Theory of biomineral hydroxyapatite

Hydroxyapatite (HA, Ca₁₀(PO₄)₆(OH)₂) is one of the most abundant materials in mammal bone. It crystallizes in an aqueous environment within spaces between tropocollagen protein chains. However, despite its abundance and possible usefulness in the medical field this complex physical system remains poorly understood to date. We present a theoretical study of the energetics of hydroxyapatite, its electronic, mechanical and thermodynamic properties. Our mechanical and thermodynamic properties from first principles are in excellent agreement with the rare available experimental data. The monoclinic and hexagonal phases are lowest in energy. A comparison of the phonon dispersions of these two phases reveals that a phase transition occurs due to a difference in vibrational free energy. The transition is of order-disorder type. Our calculated phase transition temperature is 680 K, in decent agreement with the experimentally determined 470 K. An alternative theoretical model yields 882 K. The phase transition is mediated by OH libration modes. We also report for the first time on a peculiarity in the phonon spectrum of hexagonal and monoclinic HA. When studying the Lyddane-Sachs-Teller shifts in the spectrum close to the [Gamma]-point we identify two vibration modes showing a systematically increasing Lyddane-Sachs-Teller shift in frequency with decreasing dielectric constant. In experiment, the dielectric constant varies between 5 and 20 depending on the Ca/P ratio in the sample. The frequency shifts in the affected modes are as large as 20 cm⁻¹ as one spans the range of the dielectric constant. Thus, a simple spectroscopic analysis of a sample of bone may determine the quality of the sample in a physiological sense. We also identify the chemically stable low energy surface configurations as function of the OH, PO₄ and Ca concentration. In the experimentally relevant OH-rich regime we find only two surfaces competing for lowest energy. The surface most stable over almost the entire OH-rich regime is OH-terminated, and is currently being investigated in the presence of water and atomic substitutions on the HA surface.