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Entropy compartmentalization stabilizes open host–guest colloidal clathrates

Nature Chemistry volume 15, pages 905–912 (2023)Cite this article

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Abstract

Clathrates are open crystals in which molecules are arranged in a hierarchy of polyhedral cages that encapsulate guest molecules and ions. As well as holding fundamental interest, molecular clathrates serve practical purposes, such as for gas storage, and their colloidal counterparts also appear promising for host–guest applications. Here using Monte Carlo simulations, we report the entropy-driven self-assembly of hard truncated triangular bipyramids into seven different host–guest colloidal clathrate crystals with unit cells ranging from 84 to 364 particles. The structures consist of cages that are either empty or occupied by guest particles, which can be different from or identical to the host particles. The simulations point to crystallization occurring through the compartmentalization of entropy between low- and high-entropy subsystems for the host and the guest particles, respectively. We use entropic bonding theory to design host–guest colloidal clathrates with explicit interparticle attraction, providing a route to realize such systems in the laboratory.

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Fig. 1: Particle design for host–guest colloidal clathrates.
Fig. 2: Self-assembly simulation results.
Fig. 3: Entropy compartmentalization and free energy calculation.
Fig. 4: Rotational motion of guests in clathrate cages.

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Data availability

All data that support the conclusions in this manuscript are present in the main text or the supplementary materials. Source data are provided with this paper.

Code availability

Source code for HOOMD-blue is available at https://github.com/glotzerlab/hoomd-blue. Sample codes are included in the Supplementary Information.

References

  1. Claussen, W. F. Suggested structures of water in inert gas hydrates. J. Chem. Phys. 19, 259–260 (1951).

    Article  CAS  Google Scholar 

  2. Sloan, E. D. Jr, Koh, C. A. & Koh, C. A. Clathrate Hydrates of Natural Gases (CRC Press, 2007); https://doi.org/10.1201/9781420008494

  3. Davidson, D. W. The motion of guest molecules in clathrate hydrates. Can. J. Chem. 49, 1224–1242 (1971).

    Article  CAS  Google Scholar 

  4. Lokshin, K. A. et al. Structure and dynamics of hydrogen molecules in the novel clathrate hydrate by high pressure neutron diffraction. Phys. Rev. Lett. 93, 125503 (2004).

    Article  PubMed  Google Scholar 

  5. Dolyniuk, J.-A., Owens-Baird, B., Wang, J., Zaikina, J. V. & Kovnir, K. Clathrate thermoelectrics. Mater. Sci. Eng. R. Rep. 108, 1–46 (2016).

    Article  Google Scholar 

  6. Kasper, J. S., Hagenmuller, P., Pouchard, M. & Cros, C. Clathrate structure of silicon Na8Si46 and NaxSi136 (x < 11). Science 150, 1713–1714 (1965).

    Article  CAS  PubMed  Google Scholar 

  7. Lin, H. et al. Clathrate colloidal crystals. Science 355, 931–935 (2017).

    Article  CAS  PubMed  Google Scholar 

  8. Lee, S., Teich, E. G., Engel, M. & Glotzer, S. C. Entropic colloidal crystallization pathways via fluid–fluid transitions and multidimensional prenucleation motifs. Proc. Natl Acad. Sci. USA 116, 14843–14851 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Mao, W. L. Hydrogen clusters in clathrate hydrate. Science 297, 2247–2249 (2002).

    Article  CAS  PubMed  Google Scholar 

  10. Gutt, C., Press, W., Hller, A. & Tse, J. S. Rotational dynamics in methane hydrate. Appl. Phys. A Mater. Sci. Process. 74, s1299–s1301 (2002).

    Article  CAS  Google Scholar 

  11. Patchkovskii, S. & Tse, J. S. Thermodynamic stability of hydrogen clathrates. Proc. Natl Acad. Sci. USA 100, 14645–14650 (2003).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Ratcliffe, C. I. & Ripmeester, J. A. Proton and carbon-13 NMR studies on carbon dioxide hydrate. J. Phys. Chem. 90, 1259–1263 (1986).

    Article  CAS  Google Scholar 

  13. Nguyen, N. N. et al. ‘Liquid-like’ water in clathrates induced by host-guest hydrogen bonding. J. Phys. Chem. C 125, 15751–15757 (2021).

    Article  CAS  Google Scholar 

  14. Li, Q. et al. Ab initio energetics and kinetics study of H2 and CH4 in the SI clathrate hydrate. Phys. Rev. B 84, 153103 (2011).

