Prediction of apatite lattice constants from their constituent elemental radii and artificial intelligence methods
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.
Section snippets
Computation details
In this section, we begin with the study of the visual patterns of 2D projections of apatite LCs. This was done by determining two important parameters that govern the relationship of LCs with the ionic radii. Subsequently, we present the development of ANN models for the prediction of LCs. Both patterns and ANN models were developed by using the 52 known apatites that were mainly collected from ICSD [11] and supplemented from other sources [12], [13], [14]. These compounds have the formula A5
Results of prediction
In this section, we predicted LCs of all 66 new apatites that are mentioned in the Introduction and listed in Table 2 by using the developed LC patterns and the ANN models. The classes of the 66 apatites, in terms of LCs a and c, respectively, were first determined with the obtained PR models. Then the ANN models were employed to predict the LCs a and c for all the 66 apatites. The class types of the ANN resultant LCs a and c were again determined by comparing them with the class definitions.
Conclusions
In this paper, we demonstrate that it is possible to correlate the LCs of apatite to the ionic radii of their constituent ions with the use of PR and ANN techniques. The PR helped identify the ionic radii as the predominating factors for the LCs, and the ANN technique successfully established the prediction models. The ANN models were applied to all 66 new apatites (assuming they exist) and this resulted in 54 acceptable estimates of LCs. The techniques used in the present work may provide an
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