It is widely known that even a purely hard-sphere system undergoes a fluid–solid transition, called the Alder transition, producing the hexagonal arrangement [34], whose entropy-driven ordering mechanism turns out to be universal in soft materials [35]. Above a certain critical density, the solid phase in the hexagonal arrangement gains more entropy than the fluid phase. Do regular arrangements emerge in the solid phase on the TPMSs?
Motivated by this possibility, we began an investigation on the G-surface and indeed found the Alder transition, but only for specific numbers of spheres [14]. To provide further insight into the regular arrangement and establish a unified framework, we have extended our MC simulations to the P- and D-surfaces in this study.
The well-known approximate forms [36, 37] of the surfaces are described as
$$\begin{aligned} P: s(\mathbf{q})= & \cos x+ \cos y+ \cos z=0,\\ G: s(\mathbf{q})= & \sin x \cos y+\sin y \cos z+\sin z \cos x=0,\\ D: s(\mathbf{q})= & \sin x \sin y \sin z +\sin x \cos y \cos z\\&+\cos x \sin y \cos z+\cos x \cos y \sin z=0, \end{aligned}$$
where \(\mathbf{q}=(x, y, z)\) and the lattice constant is taken to be unity; \(2\pi\) is omitted in the expressions.
In practice, the centers of spheres are confined between boundaries \(s(\mathbf{q})=-0.1\) (green) and \(s(\mathbf{q})=0.1\) (red) as shown in Fig. 1a–c. We assign \(N_{\alpha }\) to the number of spheres per cubic unit cell of the surfaces. \(\alpha\) stands for P, G, and D. We begin with a random state having a sufficiently small radius of spheres on the TPMSs, perform a MC run without any symmetry input, then increase the radius r with a small increment, and repeat the process. Details of the method of the simulations were given in Ref. [14].
The order parameter h(r) as a function of sphere radius r is defined as
$$\begin{aligned} h(r) =\left\langle \frac{1}{M}\sum _{i=1}^{M} f_{hkl} (\mathbf{q}_i)\right\rangle , \end{aligned}$$
where the sum is taken over positions of all sphere centers \(\mathbf{q}_i\), and \(\langle \cdots \rangle\) implies the MC average. The function \(f_{hkl}(\mathbf {q})\) is an invariant function under the operations of the space groups, ex. \(Im\bar{3}m\) for the P-surface [38]:
$$\begin{aligned} f_{hkl} (\mathbf{q})= \sum _\mathbf{{q} \in C_3} \cos (hx)\cos (ky)\cos (lz), \end{aligned}$$
(1)
where (h, k, l) is a set of integers and \(C_3\) is a group of all cyclic permutations of (x, y, z). Suitable choices of (h, k, l) for each \(N_{\alpha }\) give large absolute values of h(r) in the ordered phase.
Figure 1d shows a snapshot of an ordered structure for a \(N_\mathrm{P}=72\) system on the P-surface, in which alignments are observed in x, y, and z directions despite undulations. To find clear phase behavior, we have employed simulation boxes consisting of multiple cells, here for instance, \(3\times 3\times 2\) cells with periodic boundary conditions; accordingly, the number of spheres is 1296 for \(N_\mathrm{P}=72\). The existence of the fluid–solid transition is judged firstly by whether there exists a discontinuous jump or not in the acceptance ratio (AR) curve representing the movable probability of MC trial moves of spheres. If not, the simulation ends up with a frozen random state with a continuous single curve. For \(N_\mathrm{P}=72\), Fig. 1e exhibits discontinuous jumps and accordingly a hysteresis loop, indicating clear evidence of the first-order transition. Similarly, in Fig. 1f, the order parameter h(r) is plotted, where \(f_{045}(\mathbf{q})\) is chosen in Eq. (1). The transition points are exactly the same as those in Fig. 1e, implying that the ordered phase near the transition region has more movable spaces for spheres than in the disordered phase, and the ordered phase is, therefore, a high entropy phase. We have examined several system sizes to check the results.
Our observation indicates that the number of spheres in a unit cell having a definite fluid–solid transition shows discrete integers. For the P-surface, we find magic numbers \(N_\mathrm{P}=20\), 32, 56, 72, and 96; for the G-surface, we find \(N_\mathrm{G}=40\), 48, 64, 112, and 144; and for the D-surface, \(N_\mathrm{D}=96\), 128, and 224 are obtained. They can be best classified according to the hexagulation numbers H, which will be discussed in the next section.
Table 1 summarizes regular structures obtained from MC simulations, where we focus our attention on structures with cubic symmetry, since the TPMSs have inherent cubic space group symmetries, \(Im\bar{3}m\), \(Ia\bar{3}d\), and \(Pn\bar{3}m\) for P-, G- and D-surfaces, respectively. Several symmetry operations of the original space groups are broken in some cases, and the ordered arrangements are accordingly assigned to subgroups. For the \(N_\mathrm{P}=72\) system, for instance, the arrangement is assigned to the space group \(Pm\bar{3}n \subset Im\bar{3}m\). We should mention that the \(\overline{H=4}\) class denotes hypothetical structures, while the \(H=4\) class is obtained from simulations. They are very similar and the difference of the positions on the surfaces is very small, but the \(\overline{H=4}\) class is more symmetric than that of \(H=4\). Because the distances between spheres are more equivalent in \(H=4\), and consequently in simulations, the symmetry of \(\overline{H=4}\) is broken into the entropically more favorable \(H=4\).
For eager readers, in Table 2 we explicitly provide the values of coordinates of spheres obtained by our MC simulations. One can consult Wyckoff positions in International Tables for Crystallography and evaluate all coordinates. Note that the Wyckoff position 96h of \(F\overline{4}3c\) for \(N_\mathrm{D}=96\) in the International Tables is not convenient because of the different choices of the origin for symmetry operation. A part of 192h of \(Fd\overline{3}c\) is more useful to reproduce our data, which are presented in Table 3.
Polygonal tilings are constructed from the center positions of spheres, represented in Fig. 2a–i. Since the magic numbers are larger for the G- and D-surfaces, we render five results for the P-surface and two for the G- and D-surfaces. The set of integers (\(n_1, n_2, n_3,\ldots\); \(n_4, n_5, n_6,\ldots\);) denotes a tiling of multiple vertex types in the way that \(n_1\)-gon, \(n_2\)-gon, and \(n_3\)-gon, \(\cdots\) meet consecutively on each vertex, and superscripts are employed to abbreviate when possible. A set of integers like (\(3^6\); \(3^7\)) denotes a tiling composed of two vertex types \(3^6\) and \(3^7\), for instance.
The tilings are composed of triangles and quadrangles, where length of all sides is almost equal within a few percent and where quadrangles are not always flat. Unlike the hexagonal arrangement (\(3^6\)) on a flat plane, it should be noticed that vertices are not equivalent, as is the cases with icosahedral viruses. The first number of the Wyckoff positions listed in Table 1 represents the multiplicity of corresponding vertex types. All these tilings whose Euler characteristic \(\chi\) per cubic unit cell is −4, −8, and −16 for P-, G-, and D-surfaces, respectively, are hyperbolic extensions of the flat plane tessellation (\(3^6\)) with \(\chi =0\). We finally point out strong similarity for the same H classes as shown in Fig. 2d–f (\(H=4\)) and Fig. 2g–i (\(H=7\)).