Abstract
We find that 48/64 hard spheres per unit cell on the gyroid minimal surface are entropically selforganized. Striking evidence is obtained in terms of the acceptance ratio of Monte Carlo moves and order parameters. The regular tessellations of the spheres can be viewed as hyperbolic tilings on the Poincaré disc with a negative Gaussian curvature, one of which is, equivalently, the arrangement of angels and devils in Escher's Circle Limit IV.
1. Introduction
What is the regular arrangement of spheres when a surface is curved? On a flat plane, the hexagonal (equilateral–triangular) arrangement is ubiquitous from microscopic objects (e.g. electrons, atoms and molecules) to mesoscopic objects (e.g. liquid crystals, polymers and colloidal particles). The arrangement stems from the densest packing of hard spheres, space division or interactions between particles; for instance, the Coulombic force between electrons organizes the Wigner crystal. For surfaces with positive Gaussian curvatures, remarkable arrangements are observed in fullerenes and icosahedral capsids. Furthermore, the distribution problem of electrons on a sphere has been considered for many years as the ‘Thomson problem’ [1]. In contrast, a regular arrangement on a saddleshaped (hyperbolic) surface with negative curvatures has not been identified in soft materials; one regular arrangement so far discovered is in inorganic mesoporous materials [2].
Nearly 30 years ago, Rubinstein & Nelson [3] addressed the problem of hard discs on surfaces with constant negative curvatures as an analogue of frustrated sphere packing in threedimensional space. Furthermore, Modes & Kamien [4] calculated virial coefficients of hard discs on surfaces with constant negative curvatures to determine the equation of states. Unfortunately, however, a plane with a constant negative curvature mathematically cannot be realized in Euclidian threedimensional space. Instead of constant negative curvatures, triply periodic saddleshaped surfaces with nonpositive curvatures such as gyroid [5], diamond and primitive surfaces are found in various soft materials such as biological lipids, surfactants, mesoporous materials, and block copolymers [6–9]. Recently, it was found that polyisoprene spheroids spread over a polystyrene gyroidal membrane in an ABC star block copolymer melt [10]. Moreover, regulated membranes can act as scaffolds to construct more complex structures [11,12]. Therefore, it is worth studying regular arrangements on such triply periodic saddleshaped surfaces [13].
On a flat plane, a fluid–solid transition occurs even for the purely hardsphere potential because of the excess entropy of the regular arrangement [14]. Namely, above a certain critical density, the crystalline arrangement of spheres gains more entropy than the random arrangement gains. Is there a similar transition on the gyroid surface? Here, we report the results of Monte Carlo simulations of entropydriven order–disorder transitions on the surface, and propose a regular arrangement on the gyroid surface.
2. Monte Carlo simulation
2.1. Method
We consider hard spheres whose centres are confined in a rigid gyroidal membrane. The wellknown approximation form of the gyroid surface is shown by 2.1where we omit a factor 2π/a for x, y and z [15]. Hereafter, we choose the lattice constant a = 1. For simulations, we employ a gyroid membrane g(r) ≤ 0.1, whose boundaries are given by g(r) = −0.1 (green) and g(r) = 0.1 (red) as shown in figure 1.
Monte Carlo simulations are conducted with sizes (ha, ka, la), where 1 ≤ h, k, l ≤ 3. We assign N (N = 48 − 64) to the number of hard spheres per unit cell, and the total number of spheres M in a simulation box with (h, k, l) is M = hklN. Periodic boundary conditions for x, y and z directions are exerted. We prepare M spheres whose centres are within g(r) ≤ 0.1 without hard contact. This condition is easily achieved when the radius is small.
The Monte Carlo procedure is as follows. (i) Select a sphere randomly. (ii) Generate a trial move within a box whose sizes Δx, Δy and Δz =± 0.0075 by the uniform distributions. (iii) Reject the trial move if g(r) > 0.1 or there is any hard contact with other spheres; if not, accept the move. (iv) Return to (i). (v) Record the acceptance ratio and the value of an order parameter. At each radius, 10^{6} Monte Carlo steps are carried out. (vi) Change the radius of spheres upward; the increment is 0.0001. For N = 48 and N = 64, decreasing paths from regular arrangements are also performed.
Let us make a few remarks. (i) The principle of detailed balance of Monte Carlo moves is guaranteed by the above procedures. (ii) The distance between two spheres is measured in terms of the threedimensional Euclidean distance, not the geodesic distance on the hyperbolic surface, i.e. we simulate hard spheres not discs. (iii) The thickness of the membrane is not exactly uniform, but it is thinner than the radii under consideration. Thus, we presume that this does not affect the key feature of the results.
