Article Cite This: Cryst. Growth Des. 2018, 18, 449−455 pubs.acs.org/crystal Bottom-Up Design and Generation of Complex Structures: A New Twist in Reticular Chemistry ‡ ‡ ‡ § † Ashlee J. Howarth, Peng Li, Omar K. Farha,*, , and Michael O’Keeffe*, ‡ Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States § Department of Chemistry, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia † School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, United States *S Supporting Information ABSTRACT: The design and subsequent construction of complex structures using simple building blocks represent an interesting challenge in the field of chemistry. In this paper we describe complex nets of the mtn family and their relationship to the most common binary structure in chemistry, MgCu2. In this bottom-up approach we start by linking simple shapes in space and show the inevitable evolution to highly complex, low density structures. 1. INTRODUCTION was shown that a simple bottom-up principle was sufficient to The art and science of reticular chemistry is concerned with explain the occurrence of this structure. In this paper, we show that a generalization of that bottom- linking molecular modules of well-defined shapes into sym- up principle accounts for all the previously mentioned metrical frameworks. Although applied mainly to metal− structures and leads to a large family of complex low-density organic frameworks (MOFs),1 it also is relevant to the design structures derived from joining one or two simple shapes in a of related materials such as metal−organic polyhedra (MOPs)2 prescribed way. Given that such structures are clearly and covalent organic frameworks (COFs).3,4 A basic principle designable, we consider this to be an important addition to of reticular chemistry is that the underlying topologies of such the canon of reticular chemistry. materials will correspond to nets with the minimal number of topologically distinct vertices (nodes) and edges (links). In the jargon, these are nets of minimal transitivity consistent with the 2. STRUCTURAL ANALYSIS local geometry of the components.5,6 As such, minimal A salient feature of the complex structures described herein is transitivity in the context of topological analysis and/or design the presence of two kinds of pores arranged as the Mg and Cu is a “bottom-up” principle. The most important nets in this atoms of MgCu2, and as such, it is instructive to consider the connection are uninodal and binodal with just one kind of link geometry of that structure first. (i.e., transitivity 1 1 and 2 1, respectively) as these types of nets 2.1. The MgCu2 and Type II Clathrate Structures. It is are most commonly generated in practice and can serve as well-known that regular tetrahedra cannot fill Euclidean three- 15,16 templates for structural design. Recent developments empha- dimensional space. The dihedral angle is 70.5° (<72 = 360/ size the need, in some instances, to design the geometry of the 5°) meaning that five regular tetrahedra sharing a common component modules as well as consider the topology of the edge will leave a small gap, and space-filling using regular − underlying net when building new architectures.7 9 tetrahedra is only possible in curved space. To fill Euclidean Occasionally MOFs are reported with giant cubic cells that space with tetrahedra as nearly regular as possible, in some are in striking contrast to the minimal-transitivity principle. cases six sharing a common edge are necessary. Such structures are well-known in chemistry as the Frank−Kasper class of Notable examples, setting successive new records for cubic unit − ̅ intermetallic compounds.17 21 Mathematically rigorous enu- cell size (all with symmetry Fd3m), are (1) MIL-100 with unit 22 cell a = 72.9 Å10 and MIL-101 with a = 88.8 Å,11 (2) meration of possibilities shows that there are two simplest “mesoporous MOF 1” with a = 123.9 Å,12 and (3) NU-1301 such structures, and these are formed from three kinds of with a = 173.3 Å.13 This last material has the largest unit cell volume (>5 × 106 Å3) of any known nonbiological material (30 Received: October 13, 2017 times larger than the next largest porous crystal14) and has an Revised: November 13, 2017 underlying net of (vertex edge) transitivity 17 18. Strikingly, it Published: November 27, 2017 © 2017 American Chemical Society 449 DOI: 10.1021/acs.cgd.7b01434 Cryst. Growth Des. 2018, 18, 449−455 Crystal Growth & Design Article tetrahedron. These tetrahedron packings (tilings) in turn to the diamond structure. Staggered linking is not possible with correspond to the common intermetallic structure types regular tetrahedra having an angle of 109.5°, but with a small ° fi MgCu2 and Cr3Si. distortion to an angle of 108 a ring of ve is possible. In tiling with tetrahedra there is a dual structure obtained by