Environmental Engineering Reference
In-Depth Information
Ni and V. Even
T
-
T
m
= 1000 K, γ′
lv0
(
T
)/γ′
lv0
(
T
m
) values of Ni and V are only 1.09 and 1.08, or
the error range is smaller than 10%.
Note that the simulated results based on the Monte Carlo method in conjunction with the
embedded-atom method respectively show 20~60% underestimations for Al, Ni, Cu, Ag and
Au, and 20% overestimation for Co when they are compared with the experimental data [198-
200]. Thus, computer simulation methods for estimation of γ
lv0
(
T
) values of metals need to be
further improved, which indicates that the above theoretical method is a powerful and even
unique tool at present to determine γ
lv0
(
T
) function with good accuracy.
The Size Dependence of Surface Tension
γ
lv
(
D
)
A method of dividing surface pioneered by Gibbs defines that γ
lv
with a given bulk value
γ
lv0
depends on pressure
P
,
T
and the composition of the two coexisting bulk phases [201].
However, when the liquid-vapor interface is curved, γ
lv
is a function of
D
of the droplet,
γ
lv
(
D
). Guggenheim suggested that the γ
lv
(
D
) would change when
D
falls below 100 nm based
on statistical mechanical considerations [202].
A half-century ago, Tolman extended the idea of Gibbs and showed that if the radius
R
s
of the surface of tension of the droplet did not coincide with the equimolar radius
R
e
, the
surface tension must vary with droplet size [161]. Moreover, Tolman proposed that the two
surfaces must, in general, be distinct from each other. Tolman estimated the Tolman′s length
δ, or the separation between the equimolar surface and the surface of tension, δ =
R
e
-
R
s
[203].
He assumed that δ could be taken as a constant in the nanometer region, and derived the
equation [161],
γ
lv
(
D
)/γ
lv0
= 1/(1+4δ/
D
).
(5.13)
Kirkwood and Buff developed a general theory based on statistical mechanics for the
interfacial phenomena and confirmed the validity of Tolman′s approach [204]. For a
sufficiently large droplet, Eq. (5.13) may be expanded into power series. Neglecting all the
terms above the first order, the asymptotic form was obtained, which has been illustrated as
Eq. (4.9) [113]. Values for γ
lv
(
D
)/γ
lv0
determined by Eqs. (4.9) and (5.13) are close to each
other at
D
/δ ≥ 20.
Tolman predicted that γ
lv
(
D
) should decrease with decreasing particle size [161],
indicating the positive δ. The asymptotic Tolman′s length in the limit of
D
→∞, δ
∞
, is
independent of the choice of the dividing surface [205], and δ
∞
=
h
[161]. However, δ was
also predicted to be negative by a rigorous thermodynamic derivation [206], which would
lead to an increase of γ
lv
(
D
) when the size is decreased. It is generally assumed that δ > 0 for
spherical droplets and δ < 0 for bubbles in a liquid [207-209]. This consideration can also be
simply translated as that the δ value is always positive while
D
may be positive for the
droplets but negative for the bubbles. In addition to the uncertainty in the sign of δ, the
validity of Eq. (5.13) is considered to be questionable for very small particles [210].
It is known that for a planar interface at the melting point, γ
sv0
/γ
lv0
=
w
for metallic
elements with
w
= 1.18±0.03 [114]. Note that in the derivation of γ
sv
(
D
), it is assumed that the
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