N. A. Bulienkov’s Simplicial-Modular Design As a Basis for Modeling Metal Clusters

Abstract

The author considers the main principles of Bulienkov’s simplicial-modular design of metal clusters (face-centered cubic (FCC), body-centered cubic (BCC), hexagonal close-packed (HCP) metals). Models of clusters are presented along with algorithms for their construction based on operations of twinning ($\bar {1}$, 2, m) to simplices and modules of the corresponding metals.

INTRODUCTION

Due to their unique physicochemical properties, metal clusters play an important role in nanotechnology, both individually and in ensembles. Bulienkov always gave great attention to modeling the structure of clusters, since they serve as convenient objects for studying the emergence of determinate order in hierarchic noncrystalline structures. He was especially interested in the evolution of cluster structure, devising algorithms of their formation, and the reasons for the stability and limiting of growth. Bulienkov emphasized the close relationship between the formation of nanoparticles and phenomena of self-organization. It was therefore expected that he would be one of the organizers of the interinstitutional seminar “Nanoparticles and Self-Organization Phenomena” that ran from 1999 to 2010 under the direction of Yu.M. Polukarov, a Corresponding Member of the Russian Academy of Sciences at its Frumkin Institute of Physical Chemistry and Electrochemistry.

Developing theoretical models of the self-organization of atoms into hierarchic structures has in recent years been an important subject in materials science, especially where soft materials and nanotechnology are concerned. When it comes to modeling the self-assembly of stable nanoparticles, authors generally calculate the equilibrium atomic configurations through numerical experiments (e.g., molecular dynamics or Monte Carlo) [1]. The choice of the configuration of the atoms may not always be apparent, due to the complexity of describing multiparticle interaction. Bulienkov developed a local approach in crystallography that he believed was enough to explain the self-organization of a crystal. It should be noted that Bulienkov always relied on the traditions of science. He based his local approach on the works of E.S. Fedorov (on crystalline molecules), B.N. Delone and his school (N.P. Dolbilin, M.I. Shtogrin, and R.V. Galiulin, on the local theorem), and Linus Pauling (Pauling’s rules) as his predecessors. After many years of work, Bulienkov produced a new fundamental concept of modern crystallography—a crystalline module [2] based on Pauling’s rules [3, 4]. In a continuation of these works, it was shown in [5, 6] that a crystalline module can be constructed for inorganic crystals using Delone tiling.

This work is devoted to the fundamentals of simplicial-modular design developed by Bulienkov, and illustrating its use in modeling the structure of metal clusters (face-centered cubic (FCC), body-centered cubic (BCC), hexagonal close-packed (HCP) metals).¹

THEORETICAL

The structure of clusters must be modeled, since there are no direct ways of determining the positions (coordinates) of atoms. This is somewhat compensated for by data from, e.g., mass spectrometry, in which a set of maxima of intensity is recorded that corresponds to a certain number of atoms in a series of stable clusters (magic numbers of atoms). It was shown in [7, 8] that if we use a minimum nucleus consisting of atoms located at the vertices of regular polyhedra (Platonic and Archimedean solids) as the basis of a series of clusters, each magic number corresponds to the completion of the next layer according to the model of close packing of atoms. This applies to, e.g., clusters of the Chini series with such magic numbers of atoms as 13, 55, 147, 309, 561… [8]. Increasing amounts of new data from studying stable nanoparticles of different metals with other magic numbers have appeared as nanotechnology continues to develop [9, 10]. It is now clear that we need more than the model of close packing and the layered filling of the next external layer of a nanoparticle with atoms to explain their structure. Bulienkov showed the aperiodic structures to which clusters belong can be modeled using the local approach [2, 11], in which the identical coordination of all atoms of one kind in a sphere with a radius on the same order as the interatomic distances (chemical bonds) is enough for a crystal’s structure to be regular [11]. Simplifying somewhat the fundamental problem of locality as searching for the minimum cluster of atoms that can describe the crystal structure, the authors of the local theorem [11] sought this minimum cluster in the form of the environment of the central atom of a specific kind (“a stable spider of a point of an (r, R) system”). Bulienkov considered the problem of the minimum cluster of a crystal’s structure (a crystalline module) as a search for certain generalized emptiness that attracts atoms of different kinds to its surface. A crystalline module is clearly isolated from the structure of, e.g., a crystal in the form of a parallelohedron that has atoms bound by real chemical bonds at its vertices, on its edges, or on its faces. A crystalline module contains complete information on the long-range order, stoichiometric formula, and morphology of a crystal, and the structure of its most important faces [2].