Continuation with five-rings will lead to a pentagonal putting vertices in each tetrahedron and joining them to those dodecahedron. However, these dodecahedra, just like tetrahe- in adjacent tetrahedra by new edges through the faces. In this dra, cannot fill Euclidean space, and some polyhedra with six- way, we obtain a 4-c (4-coordinated) net. The nets thus rings are necessary. We have seen that the net mtn of type II obtained from MgCu2 and Cr3Si are again familiar as the nets of clathrate is the best solution to packing pentagonal the type II and type I clathrate structures with zeolite dodecahedra (least number of six-rings). This leads to the framework code MTN and MEP (RCSR symbol23 mtn and key observationstaggered linking gives the dia net with eight mep), respectively. These are the nets of tilings by simple vertices in the unit cell, whereas eclipsed linking gives the mtn polyhedra (those with trivalent vertices) with five- and six-sided net with 136 vertices in the unit cellan increase by a factor of faces. mtn is the one with the smallest fraction of six-sided 17. In what follows, we describe structures formed by linking simple shapes with tetrahedral symmetry in staggered and faces, so MgCu2 is the unique structure composed of the most nearly regular tetrahedra. It is no coincidence that it is by far the eclipsed conformation. The former lead to simple uninodal “ ” most common binary structure in chemistry. Fragments of the nets, the latter to structures 17 times more complicated . structure and its dual structure are shown in Figure 1. 2.3. Nets of the dia and mtn Families. We now describe 2.2. Eclipsed and Staggered Tetrahedral Structures. nets formed by linking cubic polyhedra into tetrahedral groups There are two symmetrical ways to link atoms with tetrahedral through triangular or hexagonal faces. To have locally just one ̅ type of node, polyhedra linked through triangular faces must coordination: staggered with symmetry 3m (D3d) or eclipsed with symmetry 6m̅2(D ). The staggered linking leads uniquely have 12 or fewer vertices tetrahedron, octahedron, truncated 3h tetrahedron, or cuboctahedron. Linking them directly or through trigonal prisms will give the eclipsed conformation as illustrated in Table 1. Linking through octahedra will give the staggered conformation and uninodal nets of the dia (diamond) family. Similarly, linking polyhedra through hexagonal faces with locally one kind of vertex can be achieved with truncated tetrahedra and truncated octahedra as also shown in Table 1. Note that staggered linking of truncated octahedra generates other truncated octahedra as “cavities”, and the whole ensemble is simply the familiar sodalite (sod) structure with symmetry Im3m.̅ We have also included the example of linking truncated cubes through four of their eight triangular faces as this leads to important (3,6)-c nets found in MOFs. The staggered dia (diamond) family of uninodal structures ranges from diamond itself, also of course the structure of silicon, to fau, the net of the zeolite faujasite. We note that these last two are economically the most important inorganic materials. The eclipsed mtn family ranges from mtn itself found − in forms of silicon and germanium24 26 and proposed for carbon27,28 to fav, the analogue of fau, which now has a 17- nodal net and presents a nice challenge to zeolite chemists, but not an unreasonable one given the ease with which the FAU materials can be made. Actually, the anion net of the fav zeolite would be 43-nodal. The mtn family is illustrated in Figure 2 where for clarity just the framework about a large (“Mg” in “ ” MgCu2 ) cavity is shown. The corresponding (smaller) dia family is illustrated in Figure 3 now showing the configuration about the single cavity which is an adamantane unit in dia itself (see figure). 3. MOFs OF THE mtn FAMILY 3.1. MIL-100 and MIL-101. In an earlier discussion29,30 these structures were discussed in terms of frameworks of eclipsed linked tetrahedra. This is clear from the fragments of the structures shown in Figure 4. Considered more carefully, it may be seen that the units in MIL-10111 are truncated tetrahedra linked by trigonal prisms as in mtn-e-a (Table 1). Similarly with the tritopic linker in MIL-100,10 the augmented Figure 1. Fragments of the MgCu2 structure: (a) the coordination net consists of (topological) truncated cubes linked by trigonal prisms as in moo-a (Table 1). Interestingly, MOFs isoreticular around Mg (orange), (b) the coordination around Cu (blue), and (c) − the polyhedra of the dual clatharate structure (mtn). to MIL-100 and MIL-101 have been made.31 33 These have 450 DOI: 10.1021/acs.cgd.7b01434 Cryst. Growth Des. 2018, 18, 449−455 Crystal Growth & Design Article a Table 1. Basic Units of the Nets Described in This Paper 451 DOI: 10.1021/acs.cgd.7b01434 Cryst.