HEXAGONAL STRUCTURES IN PHYSICAL

CHEMISTRY AND PHYSIOLOGY

G.A. Korablev*, Yu.G. Vasiliev*, G.E. Zaikov**

* Izhevsk State Agricultural Academy

** Institute of Biochemical Physics, RAS

Abstract

Some principles of forming carbon cluster nanosystems are analyzed based on spatial-energy ideas. The dependence nomogram of the degree of structural interactions on coefficient α is given, the latter is considered as an analog of entropic characteristic. The attempt is made to explain the specifics of forming hexagonal cell clusters in biosystems.

Keywords: spatial-energy parameter, hexagonal clusters, cell systems, entropy

1. Introduction

Main components of organic compounds constituting 98% of cell elemental composition are: carbon, oxygen, hydrogen and nitrogen. The polypeptide bond formed by COOH and NH₂ groups of amino acid CONH acts as the binding base of cell protein biopolymers.

Thus, carbon is the main conformation center of different structural ensembles, including the formation of cluster compounds. Анализируя механизм формирования углеродных кластеров можно по аналогии понять и гексагональную геометрию клеточных структур.

In the Nobel lecture in physiology Edvard Moser [1] pointed out such analogy and presented some trial data, which, probably need to have additional theoretical confirmation. For further discussion of these problems the idea of spatial-energy parameter (P-parameter) is introduced in this paper.

2. Initial criteria

The idea of spatial-energy parameter (P-parameter) which is the complex characteristic of the most important atomic values responsible for interatomic interactions and having the direct bond with the atom electron density is introduced based on the modified Lagrangian equation for the relative motion of two interacting material points [2].

The value of the relative difference of P-parameters of interacting atoms-components – the structural interaction coefficient α is used as the main numerical characteristic of structural interactions in condensed media:

(1)

Fig. 1

Nomogram of structural interaction degree dependence (ρ) on coefficient α

Applying the reliable experimental data we obtain the nomogram of structural interaction degree dependence (ρ) on coefficient α, the same for a wide range of structures (Fig.1). This approach gives the possibility to evaluate the degree and direction of the structural interactions of phase formation, isomorphism and solubility processes in multiple systems, including molecular ones.

Such nomogram can be demonstrated [2] as a logarithmic dependence:

, (2)

where coefficient β – the constant value for the given class of structures. β can structurally change mainly within ± 5% from the average value. Thus coefficient α is reversely proportional to the logarithm of the degree of structural interactions and therefore, by analogy with Boltzmann equation, can be characterized as the entropy of spatial-energy interactions of atomic-molecular structures [3].

Actually the more is ρ, the more probable is the formation of stable ordered structures (e.g. the formation of solid solutions), i.e. the less is the process entropy. But also the less is coefficient α.

The equation (2) does not have the complete analogy with entropic Boltzmann’s equation as in this case not absolute but only relative values of the corresponding characteristics of the interacting structures are compared, which can be expressed in percent. This refers not only to coefficient α but also to the comparative evaluation of structural interaction degree (ρ), for example – the percent of atom content of the given element in the solid solution relatively to the total number of atoms.

Conclusion: the relative difference of spatial-energy parameters of the interacting structures can be a quantitative characteristic of the interaction entropy:

3. Formation of carbon nanostructures

After different allotropic modifications of carbon nanostructures (fullerenes, tubules) have been discovered, a lot of papers dedicated to the investigations of such materials, for instance were published, determined by the perspectives of their vast application in different fields of material science.

The main conditions of stability of these structures formulated based on modeling the compositions of over thirty carbon clusters are given [4]:

1) Stable carbon clusters look like polyhedrons where each carbon atom is three-coordinated.

2) More stable carbopolyhedrons containing only 5- and 6-term cycles.

3) 5-term cycles in polyhedrons – isolated.

4) Carbopolyhedron shape is similar to spherical.

