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2008, 112, 16186–16188 Published on Web 09/27/2008
Optimizing the Number of Components in a Molecular Quasicrystal: A Three-Component Material Based on the Penrose Tiling Zhongfu Zhou and Kenneth D. M. Harris* School of Chemistry, Cardiff UniVersity, Park Place, Cardiff CF10 3AT, Wales ReceiVed: August 18, 2008
We report the design of a new quasicrystalline material constructed from discrete molecular building units and based on the Penrose tiling as the basic structural template. The quasicrystal comprises three different molecular components, which is shown to represent the minimum number of components required for a molecular representation of the Penrose tiling. The density of this molecular quasicrystal is comparable to typical densities of crystalline organic materials. With regard to both the number of molecular components and density, the experimental realization of the molecular quasicrystal reported in this paper is considerably more promising than a seven-component, low-density representation of the Penrose tiling that represents the only previous reported example of a molecular quasicrystal. Quasicrystals were discovered in 1984 when it was observed1 that some metal alloys exhibit diffraction patterns comprising sharp Bragg reflections characteristic of crystalline materials but based on symmetries (e.g., 10-fold or 5-fold) that are forbidden2 for crystals with long-range periodic order. Such diffraction patterns3 can be understood by recognizing that the Fourier transforms of quasiperiodic tilings, such as the Penrose tiling,4 give rise to patterns of sharp maxima5 (Bragg reflections) based on forbidden symmetries. Such quasiperiodic tilings6 are constructed from a set of geometrically well-defined tiles, assembled according to well-defined rules, but do not have translational periodicity. A number of approaches have been advanced to rationalize the structures of quasicrystals, including analogies to the Penrose tiling7 and other quasiperiodic models such as the cluster model,8 and symmetry properties have been rationalized in higher dimensional superspaces.9 In spite of huge interest in quasicrystals, examples reported to date have been dominated by metal alloys, and no quasicrystal based on discrete molecular building units was proposed or reported until recently.10 The proposed design of a molecular quasicrystal was based on the Penrose tiling as the structural template and employed similar “crystal engineering” design principles11 to those that are used to design crystalline molecular solids. The Penrose tiling (Figure 1) is constructed from two types of tile: a thick rhombus (vertices 108 and 72°) and a thin rhombus (vertices 144 and 36°). In the designed molecular quasicrystal,10 the molecules represent the nodes of the tiling and the two types of tile arise from the region of space between groups of four molecules. Linear intermolecular linkages between adjacent molecules represent the lines between adjacent nodes on the tiling. The Penrose tiling has seven different types of node, each with different local geometry (Figure 1), and a set of seven molecules with the same geometric properties as each type of node was proposed.10 As the angles between the * To whom correspondence
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Figure 1. A section of Penrose tiling. The red circles indicate an example of each of the seven different types of node.
lines at each node are integer multiples of 36°, a molecular core with approximate 10-fold symmetry is required, and 10,5coronene (C30H10) is appropriate for this purpose. Linear intermolecular linkages are formed by substituents of the type -CtC-CO2H on the 10,5-coronene core. The substituents of adjacent molecules interact via the carboxylic acid “dimer” motif involving two O-H · · · O hydrogen bonds. The seven molecules that match the local geometries of the seven types of node on the Penrose tiling are members of the general family C30H10-m(CtC-CO2H)m. The molecular quasicrystal constructed from this strategy was shown to be energetically stable and to have an X-ray diffraction pattern comprising sharp Bragg maxima based on a 10-fold symmetric reciprocal space, resembling the diffraction patterns observed experimentally for quasicrystalline metal alloys. Although the previous work10 led to the successful design of a quasicrystal constructed from molecular building units, it 2008 American Chemical Society
