¹Institute of Mathematical Problems of Biology, Keldysh Institute of Applied Mathematics RAS, Pushchino, Russia
²Faculty of Physics, Southern Federal University, Rostov-on-Don, Russia
³Institute of Solid State Chemistry and Mechanochemistry SB RAS, Novosibirsk, Russia

Abstract. The results of calculations of the substitution of calcium atoms for zinc in the structure of hydroxyapatite using density functional theory methods using hybrid functionals in the supercell model are presented. Changes in the parameters and volume of the unit cell, energy bands and energy of formation of substitutions with increasing number of substitutions in different positions of calcium (Ca1 and Ca2) are analyzed in comparison with experimental data. A proportional decrease in the parameters and volume of the cell with an increase in the number of substitutions has been established, and a more complex behavior of various cell parameters has been revealed, which is a consequence of the violation of the original symmetry. Electronic energy levels were found to depend on the zinc concentration and the positions of the calcium ions being replaced. In this case, the band gap Eg of hydroxyapatite experiences a jump of 0.6–0.8 eV with the introduction of one zinc ion per supercell, and then decreases and reaches values below the initial Eg value by 0.5–0.6 eV for substitutions in Ca1 positions, and by 0.8–0.9 eV for substitutions in Ca2 positions. It has been shown that the energy of substitution has a complex dependence on the concentration of the substituent and the replacement of calcium ions with zinc occurs predominantly in the Ca2 position over the entire concentration range. An analysis of changes in interatomic distances during the process of relaxation to the equilibrium state at different zinc concentrations was carried out. We revealed formation of bonds between zinc atoms and nearby oxygen anions, which violates the original symmetry of hydroxypatite structures. The data obtained are important for understanding the structural changes that occur during substitution, as well as for understanding and predicting the properties of synthesized biocompatible materials.

Key words: hydroxyapatite, substitutions, cations, zinc, modeling, density functional theory, cell parameters, electronic level energies, substitution formation energy, band gap.