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Modeling of atomic-molecular structures by contiguous filling of space with Frank-Kasper atomic domains
Journal of Molecular Graphics and Modelling 90 (2019) 9e17
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Topical Perspectives
Modeling of atomic-molecular structures by contiguous filling of space with Frank-Kasper atomic domains Oleh H. Havrysh a, *, Vladyslav Kholodovych b, Evgen A. Andreev c a
NanoComb Ltd, Artema Str, 10, Kyiv, 04053, Ukraine Office of Advanced Research Computing, Rutgers University, Piscataway, NJ, USA c Institute of Physics of NAS of Ukraine, Prospect Nauky, 46, Kyiv, 03028, Ukraine b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 November 2018 Received in revised form 4 March 2019 Accepted 5 March 2019 Available online 18 March 2019
An application of contiguous filling of space with convex polyhedra, also known as Frank-Kasper (FK) atomic domains is demonstrated here for modeling of atomic molecular structures. Both regular, when all polyhedron edges have equal length, and strained, depending on the topology of the polyhedron the length of its edges may slightly fluctuate from the common length, polyhedra are used. Polyhedra are connected to each other in agreement with Plateau's laws to form a contiguous uninterrupted space. An application of a new approach is demonstrated for a modeling of structures of graphite, graphene, graphane, diamond and two types of ice. The proposed approach allows to demonstrate a mutual arrangement of atoms in graphite layers, transitions between allotropic states of carbon atoms, to calculate the distances between layers in graphene and positions of water molecules in a square ice. © 2019 Elsevier Inc. All rights reserved.
Keywords: Contiguous filling of space Frank-kasper polyhedra Atomic domain Plateau's laws FSP model
1. Introduction Polyhedral structures exist in diverse complexity in nature. They vary in number of atoms from very simple crystals [1] to hundreds of carbons in fullerenes [2], elaborated frameworks of clathrates [3] and crystal envelope of phytoplanktonic algae [4], a great variety of packing in biomolecules and virus capsids and large biological formations associated with a vital activity of organisms [5]. Organized in the form of polyhedral frameworks they mutually penetrate and complement each other to form a continuous discrete space. The first model of space partitioned into cells was published by William Thomson Kelvin [6]. Later known as Kelvin problem, this model described the division of three-dimensional space into polyhedral cellular units of equal volume with a minimal surface area. This model was based on the laws formulated by Joseph Plateau in his study of foam formation [7]. To obey the Plateau's laws, i.e. only four lines meet symmetrically at one vertex and only three faces meet symmetrically at one line, Kelvin used a slightly deformed truncated octahedron and postulated that the most stable foam consists of bitruncated cubic honeycombs.
* Corresponding author. E-mail addresses: [email protected] (O.H. Havrysh), [email protected] (V. Kholodovych), [email protected] (E.A. Andreev). https://doi.org/10.1016/j.jmgm.2019.03.004 1093-3263/© 2019 Elsevier Inc. All rights reserved.
Later, in the study of possibility to partition the Euclidean space with identical polyhedra, Fedorow E [8]. found that only five types of polyhedra can fully fill the space. Among these five, only truncated octahedron, fills the space in agreement with Plateau's laws. Interestingly, it is the same polyhedron that was used by Kelvin in his model [6]. In further development of Kelvin's work, new models of space filling with two or more types of polyhedra [9] were proposed and one of these models [10] was experimentally confirmed in foam [11]. At some point, there was a need to describe a spatial organization of space that was filled with polyhedra. This was done by Frank and Kasper and generalized on the example of metallic alloys [12,13]. Frank and Kasper defined the concepts of “coordination polyhedron” and “atomic domain”. The set of atoms closest to the central atom, they called the coordination shell, and the volume of space around the central atom, limited to coordination shell, called coordination polyhedron. Implying that Z is a coordination number the authors named coordination polyhedra as Z12, Z14, Z15 and Z16, respectively. An atomic domain is then the area around the central atom, any point of which is closer to the center of a given atom than to the center of any other atom. Atomic domains are convex polyhedra. A coordination polyhedron and an atomic domain for a given atom, as noted by Frank and Kasper, are in a state of duality. This means that the vertices of the coordination
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polyhedron correspond to the faces of the atomic domain, and the faces of the coordination polyhedron correspond to the vertices of the atomic domain. The coordination polyhedra completely fill the space of the crystal lattice. Since each atom forms the coordination shells with neighboring atoms, coordination shells of the neighboring atoms have a common space. Atomic domains are built on the principle of Voronoi-Dirichlet, i.e. the polyhedron of the atomic domain is formed as a result of the intersection of planes drawn perpendicular to the middle of the segments laying between the central atom and the atoms of its coordination shell. Thus, atomic domains are not intersecting, i.e. do not enter the space of each other, and continuously filling the volume of the lattice. Neighboring atomic domains only have one common face, as well as common edges and vertices belonging to this face. We perform modeling by using polyhedra of atomic domains, the duals of coordination polyhedra Z12, Z14, Z15 and Z16. These polyhedra have pentagonal and hexagonal faces and for simplicity we named them as p12, p12h2, p12h3 and p12h4, where indices p and h denote pentagonal and hexagonal faces, respectively and the number after the index sums the number of such faces in polyhedra. The space of coordination polyhedra can be continuously filled with the atomic domains of the central atom and atoms from the coordination shell. Polyhedra are commonly used for modeling of chemical structures. Thus, structures of fullerenes were described in terms of Frank-Kasper polyhedra [14]. Eighty four new structures were generated by computer modeling and then compared with twenty seven known structures. The authors considered a symmetry in modeling of structural fragments, but the complete spheroidal particles of a fullerene as a whole were not studied. In Ref. [2] the similarity between icosahedral structure of certain alloys and structures of fullerenes was investigated with the help of FrankKasper (FK) polyhedra. In general, a crystal lattice of clathrates can be modeled by a contiguous framework of polyhedra with atoms are at their vertices, and chemical bonds connecting atoms are described as the edges of polyhedra, wherein all host atoms are located inside polyhedral cells of clathrates. There are several attempts to describe structure of clathrates with polyhedra, including Frank-Kasper polyhedra [14,15]. Here, we propose a new method for visualization of a complete structure of a crystal lattice, based on the idea of uniformly contiguous filling of space. We fill a lattice space with different types of predefined polyhedra, that have approximately the same length of their edges. Filling a space with polyhedra is guided by an initial arrangement of atoms obtained from experimental data of their crystal structures. Our approach differs from other geometric methods in that we fill the entire crystal space with polyhedral cells continuously. Some cells may contain atoms while others stay empty (dummy). Though the framework topology remains the same, the occupancy of the framework cells changes dynamically over time and depends on the mutual arrangement of atoms in the crystal. For example, some cells may transfer from empty to filled with atoms during allotropic transitions or upon formation of clathrates. Continuous filling of space by polyhedra allows to determine and to visualize not just the neighboring covalent interactions between atoms but also long-distance interactions like coordinate, hydrogen and vander-Waals bonds. A structure of a polyhedral framework in our approach, like a structure of foam, is formed by polyhedra that have common facets, edges and vertices. Therefore, it seems logical that for such polyhedral frameworks Plateau's laws formulated to describe the structure of foam, should be also satisfied.
