Letter | Published:

Predicting crystal growth via a unified kinetic three-dimensional partition model

Nature volume 544, pages 456459 (27 April 2017) | Download Citation

Abstract

Understanding and predicting crystal growth is fundamental to the control of functionality in modern materials. Despite investigations for more than one hundred years1,2,3,4,5, it is only recently that the molecular intricacies of these processes have been revealed by scanning probe microscopy6,7,8. To organize and understand this large amount of new information, new rules for crystal growth need to be developed and tested. However, because of the complexity and variety of different crystal systems, attempts to understand crystal growth in detail have so far relied on developing models that are usually applicable to only one system9,10,11. Such models cannot be used to achieve the wide scope of understanding that is required to create a unified model across crystal types and crystal structures. Here we describe a general approach to understanding and, in theory, predicting the growth of a wide range of crystal types, including the incorporation of defect structures, by simultaneous molecular-scale simulation of crystal habit and surface topology using a unified kinetic three-dimensional partition model. This entails dividing the structure into ‘natural tiles’ or Voronoi polyhedra that are metastable and, consequently, temporally persistent. As such, these units are then suitable for re-construction of the crystal via a Monte Carlo algorithm. We demonstrate our approach by predicting the crystal growth of a diverse set of crystal types, including zeolites, metal–organic frameworks, calcite, urea and l-cystine.

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Acknowledgements

V.A.B. is grateful to the Russian Science Foundation (Grant No. 16-13-10158) for support. The Research Council of Norway, through the project Catlife, ‘Catalyst transformation and lifetimes by in-situ techniques and modelling’, P#233848, is acknowledged for financial support. A.R.H. and J.T.G.-R. are grateful for part funding from EPSRC through CASE awards. J.D.G. thanks the Australian Research Council for support through the Discovery Programme, and the Pawsey Supercomputing Centre and National Computational Infrastructure for provision of computing resources. We also acknowledge the Leverhulme Trust and the Royal Society for financial support.

Author information

Author notes

    • James T. Gebbie-Rayet
    •  & Pablo Cubillas

    Present addresses: Scientific Computing Department, STFC Daresbury Laboratory, Warrington WA4 4AD, UK (J.T.G.-R.); Earth Sciences Department, Durham University, Lower Mountjoy, South Road, Durham DH1 3LE, UK (P.C.).

Affiliations

  1. Centre for Nanoporous Materials, School of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, UK

    • Michael W. Anderson
    • , James T. Gebbie-Rayet
    • , Adam R. Hill
    • , Nani Farida
    • , Martin P. Attfield
    •  & Pablo Cubillas
  2. Samara Center for Theoretical Materials Science (SCTMS), Samara University, Academician Pavlov Street 1, Samara 443011, Russia

    • Vladislav A. Blatov
    •  & Davide M. Proserpio
  3. School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

    • Vladislav A. Blatov
  4. Università degli Studi di Milano, Dipartimento di Chimica, Via Camillo Golgi 19, 20133 Milano, Italy

    • Davide M. Proserpio
  5. SINTEF Materials and Chemistry, PO Box 124, Blindern, 0314 Oslo, Norway

    • Duncan Akporiaye
    •  & Bjørnar Arstad
  6. Curtin Institute for Computation, Department of Chemistry, Curtin University, GPO Box U1987, Perth, Western Australia 6845, Australia

    • Julian D. Gale

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Contributions

M.W.A. conceived ideas, wrote the CrystalGrower growth code and performed simulations, J.T.G.-R. wrote an early version of the growth and visualization code and performed simulations, A.R.H. wrote the CrystalGrower visualization code and performed simulations, N.F. and P.C. recorded AFM images, M.P.A. coordinated MOF work, V.A.B. modified the ToposPro code to interface with CrystalGrower, D.M.P. developed ideas to integrate tiling methodology, D.A. and B.A. funded A.R.H. and contributed to discussions about the mechanism of crystal growth, and J.D.G. computed energetics for the calcite and l-cystine systems. All authors contributed to writing the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Michael W. Anderson.

Reviewer Information Nature thanks M. Deem, M. Tuckerman and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Supplementary information

Videos

  1. 1.

