Complexity in surfaces of densest packings for families of polyhedra

Physical Review X

Volumn 4, Issue 1, 2014, Pages

  Chen, Elizabeth R ^a,b   Klotsa, Daphne ^b   Engel, Michael ^b   Damasceno, Pablo F ^b   Glotzer, Sharon C ^b  

^a HARVARD UNIVERSITY   (United States)

^b UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ANALYTIC CALCULATIONS; COLLOIDAL MATTER; DIFFERENT STRUCTURE; HIGHER-DIMENSIONAL; STATIC PROPERTIES; STRUCTURE AND PROPERTIES; SYSTEMATIC ANALYSIS; TRIANGLE GROUPS;

GEOMETRY;

EID: 84909591837     PISSN: None     EISSN: 21603308     Source Type: Journal    
DOI: 10.1103/PhysRevX.4.011024     Document Type: Article

References (62)

1. T. C. Hales, Historical Overview of the Kepler Conjecture, Discrete Comput. Geom. 36, 5 (2006).
2. A. Bezdek and W. Kuperberg, Dense Packing of Space with Various Convex Solids, arXiv:1008.2398.
3. T. Aste and D. Weaire, The Pursuit of Perfect Packing (Institute of Physics, Bristol, PA, 2000).
4. E. D. Demaine and M. L. Demaine, Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity, Graphs Combinator. 23, 195 (2007).
5. U. Agarwal and F. A. Escobedo, Mesophase Behavior of Polyhedral Particles, Nat. Mater. 10, 230 (2011).
6. N. Arkus, V. N. Manoharan, and Michael P. Brenner, Deriving Finite Sphere Packings, SIAM J. Discrete Math. 25, 1860 (2011).
7. J. Henzie, M. Grünwald, A. Widmer-Cooper, P. L. Geissler, and P. Yang, Self-Assembly of Uniform Polyhedral Silver Nanocrystals into Densest Packings and Exotic Superlattices, Nat. Mater. 11, 131 (2011).
8. P. Song, B. K. Olmsted, P. Chaikin, and M. D. Ward, Crystallization of Micrometer-Sized Particles with Molecular Contours, Langmuir 29, 13 686 (2013).
9. J.-W. Yoo and S. Mitragotri, Polymer Particles That Switch Shape in Response to a Stimulus, Proc. Natl. Acad. Sci. U.S.A. 107, 11 205 (2010).
10. M. M. Maye, M. T. Kumara, D. Nykypanchuk, W. B. Sherman, and O. Gang, Switching Binary States of Nanoparticle Superlattices and Dimer Clusters by DNA Strands, Nat. Nanotechnol. 5, 116 (2009).
11. T. D. Nguyen and S. C. Glotzer, Reconfigurable Assemblies of Shape-Changing Nanorods, ACS Nano 4, 2585 (2010).
12. T. D. Nguyen, E. Jankowski, and S. C. Glotzer, Self-Assembly and Reconfigurability of Shape-Shifting Particles, ACS Nano 5, 8892 (2011).
13. Y. Zhang, F. Lu, D. van der Lelie, and O. Gang, Continuous Phase Transformation in Nanocube Assemblies, Phys. Rev. Lett. 107, 135701 (2011).
14. O. Gang and Y. Zhang, Shaping Phases by Phasing Shapes, ACS Nano 5, 8459 (2011).
15. K. J. Leea, J. Yoon, S. Rahmani, S. Hwang, S. Bhaskar, S. Mitragotri, and J. Lahann, Spontaneous Shape Reconfigurations in Multicompartmental Microcylinders, Proc. Natl. Acad. Sci. U.S.A. 109, 16 057 (2012).
16. K. L. Kohlstedt and S. C. Glotzer, Polymer Particles That Switch Shape in Response to a Stimulus, Phys. Rev. E 87, 032305 (2013).
17. R.Guo, Z.Liu, X.-M.Xie, and L.-T.Yan,HarnessingDynamic Covalent Bonds in Patchy Nanoparticles: Creating Shape-Shifting Building Blocks for Rational and Responsive Self-Assembly, J.Phys.Chem.Lett. 4, 1221(2013).
