Environmental Engineering Reference
In-Depth Information
−0.34, and −0.43 for
T
= 508, 530, and 554 K, respectively. In other words, the RHS of Equation
19.5 is independent of
R
S
, that is, δ/
R
S
=
const
, or δ =
const · R
S
. Then, the numerical solution of the
GTKB differential equation (19.5) gives δ/
R
S
=
d
δ/
dR
S
= −0.11, −0.15, −0.18 for
T
= 508, 530, and
554 K, respectively. Substituting the values of σ
S
(
R
S
) and
R
S
from Figure 19.4 to Equation 19.65
we got the pressure Δ
P
≅
P
β
in the range (1.16-1.27) × 10
8
Pa for 508 <
T
< 554 K. This pressure
gives the sound velocity
u
= 999 − 986 m/s and the Debye frequency ν
max
= (1.99 − 1.97) × 10
12
s
−1
,
respectively.
It is possible now to calculate the factor [1 + (δ/
R
S
)
2
f
(δ/
R
S
)]
−1/2
(1 + (
d
δ/
dR
S
))
−1/2
, and determine
σ
S
(
R
S
) using the exact Equation 19.60 and the newly determined values of ν
max
(second itera-
tion). As shown in Figure 19.4 the magnitudes of σ
S
(
R
S
) as determined in the irst and second
iterations differ by 0.2% but the derivatives
d
ln(σ
S
(
R
S
)/σ
∞
)/
d
ln(
R
S
) coincide absolutely which
means that both the iterations give the same quantity δ/
R
S
. Thus, the approximate Equation
19.61 allows determining the surface tension of the critical nucleus with high accuracy. One
should note that the substitution of ν
max
for the pressure of 1 atm instead of that for
P
β
gives neg-
ligible error for the drop surface tension but will result in the error of about 30% for the nucle-
ation rate. The accuracy of the Equation 19.61 for the nucleation rate is given by the factor [1 +
(δ/
R
S
)
2
f
(δ/
R
S
)]
−1/2
(1 + (
d
δ/
dR
S
) )
−1/2
and the ratio between the classical and quantum forms of the
partition function
Q
rot l
K
,
which are in the ranges 1.05-1.08 and 1.0012-1.0015, respectively, for
the experimental range of Ref. [46]. Thus, the total cost of simpliications for the Equation 19.61
for the rate of nucleation does not exceed 8%. Such a good accuracy in Equation 19.61 is due to
the small value of δ/
R
S
. It must be noted that both the surface of tension and the equimolar divid-
ing surface lie somewhere in the interfacial region [12] and, hence, the value |δ/
R
S
| = |(
R
e
/
R
S
) − 1|
is expected to be less than unity in most cases. Therefore, one can hope that the approximate
Equation 19.61 is suitable to evaluate the drop surface tension for a wide range of systems and
nucleation conditions.
In the following section of the article, we will evaluate the surface tension of critical nucleus for
different monovalent and bivalent metals using the Equations 19.61 through 19.64 and the experi-
mentally measured nucleation rate and supersaturation for different temperatures (see Figure 19.5).
The important parameters used for evaluation of σ
S
/σ
∞
are summarized in Table 19.1. The tempera-
ture dependence of the planar interface surface tension is presented by
σ
∞
=
ψ
(
T
−
T
)
(19.67)
0
6
Ag
15
I
= 10
9
cm
-3
s
-1
5
10
5
4
0
500
600
700
800
T
( K)
3
Li
I
= 10
9
cm
-3
s
-1
Na
Mg
2
I
= 3 × 10
9
cm
-3
s
-1
I
= 10
10
cm
-3
s
-1
Hg
1
I
= 1 cm
-3
s
-1
Zn
I
= 10
10
cm
-3
s
-1
0
200
300
400
500
600
700
800
900 1000 1100
T
(K)
FIGURE 19.5
log
10
S
versus nucleation temperature for Li
43
, Na
44
, Ag
45
, Mg
6
, Zn
7
, Hg
5
for invariant nucle-
ation rates (shown in the plot for each metal).
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