Letters suffers from two important limitations with regard to the ultimate experimental realization of the concept. First, the material comprises seven different molecular components, based on matching the local geometries of the seven types of node on the Penrose tiling. The requirement to self-assemble a sevencomponent material would be associated with significant experimental challenges, not least the fact that there are a huge number of competing pathways to form crystalline phases containing one or more components. Second, the density (0.85 g cm-3) of the proposed quasicrystalline material (based on stacking the two-dimensional planes of Penrose tiling) is significantly lower than typical densities for organic crystalline materials. Here we address both of these limitations by proposing a new molecular quasicrystal, for which the design is focused on identifying the minimum number of molecular components required to construct a stable molecular representation of the Penrose tiling. Initially, we consider using only the molecule C30(CtC-CO2H)10, based on 10,5-coronene with 10 substituents of the type -CtC-CO2H. This molecule could occupy any of the seven types of node on the Penrose tiling and establish all the required intermolecular linkages to adjacent molecules, but some of the substituents would be “redundant” as they would not form linkages to adjacent molecules (the highest number of neighbors for any of the seven types of node on the Penrose tiling is seven; see Figure 1). Such redundant substituents would point inside the thick or thin rhombuses of the tiling, rather than forming lines between adjacent nodes. However, a problem arises for redundant substituents that would point inside the thin rhombus, as such substituents would “clash” with the redundant substituents of the molecule at the opposite vertex of the thin rhombus. On the other hand, redundant substituents that point inside the thick rhombus would not cause such clashes, as there is sufficient space inside the thick rhombus to accommodate these substituents. Thus, the problem with using only molecules of the type C30(CtC-CO2H)10 arises from the redundant substituents that point inside the thin rhombus. Such redundant substituents arise only at the 144° vertices of the thin rhombus (there are no redundant substituents at the 36° vertices), and thus the 144° vertices of the thin rhombus require special consideration. The seven types of node on the Penrose tiling can be subdivided into three categories depending on the number of 144° vertices at the node12 (Figure 2): (A) four types of node have no 144° vertices, (B) two types of node have one 144° vertex, and (C) one type of node has two 144° vertices. On this basis, we propose that the Penrose tiling may be constructed from three types of molecule, shown in Figure 2; the three types of molecule are suitable for occupying nodes of types (A), (B), and (C), respectively. The three molecules are based on C30(CtC-CO2H)10 but with removal of any redundant substituents that would be present at the 144° vertex of a thin rhombus. Thus, as shown in Figure 2, the C30(CtC-CO2H)10 molecule can occupy any of the four nodes of type (A), the molecule based on C30(CtC-CO2H)10 but with one set of three adjacent -CtC-CO2H substituents removed (and replaced by H) can occupy either of the two nodes of type (B), and the molecule based on C30(CtC-CO2H)10 but with two sets of three adjacent -CtC-CO2H substituents removed (and replaced by H) can occupy the node of type (C). On the basis of the above design strategy, a molecular Penrose tiling constructed13 using only the three molecules defined above is shown in Figure 3. The X-ray diffraction pattern calculated for this molecular quasicrystal is shown in Figure 4 and clearly
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Figure 2. (a) The four nodes of type (A), (b) the two nodes of type (B), and (c) the node of type (C), together with the corresponding molecular representation (shown on the right side) based on 10,5coronene with an appropriate arrangement of -CtC-CO2H substituents.
Figure 3. Molecular quasicrystal constructed from the three types of molecule shown in Figure 2 and showing the same region of Penrose tiling as Figure 1. Examples of the thick and thin rhombuses are shaded red and green respectively. Note that the thick rhombus is filled with six redundant substituents.
comprises sharp Bragg-like maxima, the positions and intensities of which define a 10-fold symmetric reciprocal space similar
16188 J. Phys. Chem. C, Vol. 112, No. 42, 2008
Letters In summary, the new molecular quasicrystal proposed here is based on just three different types of molecule, which, as discussed above, represents the minimum number of components required for a molecular representation of the Penrose tiling. Furthermore, the density of this quasicrystal is predicted to be comparable to typical densities of crystalline organic materials. In both respects, the experimental realization of the molecular quasicrystal reported here is considerably more promising than the seven-component,15 lower-density molecular quasicrystal proposed previously.10 Acknowledgment. We are grateful to Cardiff University for financial support. References and Notes
Figure 4. X-ray diffraction pattern calculated for the new threecomponent molecular quasicrystal shown in Figure 3 (in practice, the diffraction pattern is calculated for a much larger section of the quasicrystal than that shown in Figure 3). Note: 1/a ) 0.0468 Å-1, where a is the distance between the centers of adjacent molecules on the tiling.
