Electronic Supplementary Material (ESI) for Soft Matter. This journal is © The Royal Society of Chemistry 2016

Supplementary Information: Self-assembly of a space-tessellating struc- ture in the binary system of hard tetrahedra and octahedra†

A. T. Cadotte,a‡, J. Dshemuchadse,b‡ P. F. Damasceno,a§ R. S. Newman,b and S. C. Glotzer∗a,b,c,d

Structure analysis with bond order diagrams 4-fold 3-fold 2-fold The crystal structures observed in this study were an- alyzed primarily by visual inspection. However, bond- orientational order diagrams (BODs) 1 were also found to 2 be instructive, especially when investigating structural sub- c -OT tleties. In the binary structure, the BODs were also gener- ated both for the mixture of polyhedra and for tetrahedra and octahedra separately. In the binary crystal, the centers of the tetrahedra form a simple cubic lattice, and the octa- (T)

hedra form a -centered cubic lattice. The BODs of the 2 binary structure and the octahedra packings are shown in

Fig. S1, the ones of the dodecagonal quasicrystal in Fig. S2. c -OT The quasicrystal of tetrahedra has a particular signature in the BOD, resembling a cylinder and resulting in dark spots aligned with the 12-fold axis of the dodecagonal structure. (O) Quasicrystal binary-crystal coexistence 2

Small numbers of octahedra that are present in a sim- c -OT ulation of mainly tetrahedra have a preferred alignment within the quasicrystal. In some cases, they form five- rings, alternating with pentagonal of tetrahe- dra, similar to a model of dodecagonal Al–Mn 2–4, as shown

in Fig. S3. t -O Binary crystal structure The binary crystal consists of layers of alternating tetrahe- dra and octahedra that share faces with each other, i.e., with only O–T nearest neighbor contacts. This leads to

each having six neighbors within the layer (all m -O tetrahedra) and each having three neighbors within the layer (all octahedra). The remaining faces – two per octahedron and one per tetrahedron – form a triangular lattice on the outside of the layer and are again arranged Fig. S1: Bond-orientational order diagrams (BODs), from top to alternatingly. The ratio of octahedral to tetrahedral faces in bottom, of the full binary structure c-OT2, as well as of only the this tiling is O:T = 1:1; each octahedron contributes two tetrahedra (T), only the octahedra (O), and of the trigonal and monoclinic octahedra packings, t-O and m-O, respectively. Shown are the 4-fold / pseudo-4-fold, 3-fold / pseudo-3-fold, and 2-fold crystallographic directions. a Applied Physics Program, University of Michigan, Ann Arbor, Michigan 48109, USA. b Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA. c Department of Materials Science and Engineering, University of Michigan, Ann Arbor, and each tetrahedron contributes one, but their Michigan 48109, USA. 1:2 composition overall and in the layer results in equal d Biointerfaces Institute, University of Michigan, Ann Arbor, Michigan 48109, USA. E-mail: [email protected] numbers on the outside of the layer (see Fig. 1 in the main † Article available at DOI: 10.1039/C6SM01180B text). ‡ These authors contributed equally to this work. § Present address: Department of Cellular and Molecular Pharmacology, University The resulting triangular lattices can be aligned alternat- of California, San Francisco, California 94158, USA. ingly, joining octahedral faces with tetrahedral ones and

1 / 3 3 / 2 y ntefloigodr erhdo,aohrtetrahe- another tetrahedron, shared a order, layers, following neighboring the second between in a by, located also edge is of there kind however, ex- configuration, also this hibits hexagonal gyrated, The respectively. octahedron, another an and tetrahedron, tetrahedron, a another by octahedron, shared al- are the of honeycomb in cubic kind Edges ternated second honeycomb. gyrated the of the of in kinds results solely stacking two composed structure has A structure edges. hexagonal gyrated, al- the The edge-transitive, while also polyhedra. is convex structure honeycomb from cubic built ternated or are they uniform as also convex, are honey- manuscript the this vertex-transitive, of in Both discussed . combs hon- space-filling A a emerges. is em- honeycomb eycomb is cubic – alternated layers the ployed, between arrangements polyhedral nating in in ways. variants of combined stacking number be infinite different an can with arrangements case between both the plane sphere-packings, is mirror As a forming layers. another, different an- one from joins join faces face layers tetrahedral octahedral and face each octahedral that 60 other by so rotated vertex be occur any can already around they Alternatively, that layer. contacts the same within the creating versa, vice stacked be can that octahedra five tetrahedra. of of bipyramids ring struc- pentagonal 12-QC a with the with of octahedra consisting of motif ture, intergrowth An S3: Fig. are Shown directions. do- tetrahedra. crystallographic the of 2-fold of and up 12-fold (BODs) made the diagrams 12-QC order structure Bond-orientational decagonal S2: Fig. fol h rtkn fsakn eutn nalter- in resulting – stacking of kind first the only If