The crystalline module of the closely packed crystal structures of metals can also be divided according to real chemical bonds² into simplices in the form of regular or distorted tetrahedra and octahedra. Designing structures that consist of modules and simplices by means of binary symmetry operations ($\bar {1}$, m, 2) only guarantees the preservation of high symmetry for both the local order and an entire cluster. The simplicial-modular design is based on the idea of completing the polyhedra of the modules and simplices by adding the minimum possible number of atoms as an indispensable prerequisite of the stability of the structure. The polyhedral representation of a module allows us to choose the exact place of adding the next atom or group of atoms.

Clusters of metals are considered below. The crystalline modules for FCC, BCC, and HCP metals are in the form of rhombohedra (eight atoms at the vertices). The algorithms for assembling clusters of metals are determined by the local order characteristic of a solid body and choice of the symmetry operations with which the simplicial-modular design of the clusters is executed.

The basic principles Bulienkov formulated for the simplicial-modular design of metal clusters are:

• the evolution of a homologous series of nanoparticles when each term of the homologous series of nanoparticles serves as the nucleus of the subsequent term of the series;

• limited capabilities of growth with the maximum preservation of global symmetry;

• nanoparticles must all be the same size;

• most atoms of a nanoparticle should lie on the surface.

These principles could be rectified in the future and developed to allow for the chemistry of heterogenous clusters (e.g., charge transport and localization).

RESULTS AND DISCUSSION

Modular Design of Nanoparticles of Metals with FCC Structure (Ag, Au, and Cu)

It was shown in [12, 13] that differences in the morphology of nanoparticles with close packing is determined at the local level in a nanoparticle consisting of 13 atoms. Polyhedra with identical edges and equal volumes are known as cuboctahedrons and hexagonal cuboctahedrons. The volume of these polyhedra is 20 times that of a regular tetrahedron, as follows from the geometric ratio between the volumes of an ideal tetrahedron and octahedron (a cuboctahedron and a hexagonal cuboctahedron) with equal edges V_octahedron = 4V_tetrahedron. An icosahedron and a pentagonal bipyramid are similar to these solids with respect to volume, but their packing is not exactly close. It should be noted that if the central atom is surrounded by 12 equally sized atoms (1 + 12), the volume of the icosahedron constructed on the centers of the atoms is larger than, e.g., that of a cuboctahedron. The volume of a pentagonal bipyramid composed of five hemioctahedra and ten tetrahedra is also larger because it is composed of polyhedra with more volume than that of ideal solids. However, the free space between the atoms in these packings is still not enough to accommodate one more atom in the first coordination sphere.

Nanoparticles with the same number of particles (1 + 12) can therefore have different crystal habiti and thus different surface properties, due to the different ratios of the faces with packing in, e.g., planes (111) or (100) in the FCC structure. These nanoparticles are indistinguishable in a mass spectrometric experiment.

Molecular dynamics (MD) modeling of the magic clusters of FCC metals showed that a nanocrystal with the morphology of a cuboctahedron is unstable at temperatures below the melting point. Within several picoseconds of an MD experiment, a nanoparticle with the morphology of a cuboctahedron spontaneously transitions into a nanoparticle with the morphology of an icosahedron that has lower density and energy than a cuboctahedron [14, 15]. Estimates of the density of the icosahedral packing of atoms showed it was close to that of a melt; i.e., it differed by ~7% from the density of the solid metal. The packing of atoms in them therefore differs from the crystal packing of polysynthetic twins, up to a certain size (diameter) of the nanoparticles, which depends on the ratio of the number of surface and bulk atoms. I.I. Moiseev et al. were apparently correct in suggesting that nanoparticles can be in a special cluster state, e.g., giant clusters of palladium (Pd561) in a shell consisting of organic molecules [16]. It was shown in [12] that the icosahedral crystal habitus can be preserved upon further growth of an FCC metal nanoparticle, while the latter transitions into the form of a complex twin. The mechanism of the transition is determined by the defects of the icosahedral packing being redistributed to the edges and vertices of the icosahedral nanoparticle, due to the growth of the nanoparticles. This produces fissures (edges) and dimples (vertices) in a complex twin with FCC packing. There is no limit to the growth of these nanoparticles, and they have been recorded during, e.g., the electrochemical crystallization of metals [17].

In addition to clusters of the icosahedral habitus of FCC metals [12], Bulienkov found another possible branch of the evolutionary growth of nanoparticles³ where a particle becomes the nucleus of a subsequent particle (Fig. 1). The first term of a stable series is a cluster consisting of nine atoms, with two octahedra joined by a plane of symmetry in the polyhedral representation (Fig. 1a). The next step is the formation of a fivefold twin from⁴ octahedral simplices that contains 16 atoms (Fig. 1b). A possible point of branching appears at this step, for the development of an evolutionary series of nanoparticles. The octahedra in FCC structures should join at their edges. In a fivefold twin, they join at their faces, which is characteristic of the local order in HCP structures.

Fig. 1.

