**X-ray Diffraction Analysis and Williamson-Hall Method in USDM Model for Estimating More Accurate Values of Stress-Strain of Unit Cell and Super Cells (2** *×* **2** *×* **2) of Hydroxyapatite, Confirmed by Ultrasonic Pulse-Echo Test**

**Marzieh Rabiei 1,\*, Arvydas Palevicius 1,\*, Amir Dashti 2, Sohrab Nasiri 1, Ahmad Monshi 3, Akram Doustmohammadi 4, Andrius Vilkauskas <sup>1</sup> and Giedrius Janusas 1,\***


**Abstract:** Taking into account X-ray diffraction, one of the well-known methods for calculating the stress-strain of crystals is Williamson-Hall (W–H). The W-H method has three models, namely (1) Uniform deformation model (UDM); (2) Uniform stress deformation model (USDM); and (3) Uniform deformation energy density model (UDEDM). The USDM and UDEDM models are directly related to the modulus of elasticity (E). Young's modulus is a key parameter in engineering design and materials development. Young's modulus is considered in USDM and UDEDM models, but in all previous studies, researchers used the average values of Young's modulus or they calculated Young's modulus only for a sharp peak of an XRD pattern or they extracted Young's modulus from the literature. Therefore, these values are not representative of all peaks derived from X-ray diffraction; as a result, these values are not estimated with high accuracy. Nevertheless, in the current study, the W-H method is used considering the all diffracted planes of the unit cell and super cells (2 × 2 × 2) of Hydroxyapatite (HA), and a new method with the high accuracy of the W-H method in the USDM model is presented to calculate stress (σ) and strain (ε). The accounting for the planar density of atoms is the novelty of this work. Furthermore, the ultrasonic pulse-echo test is performed for the validation of the novelty assumptions.

**Keywords:** Williamson-Hall (W-H); uniform stress deformation model (USDM); Young's modulus; X-ray diffraction; hydroxyapatite; planar density; ultrasonic pulse-echo

### **1. Introduction**

Young's modulus (E) plays an important role in the characterization of mechanical properties, and accurate knowledge of the engineering values of elastic modulus is vital for understanding materials' stiffness [1]. Recently, ceramic materials have been favored in industrial applications, because they exhibit good mechanical properties, such as high elastic moduli and high hardness [2]. One of the well-known bio ceramics is hydroxyapatite, which has bioactive properties very close to natural bone mineral and it has special biological significance [3]. The chemical formula of hydroxyapatite is Ca10(PO4)6(OH)2 and it differs very little from bone tissue [4,5]. Understanding the mechanical properties of hydroxyapatite during the crystallization and growth stages of the synthesis processes is

**Citation:** Rabiei, M.; Palevicius, A.; Dashti, A.; Nasiri, S.; Monshi, A.; Doustmohammadi, A.; Vilkauskas, A.; Janusas, G. X-ray Diffraction Analysis and Williamson-Hall Method in USDM Model for Estimating More Accurate Values of Stress-Strain of Unit Cell and Super Cells (2 × 2 × 2) of Hydroxyapatite, Confirmed by Ultrasonic Pulse-Echo Test. *Materials* **2021**, *14*, 2949. https://doi.org/10.3390/ma14112949

Academic Editor: Thomas Walter Cornelius

Received: 28 March 2021 Accepted: 28 May 2021 Published: 30 May 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

