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From predictive modelling to machine learning and reverse engineering of colloidal self-assembly

Abstract

An overwhelming diversity of colloidal building blocks with distinct sizes, materials and tunable interaction potentials are now available for colloidal self-assembly. The application space for materials composed of these building blocks is vast. To make progress in the rational design of new self-assembled materials, it is desirable to guide the experimental synthesis efforts by computational modelling. Here, we discuss computer simulation methods and strategies used for the design of soft materials created through bottom-up self-assembly of colloids and nanoparticles. We describe simulation techniques for investigating the self-assembly behaviour of colloidal suspensions, including crystal structure prediction methods, phase diagram calculations and enhanced sampling techniques, as well as their limitations. We also discuss the recent surge of interest in machine learning and reverse-engineering methods. Although their implementation in the colloidal realm is still in its infancy, we anticipate that these data-science tools offer new paradigms in understanding, predicting and (inverse) design of novel colloidal materials.

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Fig. 1: Effective interactions between colloids and nanoparticles, with typical examples of the resulting self-assembly behaviour observed in experiments and predicted in simulations.
Fig. 2: Example applications of correlation functions and bond order parameters to classify thermodynamic phases and pathways.
Fig. 3: Examples of ML techniques applied to atomic and colloidal systems.

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References

  1. Feynman, R. P. There’s plenty of room at the bottom. Eng. Sci. 23, 22–36 (1960).

    Google Scholar 

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

    Article  Google Scholar 

  3. Boles, M. A., Engel, M. & Talapin, D. V. Self-assembly of colloidal nanocrystals: from intricate structures to functional materials. Chem. Rev. 116, 11220–11289 (2016).

    Article  CAS  Google Scholar 

  4. Likos, C. N. Soft matter with soft particles. Soft Matter 2, 478–498 (2006).

    Article  CAS  Google Scholar 

  5. Sacanna, S. & Pine, D. J. Shape-anisotropic colloids: building blocks for complex assemblies. Curr. Opin. Colloid Interface Sci. 16, 96–105 (2011).

    Article  CAS  Google Scholar 

  6. Cademartiri, L. & Bishop, K. J. Programmable self-assembly. Nat. Mater. 14, 2–9 (2015).

    Article  CAS  Google Scholar 

  7. Rovigatti, L., Gnan, N., Tavagnacco, L., Moreno, A. J. & Zaccarelli, E. Numerical modelling of non-ionic microgels: an overview. Soft Matter 15, 1108–1119 (2019).

    Article  CAS  Google Scholar 

  8. Bolintineanu, D. S. et al. Particle dynamics modeling methods for colloid suspensions. Comput. Part. Mech. 1, 321–356 (2014).

    Article  Google Scholar 

  9. Allen, M. P. & Tildesley, D. J. Computer Simulation of Liquids (Clarendon, 1987).

  10. Frenkel, D. & Smit, B. Understanding Molecular Simulation 2nd edn (Academic, 2002).

  11. Binks, B. P. & Horozov, T. S. Colloidal Particles at Liquid Interfaces (Cambridge University Press, 2006).

  12. Maciołek, A. & Dietrich, S. Collective behavior of colloids due to critical Casimir interactions. Rev. Mod. Phys. 90, 045001 (2018).

    Article  Google Scholar 

  13. Muševič, I. Nematic liquid-crystal colloids. Materials 11, 24 (2018).

    Article  CAS  Google Scholar 

  14. Dijkstra, M. Computer simulations of charge and steric stabilised colloidal suspensions. Curr. Opin. Colloid Interface Sci. 6, 372–382 (2001).

    Article  CAS  Google Scholar 

  15. Alder, B. J. & Wainwright, T. E. Phase transition for a hard sphere system. J. Chem. Phys. 27, 1208–1209 (1957).

    Article  CAS  Google Scholar 

  16. Wood, W. W. & Jacobson, J. Preliminary results from a recalculation of the Monte Carlo equation of state of hard spheres. J. Chem. Phys. 27, 1207–1208 (1957).

    Article  CAS  Google Scholar 

  17. Torquato, S. & Jiao, Y. Dense packings of the Platonic and Archimedean solids. Nature 460, 876–879 (2009).

    Article  CAS  Google Scholar 

  18. Agarwal, U. & Escobedo, F. A. Mesophase behaviour of polyhedral particles. Nat. Mater. 10, 230 (2011).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  20. Dijkstra, M. Entropy-driven phase transitions in colloids: from spheres to anisotropic particles. Adv. Chem. Phys. 156, 35 (2015).

