Environmental Engineering Reference
In-Depth Information
δ(
D
)/
h
determined by Eq. (5.16) for Na, Al and water droplets are shown as a function of
D
/
h
in figure 14. It is observed that the Tolman′s length is positive for these droplets and
decreases when the size is increased, being consistent with statistical thermodynamics
[209,215-216], computer simulations [217-218] and other approaches [219] for Lennard-
Jones fluids. However, there is an obvious difference between our model predictions and
others, namely, δ(
D
) remains positive among the whole size range in this model while it will
decrease to a negative limiting value for the planar interface in the others.
4
3
2
Al
1
Na
H
2
O
0
0
1 0
2 0
3 0
4 0
5 0
D
/
h
Figure 14.
δ
(
D
)/
h
as a function of
D
/
h
in terms of Eq. (5.16) for Na, Al and water droplets.
The value of δ in Eq. (5.16) is on the verge of infinitude when
D
reaches its lower limit
h
[162]. When
D
is sufficiently large, considering the mathematical relation of exp (-
x
) ≈ 1-
x
when
x
is small enough (e.g.
x
< 0.1), the minimal value δ
min
in terms of Eq. (5.16) can be
written as δ
min
= δ
∞
=
hS
b
/(12
R
), or
hS
b
/(12
R
) < δ
(5.17)
S
b
≈ 12
R
for metallic elements as shown in table 14 leads to δ
∞
≈
h
for Na and Al as
indicated by Tolman [161] while δ′
∞
≈ 3
h
/8 for water due to
S
b
≈ 9
R
/2. This is the reason that
the differences between the model predictions in terms of Eqs. (5.13) and (5.15) appear at
D
/
h
≤ 10 for Na and Al while at
D
/
h
≥ 20 for water. Thus, the size dependence of δ(
D
) strongly
depends on the value of
S
b
. Eq. (5.17) also implies that the decrease of the bond strength leads
to the diffusion of the liquid-vapor interface. The corresponding physical picture is that the
energetic difference of the molecule on the liquid surface and that in the vapor decreases as
the bond strength weakens. Thus, the liquid-vapor interface transition zone becomes narrow.
Search WWH ::
Custom Search