important because the Young's modulus affects the growth of hydroxyapatite crystal in mechanically strained environments directly [6]. Therefore, paying attention to the mechanical properties and structural geometry of hydroxyapatite can be helpful for research and industrial applications. Paying attention to the details of the structural geometry of hydroxyapatite is essential for employing an easy, cost-effective and reliable method to determine the Young's modulus. In this research, we have developed a method based on the linear regression of the Young's modulus of each plane of the crystal lattice versus planar density to obtain a reliable total Young's modulus of materials. Hence, in this study, for the first time, we calculated the exact planar density derived from the diffracted planes of hydroxyapatite in unit cells and super cells (2 × 2 × 2). Then, we determined and investigated the total Young's modulus of samples. Furthermore, to determine the effect of cell size on the Young's modulus, an extensive, exact calculation of the planar density of super cells (2 × 2 × 2) of hydroxyapatite, and comparing it with the result obtained from unit cell calculations, was performed. The Williamson-Hall (W-H) method is a procedure to analyse stress and strain derived from X-ray diffraction. Ultrasonic pulse-echo is a scan representation commonly used for thickness measurement and sizing of the defect in an ultrasonic test in which the signal is reflected from a discontinuity in a test of material structure. This test is performed in this study for the confirmation of the validity of the novelty assumptions. The configuration of the ultrasonic pulse echo is called the acoustic sound energy and localizes the discontinuities or defect indication. In addition, ultrasonic pulse echo measurements can be used to determine the elastic constant and elastic compliance of compounds [7]. According to the W-H method, the basic calculation for the plot can be performed by using the XRD data. The big problem with utilizing the W-H method in the USDM model is attributed to the values of Young's modulus in the equation. Because in all studies and research, the values of Young's modulus have been reported for one sharp peak of an XRD pattern or the average values of Young's modulus; both of them have an error because the values are not representative of whole diffracted planes. For example, Ratan et al. calculated the stress (σ) and strain (ε) of cadmium selenide (CdSe) nanoparticles and they considered the average value of Young's modulus from the Williamson-Hall (W-H) method in the USDM model [8]. Furthermore, Khorsand et al. reported the Young's modulus value of ZnO nanoparticles, considering a sharp peak of X-ray diffraction of ZnO [9]. In another study, Rabiei et al. presented the USDM model of the Young's modulus value for hydroxyapatite and the Young's modulus value was considered an average of diffracted planes [5]. Moreover, Madhavi et al. utilized the USDM model of the Young's modulus value for VO2 doped ZnS/CdS composite nanopowder and they calculated the Young's modulus value derived from average data. Rameshbabu et al. submitted the W-H method in the USDM model for calculating stress and strain values of hydroxyapatite and they also utilized the average value of Young's modulus [10]. In all research that has utilized the W-H method in the USDM model, the Young's modulus values of the investigated materials are reported as an average value or selected through the literature, and reported values are not inclusive of high accuracy. Interestingly, the values of the elastic compliance constant of studied materials were also derived from the literature, whereas each material has a special elastic compliances constant related to itself; therefore, with reference to the literature and utilizing the compliance constant values of studies, the accuracy of the report would be decreased. In addition, the W-H method is well suited to calculating and estimating the stress and strain of materials [10]. By applying our method and deploying the results of the W-H method, the stress-strain of a sample can be calculated with high accuracy. We have used both DFT calculations and ultrasonic measurements to compare and evaluate the validity of the proposed results (details are in the Supplementary materials). Overall, the evaluated results and the extracted values of this study were in good agreement with the theoretical, experimental and literature values.

### **2. Methods**

The synthesis route of hydroxyapatite powder is explained completely in part 2 of the Support Information (preparation of hydroxyapatite powder).

### *2.1. Structural Analyses of Synthesized Hydroxyapatite*

The XRD pattern of synthesized hydroxyapatite powder is shown in Figure 1. The XRD pattern exhibits several diffraction peaks, which can be indexed as the hexagonal hydroxyapatite. The XRD pattern was evaluated based on X'pert and the pattern was in agreement with the standard XRD peaks of hydroxyapatite (ICDD 9-432). Similar observations were reported in References [11,12]. In addition, crystallographic parameters and details of each diffracted plane of hydroxyapatite were evaluated by X'pert and the values are tabulated in Table 1 and Table S1.

**Figure 1.** X-ray diffraction pattern of hydroxyapatite synthesized at 950 ◦C.


**Table 1.** Crystallographic parameters of hydroxyapatite structure.

According to Table S1, the values of the distance between planes are calculated by Equation (1). In this equation, h, k and l are indices of each plane, and a, c and d are lattice parameters and distance of planes, respectively [13].

HCP 9.400 6.930 530.301 3.140 P63/m

$$\frac{1}{\mathbf{d}^2} = \frac{4}{3} \left( \frac{\mathbf{h}^2 + \mathbf{hk} + \mathbf{k}^2}{\mathbf{a}^2} \right) + \frac{\mathbf{l}^2}{\mathbf{c}^2} \tag{1}$$

Hydroxyapatite has a hexagonal system with a P63/m space group and has little deviation from stoichiometry [14]. Figure 2 shows a sketch of a unit cell of hexagonal hydroxyapatite and a cif file of synthesized hydroxyapatite. There are two different situations of calcium ions and, in total, 18 ions are closely packed to create the hexagonal structure. At each hexagonal corner, a calcium ion is restricted by 12 calcium ions shared with 3 hexagons. Void spaces between two hexagons are filled with three phosphate tetrahedral per unit cell. Ions in hydroxyapatite can be interchangeably replaced with biologically useful ions due to the inherent versatility of this crystal structure and can also be referred to as doping. In addition, the substitution of calcium, phosphate and/or hydroxyl ions is possible [15]. Notably, the specific feature of hydroxyapatite is related to the OH− ions forming inner channels along the c axis. This property plays an important role in its mechanical and physical properties [16]. In addition, the Edax analysis of synthesized hydroxyapatite is presented in Figure S2. According to the EDX analysis, the value of the Ca/P ratio for hydroxyapatite obtained from bovine bone was recorded to be 1.60. In addition, thermal decomposition of hydroxyapatite into tricalcium phosphate and tetra calcium phosphate was not observed during the sintering, as in References [17,18], so the hydroxyapatite was successfully synthesized.

**Figure 2.** Schematic representation of (**a**) hydroxyapatite unit cell and (**b**) the hydroxyapatite structure extracted by cif file.