    CAS  Google Scholar 

  21. Gilbert, E. G., Johnson, D. W. & Keerthi, S. S. A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE J. Robot. Autom. 4, 193–203 (1988).

    Article  Google Scholar 

  22. GAMMA Research Group at the University of North Carolina RAPID—Robust and Accurate Polygon Interference Detection http://gamma.cs.unc.edu/OBB/ (1997).

  23. Asakura, S. & Oosawa, F. On interaction between two bodies immersed in a solution of macromolecules. J. Chem. Phys. 22, 1255–1256 (1954).

    Article  CAS  Google Scholar 

  24. Vrij, A. Polymers at interfaces and the interactions in colloidal dispersions. Pure Appl. Chem. 48, 471–483 (1976).

    Article  CAS  Google Scholar 

  25. Dijkstra, M., van Roij, R., Roth, R. & Fortini, A. Effect of many-body interactions on the bulk and interfacial phase behavior of a model colloid–polymer mixture. Phys. Rev. E 73, 041404 (2006).

    Article  CAS  Google Scholar 

  26. Liu, J. & Luijten, E. Rejection-free geometric cluster algorithm for complex fluids. Phys. Rev. Lett. 92, 035504 (2004).

    Article  CAS  Google Scholar 

  27. Linse, P. Structure, phase stability, and thermodynamics in charged colloidal solutions. J. Chem. Phys. 113, 4359–4373 (2000).

    Article  CAS  Google Scholar 

  28. Hockney, R. W. & Eastwood, J. W. Computer Simulation Using Particles (McGraw-Hill, 1981).

  29. Darden, T., York, D. & Pedersen, L. Particle mesh Ewald: an N·log(N) method for Ewald sums in large systems. J. Chem. Phys. 98, 10089–10092 (1993).

    Article  CAS  Google Scholar 

  30. Greengard, L. & Moura, M. On the numerical evaluation of electrostatic fields in composite materials. Acta Numer. 3, 379–410 (1994).

    Article  Google Scholar 

  31. Hynninen, A.-P. & Dijkstra, M. Phase diagrams of hard-core repulsive Yukawa particles. Phys. Rev. E 68, 021407 (2003).

    Article  CAS  Google Scholar 

  32. van Roij, R., Dijkstra, M. & Hansen, J.-P. Phase diagram of charge-stabilized colloidal suspensions: van der Waals instability without attractive forces. Phys. Rev. E 59, 2010 (1999).

    Article  Google Scholar 

  33. Linse, P. & Lobaskin, V. Electrostatic attraction and phase separation in solutions of like-charged colloidal particles. Phys. Rev. Lett. 83, 4208–4211 (1999).

    Article  CAS  Google Scholar 

  34. Levin, Y. Strange electrostatics in physics, chemistry, and biology. Physica A 352, 43–52 (2005).

    Article  CAS  Google Scholar 

  35. Leunissen, M. E. et al. Ionic colloidal crystals of oppositely charged particles. Nature 437, 235–240 (2005).

    Article  CAS  Google Scholar 

  36. Tohver, V., Smay, J. E., Braem, A., Braun, P. V. & Lewis, J. A. Nanoparticle halos: a new colloid stabilization mechanism. Proc. Natl Acad. Sci. USA 98, 8950–8954 (2001).

    Article  CAS  Google Scholar 

  37. Liu, J. & Luijten, E. Stabilization of colloidal suspensions by means of highly charged nanoparticles. Phys. Rev. Lett. 93, 247802 (2004).

    Article  CAS  Google Scholar 

  38. Sciortino, F., Giacometti, A. & Pastore, G. Phase diagram of Janus particles. Phys. Rev. Lett. 103, 237801 (2009).

    Article  CAS  Google Scholar 

  39. Jiang, S. et al. Janus particle synthesis and assembly. Adv. Mater. 22, 1060–1071 (2010).

    Article  CAS  Google Scholar 

  40. Walther, A. & Müller, A. H. E. Janus particles: synthesis, self-assembly, physical properties, and applications. Chem. Rev. 113, 5194–5261 (2013).

    Article  CAS  Google Scholar 

  41. Smallenburg, F. & Sciortino, F. Liquids more stable than crystals in particles with limited valence and flexible bonds. Nat. Phys. 9, 554–558 (2013).

    Article  CAS  Google Scholar 

  42. Zhang, J., Luijten, E. & Granick, S. Toward design rules of directional Janus colloidal assembly. Annu. Rev. Phys. Chem. 66, 581–600 (2015).