    Article  Google Scholar 

  15. Tse, J. S. et al. Anharmonic motions of Kr in the clathrate hydrate. Nat. Mater. 4, 917–921 (2005).

    Article  CAS  PubMed  Google Scholar 

  16. Damasceno, P. F., Engel, M. & Glotzer, S. C. Predictive self-assembly of polyhedra into complex structures. Science 337, 453–457 (2012).

    Article  CAS  PubMed  Google Scholar 

  17. Frenkel, D. Order through entropy. Nat. Mater. 14, 9–12 (2015).

    Article  CAS  PubMed  Google Scholar 

  18. Sloan, E. D. Jr. Fundamental principles and applications of natural gas hydrates. Nature 426, 353–359 (2003).

    Article  CAS  PubMed  Google Scholar 

  19. Moore, T. C., Anderson, J. A. & Glotzer, S. C. Shape-driven entropic self-assembly of an open, reconfigurable, binary host-guest colloidal crystal. Soft Matter 17, 2840–2848 (2021).

    Article  CAS  PubMed  Google Scholar 

  20. Anderson, J. A. A., Eric Irrgang, M., Glotzer, S. C. C., Irrgang, M. E. & Glotzer, S. C. C. Scalable Metropolis Monte Carlo for simulation of hard shapes. Comput. Phys. Commun. 204, 21–30 (2016).

    Article  CAS  Google Scholar 

  21. Frank, F. C. & Kasper, J. S. Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures. Acta Crystallogr. 12, 483–499 (1959).

    Article  CAS  Google Scholar 

  22. Frenkel, D. & Ladd, A. J. C. New Monte Carlo method to compute the free energy of arbitrary solids. Application to the fcc and hcp phases of hard spheres. J. Chem. Phys. 81, 3188–3193 (1984).

    Article  CAS  Google Scholar 

  23. Haji-Akbari, A., Engel, M. & Glotzer, S. C. Phase diagram of hard tetrahedra. J. Chem. Phys. 135, 194101 (2011).

    Article  PubMed  Google Scholar 

  24. Shen, W. et al. Symmetries in hard polygon systems determine plastic colloidal crystal mesophases in two dimensions. Soft Matter 15, 2571–2579 (2019).

    Article  CAS  PubMed  Google Scholar 

  25. Vo, T. & Glotzer, S. C. A theory of entropic bonding. Proc. Natl Acad. Sci. USA 119, e2116414119 (2022).

    Article  PubMed  PubMed Central  Google Scholar 

  26. Ye, X. et al. Competition of shape and interaction patchiness for self-assembling nanoplates. Nat. Chem. 5, 466–473 (2013).

    Article  CAS  PubMed  Google Scholar 

  27. Ramasubramani, V., Vo, T., Anderson, J. A. & Glotzer, S. C. A mean-field approach to simulating anisotropic particles. J. Chem. Phys. 153, 084106 (2020).

    Article  CAS  PubMed  Google Scholar 

  28. Wang, S. et al. The emergence of valency in colloidal crystals through electron equivalents. Nat. Mater. 21, 580–587 (2022).

    Article  CAS  PubMed  Google Scholar 

  29. Elbert, K. C. et al. Anisotropic nanocrystal shape and ligand design for co-assembly. Sci. Adv. 7, eabf9402 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  31. Fleischer, E. B. & Janda, K. C. Prediction of clathrate structure type and guest position by molecular mechanics. J. Phys. Chem. A 117, 4001–4010 (2013).

    Article  CAS  PubMed  Google Scholar 

  32. Ripmeester, J. A., Ratcliffe, C. I. & Cameron, I. G. NMR studies of guest dynamics in clathrate hydrates: spherical tops SF6, SeF6 and CH4 in structure II hydrate. J. Phys. Chem. B 108, 929–935 (2004).

    Article  CAS  Google Scholar 

  33. White, M. A. & MacLean, M. T. Rotational freedom of guest molecules in tetrahydrofuran clathrate hydrate, as determined by heat capacity measurements. J. Phys. Chem. 89, 1380–1383 (1985).

    Article  CAS  Google Scholar 

  34. Zhu, J. et al. Encapsulation kinetics and dynamics of carbon monoxide in clathrate hydrate. Nat. Commun. 5, 4128 (2014).

    Article  CAS  PubMed  Google Scholar 

  35. Alavi, S., Susilo, R. & Ripmeester, J. A. Linking microscopic guest properties to macroscopic observables in clathrate hydrates: Guest-host hydrogen bonding. J. Chem. Phys. 130, 174501 (2009).

    Article  PubMed  Google Scholar 

  36. Jacobson, L. C. & Molinero, V. A methane-water model for coarse-grained simulations of solutions and clathrate hydrates. J. Phys. Chem. B 114, 7302–7311 (2010).

