2.1 Sphere packing and “Lennard–Jonesium” solid

Atomic close-packings are widely met in condensed matter systems. A (classical) model consists in having spherical particles interacting through a pair potential displaying isotropic long-range (van der Waals-like) attraction (to eventually get a structure) and short-range repulsion (to prevent the system from collapsing). A widely studied such atomic pair potential is the Lennard–Jones potential:

Is is well known that, varying the clusters size, a large fraction of structures with (pseudo)icosahedral symmetry are found which minimize the Lennard–Jones total energy [12]. In the early seventies, this knowledge, and the experimental strong indication of a polytetrahedral (and therefore (pseudo)icosahedral) atomic order in amorphous metals [13], opened the way for a rich scientific exchange between the amorphous and the cluster scientific communities.

For the present purpose of introducing geometrical frustration, we can further simplify this potential, and consider situations where a pair of atoms sit at a distance were they minimize the Lennard–Jones energy, and then add the contribution −ɛ to the total energy. In such hard sphere limit, the total energy is just equal to −ɛ times the number of sphere contacts. We extend this to 2 dimensions as a hard-disk close-packing problem and compare the 2- and 3-dimensional cases (Fig. 1). In this simple model, one clearly sees that with four particles, the system minimizes its energy by adding the fourth particle in such a way to close a regular tetrahedron, instead of staying on a 2-dimensional space. The best configuration for the fifth particle consists in closing a second tetrahedron sharing a face with the first one. The sphere close-packing problem is thus mapped onto a tetrahedral packing in R³. In an equivalent way, had the configuration been restricted to be planar, the hard-disk close-packing problem would translate to a triangular packing in 2d.

And there comes the main difference between close-packing in two and three dimensions. A triangular packing can be freely propagated throughout a plane, the reason being that the corner angle (60°) is a submultiple of 2π. A triangular lattice, with six equilateral triangles sharing each vertex, results from that construction, and is well known to be the closest 2d lattice packing. This situation (local “best” configuration which can be freely propagated in space) is referred to as unfrustrated. On the contrary, propagation of the tetrahedral order quickly meets a problem, due to the fact that the regular tetrahedron dihedral angle (∼70°30′) is not an integer submultiple of 2π. Note that it falls short of being the fifth of 2π. This means that adding more and more tetrahedra, there is a trend toward arranging five tetrahedra sharing a common edge, but each time with a remaining small hole. These holes accumulate, and eventually destroy the perfect tetrahedral order. This is a typical frustrated system. Notice that a set of 20 tetrahedra sharing a common vertex would define a perfect centrered icosahedron, at the price of having slightly deformed tetrahedral symmetry. This is why close-packed tetrahedral order and icosahedral order are closely linked problems.

2.2 The ideal icosahedral order on a 3-dimensional curved space

In 1979, Jean-François Sadoc proposed a very original approach to analyse this frustration effect in sphere packing problems. He observed that perfect (unfrustrated) icosahedral order can propagate freely throughout the space if the latter gets positively curved. There is an analogous problem in two dimensions associated with pentagonal packing (Fig. 2). The regular pentagon vertex angle (108°) does not divide 2π, and therefore cannot perfectly tile an Euclidean plane. But if one tries instead to tile the surface of a sphere of appropriate radius, a regular tiling can be obtained: the dodecahedron (with 20 vertices and 12 faces). Sadoc observed that a correctly scaled, 3-dimensional sphere (noted S³) can also admit a perfect tetrahedral packing. One gets a polytope with 120 vertices and 600 tetrahedra cells; each edge is shared by five tetrahedra and each vertex is surrounded by a perfect icosahedron. High-dimensional polytopes, which are well known and classified by mathematicians [14], then found a new application as ideal templates for real atomic packing problems.

An interesting computer simulation, done by Straley [15], illustrates quite well the frustration effect. Straley compares annealing of interacting particles in Euclidean and in spherical space. In the first case the structure is very slowly approaching equilibrium, while on S³, with adjusted radius, the ground state configuration, corresponding to the 120-vertex polytope, is reached very quickly.

Having this ideal template, the physical properties associated with the perfect icosahedral order can be computed (see for example [3], and references herein), keeping for a next step the analysis of how far these properties will survive in the more realistic structures.

Among the theoretical studies that took advantage of these ideal templates, one should cite the detailed analyses by David Nelson and co-workers [16–18], who elaborated on the concept of icosahedral order parameter, and proposed a Landau theory of frustrated system.

2.3 Decurving procedure and topological defects

Once the ideal structure has been defined, its relation with the real structure must be described. One can imagine several ways for that purpose. If one is interested in finite clusters of icosahedral symmetry, a simple prescription consists in directly mapping a piece of the ideal polytope onto an 3-dimensional Euclidean tangent space [19]. For infinite structures, the most valuable method consists in introducing disclination defects. Indeed, a disclination defect naturally carries curvature, as shown in a simple 2-dimensional example in Fig. 3.