In order to see order–disorder transitions for N = 48 and N = 64, we measure order parameters h(r) as a function of radius r defined by where the sum is taken over all spheres in question, and 〈···〉 implies the Monte Carlo average. The function f(r) for N = 48 is f_{48}(r), which is a function of the position of sphere centres, shown by The absolute value of this function takes a large value when the positions of the spheres are in regular arrangement. Technically, the function f_{48}(r) is generated by the Fourier transform of all generic points of the space group for the (h, k, l) = (4, 2, 2) peak, and thus are invariant under the operations of [16]. In figure 2, level set surfaces defined by f_{48}(r) =±0.8 are displayed, together with the regular arrangement. There are two choices of regular arrangement for N = 48, which can be expressed by the sign of the function. As shown in figure 2, the regular positions match well with those of the level surfaces. We note that equation (2.1) is obtained from I4_{1}32 and (h, k, l) = (2, 1, 1) in the same way as mentioned above [15]. Since the regular positions for N = 64 are divided into two symmetrically inequivalent groups, a simple function as good as f_{48}(r) could not be constructed. Thus, we temporary calculate the order parameter by using where the first three terms express Wyckoff position 16a sites and the last three terms express 48g sites for . It turns out that the function is useful near the transition region.
2.2. Results
The existence of a transition is judged by whether or not there is a jump in the acceptance–ratio curve. We have obtained sound evidence for N = 64. Figure 3a displays the acceptance ratio of Monte Carlo moves for the system with (h, k, l) = (2, 2, 3); namely, 768 (64/per unit cell) hard spheres whose centres are on the gyroid surface. Upon increasing the radius of spheres, the curve basically decreases because of the increase in contacts. However, at about r = 0.105, the ratio abruptly jumps up about 0.08. This is the transition from the disordered state to the ordered state. We render the order parameter h(r) in figure 3b, which shows a jump at the same radius. When starting from a regular structure described in §3, both upward and downward curves trace the same curve above r ∼ 0.106, then the downward curve exceeds the transition point at r = 0.105 like superheating. The curve shows an abrupt downturn at approximately r = 0.101. This hysteresis is typical evidence of the firstorder transition. We illustrate a snapshot of the result just above the transition in figure 4a. Note that we have observed jumps for the simulation boxes with (h, k, l) = (1, 1, 1), (1, 1, 2), (1, 2, 2), (2, ,2, 2), (1, 1, 3) (1, 2, 3), (2, 2, 3), and (1, 3, 3). In the series, the upward transition radius turns out to increase a little with the size of boxes tending to converge, while the downward transition radius is almost constant. The box size seems to be limited by simulation time.
In the case of N = 48, however, we have seen transitions for (h, k, l) = (1, 1, 1) and (1, 1, 3). Using our computational power, we find that the transition is not easily accessible to us. As we have mentioned, there are two possible regular configurations for N = 48, which may cause the difficulty. Therefore, we have added a solid region (green shaded box) as shown in figure 4b for (hkl) = (2, 3, 3) to enhance the transition. The spheres in the region (0 ≤ z < 1) are fixed on regular sites that acted as a solid interface. As shown in figure 3c, it exhibits an entropydriven fluid–solid transition when the radius of spheres was greater than approximately 0.118 in the unit of the lattice constant. In figure 3d, the order parameter measured in the region (1.5 ≤ z < 2.5) is a monotonically increasing function of sphere radius and the maximum value is 1.473. Figure 4b displays selforganized 864 (48/per unit cell) hard spheres. Without the solid region, we have found random configurations frustrated with two (+ and −) tilings, which end up with glassy states.
It turns out that jumps seem to exist for N = 48, 52, 59, 60, 61, 62, 63 and 64 in the case of the smallest box (1, 1, 1), although curves fluctuate considerably at the transition regions. We have conducted a systematic investigation and found that, except for (1, 1, 1), there is no order–disorder transitions in the range of 48 < N < 64.
3. Hyperbolic tiling
To analyse tessellations on the gyroid surface, we use hyperbolic tiling [17–21]. A hyperbolic tiling is the hyperbolic analogue of a tiling on a plane, and is usually depicted on the Poincaré disc representing hyperbolic geometry: a household example is Escher's enchanting series of artworks Circle Limit. More than two decades ago, Charvolin and Sadoc pointed out that the Poincaré disc in figure 5 tiled by (π/2, π/4, π/6)triangles is grouptheoretically related to the gyroid surface. The dodecagonal region within 12 thick curves in figure 5 can be conformally mapped upon the gyroid surface and cover half the area of the surface in a unit cell. A useful fact is that the symmetry operation of is lifted to the orientationpreserving group action in figure 5.
We propose three regular arrangements: (1) a monohedral triangular tiling (3^{6}; 3^{8}); (2) an Archimedean tiling (3^{3}.4.3.4) [22]; and (3) a tiling (3^{6}; 3^{2} .4.3^{2} .4) consisting of 40, 48 and 64 vertices per unit cell on the gyroid surface. The set of integers (n_{1} · n_{2} · n_{3} · …) denotes tiling of a vertex type in the way that n_{1}gon, n_{2}gon and n_{3}gon, … meet consecutively on each vertex, and superscripts are employed to abbreviate when possible. A set of integers such as (3^{6}; 3^{8}) denotes a tiling composed of two vertex types 3^{6} and 3^{8}, for instance. We presume that these tilings are hyperbolic extensions of the flat plane tessellation (3^{6}).