Simplicial-modular design of the clusters of FCC metals. (a) A twin consisting of octahedra, (b) a fivefold twin consisting of octahedra, (c) a fivefold twin consisting of hemioctahedra and addition of 5 simplices, (d) the circles in the vertices of 5 simplices indicate the positions of the atoms, (e) a column consisting of 2 fivefold twins of hemioctahedra, (f) the circles mark the vertices of 10 simplices in the lower fivefold twin which correspond to 5 atoms, (g) the addition of 10 simplices in the upper fivefold twin corresponds to 5 atoms, (h) the positions of addition of 5 octahedra are shown, (i) the result of addition of 5 more octahedra is shown, (j) the filling of the wells with 15 simplices which corresponds to the addition of 5 atoms, (k) the addition of 10 simplices (at the top and bottom) around the channel, and (l) the atomic representation of the structure of the cluster. The numerals indicate the number of atoms at the next stage of design.

Subsequent evolution with the local FCC order can proceed in the direction of the periphery of the nanoparticle or from adding fivefold twins along the C5 axis (Fig. 1e). The octahedra are joined at their edges, which is characteristic of FCC structures. The dihedral of an octahedron diverging from 72° results in gaps on the periphery of the fivefold twin, which in Bulienkov’s interpretation corresponds to a rise in the stresses in the structure and its instability upon further growth. The stresses in the structure of a nanoparticle can be compensated for by removing the located on the C5 axis. This evolutionary branch of clusters produces plate-shaped nanoparticles with atoms on their surfaces, determining the maximum activity and stability of the nanoparticle. Figures 1c–1k show steps of growth of a nanoparticle, for some of which the dots indicate the positions of the atoms after the corresponding joining of the polyhedron. The last step of growth of a plate-shaped nanoparticle is completed by tetrahedral simplices being added along the faces of the hemioctahedra located at the boundary of the central channel—five simplices at the top of the nanoparticles and five simplices at the bottom. This way of joining a tetrahedron to a face of an octahedron corresponds to the local order of FCC structures. A nanoparticle with a central channel in which all the 60 atoms lie on the surface forms as a result (Figs. 1k and 1l show polyhedral and atomic representations, respectively). Channel overgrowth requires two atoms, which can be at the vertices of the octahedra (Fig. 1b) lying on the fifth order axis.

Modular Design of the Atomic Structure of the Nanoparticles of Alkaline Metals as Exemplified by Lithium

The module of the BCC structure of lithium (Im3m group) is a distorted rhombohedron with two kinds of edges that correspond to interatomic distances of 3.039 and 3.509 Å. Eight lithium atoms are at the vertices of the edges. The coordinates of the module’s atoms, expressed in fractions of a cell, can be ((0, 1, 0); (1/2, 1/2, −1/2); (−1/2, 1/2, −1/2); (0, 0, −1); (0, 0, 0); (1/2, −1/2, −1/2); (−1/2, −1/2, −1/2); (0, −1, −1)) (eight atoms bound by bonds are depicted at the vertices of the rhombohedron on the left in Fig. 2a). The rhombohedron has two kinds of faces (planes (011) and (101)): two in the form of a rhombus with edges corresponding to short bonds with an angle of 70.53° and four in the form of a parallelogram with short and long edges at an angle of 54.74°. The atoms in the module are bound differently–two atoms have three bonds (two short and one long) each. The six remaining atoms of the module have five bonds (four short and one long) each. The division of the faces of the module by the chemical bonds isolates three simplices in the form of two distorted tetrahedra and an octahedron (one distorted tetrahedron and a distorted octahedron are on the right in Fig. 2a). By choosing different rules (binary operations) for joining the modules and simplices, we can obtain aperiodic structures that preserve the order characteristic of the crystalline state of this substance at the local level and are thus close to the minimum of free energy.

Fig. 2.

Simplicial-modular design of the clusters of BCC metals. (a) A crystalline BCC module in the (to the left) ball-and-rod and (to the right) polyhedral representation which is composed of two distorted tetrahedra and one octahedron; (b) the application of the operation of twinning of a plane of symmetry to a crystalline BCC module of a V cluster; (c) the application of the $\bar {1}$ operation to the V cluster; (d–o) two possible branches of growth of the nanoparticles of BCC metals: (d–f) linear growth of the dovetail twin and (g–o) branch of the fivefold twins based on the BCC simplices; and (p) atomic representation of the structure of the cluster of BCC metals. The numerals indicate the number of atoms in the cluster at the next stage of design.

The modular design suggests the evolutionary development of the modeled structure, according to which the structure of one term of the family of nanoparticles serves as the nucleus of that of the subsequent term of the same series of magic numbers. The ionization potentials and data from mass spectrometry [9, 10] indicate that the first stable nanoparticles are dimers and a nanoparticle consisting of eight lithium atoms (Fig. 2a). The latter coincides in shape and number of atoms with the module of the crystal structure of lithium. The next step in developing the model should result in the complete (with respect to the modules or simplices and symmetry) structure of a stable cluster.