    Article  CAS  Google Scholar 

  43. Du, J. & O’Reilly, R. K. Anisotropic particles with patchy, multicompartment and Janus architectures: preparation and application. Chem. Soc. Rev. 40, 24020–2416 (2011).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  45. 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 

  46. Hong, L., Cacciuto, A., Luijten, E. & Granick, S. Clusters of charged Janus spheres. Nano Lett. 6, 2510–2514 (2006).

    Article  CAS  Google Scholar 

  47. Sciortino, F., Giacometti, A. & Pastore, G. Phase diagram of Janus particles. Phys. Rev. Lett. 103, 237801 (2009).

    Article  CAS  Google Scholar 

  48. Zhang, J., Luijten, E., Grzybowski, B. A. & Granick, S. Active colloids with collective mobility: status and research opportunities. Chem. Soc. Rev. 46, 5551–5569 (2017).

    Article  CAS  Google Scholar 

  49. Bianchi, E., Largo, J., Tartaglia, P., Zaccarelli, E. & Sciortino, F. Phase diagram of patchy colloids: towards empty liquids. Phys. Rev. Lett. 97, 168301 (2006).

    Article  CAS  Google Scholar 

  50. Ruzicka, B. et al. Observation of empty liquids and equilibrium gels in a colloidal clay. Nat. Mater. 10, 56–60 (2011).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  52. Mirkin, C. A., Letsinger, R. L., Mucic, R. C. & Storhoff, J. J. A DNA-based method for rationally assembling nanoparticles into macroscopic materials. Nature 382, 607–609 (1996).

    Article  CAS  Google Scholar 

  53. Alivisatos, A. P. et al. Organization of ‘nanocrystal molecules’ using DNA. Nature 382, 609–611 (1996).

    Article  CAS  Google Scholar 

  54. Park, S. Y. et al. DNA-programmable nanoparticle crystallization. Nature 451, 553–556 (2008).

    Article  CAS  Google Scholar 

  55. Nykypanchuk, D., Maye, M. M., van der Lelie, D. & Gang, O. DNA-guided crystallization of colloidal nanoparticles. Nature 451, 549–552 (2008).

    Article  CAS  Google Scholar 

  56. Jones, M. R., Seeman, N. C. & Mirkin, C. A. Programmable materials and the nature of the DNA bond. Science 347, 1260901 (2015).

    Article  CAS  Google Scholar 

  57. Jones, M. R., Macfarlane, R. J., Prigodich, A. E., Patel, P. C. & Mirkin, C. A. Nanoparticle shape anisotropy dictates the collective behavior of surface-bound ligands. J. Am. Chem. Soc. 133, 18865–18869 (2011).

    Article  CAS  Google Scholar 

  58. Martinez-Veracoechea, F. J., Mladek, B. M., Tkachenko, A. V. & Frenkel, D. Design rule for colloidal crystals of DNA-functionalized particles. Phys. Rev. Lett. 107, 045902 (2011).

    Article  CAS  Google Scholar 

  59. Macfarlane, R. J., O’Brien, M. N., Petrosko, S. H. & Mirkin, C. A. Nucleic acid-modified nanostructures as programmable atom equivalents: forging a new ‘table of elements’. Angew. Chem. Int. Ed. 52, 5688–5698 (2013).

    Article  CAS  Google Scholar 

  60. McGinley, J. T., Wang, Y., Jenkins, I. C., Sinno, T. & Crocker, J. C. Crystal-templated colloidal clusters exhibit directional DNA interactions. ACS Nano 9, 10817–10825 (2015).

    Article  CAS  Google Scholar 

  61. Wang, Y. et al. Crystallization of DNA-coated colloids. Nat. Commun. 6, 7253 (2015).

    Article  CAS  Google Scholar 

  62. van der Meulen, S. A. J. & Leunissen, M. E. Solid colloids with surface-mobile DNA linkers. J. Am. Chem. Soc. 135, 15129–15134 (2013).

    Article  CAS  Google Scholar 

  63. Angioletti-Uberti, S., Mognetti, B. M. & Frenkel, D. Theory and simulation of DNA-coated colloids: a guide for rational design. Phys. Chem. Chem. Phys. 18, 6373–6393 (2016).

    Article  CAS  Google Scholar 

  64. Ouldridge, T. E., Louis, A. A. & Doye, J. P. K. Structural, mechanical, and thermodynamic properties of a coarse-grained DNA model. J. Chem. Phys. 134, 085101 (2011).