2. FSP method We propose a new approach for modeling of crystal structures by contiguous Filling of Space with convex Polyhedra (FSP) that based on the rules formulated by Joseph Plateau in his study of foam properties [7]. We consider a compliance with Plateau's laws to be an important factor for space filling, as Plateau's laws are intrinsically embedded in many natural structures. We first fill the space using a limited set of convex polyhedra that have the same length of all their edges. However later the length of edges is allowed to slightly deviate from a common length to compensate for possible contortions of polyhedral faces in models built strictly with equal edges polyhedra. A use of polyhedra with planar faces is convenient because it is easy to connect them and because it simplifies the visual perception of the construction. Frank and Kasper studied metal alloys that have a uniformly filled compact lattice space. It is not the case in the lattice of graphite or ice, where the distance in a crystal lattice between identical atoms may vary significantly. It happens because some gaps exist between carbon layers in the graphite structure and cavities within the unit cells in the structure of hexagonal ice. In our models we address these structural irregularities by modeling spatial voids together with the atomic structures. Although it is possible to fill the space of a crystal lattice with different multifaceted polyhedra, in this study we only use four types of polyhedra, p12, p12h2, p12h3 and p12h4, that are duals of FK coordination polyhedra Z12, Z14, Z15 and Z16. We use the polyhedra that are atomic domains, because they fill the space by forming a continuous net. Macroscopic analogue of such a network is the foam. Our approach is based on the following principles: 1) the lattice space is filled continuously with convex polyhedra that have edges of the same length; a length of the edge is chosen to be equal to the Bohr radius of 0.529 Å:
a0 ¼
ħ ¼ 0:529 A; me $c$a
where a0 is the Bohr radius, ħ is the reduced Planck's constant, me is the electron mass at rest, c is the speed of light in a vacuum, and a is the fine structure constant. 2) obedience with Plateau's laws of foam formation: a strict compliance with topological rules - only four lines meet symmetrically at one vertex and only three faces meet symmetrically at one line; less strict compliance - the edge length, the angles between edges and faces in polyhedra may deviate slightly from the original values due to intrinsic deformations in natural structures; 3) existence of unfilled atomic domains in the modeled structure: empty (dummy) polyhedra are legitimate components of the domain structure of the crystal lattice; later they can be filled with atoms, for example, in the formation of clathrates. Frank and Kasper defined an atomic domain of dummy polyhedra as “an area around the center of the domain, any point of which is closer to the center of the given domain than to the center of any other domain” [12]; thus, a shell of empty (dummy) polyhedra around a certain polyhedron we adherently describe as a domain envelope (or shell) of a given polyhedron. On the first stage of modeling, we determine the key atoms in the lattice. Then, based on the mutual arrangement and known distances between the key atoms, we select a set of polyhedra that
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will be used for modeling. This initial set of polyhedra that corresponds to atomic domains of key atoms are placed in the position on the lattice and the space around them continuously filled with other polyhedra until a complete topological matching to the modeled crystal structure is achieved. In nature, atomic lattices of crystals are often deformed, i.e. compressed or stretched. For the first stage of modeling we use the standard polyhedra p12, p12h2, p12h3 and p12h4 with their edges being the same length. Therefore, after the first “topological” stage of the model construction, the model needs to be corrected to encompass such natural deformations. In this study we use two types of correction: tetrahedral lattices undergo correction uniformly throughout their volume by stretching (diamond) or squeezing (hexagonal ice); the lattices consisting of parallel layers, i.e. graphite and square ice, are corrected with respect of keeping the volume of their polyhedra to be constant. See more details in Results section. 3. Results 3.1. Graphite Graphite is one of the crystalline forms of carbon that has a layered structure consisting of parallel layers of hexagonally ordered carbon atoms [16]. Each atom within the layer is in the state of sp2 hybridization and covalently bonded with three nearest atoms. In general, the carbon atoms in each layer form hexagons
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similar to honeycombs. The distance between two adjacent atoms within a plane of the layer is 1.42 Å. Layers of carbon atoms in graphite are arranged in such a way that half of the atoms of one layer are positioned exactly above/below the centers of hexagonal cells of the other layer. This packing is also known as AB stacking. Thus, alpha graphite has an identical arrangement of atoms in every other layer. The distance between graphite layers is 3.35 Å. We modeled graphite, starting with a monolayer of carbon atoms, polyhedra p12h3, colored cyan in Fig. 1. Connecting the polyhedra p12h3 by their hexagonal faces, we constructed a planar framework that consisted of hexagons and accurately reproduced a structure of the graphite monolayer. The distance between centers of neighboring polyhedra was 1.42 Å. Thus, a constructed FSP model reproduced an arrangement of carbon atoms in the alpha graphite monolayer topologically and matched interatomic distances very well. Since our method implied a contiguous space filling, then for a complete modeling of alpha graphite it was necessary to build intermediate spatial layers located between carbon layers of graphite (Fig. 1). Obviously, these layers must be flat to complement carbon layers. Moreover, the intermediate layers should be complimentary to all adjacent layers of graphite including themselves. Space filling with dummy polyhedra between layers of carbon atoms is shown in Fig. 1, C. Initially, hexagonal cells formed by polyhedra p12h3 (carbon atoms) get filled with a pair of dummy polyhedra p12h2; then dummy polyhedra p12h4 get connected through hexagonal faces with polyhedra p12h2 and a remaining
Fig. 1. FSP model of alpha graphite. A e a unit cell of alpha graphite shown in the traditional representation; B e a unit cell of alpha graphite represented by polyhedra p12h3 (cyan); C e filling the space between layers of carbon atoms with dummy polyhedra.
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free space get filled with adjacent to the carbon layers dummy layers of polyhedra p12. In our unstrained FSP model of graphite the distance between carbon monolayers was 4.05 Å. The distance between atoms of carbon within the layer was approximately equal to the length of a double bond in ethylene 1.30 Å. Experimentally determined distances between layers of alpha graphite are known to be smaller, 3.35 Å. This phenomenon was explained by a mutual attraction of layers caused by van der Waals forces between carbon atoms. Thus, to match our FSP model with experimental values we had to perform a linear correction. Correction factors for crystal lattices filled with parallel layers of polyhedra were defined as Cx*Cy*Cz ¼ 1,
1
where Cx, Cy and Cz e correction factors of individual coordinate axes. This equation implies that the overall volume occupied by all polyhedra remains the same before and after corrections. The correction factor along the z-axis (Cz) in graphite model was set to be 0.83 (3.35 Å/4.05 Å). Because layers in graphite are formed by regular hexagons of carbon atoms we decided to use equal correction factors within a planar layer, Cx ¼ Cy. Thus, from equation (1) Cx ¼ Cy ¼ sqrt(1/ Cz) ¼ sqrt(1/0.83) ¼ 1.10 After the correction, all centers of p12h3 polyhedra composing carbon monolayers fit precisely into coordinates of carbon atoms from the experimentally determined crystal structure of alpha graphite. 3.2. Graphene Graphene is just one flat monolayer of carbon atoms from graphite has been obtained for the first time in 2004 [17]. Likewise graphite, carbon atoms in graphene are in the state of sp2 hybridization. An FSP model of a single monolayer of graphene/graphite composed of polyhedra of carbon atoms surrounded by dummy polyhedra is shown in Fig. 1C. 3.3. Graphane Graphane is a fully saturated hydrocarbon obtained from a
single sheet of graphene by hydrogenation. Graphane has a planar two-dimensional organization [18]. Carbon atoms in graphane stay in sp3 hybridization, and each carbon atom forms covalent s-bonds with three neighboring carbon atoms and one hydrogen atom. Half of the hydrogen atoms are located above the plane of a carbon atom layer, and the other half is on the opposite side of it, Fig. 2. An FSP model of a monolayer of graphene was built with polyhedra p12h4 connected through hexagonal faces so that they form a hexagonal framework. The hydrogen atoms in graphane were modeled by polyhedra p12h2, that were placed on both sides of the p12h4 polyhedron layer on their hexagonal faces. The remaining voids between the carbon and hydrogen atoms in the model of graphane were filled with p12 dodecahedrons. Thus, the complete model of graphane consists of polyhedra p12h4, p12h2 and p12, Fig. 2. A segment of graphene layer is shown as three connected 6member rings that are formed by carbon atoms in sp2 hybridization, Fig. 2A. Carbon atoms are depicted as polyhedra p12h3 colored cyan. In Fig. 2B a side view of the same segment of graphene as in the panel 2A is shown with completely filled space with three layers of dummy polyhedra; carbon atoms are shown as polyhedra p12h3 in cyan; dummy polyhedra p12h2, p12h3 and p12h4 are depicted in shades of brown as indicated in the picture. After hydrogenation, a layer of graphene transformed into graphane, Fig. 2C and D. Carbon atoms changed from sp2 to sp3 hybridization are shown as polyhedra p12h4 and colored cyan, hydrogen atoms are depicted as white polyhedra p12h2. A simplified view of graphene layer with only three 6-member rings formed by carbon atoms and hydrogen atoms and stripped off all filling the inner space dummy polyhedra is added in the panel 2D for visual clarity. Topology of the full network of graphane after hydrogenation remains the same as in graphene and they both resemble the similarity to graphite (Fig. 1).
3.4. Diamond Diamond consists of carbon atoms in sp3 hybridization that covalently bonded to four other carbon atoms forming a cubic crystal structure. We modeled a diamond framework with p12h4 polyhedra connected to form a hexagonal network. Each layer of polyhedra p12h4 in diamond (Fig. 3C) is topologically identical to a layer of dummy polyhedra p12h4 in graphite (Fig. 1C) and to a carbon layer of graphane (Fig. 2D). Voids between polyhedra p12h4
Fig. 2. Transition of graphene (A, B) to graphane (C, D). А e a segment of graphene layer; B e a side view of the same segment as in A, with completely filled space with three layers of dummy polyhedra; C e a side view of graphane layer; D e a simplified view of graphane layer.