12-QC 12-fold i.e. l etcsaeeuvln,a elas well as equivalent, are vertices all , 2-fold ◦ X .L n .Frey, F. and Li Z. X. 3 Li, Z. X. 2 Denton, R. A. and Roth J. 1 References described were Gabbrielli – by voids detail pack- in tetrahedral as with introduced octahedra here of – ings lengths edge different tetrahe- with and octahedra dra of tilings of re- the The of of dra. length packing is edge [T] a the voids as where tetrahedral 1:6, of = thought O:[T] be with octahedra can phase, Minkowski The the can direction. that one plane along one shear move to perpendicular respect another with parallel sheared and a be another) for can ar- layers allow are (the that octahedra plane layers shear The trigonal stacked octahedra. in is the ranged voids of tetrahedral length the edge of = the length O:[T] edge stoichiometry the a where in 1:4, [T] voids tetrahedral with dra h w caer tutrsaesoni i.S.The S4. Fig. in shown are m structures octahedra two The structure(s) crystal Octahedra employed parameter. the order BOD, with identi- Steinhardt environment the were crystalline that a of particles having clarity of as number fied the highest by the as determined well fraction was as packing This a at and 60%. 1:2 of = O:T of stoichiometry ideal the within faults stacking these structure. be map can easily layers to between used located are that tetrahedra sharing packings between hexagonal-close analogy and the cubic-close to equivalent structure derived cubic be the can from description atomic no crystallographic has its structure equivalent; honeycomb gyrated The 1/4. 1/4, e tcigfut,wihcm ntefr ftehexag- the of form the in onal come which faults, stacking few a cubic 5/8. position Wyckoff ing is group space position is group lentdcbchnyobi urt CaF fluorite is honeycomb cubic alternated octahedron. second a and octahedron, an dron, Osrcuecnb nesoda akn foctahe- of packing a as understood be can structure -O sepce,tebnr rsa em ofr eta the at best form to seems crystal binary the expected, As ihntesaiiyrneo h iayOT binary the of range stability the Within h rsalgahcsrcuetp orsodn othe to corresponding type structure crystallographic The 1995, 270. 6857. h c -OT -OT Fm 51 4 2 a 2 caCytlorpiaScinB Section Crystallographica Acta 271–275. , tutr.Tesotbn eghbtenface- between length bond short The structure. 3 ¯ ,0 n erhdateF-position the tetrahedra and 0 0, 0, tutr sdmnn,bti nesesdwith interspersed is but dominant, is structure m n.25,othdaocp h Ca-Wyckoff the occupy octahedra 225), (no. P 6 3 / mmc 2 a tal. et ,0 n tetrahedra and 0 0, 0, 1 n.14 ihothdaoccupy- octahedra with 194) (no. caCytlorpiaScinB Section Crystallographica Acta / 3 5 h delnt fteoctahe- the of length edge the . hs e.E Rev. Phys. t Osrcue lotermed also structure, -O 2000, , ccp 1995, , 2 and h space the : 2 4 f hs,the phase, 61 /,2/3, 1/3, 51 hcp 6845– , 8 c 265– , c The . -OT 1/4, 1 / 2 2 , Fig. S4: Two packings of octahedra: the trigonal closest pack- ing of octahedra, t-O structure (top) and the monoclinic layered structure m-O (bottom).

4 E. A. Lord, A. L. Mackay and S. Ranganathan, New Ge- ometries for New Materials, Cambridge University Press, 2006. 5 R. Gabbrielli, Y. Jiao and S. Torquato, Physical Review E, 2012, 86, 041141.

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