18. A. J. Liu and S. R. Nagel, The Jamming Transition and the Marginally Jammed Solid, Annu. Rev. Condens. Matter Phys. 1, 347 (2010).
19. S. Torquato and F. H. Stillinger, Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond, Rev. Mod. Phys. 82, 2633 (2010).
20. T. Hayashi and R.W. Carthew, Surface Mechanics Mediate Pattern Formation in the Developing Retina, Nature (London) 431, 647 (2004).
21. J. A. Åström and M. Karttunen, Cell Aggregation: Packing Soft Grains, Phys. Rev. E 73, 062301 (2006).
22. J. R. Ellis, Macromolecular Crowding: Obvious but Underappreciated, Trends Biochem. Sci. 26, 597 (2001).
23. J. Balbo, P. Mereghetti, D.-P. Herten, and R. C. Wade, The Shape of Protein Crowders Is a Major Determinant of Protein Diffusion, Biophys. J. 104, 1576 (2013).
24. G. Wascher, H. Haussner, and H. Schumann, An Improved Typology of Cutting and Packing Problems, Eur. J. Oper. Res. 183, 1109 (2007).
25. J. Bennell, G. Scheithauer, Y. Stoyan, and T. Romanova, Tools of Mathematical Modeling of Arbitrary Object Packing Problems, Ann. Oper. Res. 179, 343 (2010).
26. J. Kepler, Strena seu de nive sexangula (1611); http://www.thelatinlibrary.com/kepler/strena.html.
27. T. C. Hales, A Proof of the Kepler Conjecture, Ann. Math. 162, 1065 (2005).
28. T. C. Hales and S. P. Ferguson, The Kepler Conjecture: The Hales-Ferguson Proof, edited by J. C. Lagarias (Springer-Verlag, New York, 2011).
29. M. H. Huang and P.-H. Lin, Shape-Controlled Synthesis of Polyhedral Nanocrystals and Their Facet-Dependent Properties, Adv. Funct. Mater. 22, 14 (2012).
30. C.-W. Liao, Y.-S. Lin, K. Chanda, Y.-F. Song, and M. H. Huang, Formation of Diverse Supercrystals from Self-Assembly of a Variety of Polyhedral Gold Nanocrystals, J. Am. Chem. Soc. 135, 2684 (2013).
31. U. Betke and M. Henk, Densest Lattice Packings of 3-Polytopes, Comput. Geom. 16, 157 (2000).
32. E. R. Chen, A Dense Packing of Regular Tetrahedra, Discrete Comput. Geom. 40, 214 (2008).
33. S. Torquato and Y. Jiao, Dense Packings of the Platonic and Archimedean Solids, Nature (London) 460, 876 (2009).
34. E. R. Chen, M. Engel, and S. C. Glotzer, Dense Crystalline Dimer Packings of Regular Tetrahedra, Discrete Comput. Geom. 44, 253 (2010).
35. S. Gravel, V. Elser, and Y. Kallus, Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra, Discrete Comput. Geom. 46, 799 (2011).
36. J. de Graaf, R. van Roij, and M. Dijkstra, Dense Regular Packings of Irregular Nonconvex Particles, Phys. Rev. Lett. 107, 155501 (2011).
37. P. F. Damasceno, M. Engel, and S. C. Glotzer, Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces, ACS Nano 6, 609 (2012).
38. A. Haji-Akbari, E. R. Chen, M. Engel, and S. C. Glotzer, Packing and Self-Assembly of Truncated Triangular Bipyramids, Phys. Rev. E 88, 012127 (2013).
39. Y. Kallus, V. Elser, and S. Gravel, Dense Periodic Packings of Tetrahedra with Small Repeating Units, Discrete Comput. Geom. 44, 245 (2010).
40. J. C. Lagarias, Mysteries in Packing Regular Tetrahedra, Notices Am. Math. Soc. 59, 1540 (2012).
41. A. Donev, F. H. Stillinger, P.M. Chaikin, and S. Torquato, Unusually Dense Crystal Packings of Ellipsoids, Phys. Rev. Lett. 92, 255506 (2004).