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Fig. 3. FSP model of diamond. A e a unit cell of diamond shown in the traditional representation; B e a unit cell of diamond presented by polyhedra p12h4 (cyan); panels CeE: a unit cell of diamond shown from different viewpoints partially filled with dummy polyhedra p12 (yellow) in the space between carbon atoms. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
in diamond were filled by dodecahedrons p12, Fig. 3. The final FSP model of diamond was adjusted by stretching along all three coordination axes with a coefficient of 1.12. 3.5. Square ice Recently, it has been shown that in nanocapillaries between two sheets of graphene, water molecules form unusual monomolecular planar layers [19], whereas oxygen atoms are located at the vertices of squares at a distance of 2.83 Å from each other. A single, double or triple layers of water molecules were reported to exist. Interestingly, oxygen atoms in adjacent water layers were located exactly one above the other, the phenomenon known as AA stacking. The authors named the new discovered organization of water molecules “square ice”. Square ice had a density of about 1.5 times the density of ordinary ice, that pointed to a high packing density of its structure. For an FSP model of square ice in nanocapillaries between two sheets of graphene, we strictly adhered to the experimentally established facts: a) monomolecular layers of water are flat; b) oxygen atoms of water molecules are located at the vertices of squares with a side length of 2.83 Å; c) an oxygen atom of the water molecule occupies the center of a tetrahedron and hydrogen atoms and oxygen regions of polarization are located at tetrahedron vertices; d) carbon atoms are located at the vertices of hexagons that form sheets of graphene, Fig. 4. We determined that water molecules could form a square
structure in a flat layer while retaining a topologically characteristic organization of a regular water molecule, if the oxygen atom is modeled by a polyhedron p12h4, and the hydrogen atoms and oxygen polarization regions by polyhedra p12h3. Hydrogen bond in this case is formed by a pair of polyhedra p12h3, one representing a hydrogen atom and another one being one of an oxygen atom polarization regions. Water molecules from two adjacent layers form a tetragonal Bravais lattice with a distance between layers 3.20 Å, Fig. 4B. A nanocapillary space between graphene sheets was completely filled with layers of various polyhedra representing water molecules and intermolecular space filling dummy polyhedra, Fig. 4C. Interestingly this complex structure can be split into several distinct sub-layers, i.e. a layer of graphene, a layer of water and two interlayers, one is between graphene and water and the second one is between two water layers, Fig. 4D. Noteworthy sublayers at the second graphene sheet repeat themselves in the inverse order with a mirror-like orientation, not shown (more detailed see in Supplemented materials, figure S2). To match the experimentally determined distances between oxygen atoms in water molecules that form a monolayer in the capillary [19] we had to adjust positions of carbon atoms in graphene layers. In such a stretching with a coefficient of 1.35 (Cx) was performed along the x-axis within the plane of a layer for all hexagons formed by carbon atoms and a compression with a coefficient of 1.17 (Cy) was done along the y-axis in the same plane, Fig. 4A. A correction factor along the z-axis (Cz) was calculated as
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Fig. 4. FSP model of square ice in graphene nanocapillaries. A e layer of graphene. B e Bravais lattice of adjacent layers of water molecules is tetragonal (а ¼ b ¼ 2.73 Å, с ¼ 3.2 Å). C e assembly of polyhedra layers in graphene nanocapillaries. D e separated view of polyhedra layers, as shown in the panel C.
mentioned above, equation (1). Cz ¼ 1/Cx*Cy¼1/1.35*1.17 ¼ 0.63. After a complete FSP model was assembled, a final correction was performed in the direction perpendicular to the plane of the graphene sheet with a calculated compression coefficient Cz. This compression was done for all layers located between graphene sheets. A good match between an FSP model and experimentally obtained data for square ice was achieved for a) distances between layers of water molecules and b) distances between graphene sheets with different number of layers of water molecules between them, Table 1. More details about FSP models of square ice can be found in Supplemented materials. 3.6. Hexagonal ice (Ih) In nature ice appears in many amorphous forms with hexagonal
form being a predominant one [20e22]. We modeled hexagonal ice as a tetrahedron formation by placing polyhedra in the center and into vertexes of the tetrahedron. An oxygen atom (polyhedron p12h4) was placed in the center and four polyhedra p12h2 were attached to it symmetrically through hexagonal faces. Two polyhedra p12h2 representing hydrogen atoms, and other two representing regions of polarization near the oxygen atom (lone pairs of electrons) in water molecules were added as shown in Fig. 5A. A hydrogen bond in hexagonal ice is formed by a pair of polyhedra p12h2, connected to two polyhedra p12h4 (oxygen atoms), wherein for each pair of polyhedra p12h2 there is only one hydrogen atom and the other one is a dummy atom. In an FSP model of hexagonal ice each oxygen atom of a water molecule connects through hydrogen bonds with four oxygen atoms in neighboring water molecules. Six molecules of water connected by hydrogen bonds form six-member rings in chair configuration (Fig. 5B). Two six-member rings of water molecules arranged one
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Table 1 Distances between monolayers of water and graphene in nanocapillary. Distances between monolayers of water and graphene in nanocapillary Distance betweena …
Experimental data15
FSP model
two monolayers of water graphene layers with one monolayer of water graphene layers with two monolayers of water graphene layers with three monolayers of water
3.5 6.5 9.0 11.5
3.2 6.7 9.9 13.1
a
All distances are in Angstroms.
Fig. 5. Hexagonal ice. A e a single water molecule represented by polyhedra; B e one half of a unit cell of hexagonal ice; C e a snippet of an FSP model of hexagonal ice shown from above and as a side view.
above the other form a unit cell of hexagonal ice. Notably unit cells in hexagonal ice share vertices, edges and faces. A snippet of hexagonal ice consisting of 126 water molecules is shown in Fig. 5C. The FSP model of hexagonal ice constructed with regular polyhedra needs a uniform compression across all three axes with a coefficient of 0.82. Thus, the distances between oxygen atoms in the structure of hexagonal ice and in the monolayer of water molecules in “square” ice are almost identical e 2.76 and 2.83 Å. However, due to the topologically denser packing of water molecules in square ice, its density is substantially higher than that of hexagonal ice. The difference in packing can be easily seen in FSP models, e.g. in the structure of square ice the hydrogen bond between water molecules is represented by polyhedra p12h3, whereas in hexagonal ice it is modeled with p12h2 polyhedra. More details about FSP models of hexagonal ice can be found in Supplemented material.
4. Discussion 4.1. Graphite, graphene, graphane, diamond An FSP model of graphite helps to demonstrate the arrangement of carbon atoms in adjacent layers of alpha graphite. An observation that carbon atoms of one layer are shifted half way through the layer relatively to the position of carbon atoms in the adjacent layer is determined by topology of dummy polyhedra in the interatomic layer, Fig. 1C. A model also reveals that different allotropic states of carbon coexist in the graphite lattice simultaneously. They can be selected as a formation of atomic polyhedra, as a set of dummy polyhedra or as a combination of both. In such the transition of one allotropic form to another can be easily traced and visualized. Graphene, a monolayer of graphite, consists of carbon atoms in the state of sp2 hybridization. Each carbon atom in graphene forms
three covalent s-bonds with three neighboring atoms. The fourth p-bond is formed due to a partial overlapping of p-orbitals above and below the plane in which the carbon atoms are located. The pbond forms a continuous electron cloud over the entire layer of carbon atoms, a structure similar to such observed in metals. Hydrogenation of graphene leads to a transition of carbon atoms from sp2 to sp3 hybridization and to a formation of four equivalent sbonds in a tetrahedral configuration, and graphene turns into graphane, Fig. 2. In an FSP model of graphane, a fully saturated hydrocarbon obtained from a single sheet of graphene, carbon atoms were modeled as p12h4 polyhedra and the hydrogen atoms as p12h2 polyhedra. Interestingly, the layers of graphane are structurally identical to the space filling layers of dummy polyhedra in graphite, that constructed with polyhedra p12, p12h2 and p12h2 and located between layers of carbon atoms, Fig. 1. A graphane layer of carbon atoms in sp3 hybridization topologically matches a monolayer of carbon atoms in the crystal lattice of diamond. In the polyhedral model of graphite, two alternating layers are present: flat layers of carbon atoms in sp2 hybridization (shown as p12h3 polyhedra) and dummy layers of p12h4. Topologically, the layer of p12h4 dummy polyhedra resembles a layer of carbon atoms in sp3 hybridization in diamond. These observations undeniably indicate a direct relationship between graphite, graphane and diamond.