Joining two modules by the faces of the plane of the module formed by short bonds, we obtain a cluster consisting of 12 atoms in the form of the letter V (Fig. 2b). It can be built to the final completion of a cycle of four modules in two ways. The first is to turn two clusters of the previous level (Fig. 2b) by 180° (Fig. 2c) around the axis perpendicular to the plane of symmetry. The second is to join two clusters (Fig. 2b) of two modules by the plane of symmetry parallel to the short edges (Fig. 2d). The latter is preferable, since the symmetry of such a loop is greater (${{C}_{{2{v}}}}$ rather than C_2h) and allows us to obtain nanoparticles with global pentagonal symmetry (D_5h)⁵ at subsequent levels of evolution. Such a shape of closed loops consisting of rhombohedral modules formed of 20 atoms (Fig. 2d) allows their periodic repetition (Figs. 2e, 2f) in the direction of the common axis of the loops perpendicular to the fifth order axis of symmetry of the penta-gonal cluster (Fig. 2k). The number of atoms in all of these clusters consisting of one, two, and three closed loops consisting of four modules corresponds to the maxima of intensity according to data from mass spectrometry of alkaline metal clusters (20, 30, 40).

A branching point is therefore possible in the evolutionary development of the nanoparticles of BCC metals, after which the development can follow the path of rod-shaped nanoparticles or plate-shaped nanoparticles with pentagonal symmetry.

The possibility of the evolutionary development of nanoparticles in the form of a pentagonal complex twin is determined not so much by the symmetry (${{C}_{{2{v}}}}$) of these clusters but by the vertex angle of the face of the module of lithium being limited by short bonds to 70.53°, which differs by just 1.5° from the value of 72° characteristic of structures with a fifth order axis (Fig. 2g).

Finer details of the structure of stable nanoparticles can be found if simplices in the form of distorted tetrahedra are used in the design. Removing all 5 vertices of a cluster consisting of 42 atoms or 10 tetrahedral simplices (Fig. 2k) produces a cluster consisting of 37 atoms (Fig. 2h). Adding 20 tetrahedral simplices denoted by shading in Fig. 2i corresponds to adding 10 atoms and produces a stable nanoparticle consisting of 47 atoms (Fig. 2i). A nanoparticle consisting of 57 atoms forms (Fig. 2j) upon adding 10 more tetrahedral simplices (5 at the top, 5 at the bottom) to the positions denoted by dotted shading in Fig. 2i. Nanoparticles with magic numbers of atoms of 92 (Figs. 2l, 2m), 107 (Fig. 2n), and 117 (Figs. 2o, 2p) can form if the growth of the nanoparticles is prolonged by simply adding modules in the form of distorted rhombohedra.

In the modular design of the nanoparticles of alkaline metals, features of their electronic structure can be considered indirectly using Pauling’s idea of the possibility of obtaining the rotational resonance of bonds with orders below 1 in their crystal structure [18]. For example, in the crystal structure of lithium, the bond order of eight short bonds (3.039 Å) is n₈ = 1/9. That of six long is n₆ = 1/54.6. To increase the stability of the structures of these nanoparticles, the values of the specified bond indices must be maximal, and this determines the location of most atoms on the surfaces of nanoparticles, due to which their coordination number declines. The morphology of the nanoparticles of lithium will thus be a flattened pentagonal plate two rhombohedral modules thick (Figs. 2o, 2p). Figures 2o and 2p show the structure of a nanoparticle in the polyhedral and atomic representations. The twinning or nanocrystalline character of these nanoparticles with two-dimensional periodicity complicates solving the problem of their monodispersity (with respect to size and shape).

Modular Design of Nanoparticles of Metals with HCP Structure

While the simplices were distorted tetrahedrons and octahedrons in the previous case of BCC structures, these simplices have a more regular shape in HCP structure. The difference between HCP and FCC packings is determined by the different ways in which the simplices (octahedra and tetrahedra) are joined in the structures [12]. In FCC structure, the simplices of the same type join at vertices and edges only. In HCP structure, simplices of the same type can join at vertices, edges, and faces (Fig. 3a), forming columns consisting of tetrahedra with alternating vertex–vertex and face–face joins and octahedra with joins at faces (Fig. 3c). In an ideal HCP structure with ratio of axes c/a = 1.633, the module is a rhombohedron with identical edges corresponding to the lengths of bonds in the metal. In most HCP metals, this ratio deviates by no more than 4%, resulting in slight distortion of the rhombohedron. It should be noted that Zn and Cd with a ratio of axes of 1.856 and 1.886, respectively, are an exception, resulting in a distorted rhombohedron with ten short and eight long bonds. The bond lengths for zinc are 2.6647 and 2.9085 Å. For cadmium, they are 2.9764 and 3.2937 Å.

