Daphne
Klotsa†
*^{ab},
Elizabeth R.
Chen†
^{ac},
Michael
Engel
^{ad} and
Sharon C.
Glotzer
*^{aef}
^{a}Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA. E-mail: sglotzer@umich.edu
^{b}Department of Applied Physical Sciences, University of North Carolina, Chapel Hill, North Carolina 27599, USA. E-mail: dklotsa@email.unc.edu
^{c}School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
^{d}Institute for Multiscale Simulation, Friedrich-Alexander University Erlangen-Nürnberg, 91052 Erlangen, Germany
^{e}Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA
^{f}Biointerfaces Institute, University of Michigan, Ann Arbor, Michigan 48109-2136, USA

Received
1st August 2018
, Accepted 16th August 2018

First published on 23rd August 2018

We computationally study the thermodynamic assembly of more than 40000 hard, convex polyhedra belonging to three families of shapes associated with the triangle groups 323, 423, and 523. Each family is defined by vertex and/or edge truncation of symmetric polyhedra with equal edge length, producing shapes for which the majority are intermediates of more symmetric polyhedra found among the Platonic, Archimedean, and Catalan solids. In addition to the complex crystals cI16 lithium, BC8 silicon, γ-brass, β-manganese, and a dodecagonal quasicrystal, we find that most intermediate shapes assemble distorted variants of four basic cubic crystals: face-centered cubic, body-centered cubic, simple cubic, and diamond. To quantify the degree of distortion, we developed an algorithm that extracts lattice vectors from particle positions and then evaluates closeness to the four reference cubic crystals. This analysis allows us to group together in shape space related intermediate structures that would otherwise be placed in different lattice systems had we followed the lattice systems' strict definitions for angles and lengths of lattice vectors. The resulting landscapes show, as a function of shape, regions where ordered structures assemble, what is assembled and at what density, locations of transitions between regions of ordered structures, and regions of disorder. Our results provide a guide to self-assembling a host of related colloidal crystals through systematic design, by careful tweaking of the particle shape.

In this paper, we computationally studied the thermodynamic assembly of tens of thousands of hard, symmetric, convex polyhedra that belong to three families of continuously modified shapes. We quantified the degree of distortion of colloidal crystal structures, assembled by intermediate shapes, by introducing an algorithm that extracted lattice vectors from particle positions and evaluated the similarity of the intermediate structure to four reference crystals: face-centered cubic (FCC), body-centered cubic (BCC), simple cubic (SC), and diamond (DIA). This analysis allowed us to group together in shape space related structures that would otherwise be placed in different lattice systems (triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, cubic) had we followed the lattice systems' strict definitions for angles and lengths of lattice vectors. Our results are presented as “assembly landscapes” – surfaces of the minimum density for self-assembly vs. shape parameters – that reveal regions in shape space where ordered structures assemble, locations of transitions between ordered structures, and regions of disorder. We found that the minimum assembly density systematically increases towards the edges of regions of order, indicating a tendency towards decreasing thermodynamic stability of the ordered phase relative to that of the disordered fluid. In addition to distorted crystal structures, we found complex crystal structures in isolated regions of shape space. These structures include cI16 lithium (Li), γ-brass, β-manganese (Mn), a dodecagonal quasicrystal (QC), and, here reported for the first time, BC8 silicon (Si). We conclude by comparing our assembly landscapes to previously reported putative densest packing landscapes evaluated for the same three shape families.^{14}

1. Unit cell extraction.
Starting from a set of N particle positions {r_{n}}, we analyze the peak positions of the structure factor

taking into account symmetry, extinction and interference effects. Given a reference crystal structure, we select a set of M diffraction peaks {(h_{m}k_{m}_{m})} and search for a basis in reciprocal space {b_{1},b_{2},b_{3}} that maximizes

Specifically, we use M_{SC} = 6, M_{BCC} = 12, M_{FCC} = 8, M_{DIA} = 4 diffraction peaks allowed by selection rules,