4.2. Square ice An FSP model of the polyhedral structure of square ice implies that water molecules in the capillary layer between two sheets of graphene retain their tetrahedral structure. Water molecules have a specific arrangement in which their oxygen atoms located in the points on the grid with square cells. Because water molecules in square ice packed more densely than in hexagonal ice we had to use
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p12h3 polyhedra, rather than p12h2 polyhedra for hydrogen atoms. Thus, a pair of adjacent p12h3 polyhedra between oxygen atoms, one representing a hydrogen atom and the other one being dummy, we treated as a hydrogen bond. In contrast, a hydrogen bond in the structure of hexagonal ice and in bulk water was built with polyhedra p12h2. The tetrahedral structure of a water molecule is determined by oxygen atoms, that are always modeled by p12h4 polyhedra. Thus, a packing and a mutual arrangement of water molecules depend on the position and polyhedron types selected for modeling of hydrogen atoms. A polyhedral model of a graphene sheet in contact with water differ from the model of a carbon layer in graphite. The inner layers of dummy polyhedra located between a graphene sheet and monolayers of water is complex and consists of polyhedra p12, p12h2 and p12h3, whereas the inner layer between two layers of carbon in graphite can be modeled with p12, p12h2, and p12h4, compare Figs. 1 and 4. Based on our FSP model of nanocapillaries in graphene we suggest that graphene sheets undergo some deformation. The carbon atoms are stretched within the planar layer and the entire framework, including inner layers of dummy atoms between sheets of graphene are compressed in a direction perpendicular to the carbon plane, Fig. 4. With our FSP model of square ice we could estimate distances between graphene sheets and layers of water molecules, the distance between monolayers of water, and the distance between graphene sheets in nanocapillaries. Comparison of calculated values from an FSP model with experimentally obtained measurements [19] showed a good matching, Table 1. An FSP model also helped to demonstrate why the distance between oxygen atoms in neighboring layers in nanocapollaries is greater than the distance between them within the same layer. As clearly seen in the model of the entire polyhedral structure of graphene nanocapillaries, between two monolayers of water, there is an extra inner layer of dummy atoms, Fig. 4, (C-D). Therefore, a distance between oxygen atoms from neighboring monolayers of water (3.2 Å) is greater than a distance between directly connected oxygen atoms within the layer (2.8 Å). This observation is an agreement with experimentally measured distances between water layers of 3.5 Å.(16) Based on our FSP model we posit that a crystal system of square ice belongs to a tetragonal and not to a cubic crystal family as was reported previously [19]. 4.3. Hexagonal ice A polyhedral model of hexagonal ice differs significantly from the structure of square ice. This difference arises from the type of polyhedra with which hydrogen bonds are modeled. In hexagonal ice a pair of polyhedra p12h2 are used to model a hydrogen bond, whereas in square ice it is made with polyhedra p12h3. The structure of hexagonal ice (Fig. 5) assumes a significantly lower packing density. A unit cell of hexagonal ice consisted of 12 water molecules can be encapsulated inside the cube with an edge length of 8 Å. For comparison 24 water molecules located in three parallel monolayers of square ice occupy the same volume. Another important distinction is that in hexagonal ice all water molecules form a continuous network of hydrogen bonds connecting all water molecules together in the entire volume. Contrary hydrogen bonds in square ice form a network exclusively within a plane of individual water monolayer, and different layers of water are separated from each other by an inner layer of dummy atoms. Though in both types of ice water molecules exist in a characteristic tetrahedral structure, their internal spatial organization is substantially different.
5. Conclusions Applications of the method of contiguous filling of space with a limited set of convex polyhedra (FSP) were demonstrated on models of graphite, graphene, graphane, diamond, square ice and hexagonal ice. FSP models replicated crystal structures well and helped to demonstrate the mutual arrangement of atoms and hidden transitions from one allotropic state to another observed in crystal lattices. The use of dummy polyhedra in FSP models allowed to avoid ambiguities associated with significant differences in interatomic distances between identical atoms in structures that have intrinsic gaps and cavities, as shown on examples of graphite and ice. Dummy polyhedra are legitimate components of the domain structure in molecular models. By using the FSP models it was possible to demonstrate that the mutual arrangement of carbon atoms in adjacent graphite layers is governed by an inner layer of dummy polyhedra located between the carbon layers. It was shown why hydrogenation of graphene leads to its transformation into graphane. FSP models helped to demonstrate that different allotropic forms of carbon are simultaneously present in graphite, either visible through atomic polyhedra or hidden in a form of dummy polyhedra. A presence of dummy layers of polyhedra in the graphite structure that are topologically identical to layers of carbon in diamond not only confirms a well-known relationship between graphite and diamond, but also suggests the mechanism of transition of carbon atoms from one allotropic form to another. A complete FSP model of nanocapillaries in graphene allowed to calculate distances between layers of water and graphene sheets, and distances between monolayers of water. It was shown that calculated values correlate well with experimental measurements. An FSP model of square ice clearly demonstrated why square ice has rather a tetragonal than a cubic lattice. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jmgm.2019.03.004. References [1] W. Niu, et al., Dodecahedral gold nanocrystals: the missing platonic shape, J. Am. Chem. Soc. 136 (8) (2014) 3010e3012. [2] S. Alvarez, Nesting of fullerenes and frankekasper polyhedra, Dalton Trans. (17) (2006) 2045e2051. [3] A.J. Karttunen, et al., Structural principles of semiconducting group 14 clathrate frameworks, Inorg. Chem. 50 (5) (2011) 1733e1742. [4] K. Hagino, et al., Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-a in braarudosphaera bigelowii (prymnesiophyceae), PLoS One 8 (12) (2013), e81749. [5] L. Tang, et al., The structure of Pariacoto virus reveals a dodecahedral cage of duplex RNA, Nat. Struct. Biol. 8 (2001) 77. [6] W. Thomson, On the division of space with minimum partitional area, Acta Math. 11 (1887) 121e134. [7] J.A.F. Plateau, Statique exp erimentale et th eorique des liquides soumis aux seules rimentale et the orique des liquides soumis forces mol eculaires. Statique expe culaires, Gauthier-Villars, 1873. aux seules forces mole [8] E. von Fedorow, XVII. Allgemeinste Krystallisationsgesetze und die darauf fussende eindeutige Aufstellung der Krystalle, in: Zeitschrift für Kristallographie - Crystalline Materials, 1904, pp. 321e490. [9] R. Gabbrielli, A new counter-example to Kelvin's conjecture on minimal surfaces, Phil. Mag. Lett. 89 (8) (2009) 483e491. [10] D. Weaire, R. Phelan, A counter-example to Kelvin's conjecture on minimal surfaces, Phil. Mag. Lett. 69 (2) (1994) 107e110. [11] R. Gabbrielli, et al., An experimental realization of the WeaireePhelan structure in monodisperse liquid foam, Phil. Mag. Lett. 92 (1) (2012) 1e6. [12] F.C. Frank, J.S. Kasper, Complex alloy structures regarded as sphere packings. I. Definitions and basic principles, Acta Crystallogr. 11 (3) (1958) 184e190. [13] F.C. Frank, J.S. Kasper, Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures, Acta Crystallogr. 12 (7) (1959) 483e499.
