Prediction of apatite lattice constants from their constituent elemental radii and artificial intelligence methods

Abstract

Lattice constants (LCs) of all possible 96 apatite compounds, A₅(BO₄)₃C, constituted by A=Ba²⁺, Ca²⁺, Cd²⁺, Pb²⁺, Sr²⁺, Mn²⁺; B=As⁵⁺, Cr⁵⁺, P⁵⁺, V⁵⁺; and C=F¹⁻, Cl¹⁻, Br¹⁻, OH¹⁻, are predicted from their elemental ionic radii, using pattern recognition (PR) and artificial neural networks (ANN) techniques. In particular, by a PR study it is demonstrated that ionic radii predominantly govern the LCs of apatites. Furthermore, by using ANN techniques, prediction models of LCs a and c are developed, which reproduce well the measured LCs (R²=0.98). All the literature reported on 30 pure and 22 mixed apatite compounds are collected and used in the present work. LCs of all possible 66 new apatites (assuming they exist) are estimated by the developed ANN models. These proposed new apatites may be of interest to biomedical research especially in the design of new apatite biomaterials for bone remodeling. Similarly these techniques may also be applied in the study of interface growth behaviors involving other biomaterials.

Introduction

The apatite group of minerals has been of considerable interest to biologists for many years, and has recently received increasing attention in bone repair, substitution or augmentation, and in coatings for metal implants as well as teeth enamel [1], [2]. X-ray diffraction (XRD) patterns of enamel, dentin and bone are similar to those of mineral apatites such as hydroxyapatite and fluorapatite (FA), which provide the impetus for the development of commercial calcium phosphate materials. Basic research of the formation and physico-chemical properties of synthetic apatites and biologically relevant or related calcium phosphates is widely investigated [3], mostly by experimental approaches. Computer modeling may enhance our understanding of these materials. New knowledge on apatite lattice constants (LC) may shed light on not only the crystal structure behavior but also on understanding their physico-chemical properties to facilitate the design and fabrication of specific biomaterials.

Basic apatite structure is hexagonal with space group P63/m, although a few monoclinic forms with space group P1121/b were also reported. The crystal structure of apatite was studied and was shown to be very complex [4]. A research program is carried out by the author's group to develop computing models to aid in the understanding and design of new biomaterials. As an early effort, we focused on the atomic properties and LCs of stoichiometric apatites in the present work. The emphasis was placed on the relationship of the structure (or LCs) and atomic properties of these apatites.

In general, LC can be measured using X-ray, electron or neutron diffraction techniques. These techniques, however, usually involve complicated and time-consuming processes. In some circumstances, it is very difficult, if not impossible, to prepare the form of single crystals of sufficient size and quality for conventional single-crystal XRD studies [5]. This is also the case for many of the most important apatites where suitable single crystals for study are absent [4]. Advances in high-performance computing techniques allow calculating LC based on first principles of quantum mechanism [6]. However, it remains a computation-intensive job which may not be within the reach of majority of the scientific community, due to either a lack of computing resource or computational skills. Therefore, it is of practical interest to develop efficient methods for the prediction of LCs.

Artificial intelligence method seems a suitable solution which is able to deal with complicated relationships but requires less computation effort. In this study, we employed artificial intelligence methods, namely the pattern recognition (PR) and artificial neural network (ANN). We started with the determination of the atomic parameters that may govern the LCs, and found that the ionic radii of the constituent ions play a central role. Other atomic parameters like valence electron or electronegativity do not affect the LCs very much and thus can be ignored. This finding was done by performing the PR where the 2D plot of two parameters of simple functions of the ionic radii showed very good results.

The interesting result of the PR encouraged us to explore quantitative relationship between the LCs and the ionic radii. We first tried a linear regression model for predicting the LCs from the ionic radii, but this attempt failed probably due to the strong nonlinear relationship between the LCs and the ionic radii. We then resorted to the ANN technique. Over the past few years, we have successfully applied the ANN technique to the prediction of band gap energies and LCs of binary and ternary semiconductors [7], [8], melting points of binary laves phase [9], and capacity and cycle life of electrodes [10], with the use of atomic parameters as the inputs. In the present study, the ANN was again employed to predict the LCs of apatites with only the ionic radii as the inputs. The advantages of the ANN are apparent, as the relationship of the LCs of apatites with their ionic radii is unclear and ANN has the intrinsic ability to handle systems that involve many variables or strong nonlinear relationships that are very complex and poorly understood. The ANN was proven successful in this study to accurately predict apatite LCs, with a coefficient of determination R² (the experimental versus the reproduced values) about 0.98. The R² is an indicator that ranges from 0 to 1 that would indicate how closely the estimated values, from the trendline, correspond to the actual data. A trendline is most reliable when its R² is close to 1.

The general formula of stoichiometric apatite may take the form of either A₅(BO₄)₃C or A₁₀(BO₄)₆C₂. In the present work, we studied the apatite with the following composition: the cation A can be Ba²⁺, Ca²⁺, Cd²⁺, Pb²⁺, Sr²⁺, and Mn²⁺, or a mixture of any two of these cations. For cation B, it can be As⁵⁺, Cr⁵⁺, P⁵⁺, or V⁵⁺. For anion C, it is F¹⁻, Cl¹⁻, Br¹⁻, and OH¹⁻, or a mixture of any two of these anions. A full combination of these elements results in 96 (=6×4×4) pure apatites, i.e. with one A cation at A site, one B cation at BO₄ site, and one anion at C site. Among the total 96 apatites, we collected all available LCs (LC) of 30 pure apatites and 22 mixed apatites with two cations at A site and/or two anions at C site. As a result, a total number of 52 apatites were used for the present study. In addition, we predicted the LC of all new 66 (=96−30) apatites with the developed ANN models. The results showed that the estimated LC of the 54 new apatites might be acceptable, while the rest 12 are not reliable.