Note that the choices {(hk)}_{DIA}^{+} and {(hk)}_{DIA}^{−} lead to identical quantifications, but we list both for symmetry. In our algorithm, the search is performed by testing 10^{6} random values for the reciprocal lattice basis. From each reciprocal lattice basis of a reference crystal structure, we obtain the corresponding lattice basis {ζ,ξ,ς} in real space.

taking into account symmetry, extinction and interference effects. Given a reference crystal structure, we select a set of M diffraction peaks {(h

Specifically, we use M

{(hk)}_{SC} = {〈200〉}, |

{(hk)}_{BCC} = {〈110〉}, |

{(hk)}_{FCC} = {〈111〉}, |

{(hk)}_{DIA}^{+} = {〈111〉:hk = +1}, |

{(hk)}_{DIA}^{−} = {〈111〉:hk = −1}. |

2. Color map.
We quantify how similar the obtained lattice bases are to those of the reference crystal structures. For this purpose, we measure orthogonality of vector angles,

and equality of vector lengths,

and combine them into

Note that det{ξ × ς,ς × ζ,ζ × ξ} = det{ζ,ξ,ς}^{2}. Similarity to reference crystal structures is measured by

We choose colors for the assembly landscapes and the previously reported putative densest packing landscapes using the following mapping onto the rgb color cube,

where we set p = 4 and q = 2 for contrast and

By the end of this analysis, we have reduced a high dimensional space to a single color that simultaneously tracks changes of magnitudes and/or angles of the lattice vectors for all our assembled structures with up to two particles in the primitive unit cell.

and equality of vector lengths,

and combine them into

λ^{angle} = λ^{angle}_{point}λ^{angle}_{plane}, |

λ^{length} = λ^{length}_{point}λ^{length}_{plane}, |

λ = λ^{angle}λ^{length}. |

μ_{SC}{ζ,ξ,ς} = λ{ζ,ξ,ς}, |

μ_{BCC}{ζ,ξ,ς} = λ{ξ + ς − ζ,ς + ζ − ξ,ζ + ξ − ς}, |

μ_{FCC}{ζ,ξ,ς} = λ{ξ + ς,ς + ζ,ζ + ξ}, |

μ_{DIA}{ζ,ξ,ς} = λ{ξ + ς,ς + ζ,ζ + ξ}. |

〈red,green,blue〉 = 〈(μ_{FCC})^{p},(μ_{BCC})^{p},(μ_{SC})^{p}〉, |

〈red,green,blue〉 = 〈(μ^{angle}_{DIA})^{q},0,(μ^{length}_{DIA})^{q}〉, |

μ^{angle}_{4}{ζ,ξ,ς} = λ^{angle}{ξ + ς,ς + ζ,ζ + ξ}, |

μ^{length}_{4}{ζ,ξ,ς} = λ^{length}{ξ + ς,ς + ζ,ζ + ξ}. |

In the 323 family, the intermediate cubic structures occurred primarily along and on either side of the diagonal that connects the cube and octahedron, and at and around the truncated octahedron ∼〈0.20〉, (Fig. 1g). Near the cube at 〈100, 100〉, the algorithm identified intermediate structures related to SC. Moving along the diagonal (from the cube at the top right to the octahedron at the bottom left) there is a gradual but clear transition from SC to BCC to FCC, and then again to BCC near the corner at the octahedron at 〈0, 0〉. We also observe a region that assembled DIA, including β-tin, which is a distorted version of the diamond structure. The structures with unit cells with n > 2 form a region of cI16 Lithium around ∼〈0, 15〉, a region of the dodecagonal quasicrystal around the tetrahedron ∼〈0, 100〉, as well as single points of BC8 silicon at 〈30, 10〉 and β-manganese at 〈50, 22〉. Regions of disorder were frequently found between structures that had different numbers n of particles in their primitive unit cells, the exceptions being BC8 silicon (n = 8) adjacent to DIA (n = 2) and β-manganese (n = 20) adjacent to FCC (n = 1).

