D
)/γ
sl0
function in
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