O.H. Havrysh et al. / Journal of Molecular Graphics and Modelling 90 (2019) 9e17 [14] M. Dutour Sikiri c, O. Delgado-Friedrichs, M. Deza, Space fullerenes: a computer search for new FrankeKasper structures, Acta Crystallogr. A 66 (5) (2010) 602e615. [15] J. Wang, J.-A. Dolyniuk, K. Kovnir, Unconventional clathrates with transition metalephosphorus frameworks, Acc. Chem. Res. 51 (1) (2018) 31e39. [16] B.T. Kelly, Physics of Graphite, Applied Science, London; Englewood, N.J, 1981, p. 477. [17] K.S. Novoselov, et al., Electric field effect in atomically thin carbon films, Science 306 (5696) (2004) 666e669. [18] J.O. Sofo, A.S. Chaudhari, G.D. Barber, Graphane: a two-dimensional hydrocarbon, Phys. Rev. B 75 (15) (2007) 153401. [19] G. Algara-Siller, et al., Square ice in graphene nanocapillaries, Nature 519
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(7544) (2015) 443e445. [20] A. Falenty, T.C. Hansen, W.F. Kuhs, Formation and properties of ice XVI obtained by emptying a type sII clathrate hydrate, Nature 516 (7530) (2014) 231e233. [21] V. Fuentes-Landete, et al., Crystalline and amorphous ices, in: P.G. Debenedetti, M.A. Ricci, F. Bruni (Eds.), Proceedings of the International School of Physics "Enrico Fermi", Volume 187: Water: Fundamentals as the Basis for Understanding the Environment and Promoting Technology, IOS and Bologna: SIF, Amsterdam, 2015, pp. 173e208, 2015. [22] W.F. Kuhs, M.S. Lehmann, The structure of ice-ih, in: F. Franks (Ed.), Water Science Reviews 2: Crystalline Hydrates, Cambridge University Press, Cambridge, 1986, pp. 1e66.
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Journal of Molecular Graphics and Modelling journal homepage: www.elsevier.com/locate/JMGM
Topical Perspectives
Modeling of atomic-molecular structures by contiguous filling of space with Frank-Kasper atomic domains Oleh H. Havrysh a, *, Vladyslav Kholodovych b, Evgen A. Andreev c a
NanoComb Ltd, Artema Str, 10, Kyiv, 04053, Ukraine Office of Advanced Research Computing, Rutgers University, Piscataway, NJ, USA c Institute of Physics of NAS of Ukraine, Prospect Nauky, 46, Kyiv, 03028, Ukraine b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 November 2018 Received in revised form 4 March 2019 Accepted 5 March 2019 Available online 18 March 2019
An application of contiguous filling of space with convex polyhedra, also known as Frank-Kasper (FK) atomic domains is demonstrated here for modeling of atomic molecular structures. Both regular, when all polyhedron edges have equal length, and strained, depending on the topology of the polyhedron the length of its edges may slightly fluctuate from the common length, polyhedra are used. Polyhedra are connected to each other in agreement with Plateau's laws to form a contiguous uninterrupted space. An application of a new approach is demonstrated for a modeling of structures of graphite, graphene, graphane, diamond and two types of ice. The proposed approach allows to demonstrate a mutual arrangement of atoms in graphite layers, transitions between allotropic states of carbon atoms, to calculate the distances between layers in graphene and positions of water molecules in a square ice. © 2019 Elsevier Inc. All rights reserved.
Keywords: Contiguous filling of space Frank-kasper polyhedra Atomic domain Plateau's laws FSP model
1. Introduction Polyhedral structures exist in diverse complexity in nature. They vary in number of atoms from very simple crystals [1] to hundreds of carbons in fullerenes [2], elaborated frameworks of clathrates [3] and crystal envelope of phytoplanktonic algae [4], a great variety of packing in biomolecules and virus capsids and large biological formations associated with a vital activity of organisms [5]. Organized in the form of polyhedral frameworks they mutually penetrate and complement each other to form a continuous discrete space. The first model of space partitioned into cells was published by William Thomson Kelvin [6]. Later known as Kelvin problem, this model described the division of three-dimensional space into polyhedral cellular units of equal volume with a minimal surface area. This model was based on the laws formulated by Joseph Plateau in his study of foam formation [7]. To obey the Plateau's laws, i.e. only four lines meet symmetrically at one vertex and only three faces meet symmetrically at one line, Kelvin used a slightly deformed truncated octahedron and postulated that the most stable foam consists of bitruncated cubic honeycombs.
* Corresponding author. E-mail addresses: [email protected] (O.H. Havrysh), [email protected] (V. Kholodovych), [email protected] (E.A. Andreev). https://doi.org/10.1016/j.jmgm.2019.03.004 1093-3263/© 2019 Elsevier Inc. All rights reserved.
Later, in the study of possibility to partition the Euclidean space with identical polyhedra, Fedorow E [8]. found that only five types of polyhedra can fully fill the space. Among these five, only truncated octahedron, fills the space in agreement with Plateau's laws. Interestingly, it is the same polyhedron that was used by Kelvin in his model [6]. In further development of Kelvin's work, new models of space filling with two or more types of polyhedra [9] were proposed and one of these models [10] was experimentally confirmed in foam [11]. At some point, there was a need to describe a spatial organization of space that was filled with polyhedra. This was done by Frank and Kasper and generalized on the example of metallic alloys [12,13]. Frank and Kasper defined the concepts of “coordination polyhedron” and “atomic domain”. The set of atoms closest to the central atom, they called the coordination shell, and the volume of space around the central atom, limited to coordination shell, called coordination polyhedron. Implying that Z is a coordination number the authors named coordination polyhedra as Z12, Z14, Z15 and Z16, respectively. An atomic domain is then the area around the central atom, any point of which is closer to the center of a given atom than to the center of any other atom. Atomic domains are convex polyhedra. A coordination polyhedron and an atomic domain for a given atom, as noted by Frank and Kasper, are in a state of duality. This means that the vertices of the coordination
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polyhedron correspond to the faces of the atomic domain, and the faces of the coordination polyhedron correspond to the vertices of the atomic domain. The coordination polyhedra completely fill the space of the crystal lattice. Since each atom forms the coordination shells with neighboring atoms, coordination shells of the neighboring atoms have a common space. Atomic domains are built on the principle of Voronoi-Dirichlet, i.e. the polyhedron of the atomic domain is formed as a result of the intersection of planes drawn perpendicular to the middle of the segments laying between the central atom and the atoms of its coordination shell. Thus, atomic domains are not intersecting, i.e. do not enter the space of each other, and continuously filling the volume of the lattice. Neighboring atomic domains only have one common face, as well as common edges and vertices belonging to this face. We perform modeling by using polyhedra of atomic domains, the duals of coordination polyhedra Z12, Z14, Z15 and Z16. These polyhedra have pentagonal and hexagonal faces and for simplicity we named them as p12, p12h2, p12h3 and p12h4, where indices p and h denote pentagonal and hexagonal faces, respectively and the number after the index sums the number of such faces in polyhedra. The space of coordination polyhedra can be continuously filled with the atomic domains of the central atom and atoms from the coordination shell. Polyhedra are commonly used for modeling of chemical structures. Thus, structures of fullerenes were described in terms of Frank-Kasper polyhedra [14]. Eighty four new structures were generated by computer modeling and then compared with twenty seven known structures. The authors considered a symmetry in modeling of structural fragments, but the complete spheroidal particles of a fullerene as a whole were not studied. In Ref. [2] the similarity between icosahedral structure of certain alloys and structures of fullerenes was investigated with the help of FrankKasper (FK) polyhedra. In general, a crystal lattice of clathrates can be modeled by a contiguous framework of polyhedra with atoms are at their vertices, and chemical bonds connecting atoms are described as the edges of polyhedra, wherein all host atoms are located inside polyhedral cells of clathrates. There are several attempts to describe structure of clathrates with polyhedra, including Frank-Kasper polyhedra [14,15]. Here, we propose a new method for visualization of a complete structure of a crystal lattice, based on the idea of uniformly contiguous filling of space. We fill a lattice space with different types of predefined polyhedra, that have approximately the same length of their edges. Filling a space with polyhedra is guided by an initial arrangement of atoms obtained from experimental data of their crystal structures. Our approach differs from other geometric methods in that we fill the entire crystal space with polyhedral cells continuously. Some cells may contain atoms while others stay empty (dummy). Though the framework topology remains the same, the occupancy of the framework cells changes dynamically over time and depends on the mutual arrangement of atoms in the crystal. For example, some cells may transfer from empty to filled with atoms during allotropic transitions or upon formation of clathrates. Continuous filling of space by polyhedra allows to determine and to visualize not just the neighboring covalent interactions between atoms but also long-distance interactions like coordinate, hydrogen and vander-Waals bonds. A structure of a polyhedral framework in our approach, like a structure of foam, is formed by polyhedra that have common facets, edges and vertices. Therefore, it seems logical that for such polyhedral frameworks Plateau's laws formulated to describe the structure of foam, should be also satisfied.