Let us recall the standard Volterra process to create a disclination defect. One creates a cut in the structure, and add or remove a wedge of material along the edge of the cut (this is to be contrasted with adding or removing a half plane in the dislocation case), as illustrated in Fig. 3 for a 2(3)-dimensional point (line) defect, respectively. It is then clear that a disclination “carries” curvature, either positive or negative. It is then natural to propose, as decurving procedure from the ideal template on a 3-dimensional positively curved sphere to the “flat” Euclidean space, the creation of an adequate density of disclination lines (see for example [20]). Along this line, an interesting possibility is the so-called “Iterative Flattening Method”, which gives rise to infinite closely packed structures, with a hierarchical disclination order, and atomic arrangement close to that of quasicrystals [21].

2.4 The Frank–Kasper phases

Although the present curved space approach was initially meant to address the structural description of amorphous systems, it became rapidly clear that the method applies to large cell crystalline structures with complex local order inside the unit cell. The best example is provided by the Frank–Kasper metallic phases [9]. These phases present tetrahedrally close-packed atoms with many (pseudo)icosahedral local configurations, which are repeated periodically. An inspection of the unit cell arrangements indicates a rather large fraction of atoms with an almost perfect icosahedral coordination shell (with 12 neighbours, and denoted Z₁₂ sites). Frank and Kasper showed that the remaining atoms have a higher coordination (mainly Z₁₄, Z₁₅ and Z₁₆), and form uninterrupted networks (called the “major skeleton”) along the directions where the five-fold local symmetry is replaced by a six-fold one (Fig. 4). The Frank–Kasper skeleton has been since identified to a disclination line network threading a medium with the icosahedral polytope order [16,22,23]. The discovery of icosahedral quasicrystalline phases [24] renewed the interest toward the Frank–Kasper phases. Many large cell with such structures were found stable in the vicinity of the quasicrystal phase in most phase diagrams; to some respect, the true quasicrystalline phase can be understood as the asymptotic element of an infinite set of periodic structure with increasing large cell structures, the so-called approximant phases.

3.1 The double-twist frustration

The classical cholesteric phases correspond to the case where the twist propagates along one direction only, leading to planar parallel configurations twisting in the perpendicular direction. But more complex structures, called the blue phases, are sometimes found in a narrow range of the phase diagrams. Their local order corresponds to more complex arrangements where the twist seems to propagate along all directions perpendicular to the molecular axis, the so-called “double-twist” configuration. Such a situation is frustrated, as it can be understood in Fig. 5, by following and keeping track of the director orientation along two different paths ending at the same point. It is then impossible to fill the space with a vector field which obeys everywhere the “double-twist” rule. This is therefore a new example of geometrical frustration, whose strength is characterized by the pitch of the associated rotation.

3.2 The unfrustrated double-twist blue phase on S³

This fascinating example of geometrical frustration was the first to be treated in a continuous system [5,25], using an ideal template in S³ whose radius is adjusted to the helicoidal pitch. Indeed, a vector field tangent to S³ can be defined such that the double-twist configuration is everywhere perfect. This field can be visualized, using the stereographic projection on R³, as shown in Fig. 6b.

One way to describe this vector field is in relation with the S³ great circle Hopf fibration. It is possible to fill the whole S³ with non-intersecting great circles in such a way that any S³ point belong exactly to one circle. The ideal blue phase field is nothing but the vector field tangent to the Hopf great circles. The great circle gathers on a continuous set of coaxial tori, three of which being represented in Fig. 6b.

3.3 The cholesteric blue phase as a periodic arrangement of disclination lines

Topological defects must be introduced to lower the space curvature, in order to relate the ideal double-twist structure in S³ and real structures in R³ [26]. Blue phases are, along this line, similar to Frank and Kasper phases.

Note that it is already possible to minimize the “double-twist” energy in R³, along the central axis of a cylinder, called a “double-twist tube” (Fig. 6a). A standard model for the blue phases is then a 3-dimensional periodic arrangement of these tubes [27], as displayed in Fig. 7. But the director field cannot be defined everywhere and a periodic array of disclination lines, of type “π”, is present. The two well-known examples, blue phases I and II, correspond to different tube arrangements with cubic symmetry.

The analysis of the field configuration near the tube interfaces is made easier if one remarks the close relation with two Infinite Periodic Minimal Surfaces, called P and D. In particular, a local minimum of the energy is reached if the director field is tangent to the asymptotic lines of the surface [26]. Flat points on the surface, which are singularities for this field, are threaded by axes of type [111], which corresponds to the disclinations.