A theory of entropic binding



Computer simulations of hard, polyhedrally shaped particles predict a wide variety of complex colloidal crystal structures through self-assembly. The structural similarities between colloidal crystals and atomic crystals suggest that they should be describable in analogous, albeit different, conceptual frameworks. Like the chemical bonds that hold atoms together in crystals, the statistical, evolving forces that hold hard colloidal particles together should be describable in the language of bonding. While atomic crystals can be predicted a priori by solving the Schrödinger equation, we present an entropic bonding theory that allows the prediction of colloidal crystals by solving another eigenvalue equation, facilitated by the use of mathematically constructed shape orbitals analogous to atomic orbitals.


Entropy alone can assemble hard nanoparticles into colloidal crystals of remarkable complexity, whose structures are the same as atomic and molecular crystals, but with larger lattice spacings. Molecular simulation is a powerful tool that is widely used to study the self-assembly of ordered phases from disordered liquid phases of atoms, molecules, or nanoparticles. However, it is not yet possible to predict colloidal crystal structures a priori from particle shape, as we can for atomic crystals from electronic valence. Here we present such a first-principles theory. By calculating and minimizing excluded volume within the framework of statistical mechanics, we describe the directional entropic forces that arise collectively between hard shapes in terms familiar to those used to describe chemical bonds. We validate our theory by showing that it predicts thermodynamically preferred structures for four families of hard polyhedra, which in each case agree with previous simulation results. The success of this first-principle approach to predicting the entropic colloidal crystal structure advances fundamental understanding of both entropically driven crystallization and conceptual pictures of bonding in matter.


    • Accepted December 14, 2021.
  • Author contributions: TV and SCG designed research; TV and SCG conducted research; Television contributed new reagents/analytical tools; TV and SCG analyzed data; TV and SCG wrote the newspaper; and SCG supervised research.

  • Reviewers: RK, University of Pennsylvania; JK, Ecole Normale Superieure; and HL, Debye Research Institute of Utrecht University.

  • The authors declare no competing interests.

  • This article has supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2116414119/-/DCSupplemental.

data availability

All relevant data and codes are available at the University of Michigan’s Deep Blue Repository (DOI:10.7302/1b70-7970). All other study data are contained in the article or SI Appendix.


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