Fig. 3.

Simplicial-modular design of the clusters of HCP metals. (a, b) The twinning of an octahedron by a plane of symmetry in the (a) ball-and-rod and (b) polyhedral representation, (c) obtaining a threefold twin from 19 atoms, (d, e) the formation of 2 fivefold twins from octahedra, (f) the withdrawal of atoms and formation of a through channel, (g–m) the stages of further evolution of the threefold twin by the addition of hemioctahedra, and (n) the atomic structure of the cluster of HCP metals along the C3 axis and (o) cohesiveness in the cluster. The numerals indicate the number of atoms at the next stage of design.

The dihedral angle of the octahedron at the vertex remains close to 72°. We can therefore close them into a fivefold twin with a fifth order axis passing through the common vertex of the octahedra (Figs. 3d, 3e) by successively applying twinning (plane of symmetry m) to the octahedral simplices (Fig. 3c).

The atomic structure of a nine-atom cluster is shown in Fig. 3a. The numerals in the figures denote the number of atoms in the nanoparticle at the next step of evolution (design/growth). Two octahedra joined by plane of symmetry m (a binary operation) form a column consisting of octahedra (Fig. 3b). After this operation, the number of atoms grows from six (an octahedral simplex) to nine; the positions of three newly added atoms are shown in Fig. 3c by the spheres at the vertices of the octahedron. The joining of three columns of octahedra by a threefold axis produces a cluster consisting of 19 atoms (Fig. 3c shows the polyhedral structure of the cluster; the positions of the atoms of the upper layer are marked with spheres). This fragment of the crystal HCP structure contains all combinations of the joins of simplices required for crystalline assembly. It contains columns consisting of tetrahedra joined at vertices and faces, and columns consisting of octahedra. If we continue joining the columns consisting of octahedra by edges, we will obtain a crystalline HCP packing. To exclude the possibility of the crystalline growth of the nanoparticle, we must change the algorithm of adding the modules (simplices) while preserving the maximum (in our case, trigonal) symmetry of the cluster. Two octahedra must be added at the top and bottom to the faces of the chain of tetrahedra that run along the C3 axis (Fig. 3d). As in the previous case, the atoms that are added lie at the vertices of the polyhedra (octahedra). The following terms of the evolutionary series of nanoparticles are obtained by successively adding six octahedra to the side faces of a threefold twin consisting of octahedra (Fig. 3d) via binary operation m. Fivefold twins consisting of octahedra then appear (Fig. 3e). At this step, we can design a nanoparticle with the maximum developed surface. If only octahedra are used, the end term of the series will be a compact cluster with trigonal symmetry. Since a developed surface on nanoparticles is a property important for nanotechnologies, halves of an octahedron that allow us to obtain a nanoparticle with channels must be used in the design. The axes of such channels are perpendicular to the threefold axis of a nanoparticle. Polyhedral assembly clearly dictates the place for adding the next simplex/module, and thus also determines the position of the next atom in the structure. As is seen in the figures, the polyhedra (octahedra, hemioctahedra) act as frames that are then removed, while a small number of atoms is actually added at each step (see the positions of the spheres in Fig. 3). A total of 6 fivefold twins are formed by such an algorithm, and the complete crystalline nucleus consisting of 6 octahedra (19 atoms, Fig. 3c) contains just 19 − 3 × 2 = 13 atoms, since there are channels. When all hemioctahedra have been added (these steps are depicted in Figs. 3f–3i), the nanoparticle contains 49 atoms (Fig. 3i). Further growth of the nanoparticle is possible if we add 2 octahedra at the top and bottom to the faces of the tetrahedra on the threefold axis of the nanoparticle. There are then 49 + 3 × 2 = 55 atoms after this stage. The spheres show the atoms that are actually added to the structure (Fig. 3j). The fissures between the octahedra of the previous step of assembly are then filled with octahedra (Fig. 3k). A total of six octahedra are added at this stage, and the joining of the octahedra at their faces alternates with joining at their edge. In Fig. 3k, the positions of the newly added atoms that will be included in the structure are marked with spheres in the polyhedra. A nanoparticle is formed from 65 atoms. The two remaining fissures are filled with octahedra according to the same algorithm; 10 atoms are added at each step, so intermediate nanoparticles with magic numbers of atoms 75 and 85 are obtained (Figs. 3l, 3m). Figures 3n and 3o show the end structure of the nanoparticle in the atomic representation (the threefold axis is perpendicular to the plane of the drawing, Fig. 3n) and in the form of a wire model (Fig. 3o), in which the cohesiveness in the nanoparticle is depicted (one of three channels, the axes of which are perpendicular to the threefold axis, is shown).