Vertex positions of hyperbolic monohedral (3^{6}; 3^{8}) tiling are located at the site symmetry points, and of space group (no. 230). The corresponding points are denoted as 16a and 24d sites, representing multiplicity and the Wyckoff letter. The red and yellow circles in figure 5 correspond to Wyckoff positions of , 16a and 24d, respectively, whose positions are geometrically monkey and horse saddle points of the gyroid surface. The space group of (3^{6}; 3^{8}) is (no. 230).

Each centre of shaded triangles made by one 16a and two 24d positions corresponds to a vertex of a (3^{3} .4.3.4) tiling (figure 5a). Since the mapping preserves local connectivity, every vertex of (3^{3}.4.3.4) tiling has six neighbours and the same local environment. Thus, we call it Archimedean tiling even for hyperbolic geometry. The whole area is divided into alternating white and grey regions in figure 5a, and correspondingly the inversion symmetry of is broken. This procedure enables us to find a curious coincidence between the tessellation of green circles in figure 5a and that of angels (or devils) in Escher's Circle Limit IV. In the threedimensional space, the positions are (0.576, 0.386, 0.554) and its symmetry equivalents. Remarkably, the distance between the nearest spheres is the same. Fortyeight vertex positions of the tiling in figure 6a,b are generated by symmetry operations of space group (no. 220), which is one of the cubic subgroups of , but different from (no. 214) associated with the singlegyroid phase.

The space group of (3^{6}; 3^{2}.4.3^{2}.4) is . The tiling composed of 64 vertices per unit cell is shown in figure 6c,d. Notice that the tiling consists of two types of vertices: one of which is 16a, a site symmetry point of space group ; the other type is 48g, a site symmetry point ..2; here, the position is precisely given by with y = 0.337 [23] and its symmetry equivalents. Notice that the 16a points are surrounded by six isosceles triangles (not equilateral triangles); thus, the hexagon consisting of the six triangles is not flat as shown in the central part of figure 6c.
4. Discussion
Remarkably, the entropydriven transition of hard spheres on the gyroid surface occurs. Because the acceptance ratio only reflects local entropy, it is not the total configurational entropy. However, since there is no energetic term, the jump in the ratio implies entropic ordering. This kind of entropicordering principle prevails among many softmatter systems.
Why do N = 48 and N = 64 systems order? For N = 48, the property of equidistance (0.259 in the unit of the lattice constant) seems to be advantageous to entropic ordering, because the same free volume for all spheres gives the maximum entropy when it is assumed that each sphere is confined in the Voronoi cell, although the relation between the freevolume cell and the Voronoi cell is not straightforward in the presence of a curved background [13]. Furthermore, there exist a point and its symmetry equivalents on the gyroid surface calculated even by the Weierstrass–Enneper representation such that all edge lengths take the same value [22]. The same property holds for N = 64, since the edge–length difference is within 1.1 per cent (the equidistant points are given by y = 5/16 for 48g sites, but not on the gyroid surface). Because of this geometric condition, the symmetry governs the selforganization for N = 48 and N = 64. In future, it will be interesting to see the relation between the packing fractions of spheres (or discs) at transitions and those of the densest packing for a given N.
We have elucidated that the regular tessellations of the spheres can be viewed as hyperbolic tilings on the Poincaré disc or regular tiling structures in the Euclidean space. We emphasize that, as investigated by several researchers, the Poincaré disc and threedimensional nets are useful to understand the symmetry and the regulated structures on the gyroid surface. We point out that adding squares in triangle tilings is a plausible way to form regular structures relating to the gyroid structure. Squares are not so unrealistic because square–triangle tilings have been observed in ABC star block terpolymers [24,25]; as mentioned above, the monohedral tiling (3^{6};3^{8}) is possible, though. In addition, another Archimedean (3^{7}) net is possible; however, when the distance is the same, it contains tetrahedrons and it is not on the gyroid surface.
We hope that this study may hint at why the hyperbolic tiling structure can form in an ABC star block copolymer melt, and we consider that the new tiling structures obtained are plausible structural candidates.
Acknowledgements
This work was supported by the GrantinAid for Scientific Research (C) (no. 22540375) from JSPS, Japan, and the GrantinAid for Scientific Research on Priority Areas (no. 20045006) from MEXT, Japan. T.D. is grateful to Y. Matsushita, A. Takano, K. Hayashida and N. Fujita.
Footnotes
One contribution to a Theme Issue ‘Geometry of interfaces: topological complexity in biology and materials’.
 Received October 1, 2011.
 Accepted December 12, 2011.
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