In the 423 family, there are large regions of intermediate structures related to FCC and BCC, a smaller region of SC intermediate structures, and clear transitions between them (Fig. 1h). In contrast, the 523 family mostly assembled FCC intermediates. We observed a region of β-manganese and γ-brass for intermediate shapes near the left edge of the domain, near the dodecahedron.

We also plotted the minimum assembly density as a function of the shape parameters (Fig. 1d–f), where we defined the minimum assembly density as the lowest packing fraction for which we observed crystal formation. In the 323 family, the minimum assembly density is lowest in the FCC region, 51–52%, higher for SC and BCC, 53–54%, and rapidly increased in all cases to greater than 60% as the disordered regions were approached from any structure (Fig. 1d). In the 423 family, the minimum assembly density was greater than 54% at the top left corner, as well as, along the left edge of the domain, where SC and BCC intermediates formed. We observed a large region where FCC intermediates formed at minimum assembly density between 51–52%. Towards the bottom right corner of the 423 landscape, the minimum assembly density increased as FCC-intermediates transitioned to BCC-intermediates, and peaked at 62% towards the edge of the disordered region and for cI16 Lithium (Fig. 1e). In the 523 family, most shapes assembled FCC intermediates at minimum assembly densities 51–52%, while on the upper left edge of the domain, where γ-brass and β-manganese formed, the minimum assembly densities increased to 56–57% (Fig. 1f).

As can be seen by comparing Fig. 1d–f with Fig. 1g–i, visual inspection of the minimum assembly density landscapes is all that is needed to identify regions of ordered structures and transitions between different structures, even without knowing the identity of the structures. Thus, structural signatures of the thermodynamic assemblies are reflected in the minimum assembly density, even though the two measurements are independent of one another.

In comparing visually the assembly and packing landscapes for each family, we observe a general correspondence between assembly and densest packing for the simplest crystal structures. For the more complex structures, the putative densest packings generally differ from the observed assemblies. Of most relevance to this work is that those structures identified as intermediate in the assembly landscapes and labeled via the proposed algorithm to reflect their similarity to the basic cubic crystals, match the corresponding putative densest packings of the structures they are most similar to. This observation demonstrates a major strength of our algorithm, in that it reveals potential relationships between assemblies and packings that could be missed otherwise.

As with previous works on packing, our structure and minimum assembly density landscapes can guide experiments and theory by serving as “phase diagrams for shape” – locating regions where crystals form, regions where crystals fail to form, and transitions between regions. For example, within regions that assemble crystals, our landscapes show how sensitive (or insensitive) the assembly of a crystal in a given region is to small variations in particle shape. For example, there are regions where small shape variations merely affect the assembled structure, such as near the cube in 323, and 423 families, or most shapes in the 523 family (forming FCC). Also, because experimentally synthesized shapes may deviate from perfect polyhedra, such knowledge is useful to evaluate how accurately the shape must be synthesized to achieve successful assembly. For example, in the 323 family the region of shape space over which the cI16 lithium structure forms is much smaller than the region over which BCC forms (Fig. 1g), implying that to assemble the cI16 lithium structure, shape synthesis must be more precise. The minimum assembly density landscapes also provide guidance in selecting shapes for synthesis. For example, to assemble FCC it may be advisable to use the shape with the lowest minimum assembly density shown in the interior of the FCC region, 50%, than a shape at the edge of the FCC region where the minimum assembly density can be as high as 60%. Already studies have used our landscapes to, e.g., quantify solid–solid transitions in shape space^{21} and determine why shapes in the disordered regions fail to assemble crystals.^{22} Furthermore, the correspondence we observe between two independent measures of a crystal assembled from a given shape – namely, our algorithm's structure identification and the minimum density for self assembly – indicates a connection between them that should be further investigated in future work.

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## Footnote |

† D. K. and E. R. C. contributed equally to this work. |

This journal is © The Royal Society of Chemistry 2018 |