Environmental Engineering Reference
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melting and roughening transition are directly related to its value. Thus, a quantitative
knowledge of γ
sl0
values is necessary. However, direct measurements for γ
sl0
are not at all
easy even for elements in contrast to the case of interface energy of liquid-vapor γ
lv0
[11-
13]. Some attempts are made to obtain γ
sl0
by theoretical approaches or computer
simulations [8-17]. A widely used technique in an indirect way to determine γ
sl0
value is
nucleation experiments of undercooled liquid based on the classical nucleation theory
(CNT), which was made firstly by Turnbull [9]. According to the CNT, the undercooled
liquid crystallizes at nucleation temperature
T
n
with a critical nucleation size
D
n
induced by
a localized structural and energetic fluctuation where the thermodynamic properties of
nanometric aggregates of the newly nucleated phase are to be the same as those of the
corresponding bulk one. Thus, the nucleus-liquid interface energy γ
sl
(
D
n
,
T
n
) at
T
n
is treated
as the respective value for a planar interface γ′
sl0
being temperature independent (this
assumption is known as the capillarity approximation), which makes it possible to consider
the Gibbs free-energy difference of a spherical nucleus in the liquid Δ
G
(
D
,
T
) as a sum of a
volume term and an interface term [9],
Δ
G
(
D
,
T
)
= -(1/6)π
D
3
g
m
(
T
)/
V
+π
D
2
γ′
sl0
(2.1)
where
V
denotes the gram-atom volume and
g
m
(
T
) is temperature-dependent Gibbs free
energy difference between crystal and liquid.
g
m
(
T
) for elements is assumed to be a linear
function of
T
, namely contribution of specific heat difference Δ
C
p
between solid and liquid on
g
m
(
T
) is negligible under small undercooling [9,11]. Thus, the corresponding
g
m
(
T
) function
is shown as [11],
g
m
(
T
) = (1-
T
/
T
m
)
H
m
(2.2)
where
H
m
is the bulk melting enthalpy at
bulk melting temperature
T
m
. With a consideration
on homogeneous nucleation rate
I
v
, γ′
sl0
in Eq. (2.1) can be determined, which as an empirical
relationship is proportional to
H
m
[11],
γ′
sl0
=
c
1
′
hH
m
/
V
(2.3)
where
c
1
′ is a constant to be 0.45 for metals (especially close-packed metals) and 0.32 for
nonmetallic elements,
h
is the atomic diameter. Note that γ
sl
(
D
n
,
T
n
) ≈ γ′
sl0
has been implied in
Eq. (2.3) [11,18-19]. Moreover, the appropriate value of
k
increases noticeably for molecules
having more asymmetry. Undoubtedly, Eq. (2.3) overlooks some important pieces of physics.
In addition, the existence of
c
1
′ to be determined also weakens the theoretical meaning of this
equation, and makes it only be an empirical rule.
The most powerful methods at present available for experimentally measuring γ
sl0
are to
make direct use of the so-called Gibbs-Thomson expression [12-13,17]. This thermodynamic
result shows that, if all other intensive variables (such as composition, pressure and strain
energy) remain constant, a solid bounded by an element of interface having principle
diameters of curvature
D
1
and
D
2
measured in the solid will be in equilibrium with its melt at
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