2. FSP method We propose a new approach for modeling of crystal structures by contiguous Filling of Space with convex Polyhedra (FSP) that based on the rules formulated by Joseph Plateau in his study of foam properties [7]. We consider a compliance with Plateau's laws to be an important factor for space filling, as Plateau's laws are intrinsically embedded in many natural structures. We first fill the space using a limited set of convex polyhedra that have the same length of all their edges. However later the length of edges is allowed to slightly deviate from a common length to compensate for possible contortions of polyhedral faces in models built strictly with equal edges polyhedra. A use of polyhedra with planar faces is convenient because it is easy to connect them and because it simplifies the visual perception of the construction. Frank and Kasper studied metal alloys that have a uniformly filled compact lattice space. It is not the case in the lattice of graphite or ice, where the distance in a crystal lattice between identical atoms may vary significantly. It happens because some gaps exist between carbon layers in the graphite structure and cavities within the unit cells in the structure of hexagonal ice. In our models we address these structural irregularities by modeling spatial voids together with the atomic structures. Although it is possible to fill the space of a crystal lattice with different multifaceted polyhedra, in this study we only use four types of polyhedra, p12, p12h2, p12h3 and p12h4, that are duals of FK coordination polyhedra Z12, Z14, Z15 and Z16. We use the polyhedra that are atomic domains, because they fill the space by forming a continuous net. Macroscopic analogue of such a network is the foam. Our approach is based on the following principles: 1) the lattice space is filled continuously with convex polyhedra that have edges of the same length; a length of the edge is chosen to be equal to the Bohr radius of 0.529 Å:
a0 ¼
ħ ¼ 0:529 A; me $c$a
where a0 is the Bohr radius, ħ is the reduced Planck's constant, me is the electron mass at rest, c is the speed of light in a vacuum, and a is the fine structure constant. 2) obedience with Plateau's laws of foam formation: a strict compliance with topological rules - only four lines meet symmetrically at one vertex and only three faces meet symmetrically at one line; less strict compliance - the edge length, the angles between edges and faces in polyhedra may deviate slightly from the original values due to intrinsic deformations in natural structures; 3) existence of unfilled atomic domains in the modeled structure: empty (dummy) polyhedra are legitimate components of the domain structure of the crystal lattice; later they can be filled with atoms, for example, in the formation of clathrates. Frank and Kasper defined an atomic domain of dummy polyhedra as “an area around the center of the domain, any point of which is closer to the center of the given domain than to the center of any other domain” [12]; thus, a shell of empty (dummy) polyhedra around a certain polyhedron we adherently describe as a domain envelope (or shell) of a given polyhedron. On the first stage of modeling, we determine the key atoms in the lattice. Then, based on the mutual arrangement and known distances between the key atoms, we select a set of polyhedra that
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will be used for modeling. This initial set of polyhedra that corresponds to atomic domains of key atoms are placed in the position on the lattice and the space around them continuously filled with other polyhedra until a complete topological matching to the modeled crystal structure is achieved. In nature, atomic lattices of crystals are often deformed, i.e. compressed or stretched. For the first stage of modeling we use the standard polyhedra p12, p12h2, p12h3 and p12h4 with their edges being the same length. Therefore, after the first “topological” stage of the model construction, the model needs to be corrected to encompass such natural deformations. In this study we use two types of correction: tetrahedral lattices undergo correction uniformly throughout their volume by stretching (diamond) or squeezing (hexagonal ice); the lattices consisting of parallel layers, i.e. graphite and square ice, are corrected with respect of keeping the volume of their polyhedra to be constant. See more details in Results section. 3. Results 3.1. Graphite Graphite is one of the crystalline forms of carbon that has a layered structure consisting of parallel layers of hexagonally ordered carbon atoms [16]. Each atom within the layer is in the state of sp2 hybridization and covalently bonded with three nearest atoms. In general, the carbon atoms in each layer form hexagons
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similar to honeycombs. The distance between two adjacent atoms within a plane of the layer is 1.42 Å. Layers of carbon atoms in graphite are arranged in such a way that half of the atoms of one layer are positioned exactly above/below the centers of hexagonal cells of the other layer. This packing is also known as AB stacking. Thus, alpha graphite has an identical arrangement of atoms in every other layer. The distance between graphite layers is 3.35 Å. We modeled graphite, starting with a monolayer of carbon atoms, polyhedra p12h3, colored cyan in Fig. 1. Connecting the polyhedra p12h3 by their hexagonal faces, we constructed a planar framework that consisted of hexagons and accurately reproduced a structure of the graphite monolayer. The distance between centers of neighboring polyhedra was 1.42 Å. Thus, a constructed FSP model reproduced an arrangement of carbon atoms in the alpha graphite monolayer topologically and matched interatomic distances very well. Since our method implied a contiguous space filling, then for a complete modeling of alpha graphite it was necessary to build intermediate spatial layers located between carbon layers of graphite (Fig. 1). Obviously, these layers must be flat to complement carbon layers. Moreover, the intermediate layers should be complimentary to all adjacent layers of graphite including themselves. Space filling with dummy polyhedra between layers of carbon atoms is shown in Fig. 1, C. Initially, hexagonal cells formed by polyhedra p12h3 (carbon atoms) get filled with a pair of dummy polyhedra p12h2; then dummy polyhedra p12h4 get connected through hexagonal faces with polyhedra p12h2 and a remaining
Fig. 1. FSP model of alpha graphite. A e a unit cell of alpha graphite shown in the traditional representation; B e a unit cell of alpha graphite represented by polyhedra p12h3 (cyan); C e filling the space between layers of carbon atoms with dummy polyhedra.
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free space get filled with adjacent to the carbon layers dummy layers of polyhedra p12. In our unstrained FSP model of graphite the distance between carbon monolayers was 4.05 Å. The distance between atoms of carbon within the layer was approximately equal to the length of a double bond in ethylene 1.30 Å. Experimentally determined distances between layers of alpha graphite are known to be smaller, 3.35 Å. This phenomenon was explained by a mutual attraction of layers caused by van der Waals forces between carbon atoms. Thus, to match our FSP model with experimental values we had to perform a linear correction. Correction factors for crystal lattices filled with parallel layers of polyhedra were defined as Cx*Cy*Cz ¼ 1,
1
where Cx, Cy and Cz e correction factors of individual coordinate axes. This equation implies that the overall volume occupied by all polyhedra remains the same before and after corrections. The correction factor along the z-axis (Cz) in graphite model was set to be 0.83 (3.35 Å/4.05 Å). Because layers in graphite are formed by regular hexagons of carbon atoms we decided to use equal correction factors within a planar layer, Cx ¼ Cy. Thus, from equation (1) Cx ¼ Cy ¼ sqrt(1/ Cz) ¼ sqrt(1/0.83) ¼ 1.10 After the correction, all centers of p12h3 polyhedra composing carbon monolayers fit precisely into coordinates of carbon atoms from the experimentally determined crystal structure of alpha graphite. 3.2. Graphene Graphene is just one flat monolayer of carbon atoms from graphite has been obtained for the first time in 2004 [17]. Likewise graphite, carbon atoms in graphene are in the state of sp2 hybridization. An FSP model of a single monolayer of graphene/graphite composed of polyhedra of carbon atoms surrounded by dummy polyhedra is shown in Fig. 1C. 3.3. Graphane Graphane is a fully saturated hydrocarbon obtained from a
single sheet of graphene by hydrogenation. Graphane has a planar two-dimensional organization [18]. Carbon atoms in graphane stay in sp3 hybridization, and each carbon atom forms covalent s-bonds with three neighboring carbon atoms and one hydrogen atom. Half of the hydrogen atoms are located above the plane of a carbon atom layer, and the other half is on the opposite side of it, Fig. 2. An FSP model of a monolayer of graphene was built with polyhedra p12h4 connected through hexagonal faces so that they form a hexagonal framework. The hydrogen atoms in graphane were modeled by polyhedra p12h2, that were placed on both sides of the p12h4 polyhedron layer on their hexagonal faces. The remaining voids between the carbon and hydrogen atoms in the model of graphane were filled with p12 dodecahedrons. Thus, the complete model of graphane consists of polyhedra p12h4, p12h2 and p12, Fig. 2. A segment of graphene layer is shown as three connected 6member rings that are formed by carbon atoms in sp2 hybridization, Fig. 2A. Carbon atoms are depicted as polyhedra p12h3 colored cyan. In Fig. 2B a side view of the same segment of graphene as in the panel 2A is shown with completely filled space with three layers of dummy polyhedra; carbon atoms are shown as polyhedra p12h3 in cyan; dummy polyhedra p12h2, p12h3 and p12h4 are depicted in shades of brown as indicated in the picture. After hydrogenation, a layer of graphene transformed into graphane, Fig. 2C and D. Carbon atoms changed from sp2 to sp3 hybridization are shown as polyhedra p12h4 and colored cyan, hydrogen atoms are depicted as white polyhedra p12h2. A simplified view of graphene layer with only three 6-member rings formed by carbon atoms and hydrogen atoms and stripped off all filling the inner space dummy polyhedra is added in the panel 2D for visual clarity. Topology of the full network of graphane after hydrogenation remains the same as in graphene and they both resemble the similarity to graphite (Fig. 1).