The entire evolutionary series of nanoparticles known from mass spectrometry experiments for HCP metals (e.g., Zn and Cd [9]) has thus been obtained, and the evolutionary branch of the nanoparticles has been found, in which all atoms are located on surfaces. Such forms are good because the stresses that appear at the twin boundaries of a nanoparticle can be more uniformly distributed in the structure if we remove six atoms from its crystalline nucleus (Fig. 3c) and three channels emerge perpendicular to the threefold axis.

Simplices characteristic of the crystal structure are used in the simplicial-modular design of the nanoparticles of HCP metals, so the cluster at the local level has a similar environment and energy close to the minimum of free energy.

Design of Clusters of Complex Compounds

The authors of [19] presented mass spectrometry data on the number of molecules in stable clusters consisting of C60 molecules, on which the fine structure of the distribution of the maxima of intensity was visible. The magic series composed of the number of C60 molecules in the stable clusters in [19] contained the terms: 13, 19, 23, 27, 31, 35, 39, 43, 46, 49, 55, 58, 61, 64, 67, 70, 73, 76, and 79 molecules. In [13], Bulienkov proposed an evolutionary model of growth for this series of clusters on the basis of simplicial-modular design. This allowed us to explain the interchange of the algorithm of growth of stable clusters at the initial step. An algorithm for adding 4 molecules at a time (in series of 19 to 43 molecules) is used at the first step. Results are seen in the maxima of intensity of the mass spectra, which then changes to an algorithm for adding 3 molecules at a time (in series of 43 to 79 molecules) in stable clusters consisting of C60 molecules. Figure 4d shows that for a cluster consisting of 19 molecules with the morphology of an octahedron (Fig. 4c), its further growth can be modeled by adding 8 octahedra (which corresponds to adding 3 molecules) to the corresponding positions on the edges of a nanoparticle consisting of 19 molecules. Such an algorithm is apparently not good in terms of energy, and nature chooses the option of successively adding fourfold twins consisting of simplices at the vertices of an octahedron (Fig. 4e), leading to the next stable cluster (Fig. 4f). The addition of more threefold twins in the polyhedral representation is shown in Fig. 4g and, in the final form, in Fig. 4h. The formation of a series of stable clusters is completed by adding 6 octahedral simplices (Fig. 4h) to produce a cluster with an octahedral crystal habitus consisting of 85 C60 molecules (Fig. 4i).

Fig. 4.

Simplicial-modular design of a cluster consisting of C₆₀ fullerenes. (a) The nucleus of the cluster in the form of a cuboctahedron, (b, c) addition of six hemioctahedra, (d–f) two possible branches of evolution of the clusters: by the addition of (d) octahedra and (e, f) fourfold twins of simplices which is recorded in mass spectrometry. (g‒i) The completion of the evolution of the clusters by the addition of the threefold twins of atoms represented in the (g) polyhedral form and (i) in the form of hexahedra and octahedra.

Data on a crystallized Pd145(CO_x)(PEt₃)₃₀ cluster were presented in [20], and the coordinates of the atoms were determined using X-rays. The authors of the work proposed a three-shell model based on embedded regular polyhedra to explain the self-organization of the metal nucleus of this cluster consisting of palladium atoms. Using simplicial-modular design, Bulienkov found an alternative solution that showed yet another possible evolution of the clusters of a new series for FCC metals (13, 43, 55, 115, 145, 157, 217, 229, 309 atoms) that differed from the Chini series (13, 55, 147, 309, 561, …) (Fig. 5).

Fig. 5.

(a–e) Simplicial-modular design of the metal nucleus of a cluster Pd₁₄₅(CO_x)(PEt₃)₃₀. (f–i) Possible further evolution of a cluster consisting of 145 atoms. The numerals indicate the number of atoms at the next stage of design. (c) Arrow indicates one of the 12 directions of twinning of D_5h modules which will lead to a series of Chini clusters upon further filling of the sectors with octahedra. (d) Arrow indicates the place of addition of an octahedron; in this case, a branch of clusters can be formed, a term of which is (e) the metal nucleus of a cluster Pd₁₄₅(CO_x)(PEt₃)₃₀. The numerals indicate the number of atoms at the next stage of design.