42. Y. Jiao, F. H. Stillinger, and S. Torquato, Optimal Packings of Superballs, Phys. Rev. E 79, 041309 (2009).
43. Y. Kallus and V. Elser, Dense-Packing Crystal Structures of Physical Tetrahedra, Phys. Rev. E 83, 036703 (2011).
44. M. Marechal and M. Dijkstra, Phase Behavior and Structure of Colloidal Bowl-Shaped Particles: Simulations, Phys. Rev. E 82, 031405 (2010).
45. T. Ras, R. Schilling, and M. Weigel, Regular Packings on Periodic Lattices, Phys. Rev. Lett. 107, 215503 (2011).
46. A. P. Gantapara, J. de Graaf, R. van Roij, and M. Dijkstra, Phase Diagram and Structural Diversity of a Family of Truncated Cubes: Degenerate Close-Packed Structures and Vacancy-Rich States, Phys. Rev. Lett. 111, 015501 (2013).
47. H. Minkowski, Geometrie der Zahlen (Teubner, Leipzig, 1896).
48. H. Minkowski, Dichteste gitterförmige Lagerung kongruenter Körper, Nachr. K. Ges. Wiss. Göttingen, 311 (1904); H. Minkowskiin Gesammelte Abhandlungen Vol. II (Teubner, Berlin, 1911), pp. 3-42.
49. G. Kuperberg and W. Kuperberg, Double-Lattice Packings of Convex Bodies in the Plane, Discrete Comput. Geom. 5, 389 (1990).
50. P. G. Szabó, M. C. Markót, T. Csendes, E. Specht, L. G. Casado, and I. García, New Approaches to Circle Packing in a Square with Program Codes (Springer-Verlag, New York, 2007).
51. E. R. Chen, Ph.D. Thesis, University of Michigan, 2010.
52. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.4.011024 for more details on Bravais lattice vectors, volume and face areas of our polyhedra, our numeric algorithm, and intersection equations for all regions.
53. B. Grünbaum and G. C. Shephard, Tilings and Patterns (W. H. Freeman, New York, 1987).
54. We exclude the irreducible spheric triangle groups δp,2,2 (axial dihedral, Dph) because its intermediate polyhedra (polygonal prisms, polygonal bipyramids) are very anisotropic.
55. A. Haji-Akbari, M. Engel, A. S. Keys, X. Zheng, R. G. Petschek, P. Palffy-Muhoray, and S. C. Glotzer, Disordered, Quasicrystalline and Crystalline Phases of Densely Packed Tetrahedra, Nature (London) 462, 773 (2009).
56. P. F. Damasceno, M. Engel, and S. C. Glotzer, Predictive Self-Assembly of Polyhedra into Complex Structures, Science 337, 453 (2012).
57. E. G. Gilbert, D.W. Johnson, and S. S. Keerthi, A Fast Procedure for Computing the Distance between Complex Objects in Three-Dimensional Space, IEEE Journal of robotics and automation 4, 193 (1988).
58. Y. Kallus, V. Elser, and S. Gravel, Method for Dense Packing Discovery, Phys. Rev. E 82, 056707 (2010).
59. E. Marcotte and S. Torquato, Efficient Linear Programming Algorithm to Generate the Densest Lattice Sphere Packings, Phys. Rev. E 87, 063303 (2013).
60. S. C. Glotzer and M. J. Solomon, Anisotropy of Building Blocks and Their Assembly into Complex Structures, Nat. Mater. 6, 557 (2007).
61. G. van Anders, N. K. Ahmed, D. Klotsa, M. Engel, and S. C. Glotzer, Unified Theoretical Framework for Shape Entropy in Colloids, arXiv:1304.7545.
62. G. van Anders, N. K. Ahmed, R. Smith, M. Engel, and S. C. Glotzer, Entropically Patchy Particles, arXiv:1309.1187.