3.4. Diamond Diamond consists of carbon atoms in sp3 hybridization that covalently bonded to four other carbon atoms forming a cubic crystal structure. We modeled a diamond framework with p12h4 polyhedra connected to form a hexagonal network. Each layer of polyhedra p12h4 in diamond (Fig. 3C) is topologically identical to a layer of dummy polyhedra p12h4 in graphite (Fig. 1C) and to a carbon layer of graphane (Fig. 2D). Voids between polyhedra p12h4
Fig. 2. Transition of graphene (A, B) to graphane (C, D). А e a segment of graphene layer; B e a side view of the same segment as in A, with completely filled space with three layers of dummy polyhedra; C e a side view of graphane layer; D e a simplified view of graphane layer.
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Fig. 3. FSP model of diamond. A e a unit cell of diamond shown in the traditional representation; B e a unit cell of diamond presented by polyhedra p12h4 (cyan); panels CeE: a unit cell of diamond shown from different viewpoints partially filled with dummy polyhedra p12 (yellow) in the space between carbon atoms. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
in diamond were filled by dodecahedrons p12, Fig. 3. The final FSP model of diamond was adjusted by stretching along all three coordination axes with a coefficient of 1.12. 3.5. Square ice Recently, it has been shown that in nanocapillaries between two sheets of graphene, water molecules form unusual monomolecular planar layers [19], whereas oxygen atoms are located at the vertices of squares at a distance of 2.83 Å from each other. A single, double or triple layers of water molecules were reported to exist. Interestingly, oxygen atoms in adjacent water layers were located exactly one above the other, the phenomenon known as AA stacking. The authors named the new discovered organization of water molecules “square ice”. Square ice had a density of about 1.5 times the density of ordinary ice, that pointed to a high packing density of its structure. For an FSP model of square ice in nanocapillaries between two sheets of graphene, we strictly adhered to the experimentally established facts: a) monomolecular layers of water are flat; b) oxygen atoms of water molecules are located at the vertices of squares with a side length of 2.83 Å; c) an oxygen atom of the water molecule occupies the center of a tetrahedron and hydrogen atoms and oxygen regions of polarization are located at tetrahedron vertices; d) carbon atoms are located at the vertices of hexagons that form sheets of graphene, Fig. 4. We determined that water molecules could form a square
structure in a flat layer while retaining a topologically characteristic organization of a regular water molecule, if the oxygen atom is modeled by a polyhedron p12h4, and the hydrogen atoms and oxygen polarization regions by polyhedra p12h3. Hydrogen bond in this case is formed by a pair of polyhedra p12h3, one representing a hydrogen atom and another one being one of an oxygen atom polarization regions. Water molecules from two adjacent layers form a tetragonal Bravais lattice with a distance between layers 3.20 Å, Fig. 4B. A nanocapillary space between graphene sheets was completely filled with layers of various polyhedra representing water molecules and intermolecular space filling dummy polyhedra, Fig. 4C. Interestingly this complex structure can be split into several distinct sub-layers, i.e. a layer of graphene, a layer of water and two interlayers, one is between graphene and water and the second one is between two water layers, Fig. 4D. Noteworthy sublayers at the second graphene sheet repeat themselves in the inverse order with a mirror-like orientation, not shown (more detailed see in Supplemented materials, figure S2). To match the experimentally determined distances between oxygen atoms in water molecules that form a monolayer in the capillary [19] we had to adjust positions of carbon atoms in graphene layers. In such a stretching with a coefficient of 1.35 (Cx) was performed along the x-axis within the plane of a layer for all hexagons formed by carbon atoms and a compression with a coefficient of 1.17 (Cy) was done along the y-axis in the same plane, Fig. 4A. A correction factor along the z-axis (Cz) was calculated as
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Fig. 4. FSP model of square ice in graphene nanocapillaries. A e layer of graphene. B e Bravais lattice of adjacent layers of water molecules is tetragonal (а ¼ b ¼ 2.73 Å, с ¼ 3.2 Å). C e assembly of polyhedra layers in graphene nanocapillaries. D e separated view of polyhedra layers, as shown in the panel C.
mentioned above, equation (1). Cz ¼ 1/Cx*Cy¼1/1.35*1.17 ¼ 0.63. After a complete FSP model was assembled, a final correction was performed in the direction perpendicular to the plane of the graphene sheet with a calculated compression coefficient Cz. This compression was done for all layers located between graphene sheets. A good match between an FSP model and experimentally obtained data for square ice was achieved for a) distances between layers of water molecules and b) distances between graphene sheets with different number of layers of water molecules between them, Table 1. More details about FSP models of square ice can be found in Supplemented materials. 3.6. Hexagonal ice (Ih) In nature ice appears in many amorphous forms with hexagonal
form being a predominant one [20e22]. We modeled hexagonal ice as a tetrahedron formation by placing polyhedra in the center and into vertexes of the tetrahedron. An oxygen atom (polyhedron p12h4) was placed in the center and four polyhedra p12h2 were attached to it symmetrically through hexagonal faces. Two polyhedra p12h2 representing hydrogen atoms, and other two representing regions of polarization near the oxygen atom (lone pairs of electrons) in water molecules were added as shown in Fig. 5A. A hydrogen bond in hexagonal ice is formed by a pair of polyhedra p12h2, connected to two polyhedra p12h4 (oxygen atoms), wherein for each pair of polyhedra p12h2 there is only one hydrogen atom and the other one is a dummy atom. In an FSP model of hexagonal ice each oxygen atom of a water molecule connects through hydrogen bonds with four oxygen atoms in neighboring water molecules. Six molecules of water connected by hydrogen bonds form six-member rings in chair configuration (Fig. 5B). Two six-member rings of water molecules arranged one
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Table 1 Distances between monolayers of water and graphene in nanocapillary. Distances between monolayers of water and graphene in nanocapillary Distance betweena …
Experimental data15
FSP model
two monolayers of water graphene layers with one monolayer of water graphene layers with two monolayers of water graphene layers with three monolayers of water
3.5 6.5 9.0 11.5
3.2 6.7 9.9 13.1
a
All distances are in Angstroms.
Fig. 5. Hexagonal ice. A e a single water molecule represented by polyhedra; B e one half of a unit cell of hexagonal ice; C e a snippet of an FSP model of hexagonal ice shown from above and as a side view.
above the other form a unit cell of hexagonal ice. Notably unit cells in hexagonal ice share vertices, edges and faces. A snippet of hexagonal ice consisting of 126 water molecules is shown in Fig. 5C. The FSP model of hexagonal ice constructed with regular polyhedra needs a uniform compression across all three axes with a coefficient of 0.82. Thus, the distances between oxygen atoms in the structure of hexagonal ice and in the monolayer of water molecules in “square” ice are almost identical e 2.76 and 2.83 Å. However, due to the topologically denser packing of water molecules in square ice, its density is substantially higher than that of hexagonal ice. The difference in packing can be easily seen in FSP models, e.g. in the structure of square ice the hydrogen bond between water molecules is represented by polyhedra p12h3, whereas in hexagonal ice it is modeled with p12h2 polyhedra. More details about FSP models of hexagonal ice can be found in Supplemented material.