    Article  CAS  PubMed  Google Scholar 

  37. Anderson, J. A., Glaser, J. & Glotzer, S. C. HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations. Comput. Mater. Sci. 173, 109363 (2020).

    Article  CAS  Google Scholar 

  38. Glaser, J. et al. Strong scaling of general-purpose molecular dynamics simulations on GPUs. Comput. Phys. Commun. 192, 97–107 (2015).

    Article  CAS  Google Scholar 

  39. Frenkel, D. & Smit, B. Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, 2001).

  40. Blavatska, V., von Ferber, C. & Holovatch, Y. Shapes of macromolecules in good solvents: field theoretical renormalization group approach. Condens. Matter Phys. 14, 33701 (2011).

    Article  Google Scholar 

  41. Spellings, M., Marson, R. L., Anderson, J. A. & Glotzer, S. C. GPU accelerated discrete element method (DEM) molecular dynamics for conservative, faceted particle simulations. J. Comput. Phys. 334, 460–467 (2017).

    Article  Google Scholar 

  42. Towns, J. et al. XSEDE: accelerating scientific discovery. Comput. Sci. Eng. 16, 62–74 (2014).

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported as part of the Center for Bio-Inspired Energy Science, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under award no. DE-SC0000989. T.V. was supported by the Office of the Undersecretary of Defense for Research and Engineering, Newton Award for Transformative Ideas during the COVID-19 Pandemic, Award HQ00342010030. With support from XSEDE award DMR 140129, simulations were conducted on resources provided by the Extreme Science and Engineering Discovery Environment (XSEDE)42, which is supported by National Science Foundation grant no. ACI-1053575. Computational resources and services were also supported by Advanced Research Computing at the University of Michigan, Ann Arbor. The funders had no role in the conceptualization, design, data collection, analysis, decision to publish, or preparation of the manuscript.

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, University of Michigan, Ann Arbor, MI, USA

    Sangmin Lee, Thi Vo & Sharon C. Glotzer

  2. Biointerfaces Institute, University of Michigan, Ann Arbor, MI, USA

    Sharon C. Glotzer

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Contributions

S.L. designed the research and performed HPMC simulations and related data analysis. T.V. performed EBT calculations and related data analysis. S.L. and T.V. performed MD simulations. S.C.G. directed the research. All authors contributed to data analysis and manuscript preparation.

Corresponding author

Correspondence to Sharon C. Glotzer.

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Nature Chemistry thanks Emanuela Zaccarelli and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 TBP truncation.

The truncation parameter S is defined as the length ratio of the maximum truncation length (ltr_max). The ltr_max is defined as the maximum distance of the vertex truncation without overlapping the two truncation directions. The regular TBP (S = 0.0) vertices are: (1,1,1), (−1,−1,−1), (2,2,−4), (2,−4,2) and (−4,2,2). All the TBPs were rescaled to have unit volume.

Extended Data Fig. 2 Cavity size measurement.

a, The truncated TBP tips (left) that point toward a cage center form a cavity (right). The radius of the cavity is determined by the average distance between the center of the cavity and the truncated TBP tips. b, The distribution of the radius of the cavity B obtained from a bulk crystal phase of S = 0.52 system (Clath I – A0B1) at three different pressures.

Source data

Extended Data Fig. 3 Free energy calculation.

A schematic diagram of the free energy calculation of the guest and host sublattices in a single component system. The free energies of the guest truncated TBP sublattice and the host truncated TBP sublattice were calculated along two paths (A and B). For each path, we selected multiple states (1, 2, 3 …) between the reference state and the equilibrium state that are connected to subsequent states by a continuous and reversible harmonic potential. See Methods for more details.

Extended Data Fig. 4 Free energy matching between states.

Free-energy matching between two states (B1, A2) represented in Extended Data Fig. 3. a, Potential energy U(γ) change of between the A1 and the A2 states of S = 0.56 and ϕ = 0.60. As we gradually decrease the spring constant γ of the harmonic potential holding the host particles in place, we continue tracking the translational and the rotational displacements of the host particles. b,c, When the translational and the rotational displacements of the host particles of the A2 matches with that of B1, the spring constant is prevented from further decreasing. Here, the γ of A2 is 1.0 (ln γ = 0).

Source data

Extended Data Fig. 5 Melting pressure measurement.

a, Gibbs free energy of the fluid and the solid phases of S = 0.42 system. The free energy of the fluid was calculated by thermodynamic integration and that of the solid was calculated by the Frenkel-Ladd method (Methods). The red arrow indicates the pressure where the free energy of the fluid and the solid crosses. b, The equation of state of S = 0.42 system, obtained by melting Clath II – A0D1. The crystal melts at P* ~ 9.7, which is a similar pressure obtained in a.