It should be noted that, while preserving general icosahedral symmetry, the evolutionary series of these clusters differ with respect to the algorithms of design. For example, in the simplicial-modular design, the second term of the series of clusters is obtained by arranging 20 octahedra on 20 faces of an icosahedron; in this case, the number of atoms is 13 + 3 × 20/2 = 43. At the next step, twinning D_5h modules along 12 radial directions of the icosahedron by a binary operation (the plane of symmetry m), we obtain a cluster consisting of 55 atoms (43 + 1 × 12 = 55) (the Mackay icosahedron, Fig. 5c). Since two building blocks, an octahedron and a D_5h module, were chosen for the simplicial-modular design, a branching point—two pathways of further evolution of the cluster—appears for the 55-atom cluster. With the branch of the clusters of the Chini series, the design is executed by the twinning by a binary operation (plane of symmetry m) of the D_5h modules (the vertex of the module is specified in Fig. 5c by an arrow and there are 12 such modules) along the radial direction of the icosahedron followed by the filling of the sectors with octahedra based on the FCC packing. If octahedra are placed onto the faces of the Mackay icosahedron and then to arrange octahedra in all the gaps of the shell consisting of octahedra using twinning operation (m) (one of the corresponding positions of the octahedron is indicated by an arrow in Fig. 5d), we obtain a complete 145-atom cluster. The difference in the number of atoms at this step of design (145 atoms) differs by less than 2% from the term of the Chini series (147 atoms). However, the structure of the new series of clusters is different at the local level. Bulienkov called this the “sensitivity” of simplicial-modular design. The simplicity of calculating the number of atoms at each step of the evolution of clusters should be noted, since an icosahedron (12 vertices, 20 faces, 30 edges, Fig. 5a) forms the base (the cluster nucleus):

13 (Fig. 5a) + 3 × 20/2 = 43 (Fig. 5b) + 1 × 12 = 55 (Fig. 5c) + 3 × 20 = 115 (Fig. 5d) + 1 × 30 = 145 (Fig. 5e) + 1 × 12 = 157 (Fig. 5f) + 5 × 12 = 217 (Fig. 5g) + 1 × 12 = 229 (Fig. 5h) + (3+1) × 20 = 309 (Fig. 5k).

These examples demonstrate Bulienkov’s remarkable ability to see the whole picture behind the details—the systemic self-organization of matter based on the laws of symmetry. The twinning operations of crystalline simplices and modules that preserve the local order of a solid body in their structure allow us to model and predict branches of the possible evolution of nanoparticles without sacrificing the symmetry of the cluster nucleus.

CONCLUSIONS

In his scientific research, Bulienkov attached great importance to the methodological and philosophical justification of both already used in science categories and those proposed by himself—a crystalline module, self-organization, simplicial-modular design, and some others [21, 22]. For him, one of the key categories was the propensity category introduced into science by Popper [23], which in a first approximation can be interpreted as the possibility space in the evolution of a system. Bulienkov emphasized that if the basis of modeling (e.g., building blocks, modules) is chosen correctly, important fundamental conclusions can be drawn about the structure and possible pathways of the self-organization of matter at different hierarchic levels even based on indirect experimental data.

One of Bulienkov’s fundamental achievements was the creation of three branches of generalized crystallography—the base for solving many theoretical problems of modern materials science: Penrose parquets, quasi-crystals, clusters, and structures of bound water [24]. The next generations of researchers can expect long and meticulous work to comprehend Bulienkov’s enormous theoretical heritage and develop its ideas and approaches in different fields of natural science.

APPENDIX

Computer-Aided Execution of the Simplicial-Modular Design

The simplicial-modular design assumes the use of symmetry operations not with individual atoms but with groups of atoms assembled into polyhedra. A simple coordinate description is insufficient for the modular design because the structure is modeled on the basis of a group of atoms (simplices or modules of the structure). The atoms of the structure can be arranged at the vertices, on the edges, or on the faces of the corresponding polyhedra (modules of the structure). The modular assembly of the structure of a cluster is done by finding algorithms by which an individual module or simplex can multiply. The neighboring modules (simplices) are joined by binary symmetry operations ($\bar {1}$, m, 2). To solve the problem of the modular design of a structure of clusters consisting of modules (simplices) by means of computer mathematics, we must use homogeneous coordinates [25, 26].

In a system of homogeneous coordinates, four-dimensional column matrices are used to represent the points and vectors of a three-dimensional space. In a system of coordinates defined by the set of parameters $({{v}_{1}},{{v}_{2}},{{v}_{3}},{{P}_{0}})$, any point P can be clearly represented by the correlation $P = {{P}_{0}} + {{a}_{1}}{{v}_{1}} + {{a}_{2}}{{v}_{2}} + {{a}_{3}}{{v}_{3}}$.

