8th Liquid Matter Conference September 6-10, 2011 Wien, Austria ...

P5.48Wed 711:10-14:00Hard spheres on the minimal gyroid surfaceTomonari Dotera 1 and Junichi Matsuzawa 21 Kinki University, Department of Physics, Kowakae 3-4-1 577-8502, Higashi-Osaka,Japan2 Nara Women’s University, Nara, JapanOn a flat plane, the hexagonal arrangement, namely, the triangular lattice gives the densest packingof spheres. The liquid-solid transition occurs even for purely repulsive potentials because of excessentropy of regular arrangement. The problem arises: what is the regular arrangement when thesurface is curved? For positive surfaces, this issue was studied for a number of materials such asC 60, and icosahedral capsids. In the same way, one could consider the problem of hard disks onsaddle-shaped (hyperbolic) spaces. Three decades ago, Rubinstein and Nelson addressed this issueon surfaces with constant negative curvatures as an analogue of the frustrated sphere packing in thethree-dimensional space. Furthermore, Modes and Kamien recently calculated virial coefficientsof hard disk on surfaces with constant negative curvatures. In this presentation, we consider regulararrangement of hard spheres on the periodic minimal surface called the gyroid surface. We findthat 48/64 hard spheres per unit cell on the gyroid minimal surface are entropically self-organized:The space groups of the tessellations are I4 1 32 and Ia3d, respectively. The regular tessellations ofthe spheres can be viewed as two kinds of hyperbolic tilings on the Poincare disk with a negativeGaussian curvature, one of which is equivalently, the arrangement of angels and devils in Escher’sCircle Limit IV, (3. 3. 3. 4. 3. 4) Archimedean tiling. Numerical evidences for the transitionsas a function of hard sphere radius are obtained in terms of the acceptance ratios of Monte Carlomoves and order parameters. This result suggests that regulated membranes in soft materials can bescaffolds to self-assemble more complex but ordered structures on the membranes. As an example,we discuss the recently proposed hyperbolic tiling structure in an ABC star block copolymer melt.48