Let us demonstrate some possible explanations of such experimental data based on the application of spatial-energy concepts. The approximate equality of effective energies of interacting subsystems is the main condition for the formation of stable structure in this model based on the following equation:

; (3)

where К – coordination number, R – bond dimensional characteristic.

At the same time, the phase-formation stability criterion (coefficient α) is the relative difference of parameters Р₁ and Р₂ that is calculated following the equation (1) and is αST<(20-25)% (according to the nomogram).

During the interactions of similar orbitals of homogeneous atoms we have (3а)

Let us consider these initial notions as applicable to certain allotropic carbon modifications:

1. Diamond. Modification of structure where К₁=4, К₂=4; , R₁=R₂, Р₁=Р₂ and α=0. This is absolute bond stability.

2. Non-diamond carbon modification for which , К₁=1; R₁=0,77Å; К₂=4; , α=3,82%. Absolute stability due to ionic-covalent bond.

3. Graphite. , К₁=К₂=3, R₁=R₂, α=0 – absolute bond stability.

4. Chains of hydrocarbon atoms consisting of the series of homogeneous fragments with similar values of P-parameters.

5. Cyclic organic compounds as a basic variant of carbon nanostructures. Apparently, not only inner-atom hybridization of valence orbitals of carbon atom takes place in cyclic structures, but also total hybridization of all cycle atoms.

But not only the distance between the nearest similar atoms by bond length (d) is the basic dimensional characteristic, but also the distance to geometric center of cycle interacting atoms (D) as the geometric center of total electron density of all hybridized cycle atoms.

Then the basic stabilization equation for each cycle atom will take into account the average energy of hybridized cycle atoms:

; (4) (4а)

where ΣР₀=Р₀N; N – number of homogeneous atoms, Р₀ – parameter of one cycle atom, К – coordination number relatively to geometric center of cycle atoms. Since in these cases К =K and N =N , К=N, the following simple correlation for paired bond appears:

; (5)

During the interactions of similar orbitals of homogeneous atoms , and then: (5а)

Equation (5) reflects a simple regularity of stabilization of cyclic structures:

In cyclic structures the main condition of their stability is an approximate equality of effective interaction energies of atoms along all bond directions.

The corresponding geometric comparison of cyclic structures consisting of 3, 4, 5 and 6 atoms results in the conclusion that only in 6-term cycle (hexagon) the bond length (d) equals the length to geometric center of atoms (D): d=D.

Such calculation of α following the equation analogous to (1), gives for hexagon α=0 and absolute bond stability. And for pentagon d≈1.17D and the value of α=16%, i.e. this is the relative stability of the structure being formed. For the other cases α>25% - structures are not stable. Therefore hexagons play the main role in nanostructure formation and pentagons are additional substructures, spatially limited with hexagons. Based on stabilization equation hexagons can be arranged into symmetrically located conglomerates consisting of several hexahedrons.

It is assumed that defectless carbon nanotubes (NТ) are formed as a result of rolling the bands of flat atomic graphite net. The graphite has a lamellar structure, each layer of which is composed of hexagonal cells. Under the center of hexagon of one layer there is an apex of hexagon of the next layer.

The process of rolling flat carbon systems into NT is, apparently, determined by polarizing effects of cation-anion interactions resulting in statistic polarization of bonds in a molecule and shifting of electron density of orbitals in the direction of more electronegative atoms.

Thus, the aforesaid spatial-energy notions allow characterizing in general the directedness of the process of carbon nanosystem formation [5].

4. Hexagonal structures in biosystems

In the full-on report by Edvard Moser [1] the following problem results can be pointed out:

1. Cluster structures of cells form geometrically symmetrical hexagonal systems.

2. Cells themselves statistically concentrate along coordinate axes of symmetry with deviations not exceeding 7.5% (Fig. 2).

Fig. 2. Statistic distribution of cells along coordinate axes [1]

3. For independent cluster systems in different excitation activity phases four modules which differ scale-wisely on coefficients can be pointed out: 1.4–1.421.

