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Broken symmetry and the variation of critical properties in the phase behaviour of supramolecular rhombus tilings


The tiling of surfaces has long attracted the attention of scientists, not only because it is intriguing intrinsically, but also as a way to control the properties of surfaces. However, although random tiling networks are studied increasingly, their degree of randomness (or partial order) has remained notoriously difficult to control, in common with other supramolecular systems. Here we show that the random organization of a two-dimensional supramolecular array of isophthalate tetracarboxylic acids varies with subtle chemical changes in the system. We quantify this variation using an order parameter and reveal a phase behaviour that is consistent with long-standing theoretical studies on random tiling. The balance between order and randomness is driven by small differences in intermolecular interaction energies, which can be related by numerical simulations to the experimentally measured order parameter. Significant variations occur with very small energy differences, which highlights the delicate balance between entropic and energetic effects in complex self-assembly processes.

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Figure 1: Tetracarboxylic acid supramolecular assemblies and rhombus tilings.
Figure 2: Comparison of experiments with simulations of the rhombus-tiling model.
Figure 3: Spatial variation of ordered phases.


  1. Elemans, J. A. A. W., Lei, S. & De Feyter, S. Molecular and supramolecular networks on surfaces: from two-dimensional crystal engineering to reactivity. Angew. Chem. Int. Ed. 48, 7298–7332 (2009).

    Article  CAS  Google Scholar 

  2. Bartels, L. Tailoring molecular layers at metal surfaces. Nature Chem. 2, 87–95 (2010).

    Article  CAS  Google Scholar 

  3. Blunt, M. O. et al. Random tiling and topological defects in a two-dimensional molecular network. Science 322, 1077–1081 (2008).

    Article  CAS  Google Scholar 

  4. Otero, R. et al. Elementary structural motifs in a random network of cytosine adsorbed on a gold(111) surface. Science 319, 312–315 (2008).

    Article  CAS  Google Scholar 

  5. Marschall, M. et al. Random two-dimensional string networks based on divergent coordination assembly. Nature Chem. 2, 131–137 (2010).

    Article  CAS  Google Scholar 

  6. Zhou, H. et al. Frustrated 2D molecular crystallization. J. Am. Chem. Soc. 129, 13774–13775 (2007).

    Article  CAS  Google Scholar 

  7. Fisher, M. E. Statistical mechanics of dimers on a plane lattice. Phys. Rev. 124, 1664–1672 (1961).

    Article  Google Scholar 

  8. Kasteleyn, P. W. Dimer statistics and phase transitions. J. Math. Phys. 4, 287–297 (1963).

    Article  Google Scholar 

  9. Henley, C. L. in Quasicrystals, the State of the Art (eds Di Vincenzo, D. P. & Steinhardt, P. J.) Ch. 15 (World Scientific, 1999).

  10. Destainville, N. Flip dynamics in octagonal rhombus tiling sets. Phys. Rev. Lett. 88, 030601 (2002).

    Article  CAS  Google Scholar 

  11. Wilson, D. B. Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14, 274–325 (2004).

    Article  Google Scholar 

  12. Lu, P. J. & Steinhardt, P. J. Decagonal and quasi-crystalline tilings in medieval Islamic architecture. Science 315, 1106–1110 (2007).

    Article  CAS  Google Scholar 

  13. Alet, F., Ikhlef, Y., Jacobsen, J. L., Misguich, G. & Pasquier, V. Classical dimers with aligning interactions on the square lattice. Phys. Rev. E 74, 041124 (2006).

    Article  Google Scholar 

  14. Papanikolaou, S., Luijten, E. & Fradkin, E. Quantum criticality, lines of fixed points, and phase separation in doped two-dimensional quantum dimer models. Phys. Rev. B 76, 134514 (2007).

    Article  Google Scholar 

  15. Castelnovo, C., Chamon, C., Mudry, C. & Pujol, P. Zero-temperature Kosterlitz–Thouless transition in a two-dimensional quantum system. Ann. Phys. 322, 903–934 (2007).

    Article  CAS  Google Scholar 

  16. Jacobsen, J. L. & Alet, F. Semiflexible fully packed loop model and interacting rhombus tilings. Phys. Rev. Lett. 102, 145702 (2009).

    Article  Google Scholar 

  17. Lackinger, M. & Heckl, W. M. Carboxylic acids: versatile building blocks and mediators for two-dimensional supramolecular self-assembly. Langmuir 25, 11307–11321 (2009).

