132
6 Melting of Molecular Crystals
Since this is the self-consistency equation having virtually the same form as
Eq. 6.15, this theory predicts a continuous transition between the crystal and liquid
at T LJD = zw/4k B under the constant volume. The entropy involved in this transition
is 2N k B ln 2 ≈ 1.4N k B .
Melting is usually observed under not the constant volume but the constant pressure. The discussion about the melting under the latter condition requires some
consideration of the volume dependence. There exist two terms that depend on the
volume. One is the contribution of a single atom involved in q(V ) whereas the other
is in w. For the latter, Lennard-Jones and Devonshire [29] assumed that the repulsive
part of the interatomic potential is dominant. Considering the repulsive part of the
Lennard-Jones potential as a function of interatomic distance r given by
φ(r ) = 4ε
r 0
r
12 −
r 0
r
6
,
(6.31)
they assumed
w = w 0
v 0
v
4 .
(6.32)
On the other hand, they assumed the free volume model (Eq. 6.21) with v = V /N
for q(V ) for the former. Here, it is essential to be aware that the potential energy u
sensed by each atom (considered in Sect. 6.2.2.1) does not affect any for the melting
under constant volume but becomes effective for the melting under the constant pressure. However, its introduction requires another characteristic energy in the model,
leading to more complexity. Thus, a possible contribution from its volume dependence is neglected in the following, though this contribution may affect the location
of equilibrium pressure between two phases. Under this condition, the pressure of
the system is calculated as
p =
N
V
k B T +
zwξ(1 − ξ)
4
(6.33)
with ξ that minimizes the Helmholtz energy. The last term is the pressure arising from
the “disorder” and does not vanish even in the complete disorder (ξ =
1
2
). Figure 6.4
shows isotherms on the p − V plane. The isotherms at high temperatures are not
monotonic. This behavior indicates that, under such pressures, two phases having
different densities, i.e., a crystal and a liquid, coexist, as seen in the melting in daily
life. Indeed, they were successful in explaining the melting properties of argon by
suitably choosing some parameters to reproduce the melting temperature [29].
6 Melting of Molecular Crystals
Since this is the self-consistency equation having virtually the same form as
Eq. 6.15, this theory predicts a continuous transition between the crystal and liquid
at T LJD = zw/4k B under the constant volume. The entropy involved in this transition
is 2N k B ln 2 ≈ 1.4N k B .
Melting is usually observed under not the constant volume but the constant pressure. The discussion about the melting under the latter condition requires some
consideration of the volume dependence. There exist two terms that depend on the
volume. One is the contribution of a single atom involved in q(V ) whereas the other
is in w. For the latter, Lennard-Jones and Devonshire [29] assumed that the repulsive
part of the interatomic potential is dominant. Considering the repulsive part of the
Lennard-Jones potential as a function of interatomic distance r given by
φ(r ) = 4ε
r 0
r
12 −
r 0
r
6
,
(6.31)
they assumed
w = w 0
v 0
v
4 .
(6.32)
On the other hand, they assumed the free volume model (Eq. 6.21) with v = V /N
for q(V ) for the former. Here, it is essential to be aware that the potential energy u
sensed by each atom (considered in Sect. 6.2.2.1) does not affect any for the melting
under constant volume but becomes effective for the melting under the constant pressure. However, its introduction requires another characteristic energy in the model,
leading to more complexity. Thus, a possible contribution from its volume dependence is neglected in the following, though this contribution may affect the location
of equilibrium pressure between two phases. Under this condition, the pressure of
the system is calculated as
p =
N
V
k B T +
zwξ(1 − ξ)
4
(6.33)
with ξ that minimizes the Helmholtz energy. The last term is the pressure arising from
the “disorder” and does not vanish even in the complete disorder (ξ =
1
2
). Figure 6.4
shows isotherms on the p − V plane. The isotherms at high temperatures are not
monotonic. This behavior indicates that, under such pressures, two phases having
different densities, i.e., a crystal and a liquid, coexist, as seen in the melting in daily
life. Indeed, they were successful in explaining the melting properties of argon by
suitably choosing some parameters to reproduce the melting temperature [29].