    Article  CAS  Google Scholar 

  65. Li, T. I. N. G., Sknepnek, R., Macfarlane, R. J., Mirkin, C. A. & Olvera de la Cruz, M. Modeling the crystallization of spherical nucleic acid nanoparticle conjugates with molecular dynamics simulations. Nano Lett. 12, 2509–2514 (2012).

    Article  CAS  Google Scholar 

  66. Hinckley, D. M., Freeman, G. S., Whitmer, J. K. & de Pablo, J. J. An experimentally-informed coarse-grained 3-site-per-nucleotide model of DNA: structure. J. Chem. Phys. 139, 144903 (2013).

    Article  CAS  Google Scholar 

  67. Markegard, C. B., Gallivan, C. P., Cheng, D. D. & Nguyen, H. D. Effects of concentration and temperature on DNA hybridization by two closely related sequences via large-scale coarse-grained simulations. J. Phys. Chem. B 120, 7795–7806 (2016).

    Article  CAS  Google Scholar 

  68. Fong, L.-K., Wang, Z., Schatz, G. C., Luijten, E. & Mirkin, C. A. The role of structural enthalpy in spherical nucleic acid hybridization. J. Am. Chem. Soc. 140, 6226–6230 (2018).

    Article  CAS  Google Scholar 

  69. Girard, M. et al. Particle analogs of electrons in colloidal crystals. Science 364, 1174–1178 (2019).

    Article  CAS  Google Scholar 

  70. Hynninen, A.-P., Christova, C., van Roij, R., van Blaaderen, A. & Dijkstra, M. Prediction and observation of crystal structures of oppositely charged colloids. Phys. Rev. Lett. 96, 138308 (2006).

    Article  CAS  Google Scholar 

  71. Fornleitner, J., LoVerso, F., Kahl, G. & Likos, C. N. Genetic algorithms predict formation of exotic ordered configurations for two-component dipolar monolayers. Soft Matter 4, 480–484 (2008).

    Article  CAS  Google Scholar 

  72. Bianchi, E., Doppelbauer, G., Filion, L., Dijkstra, M. & Kahl, G. Predicting patchy particle crystals: variable box shape simulations and evolutionary algorithms. J. Chem. Phys. 136, 214102 (2012).

    Article  CAS  Google Scholar 

  73. Fornleitner, J. & Kahl, G. Lane formation vs. cluster formation in two-dimensional square-shoulder systems—a genetic algorithm approach. Europhys. Lett. 82, 18001 (2008).

    Article  CAS  Google Scholar 

  74. Stucke, D. P. & Crespi, V. H. Predictions of new crystalline states for assemblies of nanoparticles: perovskite analogues and 3-D arrays of self-assembled nanowires. Nano Lett. 3, 1183–1186 (2003).

    Article  CAS  Google Scholar 

  75. Filion, L. et al. Efficient method for predicting crystal structures at finite temperature: variable box shape simulations. Phys. Rev. Lett. 103, 188302 (2009).

    Article  CAS  Google Scholar 

  76. Haji-Akbari, A. et al. Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra. Nature 462, 773–777 (2009).

    Article  CAS  Google Scholar 

  77. de Graaf, J., Filion, L., Marechal, M., van Roij, R. & Dijkstra, M. Crystal-structure prediction via the floppy-box Monte Carlo algorithm: method and application to hard (non)convex particles. J. Chem. Phys. 137, 214101 (2012).

    Article  CAS  Google Scholar 

  78. Ladd, A. & Woodcock, L. Interfacial and co-existence properties of the Lennard-Jones system at the triple point. Mol. Phys. 36, 611–619 (1978).

    Article  CAS  Google Scholar 

  79. Kofke, D. A. Gibbs–Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation. Mol. Phys. 78, 1331–1336 (1993).

    Article  CAS  Google Scholar 

  80. Bolhuis, P. G. & Kofke, D. A. Monte Carlo study of freezing of polydisperse hard spheres. Phys. Rev. E 54, 634 (1996).

    Article  CAS  Google Scholar 

  81. Torrie, G. M. & Valleau, J. P. Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J. Comp. Phys. 23, 187–199 (1977).

    Article  Google Scholar 

  82. Allen, R. J., Frenkel, D. & ten Wolde, P. R. Simulating rare events in equilibrium or nonequilibrium stochastic systems. J. Chem. Phys. 124, 024102 (2006).