4. Discussion 4.1. Graphite, graphene, graphane, diamond An FSP model of graphite helps to demonstrate the arrangement of carbon atoms in adjacent layers of alpha graphite. An observation that carbon atoms of one layer are shifted half way through the layer relatively to the position of carbon atoms in the adjacent layer is determined by topology of dummy polyhedra in the interatomic layer, Fig. 1C. A model also reveals that different allotropic states of carbon coexist in the graphite lattice simultaneously. They can be selected as a formation of atomic polyhedra, as a set of dummy polyhedra or as a combination of both. In such the transition of one allotropic form to another can be easily traced and visualized. Graphene, a monolayer of graphite, consists of carbon atoms in the state of sp2 hybridization. Each carbon atom in graphene forms
three covalent s-bonds with three neighboring atoms. The fourth p-bond is formed due to a partial overlapping of p-orbitals above and below the plane in which the carbon atoms are located. The pbond forms a continuous electron cloud over the entire layer of carbon atoms, a structure similar to such observed in metals. Hydrogenation of graphene leads to a transition of carbon atoms from sp2 to sp3 hybridization and to a formation of four equivalent sbonds in a tetrahedral configuration, and graphene turns into graphane, Fig. 2. In an FSP model of graphane, a fully saturated hydrocarbon obtained from a single sheet of graphene, carbon atoms were modeled as p12h4 polyhedra and the hydrogen atoms as p12h2 polyhedra. Interestingly, the layers of graphane are structurally identical to the space filling layers of dummy polyhedra in graphite, that constructed with polyhedra p12, p12h2 and p12h2 and located between layers of carbon atoms, Fig. 1. A graphane layer of carbon atoms in sp3 hybridization topologically matches a monolayer of carbon atoms in the crystal lattice of diamond. In the polyhedral model of graphite, two alternating layers are present: flat layers of carbon atoms in sp2 hybridization (shown as p12h3 polyhedra) and dummy layers of p12h4. Topologically, the layer of p12h4 dummy polyhedra resembles a layer of carbon atoms in sp3 hybridization in diamond. These observations undeniably indicate a direct relationship between graphite, graphane and diamond.
4.2. Square ice An FSP model of the polyhedral structure of square ice implies that water molecules in the capillary layer between two sheets of graphene retain their tetrahedral structure. Water molecules have a specific arrangement in which their oxygen atoms located in the points on the grid with square cells. Because water molecules in square ice packed more densely than in hexagonal ice we had to use
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p12h3 polyhedra, rather than p12h2 polyhedra for hydrogen atoms. Thus, a pair of adjacent p12h3 polyhedra between oxygen atoms, one representing a hydrogen atom and the other one being dummy, we treated as a hydrogen bond. In contrast, a hydrogen bond in the structure of hexagonal ice and in bulk water was built with polyhedra p12h2. The tetrahedral structure of a water molecule is determined by oxygen atoms, that are always modeled by p12h4 polyhedra. Thus, a packing and a mutual arrangement of water molecules depend on the position and polyhedron types selected for modeling of hydrogen atoms. A polyhedral model of a graphene sheet in contact with water differ from the model of a carbon layer in graphite. The inner layers of dummy polyhedra located between a graphene sheet and monolayers of water is complex and consists of polyhedra p12, p12h2 and p12h3, whereas the inner layer between two layers of carbon in graphite can be modeled with p12, p12h2, and p12h4, compare Figs. 1 and 4. Based on our FSP model of nanocapillaries in graphene we suggest that graphene sheets undergo some deformation. The carbon atoms are stretched within the planar layer and the entire framework, including inner layers of dummy atoms between sheets of graphene are compressed in a direction perpendicular to the carbon plane, Fig. 4. With our FSP model of square ice we could estimate distances between graphene sheets and layers of water molecules, the distance between monolayers of water, and the distance between graphene sheets in nanocapillaries. Comparison of calculated values from an FSP model with experimentally obtained measurements [19] showed a good matching, Table 1. An FSP model also helped to demonstrate why the distance between oxygen atoms in neighboring layers in nanocapollaries is greater than the distance between them within the same layer. As clearly seen in the model of the entire polyhedral structure of graphene nanocapillaries, between two monolayers of water, there is an extra inner layer of dummy atoms, Fig. 4, (C-D). Therefore, a distance between oxygen atoms from neighboring monolayers of water (3.2 Å) is greater than a distance between directly connected oxygen atoms within the layer (2.8 Å). This observation is an agreement with experimentally measured distances between water layers of 3.5 Å.(16) Based on our FSP model we posit that a crystal system of square ice belongs to a tetragonal and not to a cubic crystal family as was reported previously [19]. 4.3. Hexagonal ice A polyhedral model of hexagonal ice differs significantly from the structure of square ice. This difference arises from the type of polyhedra with which hydrogen bonds are modeled. In hexagonal ice a pair of polyhedra p12h2 are used to model a hydrogen bond, whereas in square ice it is made with polyhedra p12h3. The structure of hexagonal ice (Fig. 5) assumes a significantly lower packing density. A unit cell of hexagonal ice consisted of 12 water molecules can be encapsulated inside the cube with an edge length of 8 Å. For comparison 24 water molecules located in three parallel monolayers of square ice occupy the same volume. Another important distinction is that in hexagonal ice all water molecules form a continuous network of hydrogen bonds connecting all water molecules together in the entire volume. Contrary hydrogen bonds in square ice form a network exclusively within a plane of individual water monolayer, and different layers of water are separated from each other by an inner layer of dummy atoms. Though in both types of ice water molecules exist in a characteristic tetrahedral structure, their internal spatial organization is substantially different.
5. Conclusions Applications of the method of contiguous filling of space with a limited set of convex polyhedra (FSP) were demonstrated on models of graphite, graphene, graphane, diamond, square ice and hexagonal ice. FSP models replicated crystal structures well and helped to demonstrate the mutual arrangement of atoms and hidden transitions from one allotropic state to another observed in crystal lattices. The use of dummy polyhedra in FSP models allowed to avoid ambiguities associated with significant differences in interatomic distances between identical atoms in structures that have intrinsic gaps and cavities, as shown on examples of graphite and ice. Dummy polyhedra are legitimate components of the domain structure in molecular models. By using the FSP models it was possible to demonstrate that the mutual arrangement of carbon atoms in adjacent graphite layers is governed by an inner layer of dummy polyhedra located between the carbon layers. It was shown why hydrogenation of graphene leads to its transformation into graphane. FSP models helped to demonstrate that different allotropic forms of carbon are simultaneously present in graphite, either visible through atomic polyhedra or hidden in a form of dummy polyhedra. A presence of dummy layers of polyhedra in the graphite structure that are topologically identical to layers of carbon in diamond not only confirms a well-known relationship between graphite and diamond, but also suggests the mechanism of transition of carbon atoms from one allotropic form to another. A complete FSP model of nanocapillaries in graphene allowed to calculate distances between layers of water and graphene sheets, and distances between monolayers of water. It was shown that calculated values correlate well with experimental measurements. An FSP model of square ice clearly demonstrated why square ice has rather a tetragonal than a cubic lattice. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jmgm.2019.03.004. References [1] W. Niu, et al., Dodecahedral gold nanocrystals: the missing platonic shape, J. Am. Chem. Soc. 136 (8) (2014) 3010e3012. [2] S. Alvarez, Nesting of fullerenes and frankekasper polyhedra, Dalton Trans. (17) (2006) 2045e2051. [3] A.J. Karttunen, et al., Structural principles of semiconducting group 14 clathrate frameworks, Inorg. Chem. 50 (5) (2011) 1733e1742. [4] K. Hagino, et al., Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-a in braarudosphaera bigelowii (prymnesiophyceae), PLoS One 8 (12) (2013), e81749. [5] L. Tang, et al., The structure of Pariacoto virus reveals a dodecahedral cage of duplex RNA, Nat. Struct. Biol. 8 (2001) 77. [6] W. Thomson, On the division of space with minimum partitional area, Acta Math. 11 (1887) 121e134. [7] J.A.F. Plateau, Statique exp erimentale et th eorique des liquides soumis aux seules rimentale et the orique des liquides soumis forces mol eculaires. Statique expe culaires, Gauthier-Villars, 1873. aux seules forces mole [8] E. von Fedorow, XVII. Allgemeinste Krystallisationsgesetze und die darauf fussende eindeutige Aufstellung der Krystalle, in: Zeitschrift für Kristallographie - Crystalline Materials, 1904, pp. 321e490. [9] R. Gabbrielli, A new counter-example to Kelvin's conjecture on minimal surfaces, Phil. Mag. Lett. 89 (8) (2009) 483e491. [10] D. Weaire, R. Phelan, A counter-example to Kelvin's conjecture on minimal surfaces, Phil. Mag. Lett. 69 (2) (1994) 107e110. [11] R. Gabbrielli, et al., An experimental realization of the WeaireePhelan structure in monodisperse liquid foam, Phil. Mag. Lett. 92 (1) (2012) 1e6. [12] F.C. Frank, J.S. Kasper, Complex alloy structures regarded as sphere packings. I. Definitions and basic principles, Acta Crystallogr. 11 (3) (1958) 184e190. [13] F.C. Frank, J.S. Kasper, Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures, Acta Crystallogr. 12 (7) (1959) 483e499.
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