Equation of state of hexagonal hydroxylapatite (P6₃) as obtained from density functional theory simulations

Abstract

Hydroxylapatite is an important calcium phosphate phase whose knowledge is useful in different fields, for instance in bone biology, development of biomaterials and even cultural heritage. In this work, the equation of state of this mineral was calculated by using the quasi-harmonic approximation, also including for the first time the temperature effect. In athermal conditions (0 K), we found that the pressure dependence of the hydroxylapatite unit cell volume is well described by a third-order Birch-Murnaghan formulation, with parameters K 0 5 115.9(1), K' 5 4.47(6), and V 0 5 524.323 (27). The inclusion of temperature led to a lower bulk modulus, for example, KT 0 5 109.55 GPa at 300 K. The thermal expansion coefficient between 0 and 1000 K was also reported. The results are in good agreement with the few available experimental data reported in literature and further extend the knowledge of the mechanical and thermal behavior of this important mineral. K E Y W O R D S DFT/B3LYP-D*, equation of state, hydroxylapatite, quasi-harmonic approximation, thermal expansion coefficient | I N TR ODU C TI ONThe growing interest in biomaterials and biominerals led, in recent years, to several researches aimed to unravel the mechanical properties of calcium phosphates among which hydroxylapatite (OHAp, space group P6 3 /m, Figure 1) is one of the most investigated because of its similarities to the mineral phase of bone and dental tissues. For instance, specific and peculiar experimental Raman piezo-spectroscopy were employed to better understand the microscopic processes governing the behavior of natural bone tissues (sane and osteoporotic) as well as the mechanochemical framework describing how the microstructure of synthetic biomaterials affects the failure of artificial joints. [1,2] This method can analyse the mechanical stress as stored in the apatite crystals belonging to both natural bone (sane and osteoporotic) and synthetic biomaterials by the Raman shift of the apatite (compared to unstressed apatite). Another example regards the mechanical stability and failure of hydroxylapatite/collagen fibrils nano-composite, which was simulated by molecular mechanics methods in several works of Dubey and Tomar. [3][4][5] Finally, the effects of carbonate ion on the OHAp unit cell was studied from both the structural [6][7][8][9][10] and vibrational [11][12][13][14] points of view, to better understand the differences between stoichiometric hydroxylapatite and biological apatite (which contains several CO 22 3 and other ionic substitutions). Hexagonal OHAp is known to be a piezoelectric material, a property related to the ordered hydroxyl group alignment in the calcium channel.According to recent studies in literature, [15][16][17][18] OHAp exhibits also a ferroelectric behavior. Being a piezoelectric compound, a detailed knowledge on the mechanical properties of hydroxylapatite could be important.Hydroxylapatite is also a valued material for cultural heritage purposes. Recently, it was found that this mineral could be used to...

First principle investigation of the thermomechanical properties of type A carbonated apatite

“…The well‐known hybrid DFT Hamiltonian B3LYP was selected as it was very suitable for the investigation of structural, vibrational, thermodynamics, and mechanical properties of apatite minerals . A correction to take into account dispersive forces using the DFT‐D2 approach was included in its recently adjusted form for the hybrid B3LYP functional (B3LYP‐D* approach) …”

“…The all‐electron Gaussian‐type orbital basis sets adopted were the same used in previous works on OHAp and type A Cap . Calcium has been described with an 86‐511G(2d), with outer shell exponents α sp = 0.453 bohr −2 , α d1 = 3.1910 and 0.8683 bohr −2 , and α d2 = 0.2891 bohr −2 .…”

“…However, there are very few studies on the fundamentals of apatite mechanics in literature. Most of them are related to stoichiometric hydroxylapatite, reporting either the second‐order elastic constants or the equation of state (EoS), with one paper reporting the EoS of chloroapatite . Regarding carbonated hydroxylapatite, only the study by Liu et al reported the EoS of the mineral up to 10 GPa but only at ambient temperature (~300 K).…”

“…The present work aims to further extend the knowledge of the thermoelastic and thermodynamic properties of apatite minerals, started on the stoichiometric OHAp, and to provide the EoS of type A carbonated apatite [CAp, Ca 10 (PO 4 ) 6 CO 3 , space group P 1, Figure ] calculated at different P ‐ T conditions by using the well‐known quasiharmonic approximation (QHA). The aim of this study is fill in the knowledge gap on the thermomechanical behavior of this mineral phase, which could be useful, as previously mentioned, in biomaterials and other technological applications.…”

The structural, mechanical, and thermodynamic properties of B2‐type TMZr (TM = Ru, Mo, Rh, Os, and Re) compounds from first‐principles calculations

“…Naturally, the mechanical properties of high‐temperature materials are estimated by the stress‐strain method . Here, the mechanical properties of TMZr compounds were estimated by the elastic modulus: bulk modulus, shear modulus, and Young's modulus .…”