    Article  CAS  Google Scholar 

  83. Dellago, C., Bolhuis, P. G., Csajka, F. S. & Chandler, D. Transition path sampling and the calculation of rate constants. J. Chem. Phys. 108, 1964–1977 (1998).

    Article  CAS  Google Scholar 

  84. Earl, D. J. & Deem, M. W. Parallel tempering: theory, applications, and new perspectives. Phys. Chem. Chem. Phys. 7, 3910–3916 (2005).

    Article  CAS  Google Scholar 

  85. Dress, C. & Krauth, W. Cluster algorithm for hard spheres and related systems. J. Phys. A 28, L597–L601 (1995).

    Article  CAS  Google Scholar 

  86. Heringa, J. R. & Blöte, H. W. J. Geometric cluster Monte Carlo simulation. Phys. Rev. E 57, 4976–4978 (1998).

    Article  CAS  Google Scholar 

  87. Whitelam, S. & Geissler, P. L. Avoiding unphysical kinetic traps in Monte Carlo simulations of strongly attractive particles. J. Chem. Phys. 127, 154101 (2007).

    Article  CAS  Google Scholar 

  88. Liu, J., Wilding, N. B. & Luijten, E. Simulation of phase transitions in highly asymmetric fluid mixtures. Phys. Rev. Lett. 97, 115705 (2006).

    Article  CAS  Google Scholar 

  89. Sinkovits, D. W., Barr, S. A. & Luijten, E. Rejection-free Monte Carlo scheme for anisotropic particles. J. Chem. Phys. 136, 144111 (2012).

    Article  CAS  Google Scholar 

  90. Bernard, E. P., Krauth, W. & Wilson, D. B. Event-chain Monte Carlo algorithms for hard-sphere systems. Phys. Rev. E 80, 056704 (2009).

    Article  CAS  Google Scholar 

  91. Michel, M., Kapfer, S. C. & Krauth, W. Generalized event-chain Monte Carlo: constructing rejection-free global-balance algorithms from infinitesimal steps. J. Chem. Phys. 140, 054116 (2014).

    Article  CAS  Google Scholar 

  92. Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784 (1983).

    Article  CAS  Google Scholar 

  93. van Meel, J. A., Filion, L., Valeriani, C. & Frenkel, D. A parameter-free, solid-angle based, nearest-neighbor algorithm. J. Chem. Phys. 136, 234107 (2012).

    Article  CAS  Google Scholar 

  94. Lechner, W. & Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. J. Chem. Phys. 129, 114707 (2008).

    Article  CAS  Google Scholar 

  95. Mickel, W., Kapfer, S. C., Schröder-Turk, G. E. & Mecke, K. Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter. J. Chem. Phys. 138, 044501 (2013).

    Article  CAS  Google Scholar 

  96. Auer, S. & Frenkel, D. Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature 409, 1020–1023 (2001).

    Article  CAS  Google Scholar 

  97. Malins, A., Williams, S. R., Eggers, J. & Royall, C. P. Identification of structure in condensed matter with the topological cluster classification. J. Chem. Phys. 139, 234506 (2013).

    Article  CAS  Google Scholar 

  98. Gantapara, A. P., de Graaf, J., van Roij, R. & Dijkstra, M. Phase diagram and structural diversity of a family of truncated cubes: degenerate close-packed structures and vacancy-rich states. Phys. Rev. Lett. 111, 015501 (2013).

    Article  CAS  Google Scholar 

  99. Klotsa, D., Chen, E. R., Engel, M. & Glotzer, S. C. Intermediate crystalline structures of colloids in shape space. Soft Matter 14, 8692–8697 (2018).

    Article  CAS  Google Scholar 

  100. Geiger, P. & Dellago, C. Neural networks for local structure detection in polymorphic systems. J. Chem. Phys. 139, 164105 (2013).

    Article  CAS  Google Scholar 

  101. Dietz, C., Kretz, T. & Thoma, M. Machine-learning approach for local classification of crystalline structures in multiphase systems. Phys. Rev. E 96, 011301 (2017).

    Article  CAS  Google Scholar 

  102. Boattini, E., Ram, M., Smallenburg, F. & Filion, L. Neural-network-based order parameters for classification of binary hard-sphere crystal structures. Mol. Phys. 116, 3066–3075 (2018).

    Article  CAS  Google Scholar 

  103. DeFever, R. S., Targonski, C., Hall, S. W., Smith, M. C. & Sarupria, S. A generalized deep learning approach for local structure identification in molecular simulations. Chem. Sci. 10, 7503–7515 (2019).

