1.3 What the Interaction Brings
27
1.3.3 Crystallization
In contrast to the liquefaction by the attractive part of the intermolecular interaction, crystallization is not easy to understand. Suppose spherical molecules, between
which the interaction works only for neighbors. Since the intermolecular interaction
energy generally has a minimum, as shown in Fig. 1.5, molecular arrangements that
guarantee minimum energy for all neighboring pairs correspond to the minimum
energy of the whole system. This arrangement is the most stable at the absolute zero.
The closest packing structures certainly offer a suitable situation for all particles.
Note, however, that the so-called closest packing covers not only crystalline structures but also other disordered structures. Two simplest structures with the crystalline
order consist of the closely packed layers, which is just the triangle lattice. The two
crystalline structures are different only in relative stacking of the layers. The stacking in the face-centered cubic (FCC) structure is a repetition of three layers ABC
(...ABCABC...)
9 while that in the hexagonal closest packing (HCP) structure is of
...ABAB... The possibility of such descriptions of two crystalline structures means
any aperiodic stacking can be a closest packing structure if neighboring layers are
of different kinds. This simple example suggests that the origin of the periodicity is
not merely the energetic effect but also from other sources.
Even apart from the non-uniqueness of the aggregation structure, the crystallization process also put forward a problem. Consider a stable structure of clusters
consisting of a small number of particles. With the increase of the number of particles (n) form 3 to 5, the structures of most stable clusters (with the minimum
energy) changes from the equilateral triangle (n = 3), the regular tetrahedron (4) to
the trigonal bipyramid (5). These are partial structures of the closest packing structures. For n = 6, regular octahedron has the minimum energy due to the formation
of the equilateral triangles for the shortest bonds between neighboring particles.
However, the situation changes for n = 7. Although the cluster shown in Fig. 1.7a
has the same structure as in the FCC closest packing, it has a higher energy of
−15.10 than the pentagonal bipyramid (b) with −15.94 for the Lennard-Jones
(6,12)–potential (Eq. 1.63). Since the five-fold symmetry of the cluster (b) is incompatible with any translational periodicities, the growth of clusters cannot result in
translationally symmetric structures (crystalline order). This also suggests that the
origin of the crystalline orders is to be attributed to other sources.
The indispensable source of the crystalline order was revealed as the entropic
effect through computer simulations on classical particles by Alder and Wainwright
[20]. They performed the simulations assuming only repulsive interactions (rigid
spheres) while confining particles inside a box. They found that the equation of states
consists of two branches, which correspond to fluid and crystal. The periodicity of the
crystalline phase was clearly demonstrated by snapshots of trajectories of particles.
Since the simulations are classical, the energy of the system is always the classical
value (
3
2
k B T per particle) of kinetic energy because of the prohibition of overlap of
particles, only the source that can bring about the crystalline order is the so-called
9 The FCC structure can be represented as alternating stacks of square plane lattices as ...A’B’A’B’...
27
1.3.3 Crystallization
In contrast to the liquefaction by the attractive part of the intermolecular interaction, crystallization is not easy to understand. Suppose spherical molecules, between
which the interaction works only for neighbors. Since the intermolecular interaction
energy generally has a minimum, as shown in Fig. 1.5, molecular arrangements that
guarantee minimum energy for all neighboring pairs correspond to the minimum
energy of the whole system. This arrangement is the most stable at the absolute zero.
The closest packing structures certainly offer a suitable situation for all particles.
Note, however, that the so-called closest packing covers not only crystalline structures but also other disordered structures. Two simplest structures with the crystalline
order consist of the closely packed layers, which is just the triangle lattice. The two
crystalline structures are different only in relative stacking of the layers. The stacking in the face-centered cubic (FCC) structure is a repetition of three layers ABC
(...ABCABC...)
9 while that in the hexagonal closest packing (HCP) structure is of
...ABAB... The possibility of such descriptions of two crystalline structures means
any aperiodic stacking can be a closest packing structure if neighboring layers are
of different kinds. This simple example suggests that the origin of the periodicity is
not merely the energetic effect but also from other sources.
Even apart from the non-uniqueness of the aggregation structure, the crystallization process also put forward a problem. Consider a stable structure of clusters
consisting of a small number of particles. With the increase of the number of particles (n) form 3 to 5, the structures of most stable clusters (with the minimum
energy) changes from the equilateral triangle (n = 3), the regular tetrahedron (4) to
the trigonal bipyramid (5). These are partial structures of the closest packing structures. For n = 6, regular octahedron has the minimum energy due to the formation
of the equilateral triangles for the shortest bonds between neighboring particles.
However, the situation changes for n = 7. Although the cluster shown in Fig. 1.7a
has the same structure as in the FCC closest packing, it has a higher energy of
−15.10 than the pentagonal bipyramid (b) with −15.94 for the Lennard-Jones
(6,12)–potential (Eq. 1.63). Since the five-fold symmetry of the cluster (b) is incompatible with any translational periodicities, the growth of clusters cannot result in
translationally symmetric structures (crystalline order). This also suggests that the
origin of the crystalline orders is to be attributed to other sources.
The indispensable source of the crystalline order was revealed as the entropic
effect through computer simulations on classical particles by Alder and Wainwright
[20]. They performed the simulations assuming only repulsive interactions (rigid
spheres) while confining particles inside a box. They found that the equation of states
consists of two branches, which correspond to fluid and crystal. The periodicity of the
crystalline phase was clearly demonstrated by snapshots of trajectories of particles.
Since the simulations are classical, the energy of the system is always the classical
value (
3
2
k B T per particle) of kinetic energy because of the prohibition of overlap of
particles, only the source that can bring about the crystalline order is the so-called
9 The FCC structure can be represented as alternating stacks of square plane lattices as ...A’B’A’B’...