Source data

Extended Data Fig. 6 Reorganization of clathrate structure by the guest removal.

(left) Guest particles are removed from Clath I – A0B1 (S = 0.52), and the system is equilibrated at P* = 11.0. (middle and right) Part of host particles (red particles) spontaneously fill the empty cavity. In these simulation snapshots, all blue and red spheres represent the centers of truncated TBPs, and grey bonds connect nearest neighbors.

Extended Data Fig. 7 Clathrate Phase Diagram from EBT.

Phase diagram of various clathrate morphologies as computed from EBT. Colors correspond to same phases as in Fig. 2 of the main text. Scatter points are from simulations.

Extended Data Fig. 8 Histogram of Orientation Free Energy per Particle from EBT.

(left) D-cage of Clath II – A0D1. (right) B-cage of Clath I – A0B1. Distribution of D-cage is continuous, but this is a distinct set of lower free energy configurations for the B-cage.

Source data

Extended Data Fig. 9 Cage Particle Bonding Free Energy.

Computed bonding free energy between particles making up the clathrate cages. σ is the in-sphere diameter of the truncated TBP. Free energy well sits at ~ 3 kT.

Source data

Extended Data Fig. 10 Rotational motion of guest in binary clathrate systems.

a-b, Accumulated points of a direction vector (from particle center to a vertex) of (a) a tetrahedron guest in a A-cage and (b) a truncated TBP guest in a B-cage of a Clath I – A1B1 crystal phase (S = 0.52) at a constant volume fraction (ϕ = 0.62) for 3 × 107 MC sweeps, represented on a unit sphere (left) and spherical coordinates (r = 1.0σ) (right). c, Rotational displacement of the host truncated TBPs (blue), the guest truncated TBPs (green) and the guest tetrahedra (yellow) of the Clath I – A1B1 (S = 0.52) at ϕ = 0.62 for 107 MC sweeps. d, Accumulated points of direction vectors of (left) a dodecahedron guest in a A-cage and (right) a truncated TBP guest in a B-cage of a Clath I – A1B1 crystal phase (S = 0.52) at a constant volume fraction (ϕ = 0.62) for 106 MC sweeps, represented on a unit sphere. (left) For the dodecahedron guest, the direction vectors are from particle center to six randomly selected vertices of the particle, so there are six different colors. (right) For the truncated TBP guest, the direction vector is the same vector with Fig. 4a.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–4, Tables 1–3 and Videos.

Supplementary Video 1

HPMC simulation of the self-assembly of a one-component system of hard truncated TBPs from the fluid to the host–guest clathrate crystal phase, with S = 0.52 at ϕ = 0.55. The duration of the movie spans 9 × 107 MC sweeps. The resulting crystal is Clath I–A0B1. Red truncated TBPs are guest particles in B-cages. Host TBPs are shown as a network representation (grey bonds; Methods).

Supplementary Video 2

Single guest particles (red) rotating continuously around an axis in the B-cages of the Clath I–A0B1 colloidal crystal of Supplementary Video 1 at P* = 13.

Supplementary Video 3

Freely rotating truncated TBP single guest particles (red) in the Clath II–A0D1 colloidal crystal with S = 0.42 at P* = 11.

Supplementary Video 4

Truncated TBP tetramer guest particles (blue) discretely rotating among four orientations in the Clath II–A1D4 colloidal crystal with S = 0.64 at P* = 15. The blue spheres represent the centres of the guest truncated TBPs.

Supplementary Video 5

Stabilization of B-cage made of truncated TBPs. Simulations are performed using a recently developed anisotropic Lennard–Jones potential implemented in HOOMD-blue27. Host particles comprising the cage are mutually attractive with a well depth ε of 5 kT. The guest particle is modelled using a WCA, repulsive interaction with all host particles. Simulations were performed in the NVT ensemble at T = 1.0 kT/ε for 107 timesteps with dt = 0.0005σ(m/ε)0.5 in simulation units, where m is mass.

Supplementary Data

Source data for Supplementary Figs. 2 and 4b.

Supplementary Code

Sample codes; source code for HOOMD-blue is available at https://github.com/glotzerlab/hoomd-bluemp4.

Source data

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Lee, S., Vo, T. & Glotzer, S.C. Entropy compartmentalization stabilizes open host–guest colloidal clathrates. Nat. Chem. 15, 905–912 (2023). https://doi.org/10.1038/s41557-023-01200-6

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