    Article  CAS  Google Scholar 

  104. Terao, T. A machine learning approach to analyze the structural formation of soft matter via image recognition. Soft Mater. 18, 215–227 (2020).

  105. Schoenholz, S. S., Cubuk, E. D., Sussman, D. M., Kaxiras, E. & Liu, A. J. A structural approach to relaxation in glassy liquids. Nat. Phys. 12, 469–471 (2016).

    Article  CAS  Google Scholar 

  106. Bapst, V. et al. Unveiling the predictive power of static structure in glassy systems. Nat. Phys. 16, 448–454 (2020).

    Article  CAS  Google Scholar 

  107. Behler, J. Perspective: Machine learning potentials for atomistic simulations. J. Chem. Phys. 145, 170901 (2016).

    Article  CAS  Google Scholar 

  108. Boattini, E., Bezem, N., Punnathanam, S. N., Smallenburg, F. & Filion, L. Modeling of many-body interactions between elastic spheres through symmetry functions. J. Chem. Phys. 153, 064902 (2020).

  109. Dai, C. & Glotzer, S. C. Efficient phase diagram sampling by active learning. J. Phys. Chem. B 124, 1275–1284 (2020).

    Article  CAS  Google Scholar 

  110. Reinhart, W. F., Long, A. W., Howard, M. P., Ferguson, A. L. & Panagiotopoulos, A. Z. Machine learning for autonomous crystal structure identification. Soft Matter 13, 4733–4745 (2017).

    Article  CAS  Google Scholar 

  111. Reinhart, W. F. & Panagiotopoulos, A. Z. Automated crystal characterization with a fast neighborhood graph analysis method. Soft Matter 14, 6083–6089 (2018).

    Article  CAS  Google Scholar 

  112. Jadrich, R., Lindquist, B. & Truskett, T. Unsupervised machine learning for detection of phase transitions in off-lattice systems. I. Foundations. J. Chem. Phys. 149, 194109 (2018).

    Article  CAS  Google Scholar 

  113. Jadrich, R., Lindquist, B., Piñeros, W., Banerjee, D. & Truskett, T. Unsupervised machine learning for detection of phase transitions in off-lattice systems. II. Applications. J. Chem. Phys. 149, 194110 (2018).

    Article  CAS  Google Scholar 

  114. Spellings, M. & Glotzer, S. C. Machine learning for crystal identification and discovery. AIChE J. 64, 2198–2206 (2018).

    Article  CAS  Google Scholar 

  115. Boattini, E., Dijkstra, M. & Filion, L. Unsupervised learning for local structure detection in colloidal systems. J. Chem. Phys. 151, 154901 (2019).

    Article  CAS  Google Scholar 

  116. Adorf, C. S., Moore, T. C., Melle, Y. J. & Glotzer, S. C. Analysis of self-assembly pathways with unsupervised machine learning algorithms. J. Phys. Chem. B 124, 69–78 (2019).

    Article  CAS  Google Scholar 

  117. Bereau, T., Andrienko, D. & Kremer, K. Research update: Computational materials discovery in soft matter. APL Mater. 4, 053101 (2016).

    Article  CAS  Google Scholar 

  118. Ferguson, A. L. Machine learning and data science in soft materials engineering. J. Phys.: Condens. Matter 30, 043002 (2017).

    Google Scholar 

  119. Wang, J. & Ferguson, A. Nonlinear machine learning in simulations of soft and biological materials. Mol. Simul. 44, 1090–1107 (2018).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  121. Lindquist, B. A., Jadrich, R. B. & Truskett, T. M. Communication: Inverse design for self-assembly via on-the-fly optimization. J. Chem. Phys. 145, 11110 (2016).

    Google Scholar 

  122. Shell, M. S. The relative entropy is fundamental to multiscale and inverse thermodynamic problems. J. Chem. Phys. 129, 144108 (2008).

    Article  CAS  Google Scholar 

  123. Piñeros, W. D., Lindquist, B. A., Jadrich, R. B. & Truskett, T. M. Inverse design of multicomponent assemblies. J. Chem. Phys. 148, 104509 (2018).

    Article  CAS  Google Scholar 

  124. Lindquist, B. A., Jadrich, R. B., Piñeros, W. D. & Truskett, T. M. Inverse design of self-assembling Frank–Kasper phases and insights into emergent quasicrystals. J. Phys. Chem. B 122, 5547–5556 (2018).