    LTA – Growth via Screw Dislocation

    An LTA crystal grown with a screw dislocation running in the [100] direction through the crystal. Similar to the FAU simulation, the crystal is first grown under high supersaturation conditions (90 kcal/mol) for 5 million iterations, followed by a drop in supersaturation to slightly above equilibrium (1.5 kcal/mol) to allow a more detailed study of the growing process. During the slow growth, the propagation of the spiral from the screw dislocation is clear. The crystal is then rotated to the opposing face, demonstrating the elongation of the morphology in the [100] as expected as the screw dislocation creates a low energy surface for the growth units to attach to. After arriving at the opposing face, the slow growth process is reversed then replayed to again demonstrate the spiral-growth of the crystal.

  2. 2.

    FAU – Layer Growth

    A FAU crystal is grown under high supersaturation conditions (50 kcal/mol) for 5 million iterations, showing rapid growth and the adoption of an octahedral crystal morphology. The second 5 million iterations occur after the supersaturation is lowered to slightly above equilibrium (1 kcal/mol), allowing the growth process to continue slowly. At this point the triangular shape of the terraces can be observed, growing in the opposing direction to the facet where the growth is occurring. The final portion of the video shows the surface structure viewed from the top, followed by the side of a grown terrace.

  3. 3.

    UOV – Layer Growth

    Growth of a UOV crystal is demonstrated using similar conditions to the previous videos: 5 million iterations at high supersaturation (70 kcal/mol) followed by 5 million iterations at a supersaturation value slightly above equilibrium (1 kcal/mol). The diamond-like morphology of the UOV crystal can clearly be seen during panning and rotation. Rounded / isotropic layer growth can be observed during the growth process, along with the fusion of terraces once the growth process is slowed. Close-up shots are then shown of the surface structure on the (100) face, followed by the side-wall of the crystal: the (013) face.

  4. 4.

    MFI – Hourglass Growth and Tile Type Relation

    To begin, the t-mfi-1 tiles are shown during the growth process of an MFI crystal to allow a view into the internal structure of the crystal as it grows, showing the formation of the hourglass structure mentioned previously. Following this growth, a series of frames are shown where the tile type shown is cycled on the (001), (100) and (001) faces respectively, demonstrating how different tile types contribute to this internal feature in varying degrees.

  5. 5.

    ETS-10 – Orthogonal Rod Growth

    Demonstration of the growth mechanism of ETS-10 via titanate rods in alternating directions, orthogonal to each other. Each alternating layer is represented by different colours to draw attention to the orthogonal directions. The conditions for this growth process are changed slightly compared to the previous videos. The crystal is first grown at high supersaturation (140 kcal/mol) to ensure the growth process begins, however no frames are recorded. The supersaturation is then lowered to 1 kcal/mol, and frames are taken every 200,000 iterations for the first 20 frames, followed by 40 frames recorded at every 10,000 iterations. During these slowed down frames, a closer view of the surface is shown, with arrows overlaid following the direction of rod growth that can be observed. A final 20 frames with a spacing of 20000 iterations are shown, along with close-ups of the surface structures adopted by the vertical and horizontal rods that express the structure.

  6. 6.

    HKUST-1 – Growth via Screw Dislocation

    The growth of an HKUST-1 crystal. The first 5 million iterations are run at high supersaturation conditions (100 kcal/mol) followed by a drop to a lower supersaturation (1 kcal/mol). Slowing down the growth clearly demonstrates how the screw dislocation running along the [110] direction completely alters the final morphology of the crystal. The growth is then reversed, and the screw dislocation is followed to the point where it migrates back onto the (111) that it originally appeared from. Rotating the crystal then allows the viewer to observe the elongation of the crystal shape caused by the spiral growth on both sides. Rotating the crystal to view the opposite side of the screw dislocation, the location of the core is again followed whilst re-growing the crystal at low supersaturation (1 kcal/mol) ending with the final growth frame, and a panned out view of the entire crystal.

  7. 7.

    L-cystine – Growth via Screw Dislocation

    The growth of l-cystine at a supersaturation of 0.6 kcal mol-1. The optimal binding free energies where: (i) strong binding 3.5 kcal mol-1; (ii) and (iii) weak binding 0.78 kcal mol-1 and 0.98 kcal mol-1; (iv) binding in the c-direction 1.75 kcal mol-1. Advancement of the step bunches is through birth-and-spread growth on the <100> side faces, which is encouraged through the strong binding direction. This birth-and-spread growth precipitates growth at the single slow growth edges. Growth is further complicated by a small amount of birth-and-spread growth on the <001> face.

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https://doi.org/10.1038/nature21684

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