Environmental Engineering Reference
In-Depth Information
γ
sl0
(T
m
) in Metals: fcc Versus bcc
The dependence of γ
sl0
on crystal structure is crucial to understanding the role of
metastable structures in nucleation pathways. In 1897, Ostwald [63] formulated his ″step
rule″, which states that nucleation from the melt occurs to the phase with the lowest activation
barrier, which is not necessarily the thermodynamically most stable bulk phase. In the case of
the nucleation of fcc crystals, there is evidence that crystallization often proceeds first through
the formation of bcc nuclei, which transform to fcc crystals later in the growth process. This
phenomenon has been observed in experiments on metal alloys [64], in computer simulations
of Lennard-Jones particles [65] and weakly charged collides [66], and in classical density-
functional theory studies of nucleation in Lennard-Jones [67]. These results could be
explained if γ
sl0
for bcc crystals were significantly lower than those for fcc crystals in these
systems as that would lead to substantially lower activation barriers [68]. Using a simple
model of interfacial structure, Spaepen and Meyer [69] predicted that γ
sl0
for bcc-melt
interfaces should be about 20% lower than that for fcc-melt interfaces, based on packing
considerations. In a recent paper, Sun
et al
. [34,70] determined γ
sl0
for bcc- and fcc-melt
interfaces for several models for iron by molecular-dynamics simulations, obtaining values of
γ
sl0
that were about 30~35% smaller for bcc-melt interfaces than those for fcc-melt interfaces.
However, it is also easy to understand the difference of γ
sl0
between bcc- and fcc-
interfaces in terms of Eq. (2.13). According to Goldschmidt premise for lattice contraction
[71],
h
contracts 3% if the coordination number (
CN
) of the atom reduces from 12 to 8. Thus,
the effect of the change of
h
on γ
sl0
can be neglected as a first order approximation. Similarly,
the effect of the change of
V
can also be neglected since the difference is only 1.3% [34].
Since Δ
V
values for bcc and fcc crystals are also very close, γ
bcc
/γ
fcc
≈
(
H
m
bcc
/
H
m
fcc
)
2
/(
T
m
bcc
/
T
m
fcc
) in terms of Eq. (2.13) with the simplification of
S
vib
≈
H
m
/
T
m
. With
the known values of
H
m
bcc
/
H
m
fcc
= 0.715 and
T
m
bcc
/
T
m
fcc
= 0.805 [34], γ
bcc
/γ
fcc
≈ 0.64, which is
consistent with the reported values 30~35% [34,70].
The Size Dependence of Solid-liquid Interface Energy
γ
sl
(
D
)
For comparison, γ′
sl0
values for elemental crystals Au, Al, Sn, Pb and Bi in terms of Eq.
(2.3) are 132, 93, 33.6, 54.5 and 54.4 mJ/m
2
[11], which correspond to the lower limits of the
corresponding experimental data for Sn, Pb and Bi [12] while are by far lower than the
corresponding experimental data for Au and Al [12]. This disagreement results from the two
approximations in the CNT: 1. The specific heat difference between solid and liquid Δ
C
p
is
assumed to be zero. Namely, the influence of Δ
C
p
is neglected, which makes the neglect of
temperature dependence of
H
m
(
T
) in terms of
H
m
(
T
) =
g
m
(
T
)-
T
d
g
m
(
T
)/d
T
(Helmholtz
function); 2. The nucleus-liquid interface energy γ
sl
(
D
n
,
T
n
) at
T
n
is treated as the value for a
planar interface γ′
sl0
and γ′
sl0
is assumed to be temperature independent, i.e. the capillarity
approximation [11]. The
g
m
(
T
) functions have been improved theoretically and confirmed
experimentally by Hoffman [72], Spaepen [73] and Singh [74]. Since the values of γ′
sl0
in Eq.
(2.3) are initially obtained for nuclei-liquid interface [11,18-19] while any nucleus during
solidification is in nanometer size range, γ
sl
(
D
), not γ
sl0
, has to be considered for theoretical
prediction of actual nucleation and growth processes.
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