Numerical Estimation of Elastic Constants of Hydroxyapatite at Finite Temperatures: A Comparisons of Different Force Fields

Hydroxyapatite (HAP) is a naturally occurring calcium phosphate mineral that resembles human hard tissue in structure and composition. Its unique structure makes it suitable for a variety of applications such as biomedical implants, pollution control, nuclear waste management. The hexagonal structure of HAP endows it with five independent elastic constants, and a comprehensive understanding of their role and properties is essential for large scale adoption of HAP in various applications. However, the limited experimental and computational estimates currently available exhibit a wide scatter and occasional inconsistencies, and little effort has been made to quantify the influence of operating temperature on HAP elastic response. In this paper, we calculate the five elastic constants of hexagonal HAP first from Density Functional Theory (DFT)-based stress-strain relations derived using ultrasoft pseudopotential with Perdew-Burke-Ernzerho (PBE) exchangecorrelation functional under generalized gradient approximation (GGA) on the 44 atom single unit cell, and then with molecular dynamics (MD)-based stress-strain relations derived from 4 x 4 x 6 unit cells using three distinct families of force fields. These force fields are differentiated by how they model the non-bonded interactions: Lennard-Jones, Born-Mayer-Huggins and Buckingham types. We conduct the MD studies in the temperature range of 10-500 K. We find DFT to slightly overestimate the unit cell volume (a known consequence of using GGA) compared to X-ray powder diffraction-based experimental values reported in the literature. The predicted elastic constants satisfy Born's criterion for mechanical stability. All potential models agree with DFT that HAP exhibits higher stiffness along c-axis. Lattice parameters are found to increase with temperature, while the elastic constants decrease. Their rate of change, however, differs based on the force field. Among the three, the force field based on Buckingham potential appears to perform the best and agrees qualitatively with DFT and experimental results.