    Article  CAS  Google Scholar 

  18. Zhao, J. F. et al. Molecule length directed self-assembly behaviour of tetratopic oligomeric phenylene–ethynylenes end-capped with carboxylic groups by scanning tunneling microscopy. J. Phys. Chem. C 114, 9931–9937 (2010).

    Article  CAS  Google Scholar 

  19. Blunt, M. et al. Directing two-dimensional molecular crystallization using guest templates. Chem. Commun. 2304–2306 (2008).

  20. Kampschulte, L. et al. Solvent induced polymorphism in supramolecular 1,3,5-benzenetribenzoic acid monolayers. J. Phys. Chem. C 110, 10829–10836 (2006).

    Article  CAS  Google Scholar 

  21. Tahara, K. et al. Two-dimensional porous molecular networks of dehydrobenzo[12]annulene derivatives via alkyl chain interdigitation. J. Am. Chem. Soc. 128, 16613–16625 (2006).

    Article  CAS  Google Scholar 

  22. Furukawa, S. et al. Structural transformation of a two-dimensional molecular network in response to selective guest inclusion. Angew. Chem. Int. Ed. 46, 2831–2834 (2007).

    Article  CAS  Google Scholar 

  23. Griessl, S. J. H. et al. Incorporation and manipulation of coronene in an organic template structure. Langmuir 20, 9403–9407 (2004).

    Article  CAS  Google Scholar 

  24. Wu, D., Deng, K., He, M., Zeng, Q. & Wang, C. Coadsorption-induced reconstruction of supramolecular assembly characteristics. ChemPhysChem 8, 1519–1523 (2007).

    Article  CAS  Google Scholar 

  25. Garrahan, J. P., Stannard, A., Blunt, M. O. & Beton, P. H. Molecular random tilings as glasses. Proc. Natl Acad. Sci. USA 106, 15209–15213 (2009).

    Article  CAS  Google Scholar 

  26. Stannard, A., Blunt, M. O., Beton, P. B. & Garrahan, J. P. Entropically stabilized growth of a two-dimensional random tiling. Phys. Rev. E 82, 041109 (2010).

    Article  Google Scholar 

  27. Gutzler, R. et al. Reversible phase transitions in self-assembled monolayers at the liquid–solid interface: temperature-controlled opening and closing of nanopores. J. Am. Chem. Soc. 132, 5084–5090 (2010).

    Article  CAS  Google Scholar 

  28. Yang, Y. and Wang, C. Curr. Opin. Colloid Interface Sci. 14, 135–147 (2009).

    Article  CAS  Google Scholar 

  29. Blunt, M. O. et al. Guest-induced growth of a surface-based supramolecular bilayer. Nature Chem. 3, 74–78 (2011).

    Article  CAS  Google Scholar 

  30. Cook, J. L., Hunter, C. A., Low, C. M. R., Perez-Velasco, A. & Vinter, J. G. Solvent effects on hydrogen bonding. Angew. Chem. Int. Ed. 46, 3706–3709 (2007).

    Article  CAS  Google Scholar 

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We thank the UK Engineering and Physical Sciences Research Council (EPSRC) for financial support under grant EP/D048761/01. J.P.G. was supported by EPSRC Grant No. GR/S54074/01. A.S. was supported by the Leverhulme Trust (ECF/2010/0380) and the EPSRC (EP/P502632/1). M.S. acknowledges receipt of a Royal Society Wolfson Merit Award and an European Research Council Advanced Grant. N.R.C. acknowledges receipt of a Royal Society Leverhulme Trust Senior Research Fellowship.

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A.S., J.C.R., M.O.B., J.P.G., N.R.C. and P.H.B. designed and conceived the experiment, A.S., J.C.R. and M.O.B. performed the experiments, A.S. and J.P.G. performed the numerical simulations, C.S., M.C.G-L., N.T., M.S. and N.R.C. prepared the materials, A.S., J.P.G., M.O.B., J.C.R. and P.H.B. analysed the data, A.S., J.P.G. and P.H.B. co-wrote the paper and all authors provided revisions and comments on the manuscript.

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Correspondence to Andrew Stannard or Peter H. Beton.

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Stannard, A., Russell, J., Blunt, M. et al. Broken symmetry and the variation of critical properties in the phase behaviour of supramolecular rhombus tilings. Nature Chem 4, 112–117 (2012).

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