If the multiplication by scalars 0 and 1 on a set of points is determined as

$$\begin{gathered} 0 \times P = 0, \hfill \\ 1 \times P = P, \hfill \\ \end{gathered} $$

the above correlation can be written in matrix form using matrix multiplication:

$$P = \left[ {{{a}_{1}}\;{{a}_{2}}\;{{a}_{3}}\;1} \right]\left[ {\begin{array}{*{20}{c}} {{{v}_{1}}} \\ {{{v}_{2}}} \\ {{{v}_{3}}} \\ {{{P}_{0}}} \end{array}} \right].$$

This expression is generally not a scalar product because the elements of the matrix are inhomogeneous, but in a computer such an expression is implemented using the same procedure as for a scalar product. The four-dimensional row matrix in the right part of the equation is a representation in homogenous coordinates of point P in a system of coordinates defined by parameters $({{v}_{1}},{{v}_{2}},{{v}_{3}},{{P}_{0}})$. Point P can be represented by the column matrix $P = \left[ {\begin{array}{*{20}{c}} {{{a}_{1}}} \\ {{{a}_{2}}} \\ {{{a}_{3}}} \\ 1 \end{array}} \right].$

In the same system of coordinates, any vector v can be presented in the form $P = \left[ {{{b}_{1}}\;{{b}_{2}}\;{{b}_{3}}\;0} \right]\left[ {\begin{array}{*{20}{c}} {{{v}_{1}}} \\ {{{v}_{2}}} \\ {{{v}_{3}}} \\ {{{P}_{0}}} \end{array}} \right].$ Vector v can therefore be represented by column matrix $a = \left[ {\begin{array}{*{20}{c}} {{{b}_{1}}} \\ {{{b}_{2}}} \\ {{{b}_{3}}} \\ 0 \end{array}} \right].$

When applying symmetry operations to polyhedra (simplices and modules), we can use homogeneous coordinates for which the operations on points and vectors are performed using common matrix algebra operations. It is sufficient to define the following operations for the representation of twinning symmetry operations: translation, scaling, and turn by a certain angle. Note that in any system of coordinates, an affine transformation can be represented by a 4 × 4 matrix of the type $A = \left[ {\begin{array}{*{20}{c}} {{{a}_{{11}}}}&{{{a}_{{12}}}}&{{{a}_{{13}}}}&{{{a}_{{14}}}} \\ {{{a}_{{21}}}}&{{{a}_{{22}}}}&{{{a}_{{23}}}}&{{{a}_{{24}}}} \\ {{{a}_{{31}}}}&{{{a}_{{32}}}}&{{{a}_{{33}}}}&{{{a}_{{34}}}} \\ 0&0&0&1 \end{array}} \right].$

Shift (Translation)

Shift (translation) transformation shifts a point to new positions in accordance with set translation vector p' = p + d. Using the multiplication of matrices, the shift (translation) is presented as p' = Tp, where ${\mathbf{T}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&{{{a}_{x}}} \\ 0&1&0&{{{a}_{y}}} \\ 0&0&1&{{{a}_{z}}} \\ 0&0&0&1 \end{array}} \right].$

Scaling

In homogeneous coordinates, scaling can be expressed by the matrix equation p' = Sp, where ${\mathbf{S}} = \left[ {\begin{array}{*{20}{c}} {{{b}_{x}}}&0&0&0 \\ 0&{{{b}_{y}}}&0&0 \\ 0&0&{{{b}_{z}}}&0 \\ 0&0&0&1 \end{array}} \right]$.

Turn

In the matrix form, a turn around axis z by angle θ can be represented by the matrix equation p' = R_zp, where ${{{\mathbf{R}}}_{{\mathbf{z}}}}(\boldsymbol{\theta} ) = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ - \sin \theta }&0&0 \\ {\sin \theta }&{\cos \theta }&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]$. The matrices of the turn around the x and y axes can be formed in a similar manner:

$${{{\mathbf{R}}}_{{\mathbf{x}}}}(\boldsymbol{\theta} ) = \left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&{\cos \theta }&{ - \sin \theta }&0 \\ 0&{\sin \theta }&{\cos \theta }&0 \\ 0&0&0&1 \end{array}} \right],$$

$${{{\mathbf{R}}}_{{\mathbf{y}}}}(\boldsymbol{\theta} ) = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&0&{ - \sin \theta }&0 \\ 0&1&0&0 \\ {\sin \theta }&0&{\cos \theta }&0 \\ 0&0&0&1 \end{array}} \right].$$

Any complex transformation of a module can thus be presented as the superposition of the base transformations that were considered above. For example, a turn around an arbitrary axis passing through the geometric center of the module p₀ can be accomplished by multiplying translations and turns:

M = T(p₀)R_x(−θ_x) R_y(−θ_y) R_z(θ_z) R_y(θ_y) R_x(θ_x)T(−p₀). The transformation sequence is from right to left.

The use of homogeneous coordinates in the modular design therefore allow us to transform the coordinates of groups of atoms (modules) and obtain objects with regular aperiodic structures. Since the modular design of aperiodic structures produces gaps in a structure, we must eliminate them after assembling it. This elongates the bonds and creates local nonequilibrium stresses. To relax these we can use, e.g., a three-particle potential, as in molecular mechanics [27]. We must then calculate the table of connectivity among the atoms in the cluster and geometrically optimize the structure of the cluster, after which its internal structure and external shape will be more symmetrical.