Figure 5. An example of a thick rhombus in the new molecular quasicrystal, demonstrating the filling of the thick rhombus with redundant substituents.
to the diffraction patterns observed for experimentally known quasicrystalline materials. A feature of this new molecular quasicrystal is that six redundant substituents point inside each thick rhombus (Figure 5),14 whereas in the molecular quasicrystal proposed previously10 the thick rhombus is empty. An advantageous feature of filling the thick rhombus in this way is that the density of the tiling is increased. The density of the three-dimensional quasicrystal formed by stacking the planes of Penrose tiling in the new molecular quasicrystal (assuming 3.4 Å stacking of planes) is 1.29 g cm-3 and is significantly higher than the density (0.85 g cm-3) of the molecular quasicrystal proposed previously.10
(1) Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. W. Phys. ReV. Lett. 1984, 53, 1951. (2) Bravais, A. J. Ecole Polytech. (Paris) 1850, 19, 1. (3) (a) Levine, D.; Steinhardt, P. J. Phys. ReV. Lett. 1984, 53, 2477. (b) Levine, D.; Steinhardt, P. J. Phys. ReV. B 1986, 34, 596. (c) Socolar, J. E. S.; Steinhardt, P. J. Phys. ReV. B 1986, 34, 617. (4) Penrose, R. Bull. Inst. Math. Appl. 1974, 10, 266. (5) Mackay, A. L. Physica A 1982, 114, 609. (6) (a) de Bruijn, N. G. Proc. K. Ned. Akad. Wet. Ser. 1981, 84, 39– 53. (b) Senechal, M. Quasicrystals and Geometry; Cambridge University Press: Cambridge, 1996. (7) (a) Introduction to Quasicrystals; Jaric´, M. V., Ed.; Academic Press: San Diego, CA, 1988. (b) Janot, C. Quasicrystals: A Primer, 2nd Edition; Oxford University Press: Oxford, 1994. (c) Bursill, L. A.; Lin, P. J. Nature 1985, 316, 50. (8) (a) Steinhardt, P. J.; Jeong, H.-C.; Saitoh, K.; Tanaka, M.; Abe, E.; Tsai, A. P. Nature 1998, 396, 55. (b) Steinhardt, P. J.; Jeong, H.-C. Nature 1996, 382, 431. (c) Steinhardt, P. J. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 14267. (d) Jeong, H.-C.; Steinhardt, P. J. Phys. ReV. B 1997, 55, 3520. (9) (a) Janssen, T. Acta Crystallogr., Sect. A 1986, 42, 261. (b) van Smaalen, S. Phys. ReV. B 1989, 39, 5850. (c) Elser, V. Phys. ReV. B 1985, 32, 4892. (d) Prince, E. Acta Crystallogr., Sect. A 1987, 43, 393. (e) Caspar, D. L. D.; Fontano, E. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 14271. (f) Lifshitz, R. Physica A 1996, 232, 633. (g) Takakura, H.; Pay Go´mez, C.; Yamamoto, A.; de Boissieu, M.; Tsai, A. P. Nat. Mater. 2007, 6, 58. (10) Zhou, Z.; Harris, K. D. M. ChemPhysChem 2006, 7, 1649. (11) (a) Schmidt, G. M. J. Pure Appl. Chem. 1971, 27, 647. (b) Thomas, J. M. Philos. Trans. R. Soc. London 1974, 277, 251. (c) Adams, J. M.; Pritchard, R. G.; Thomas, J. M. J. Chem. Soc., Chem. Commun. 1976, 358. (e) Desiraju, G. R. Crystal Engineering: The Design of Organic Solids; Elsevier: Amsterdam, 1989. (f) Etter, M. C. Acc. Chem. Res. 1990, 23, 120. (g) Harris, K. D. M.; Hollingsworth, M. D. Nature 1989, 341, 19. (h) Desiraju, G. R. Angew. Chem., Int. Ed. 2007, 46, 8342. (12) A node cannot have three (or more) vertices of this type as 3 × 144° ) 432°, and the sum of angles for all vertices at a node cannot exceed 360°. (13) The general method for constructing the Penrose tiling and inserting molecules at appropriate nodes was the same as that used previously10 (the present work differs in the number and type of molecules used to represent nodes on the tiling). The molecular quasicrystalline tiling was subjected to energy minimization using the method and parameterization discussed previously10 and was shown to be energetically stable. (14) The distances between the carboxylic acid groups of the six redundant substituents within the thick rhombus are not sufficiently close to form significant intermolecular hydrogen bonding interactions. (15) For crystalline organic materials, there are many examples of threecomponent cocrystals, but to our knowledge there has been no reported case of a cocrystal composed of seven different types of molecule.
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