A theory of entropic bonding


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


Entropy alone can self-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 widely used to study the self-assembly of ordered phases from disordered fluid 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 theory from first principles. By calculating and minimizing the excluded volume within the framework of statistical mechanics, we describe the directional entropic forces that emerge collectively between the hard forms, in the colloquial terms used to describe chemical bonds. We validate our theory by showing that it predicts thermodynamically preferred structures for four families of hard polyhedra which correspond, in all cases, to previous simulation results. The success of this first-principles approach to the prediction of entropic colloidal crystal structure promotes fundamental understanding of both entropic crystallization and conceptual images of bonding in matter.


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

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

  • The authors declare no competing interests.

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

Data availability

All relevant data and codes are available from the University of Michigan Deep Blue Repository (DOI: 10.7302/1b70-7970). All other study data is included in the article and/or IS Annex.

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