Environmental Engineering Reference
In-Depth Information
dependence of Φ/
N
1
is in excellent agreement with that of Φ
K
/
N
1
. As one can see from Figure
19.1 the dominant contribution to the temperature dependence comes from the Reiss factor Φ
R
/
N
1
.
However, the last factor (
Q
v
)
−3
in Equation 19.59 also depends signiicantly on temperature. The
temperature dependence for (
Q
v
)
−3
is mainly governed by the irst factor in Equation 19.53 which
takes into account the zero-point vibrations. Without considering this factor one will lose the agree-
ment between the temperature dependencies of Φ/
N
1
and Φ
K
/
N
1
.
The excellent agreement between the temperature dependencies for Φ/
N
1
and Φ
K
/
N
1
supports our
assumption that the rotational degrees of freedom of embedded cluster which must to be deactivated
are, actually, the torsion vibrations.
19.8 APPROXIMATE ANALYTICAL FORMULA FOR THE NUCLEATION RATE:
ITS APPLICATION TO THE ESTIMATION OF SURFACE TENSION OF
CRITICAL NUCLEUS FROM THE EXPERIMENTAL SUPERSATURATION
RATIO AND NUCLEATION RATE
The thermodynamic properties of the interphase surface are fully characterized by a state function
which is called surface tension σ
S
. Therefore, the knowledge of the surface tension is necessary to
understand the mechanism of nanoparticle nucleation, and stability and evolution of nanosystems.
In this section we will outline an application of the analytical formula for the nucleation rate to
determine the surface tension of the critical drop. The metal vapor nucleation will be considered as
an example.
There are experimental contributions studying the nucleation of Li, Na, Ag, Hg, Mg, Zn where
the supersaturation versus temperature was measured for the nucleation rate considered as constant
and being evaluated rather approximately [5-7,43-45]. However, the surface tension is a func-
tion of both radius and temperature, and to understand the interfacial thermodynamics one must
determine irst the dependence of surface tension on the drop radius (at constant temperature) and
then the temperature dependence of surface tension. Therefore, the experimental measurements of
the nucleation rate at different supersaturation ratios and ixed temperature are important. To our
knowledge there is only one such investigation in the ield of nucleation of metals [46]. In Ref. [46]
the rate of nucleation from the vapor of Cs was measured as a function of supersaturation for three
temperatures 508, 530, and 554 K. Therefore, we will start from the consideration of the experi-
mental data on Cs vapor nucleation which are plotted in Figure 19.3. The original experimental data
are shown as circles. For the convenience of presentation we have averaged the neighboring points
(diamonds).
Substituting the factor
K
in Equation 19.24 by Φ/
N
1
(Equation 19.59) and
W
crit
by Equation 19.8
and assuming χ ≈ 1 we get the inal formula for the nucleation rate:
3 2
/
3
⎛
⎜
5
β
⎞
⎟
⎧
⎨
⎩
⎫
⎬
⎭
1
2
64
R
ρ
k T
⎛
3
2
h
k T
ν
⎞
⎟
⎛
⎜
h
k T
ν
⎞
⎟
5
S
B
max
max
I
≈
×
π
exp
1
−
exp
−
⎜
sat
(
3 2
/
3
2
Sn
π
)
σ
15
h
1
B
B
−
1 2
/
⎡
2
⎤
−
1 2
/
⎛
⎜
2
⎞
2
m
σ
π
(
R
)
1
⎛
⎜
δ
⎞
⎟
⎛
⎜
δ
⎞
⎟
⎛
⎜
d
dR
δ
⎞
⎟
4
π
R
σ
S
(
R
)
×
(
)
2
⎟
(19.60)
sat
S
S
S
S
Sn
⎢
⎢
1
+
f
⎥
⎥
1
+
exp
−
1
β
ρ
R
R
3
k T
⎣
S
S
⎦
S
B
It will be shown in the following that the factor [1 + (δ/
R
S
)
2
f
(δ/
R
S
)]
−1/2
(1 + (
d
δ/
dR
S
) )
−1/2
can be
considered as equal to unity with adequate accuracy, and the quantum form of the vibrational
partition function
Q
rot l
K
,
(see Equations 19.53 and 19.59) can be substituted by the classical form:
K
3
at high temperatures. Then, accounting Equation 19.48 one can rewrite
Q
≈
(
k
T
/
h
ν
)
rot
,
l
B
max
Equation 19.60 as
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