Cluster С₆₀ containing 60 three-coordinated atoms and 180 effective bonds is the smallest stable carbon cluster. The similar structure is most probable in biosystems even with the availability of three-coordinated bonds of nitrogen atoms due to its 2Р³-orbitals. This is cluster К₆₀. The second module of clusters with the coefficient 1.4 has 252 bonds, which means 72 additional bonds and corresponds to 12 new hexagons or 24 new atoms of the system forming cluster К₈₄. By the way, such cluster as С₈₄ is among stable ones in carbon systems. If the rectified coefficient 1.421 is used in calculations, four more additional bonds are formed, which, apparently, play a binding role between the cluster subsystems.

The carbon cluster С₆₀ contains 12 pentagons separated with 20 hexagons. The pentagons can be considered as a defect of graphite plane but structurally stabilizing the whole system. Still it is unknown if there are similar formations in biosystems.

It can be assumed that entropic statistics of distribution of activity degree of structural interactions given in section 2 based on the nomogram (Fig. 1) is also fulfilled in biosystems. Thus, according to the nomogram if α < 7%, the maximum of structural interactions is observed, and their sharp decrease – if > 7%.

Therefore the maximal deviation angle of cell statistic distribution from coordinate axes equal to 7.5⁰ can be considered the demonstration of entropic regularity.

Apart from the aforesaid, a number of facts of hexagonal formation of biological systems can be also given as examples. For instance, the collocation of thin and thick myofilaments in skeletal muscle fibers and cardiomyocytes. At the same time, 6 thin myofilaments are revealed around each thick one. This system of functionally linked macromolecular complexes is constituted of calcium-dependent transient bonds between myosins and actins.

Also mechanotoropic interactions in surface layers of multilayer flat epitheliums conjugated with ample desmosomal contacts creating force fields on the background of the available hydrostatic pressure in epithelial cells are naturally followed by the formation of ordered epidermal columns with flat cells in the surface mainly of hexagonal shape and more rarely – of pentagonal one. There are also other options of revealing the aforesaid regularities.

General conclusion

The comparisons and calculations carried out based on spatial-energy ideas allow explaining some features of forming hexagonal structures in biosystems.

References

1. Nobel lecture by E. Moser in physiology: 11.03.2015, TV channel “Science”.

2. Korablev G.A. Spatial-Energy Principles of Complex Structures Formation//Brill Academic Publishers and VSP, Netherlands, 2005, 426pр.

3. Korablev G.A., Petrova N.G., Korablev R.G., Osipov A.K., Zaikov G.E. On Diversified Demonstration of Entropy // Polymers Research Journal. – 2014. - Vol. 8, № 3. – P. 145–153.

4. Sokolov V.I., Stankevich I.F. Fullerenes – new allotropic forms of carbon: structure, electron composition and chemical properties // Successes in chemistry, 1993, v. 62, №5, p. 455-473.

5. Korablev G.A., Zaikov G.E. Formation of carbon nanostructures and spatial-energy criterion of stabilization // Mechanics of composite materials and structures: RAS – IPM. – 2009. – V. 15, № 1. – P. 106-118.

Information about the authors

1. Grigory Andreevich Korablev, Doctor of Science in Chemistry, Professor, Head of Department of Physics at Izhevsk State Agricultural Academy.

E-mail: korablevga@mail.ru.

2. Yury Gennadievich Vasiliev, Doctor of Science in Medicine, Head of Department of Physiology and Animal Sanitation at Izhevsk State Agricultural Academy, Professor of Department of Histology, Cytology and Embryology at Izhevsk State Agricultural Academy.

E-mail: devugen@mail.ru

3. Gennady Efremovich Zaikov – Doctor of Science in Chemistry, Professor of N.M. Emmanuel Institute of Biochemical Physics, RAS.

E-mail: chembio@sky.chph.ras.ru.