    Article  CAS  Google Scholar 

  125. Florescu, M., Torquato, S. & Steinhardt, P. J. Designer disordered materials with large, complete photonic band gaps. Proc. Natl Acad. Sci. USA 106, 20658–20663 (2009).

    Article  CAS  Google Scholar 

  126. Geng, Y., van Anders, G., Dodd, P. M., Dshemuchadse, J. & Glotzer, S. C. Engineering entropy for the inverse design of colloidal crystals from hard shapes. Sci. Adv. 5, eaaw0514 (2019).

    Article  CAS  Google Scholar 

  127. Miskin, M. Z., Khaira, G., de Pablo, J. J. & Jaeger, H. M. Turning statistical physics models into materials design engines. Proc. Natl Acad. Sci. USA 113, 34–39 (2016).

    Article  CAS  Google Scholar 

  128. Kumar, R., Coli, G. M., Dijkstra, M. & Sastry, S. Inverse design of charged colloidal particle interactions for self assembly into specified crystal structures. J. Chem. Phys. 151, 084109 (2019).

    Article  CAS  Google Scholar 

  129. Long, A. W. & Ferguson, A. L. Rational design of patchy colloids via landscape engineering. Mol. Syst. Des. Eng. 3, 49–65 (2018).

    Article  CAS  Google Scholar 

  130. Ma, Y. & Ferguson, A. L. Inverse design of self-assembling colloidal crystals with omnidirectional photonic bandgaps. Soft Matter 15, 8808–8826 (2019).

    Article  CAS  Google Scholar 

  131. Sherman, Z. M., Howard, M. P., Lindquist, B. A., Jadrich, R. B. & Truskett, T. M. Inverse methods for design of soft materials. J. Chem. Phys. 152, 140902 (2020).

    Article  CAS  Google Scholar 

  132. Ou, Z., Wang, Z., Luo, B., Luijten, E. & Chen, Q. Kinetic pathways of crystallization at the nanoscale. Nat. Mater. 19, 450–455 (2020).

    Article  CAS  Google Scholar 

  133. Wang, J. et al. Magic number colloidal clusters as minimum free energy structures. Nat. Commun. 9, 5259 (2018).

    Article  CAS  Google Scholar 

  134. Henzie, J., Grünwald, M., Widmer-Cooper, A., Geissler, P. L. & Yang, P. Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices. Nat. Mater 11, 131–137 (2012).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  136. Haji-Akbari, A. et al. Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra. Nature 462, 773–777 (2009).

    Article  CAS  Google Scholar 

  137. Ramananarivo, S., Ducrot, E. & Palacci, J. Activity-controlled annealing of colloidal monolayers. Nat. Commun. 10, 3380 (2019).

    Article  CAS  Google Scholar 

  138. Sharp, T. A. et al. Machine learning determination of atomic dynamics at grain boundaries. Proc. Natl Acad. Sci. USA 115, 10943–10947 (2018).

    Article  CAS  Google Scholar 

  139. Chen, W., Tan, A. R. & Ferguson, A. L. Collective variable discovery and enhanced sampling using autoencoders: innovations in network architecture and error function design. J. Chem. Phys. 149, 072312 (2018).

    Article  CAS  Google Scholar 

  140. Gan, Z. & Xu, Z. Multiple-image treatment of induced charges in Monte Carlo simulations of electrolytes near a spherical dielectric interface. Phys. Rev. E 84, 016705 (2011).

    Article  CAS  Google Scholar 

  141. Freed, K. F. Perturbative many-body expansion for electrostatic energy and field for system of polarizable charged spherical ions in a dielectric medium. J. Chem. Phys. 141, 034115 (2014).

    Article  CAS  Google Scholar 

  142. Qin, J., de Pablo, J. J. & Freed, K. F. Image method for induced surface charge from many-body system of dielectric spheres. J. Chem. Phys. 145, 124903 (2016).

    Article  CAS  Google Scholar 

  143. Maggs, A. & Rossetto, V. Local simulation algorithms for Coulomb interactions. Phys. Rev. Lett. 88, 196402 (2002).

    Article  CAS  Google Scholar 

  144. Levitt, D. G. Electrostatic calculations for an ion channel. I. Energy and potential profiles and interactions between ions. Biophys. J. 22, 209–219 (1978).

