Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Entropy favours open colloidal lattices


Burgeoning experimental and simulation activity seeks to understand the existence of self-assembled colloidal structures that are not close-packed1,2,3,4,5,6,7,8,9. Here we describe an analytical theory based on lattice dynamics and supported by experiments that reveals the fundamental role entropy can play in stabilizing open lattices. The entropy we consider is associated with the rotational and vibrational modes unique to colloids interacting through extended attractive patches10. The theory makes predictions of the implied temperature, pressure and patch-size dependence of the phase diagram of open and close-packed structures. More generally, it provides guidance for the conditions at which targeted patchy colloidal assemblies in two and three dimensions are stable, thus overcoming the difficulty in exploring by experiment or simulation the full range of conceivable parameters.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Examples of 2D open lattices.
Figure 2: Predicted entropy in 2D open colloidal structures and comparison with experiment.
Figure 3: Predicted phase diagram and comparison with experiment.
Figure 4: Predicted 3D structures.


  1. Antlanger, M., Doppelbauer, G. & Kahl, G. On the stability of Archimedean tilings formed by patchy particles. J. Phys. Condens. Matter 23, 404206 (2011).

    Article  Google Scholar 

  2. Torquato, S. Inverse optimization techniques for targeted self-assembly. Soft Matter 5, 1157–1173 (2009).

    Article  CAS  Google Scholar 

  3. Cohn, H. & Kumar, A. Algorithmic design of self-assembling structures. Proc. Natl Acad. Sci. USA 106, 9570–9575 (2009).

    Article  CAS  Google Scholar 

  4. Edlund, E., Lindgren, O. & Jacobi, M. N. Novel self-assembled morphologies from isotropic interactions. Phys. Rev. Lett. 107, 085501 (2011).

    Article  CAS  Google Scholar 

  5. Chen, Q., Bae, S. & Granick, S. Directed self-assembly of a colloidal kagome lattice. Nature 469, 381–384 (2011).

    Article  CAS  Google Scholar 

  6. Chen, Q. et al. Triblock colloids for directed self-assembly. J. Am. Chem. Soc. 133, 7725–7727 (2011).

    Article  CAS  Google Scholar 

  7. Romano, F. & Sciortino, F. Two-dimensional assembly of triblock Janus particles into crystal phases in the two bond per patch limit. Soft Matter 7, 5799–5804 (2011).

    Article  CAS  Google Scholar 

  8. Romano, F. & Sciortino, F. Patterning symmetry in the rational design of colloidal crystals. Nature Commun. 3, 975 (2012).

    Article  Google Scholar 

  9. Khalil, K. S. et al. Binary colloidal structures assembled through Ising interactions. Nature Commun. 3, 794 (2012).

    Article  Google Scholar 

  10. Glotzer, S. C. & Solomon, M. J. Anisotropy of building blocks and their assembly into complex structures. Nature Mater. 6, 557–562 (2007).

    Article  Google Scholar 

  11. Galisteo-López, J. F. et al. Self-assembled photonic structures. Adv. Mater. 23, 30–69 (2011).

    Article  Google Scholar 

  12. Su, B-L., Sanchez, C. & Yang, X-Y. Hierarchically Structured Porous Materials: From Nanoscience to Catalysis, Separation, Optics, Energy, and Life Science 55–129 (Wiley, 2012).

    Google Scholar 

  13. Maxwell, J. C. On the calculation of the equilibrium and stiffness of frames. Phil. Mag. 27, 294 (1864).

    Article  Google Scholar 

  14. Souslov, A., Liu, A. J. & Lubensky, T. C. Elasticity and response in nearly isostatic periodic lattices. Phys. Rev. Lett. 103, 205503 (2009).

    Article  Google Scholar 

  15. Kapko, V., Treacy, M., Thorpe, M. & Guest, S. On the collapse of locally isostatic networks. Proc. R. Soc. Lond. A 465, 3517 (2009).

    Article  Google Scholar 

  16. Hammonds, K. D., Dove, M. T., Giddy, A. P., Heine, V. & Winkler, B. Rigid-unit phonon modes and structural phase transitions in framework silicates. Am. Mineral. 81, 1057–1079 (1996).

    Article  CAS  Google Scholar 

  17. Sun, K., Souslov, A., Mao, X. & Lubensky, T. C. Surface phonons, elastic response, and conformational invariance in twisted kagome lattice. Proc. Natl Acad. Sci. USA 31, 12369–12374 (2012).

    Article  Google Scholar 

  18. Guest, S. D. & Hutchinson, J. W. On the determinacy of repetitive structures. J. Mech. Phys. Solids 51, 383–391 (2003).

    Article  Google Scholar 

  19. Yodh, A. et al. Entropically driven self-assembly and interaction in suspension. Phils. Trans. R. Soc. Lond. A 359, 921–937 (2001).

    Article  CAS  Google Scholar 

  20. Meng, G. et al. The free-energy landscape of clusters of attractive hard spheres. Science 327, 560–563 (2010).

    Article  CAS  Google Scholar 

  21. Chen, Q. et al. Supracolloidal reaction kinetics of Janus spheres. Science 331, 199–202 (2011).

    Article  CAS  Google Scholar 

  22. Kern, N. & Frenkel, D. Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction. J. Chem. Phys. 118, 9882–9889 (2003).

    Article  CAS  Google Scholar 

  23. Born, M. & Huang, K. Dynamic Theory of Crystal Lattices (Clarendon, 1998).

    Google Scholar 

  24. Keim, P., Maret, G., Herz, U. & von Grünberg, H. H. Harmonic lattice behavior of two-dimensional colloidal crystals. Phys. Rev. Lett. 92, 215504 (2004).

    Article  CAS  Google Scholar 

  25. Ghosh, A. et al. Density of states of colloidal glasses. Phys. Rev. Lett. 104, 248305 (2010).

    Article  Google Scholar 

  26. Ho, K. M., Chan, C. T. & Soukoulis, C. M. Existence of a photonic band gap in periodic dielectric structures. Phys. Rev. Lett. 65, 3152–3155 (1990).

    Article  CAS  Google Scholar 

  27. Romano, F., Sanz, E. & Sciortino, F. Role of the range in the fluid-crystal coexistence for a patchy particle model. J. Phys. Chem. B 113, 15133 (2009).

    Article  CAS  Google Scholar 

  28. Mao, X. & Lubensky, T. C. Coherent potential approximation of random nearly isostatic kagome lattice. Phys. Rev. E 83, 011111 (2011).

    Article  Google Scholar 

Download references


X.M. was supported by NSF under grant DMR-1104707. Q.C. and S.G. were supported by the US Department of Energy Division of Materials Science, under award number DE-FG02-07ER46471 through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign. For equipment, we acknowledge the National Science Foundation, CBET-0853737. We thank K. Chen for help with particle tracking, and J. Whitmer and T. C. Lubensky for useful discussions.

Author information

Authors and Affiliations



X.M. and Q.C. initiated this work; X.M. developed the theory and did calculations based on the theory; Q.C. and S.G. designed and performed the experiment; X.M., Q.C. and S.G. wrote the paper.

Corresponding author

Correspondence to Xiaoming Mao.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 416 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mao, X., Chen, Q. & Granick, S. Entropy favours open colloidal lattices. Nature Mater 12, 217–222 (2013).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing