Chapter 4
Molecular Crystals
4.1 Crystallization
4.1.1 Alder Transition
As seen in Sect. 1.3.2, attractive interaction causes the condensation of gas to fluid,
the volume of which is mainly kept by the volume of molecules. In the van der
Waals treatment, no mechanism is embedded for the crystallization. Here, we naïvely
assume that crystals are condensed phase having a three-dimensional spatial periodicity, though the International Union of Crystallography (IUCr) has used a more
general definition [1] paying attention to the emergence of a sharp (the so-called
δ-function type) Bragg reflection if irradiated by a suitable ray (see Sect. 4.2). Since
the volume of the condensate vanishes unless the molecular volume is taken into
account, it deserves consideration whether the repulsive interaction can lead the
crystallization.
In this context, it is worse to consider an ensemble of N identical rigid spheres
without any attractive interaction. The container should maintain the volume of the
system. Assume the system is classical in nature. The average energy of the system is
always
3
2
N k B T because the infinitely large energy completely prohibits the overlap
of two spheres. This situation means that energy is not a factor governing the state
of the system. The key quantity should be the entropy, accordingly. That is, larger
the number of microscopic states, more stable the macroscopic state.
The packing density of close-packed crystals of hard spheres is ρ cp = π/3
√
2
(≈ 0.74), while the maximum density for their random packing has been believed
to be around ρ jam = 0.64 [2], though the closest random packing was claimed to
be ill-defined [3]. It is essential to recognize what the former density is larger than
the latter. Ideally, each sphere can move in neither the close-packed crystals nor the
jammed state. However, if the crystal inflates isotropically but slightly while keeping
the spheres unchanged, each sphere gets space around it, leading to the mobility.
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2020
K. Saito, Chemical Physics of Molecular Condensed Matter,
Lecture Notes in Chemistry 104,
https://doi.org/10.1007/978-981-15-9023-8_4
65
Molecular Crystals
4.1 Crystallization
4.1.1 Alder Transition
As seen in Sect. 1.3.2, attractive interaction causes the condensation of gas to fluid,
the volume of which is mainly kept by the volume of molecules. In the van der
Waals treatment, no mechanism is embedded for the crystallization. Here, we naïvely
assume that crystals are condensed phase having a three-dimensional spatial periodicity, though the International Union of Crystallography (IUCr) has used a more
general definition [1] paying attention to the emergence of a sharp (the so-called
δ-function type) Bragg reflection if irradiated by a suitable ray (see Sect. 4.2). Since
the volume of the condensate vanishes unless the molecular volume is taken into
account, it deserves consideration whether the repulsive interaction can lead the
crystallization.
In this context, it is worse to consider an ensemble of N identical rigid spheres
without any attractive interaction. The container should maintain the volume of the
system. Assume the system is classical in nature. The average energy of the system is
always
3
2
N k B T because the infinitely large energy completely prohibits the overlap
of two spheres. This situation means that energy is not a factor governing the state
of the system. The key quantity should be the entropy, accordingly. That is, larger
the number of microscopic states, more stable the macroscopic state.
The packing density of close-packed crystals of hard spheres is ρ cp = π/3
√
2
(≈ 0.74), while the maximum density for their random packing has been believed
to be around ρ jam = 0.64 [2], though the closest random packing was claimed to
be ill-defined [3]. It is essential to recognize what the former density is larger than
the latter. Ideally, each sphere can move in neither the close-packed crystals nor the
jammed state. However, if the crystal inflates isotropically but slightly while keeping
the spheres unchanged, each sphere gets space around it, leading to the mobility.
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2020
K. Saito, Chemical Physics of Molecular Condensed Matter,
Lecture Notes in Chemistry 104,
https://doi.org/10.1007/978-981-15-9023-8_4
65