    Article  CAS  Google Scholar 

  145. Hoshi, H., Sakurai, M., Inoue, Y. & Chûjô, R. Medium effects on the molecular electronic structure. I. The formulation of a theory for the estimation of a molecular electronic structure surrounded by an anisotropic medium. J. Chem. Phys. 87, 1107–1115 (1987).

    Article  CAS  Google Scholar 

  146. Bharadwaj, R., Windemuth, A., Sridharan, S., Honig, B. & Nicholls, A. The fast multipole boundary element method for molecular electrostatics: an optimal approach for large systems. J. Comput. Chem. 16, 898–913 (1995).

    Article  CAS  Google Scholar 

  147. Allen, R., Hansen, J.-P. & Melchionna, S. Electrostatic potential inside ionic solutions confined by dielectrics: a variational approach. Phys. Chem. Chem. Phys. 3, 4177–4186 (2001).

    Article  CAS  Google Scholar 

  148. Boda, D., Gillespie, D., Eisenberg, B., Nonner W., & Henderson, D. in Ionic Soft Matter: Modern Trends in Theory and Applications (eds Henderson, D. et al.) 19–43 (NATO Science Series II: Mathematics, Physics and Chemistry Vol. 206, Springer, 2005).

  149. Tyagi, S. et al. An iterative, fast, linear-scaling method for computing induced charges on arbitrary dielectric boundaries. Phys. Chem. Chem. Phys. 3, 4177–4186 (2001).

    Article  CAS  Google Scholar 

  150. Jadhao, V., Solis, F. J. & Olvera de la Cruz, M. Simulation of charged systems in heterogeneous dielectric media via a true energy functional. Phys. Rev. Lett. 109, 223905 (2012).

    Article  CAS  Google Scholar 

  151. Barros, K., Sinkovits, D. & Luijten, E. Efficient and accurate simulation of dynamic dielectric objects. J. Chem. Phys. 140, 064903 (2014).

    Article  CAS  Google Scholar 

  152. Barros, K. & Luijten, E. Dielectric effects in the self-assembly of binary colloidal aggregates. Phys. Rev. Lett. 113, 017801 (2014).

    Article  CAS  Google Scholar 

  153. Gan, Z., Wang, Z., Jiang, S., Xu, Z. & Luijten, E. Efficient dynamic simulations of charged dielectric colloids through a novel hybrid method. J. Chem. Phys. 151, 024112 (2019).

    Article  CAS  Google Scholar 

  154. Holland, J. H. Adaptation in Natural and Artificial Systems (MIT Press, 1992).

  155. Oganov, A. R. & Glass, C. W. Crystal structure prediction using ab initio evolutionary techniques: principles and applications. J. Chem. Phys. 124, 244704 (2006).

    Article  CAS  Google Scholar 

  156. Wang, Y., Lv, J., Zhu, L. & Ma, Y. Crystal structure prediction via particle-swarm optimization. Phys. Rev. B 82, 094116 (2010).

    Article  CAS  Google Scholar 

  157. Amsler, M. & Goedecker, S. Crystal structure prediction using the minima hopping method. J. Chem. Phys. 133, 224104 (2010).

    Article  CAS  Google Scholar 

  158. Wales, D. J. & Scheraga, H. A. Global optimization of clusters, crystals, and biomolecules. Science 285, 1368–1372 (1999).

    Article  CAS  Google Scholar 

  159. Martoňák, R. et al. Simulation of structural phase transitions by metadynamics. Z. Kristallogr. Cryst. Mater. 220, 489–498 (2009).

    Google Scholar 

  160. Panagiotopoulos, A. Z. Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble. Mol. Phys. 61, 813–826 (1987).

    Article  CAS  Google Scholar 

  161. Ferrenberg, A. M. & Swendsen, R. H. Optimized Monte Carlo data analysis. Phys. Rev. Lett. 63, 1195–1198 (1989).

    Article  CAS  Google Scholar 

  162. Potoff, J. J. & Panagiotopoulos, A. Z. Surface tension of the three-dimensional Lennard-Jones fluid from histogram-reweighting Monte Carlo simulations. J. Chem. Phys. 112, 6411–6415 (2000).

    Article  CAS  Google Scholar 

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Acknowledgements

We thank Z. Wang for designing the figures. E.L. is supported by the Center for Bio-Inspired Energy Science (CBES), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences BES, under award no. DE-SC0000989.

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Dijkstra, M., Luijten, E. From predictive modelling to machine learning and reverse engineering of colloidal self-assembly. Nat. Mater. 20, 762–773 (2021). https://doi.org/10.1038/s41563-021-01014-2

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