**1. Introduction**

Perovskites have a general formula of ABO3. In these structures, the A site cation is a typical lanthanide, alkaline or alkaline-earth metal with 12-fold oxygen coordination, and the B-site is any one of a variety of transition metal cations [1]. Calcium titanate (CaTiO3) was established in 1839 by a Russian mineralogist Perovski, and materials with the same type of CaTiO3 were introduced as the perovskite structure. CaTiO3 has ionic bonds, as well as the ionic radii of Ca2+, O2<sup>−</sup> and Ti4+ are 1 Å, 1.40 Å and 0.6 Å, respectively [2]. In recent years, researchers have focused on developing perovskites and their mechanical properties in order to obtain a high yield. Furthermore, CaTiO3 is a well-known component in ferroelectric perovskite category, which has been considerably utilized as a dopant/doping in electronic materials due to its dielectric manner and flexibility in structural transformations [3,4]. The modulus of elasticity (E) or Young's modulus is defined as the proportion of the stress to the strain, created by the stress on the body when the body is in the elastic region [5]. The elastic constants are specified from the lattice crystal deformation against force. These elastic moduli are: Young's modulus, shear modulus and volumetric modulus. These modules are registered via inherent elastic properties of materials and their resistance to deformation due to loading. Elastic behavior of materials is described by models such

**Citation:** Rabiei, M.; Palevicius, A.; Nasiri, S.; Dashti, A.; Vilkauskas, A.; Janusas, G. Relationship between Young's Modulus and Planar Density of Unit Cell, Super Cells (2 × 2 × 2), Symmetry Cells of Perovskite (CaTiO3) Lattice. *Materials* **2021**, *14*, 1258. http://doi.org/10.3390/ ma14051258

Academic Editor: Thomas Walter Cornelius and Souren Grigorian

Received: 5 January 2021 Accepted: 2 March 2021 Published: 6 March 2021

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as Cauchy elastic, hypo-elastic and hyper-elastic. A hyper-elastic is a constitutive model for ideally elastic material that responds against stress gain from a strain energy density function, while for hypo-elastic material, their governing equation is independent of finite strain quantity except in the linearized state [6]. The elastic properties are intimately connected to the crystal structure, the intrinsic character of bonding between the atoms and the anisotropic nature of materials [7,8]; therefore, elastic constants can be derived from crystal lattice calculations [9]. There are several studies on the relationship between elastic constants and planes/directions in a lattice structure, for example, in [10–12]. One of the most accurate methods to measure the elastic stiffness constants and Young's modulus is to determine the velocity of long-wavelength acoustic waves through the ultrasonic pulse-echo technique [13]. In a crystal structure, points, directions and planes are described with an indexing scheme, and planar density is obtained as the number of atoms per unit area, which are centered on a specific crystallographic plane with a defined index [14]. Since the discovery of X-rays at the end of the 19th century, this method has been often used for material characterization [15]. It is used to identify the atomic-scale structure of different materials in a variety of states [16]. X-ray diffraction is the only method that provides the specification of both the mechanical and microstructural character of each diffracted plane. These planes are used as a strain to quantify Young's modulus in one or more planes/directions of the diffraction vector [17]. In forming, designing and manufacturing equipment industries, the use of non-destructive, accurate and convenient methods to determine the mechanical properties of materials is particularly important. Mechanical tests, such as tensile, strike and collision tests, are destructive. Ultrasonic methods are very time-consuming and require operator expertise in this area, and theoretical methods require time-consuming density functional theory (DFT) calculation and may need verification with experimental tests. Our proposed method only needs the XRD analysis, which is a routine test and calculation of planar density; therefore, it can be very significant in terms of industrial application. In this study, the effects of cell size on the accuracy of Young's modulus calculation were considered. Locations of atoms and diffracted planes of unit cell, super cells (2 × 2 × 2) and symmetry cells of CaTiO3 are designed and investigated. The super cell is a cell that describes the same crystal but has a larger volume than a unit cell. By extension of a unit cell proportional to the lattice vectors, the super cells are generated. In super cells (2 × 2 × 2), the extension is twice of unit cell length in each direction; likewise, for super cells (8 × 8 × 8), the extension is 8 times. The result extracted by symmetry cells was in good agreement with results recorded via ultrasonic technique. Therefore, this new approach of exploration of reliable Young's modulus quantity based on XRD is proposed for either single crystal or polycrystalline of CaTiO3.

### **2. Experimental**

### *2.1. Materials*

In this study, for synthesis CaTiO3, titanium (IV) butoxide, calcium chloride dehydrate, sodium hydroxide and ethanol reagents were purchased from Sigma Aldrich (Taufkirchen, Germany) and deionized water as the solvent for dispersions was prepared.

### *2.2. Instrumentation*

In this research, a Bruker D8 Advance X-ray diffractometer (Kaunas, Lithuania) with CuK<sup>α</sup> radiation was used. The powder X-ray diffraction was taken at 40 kV and 40 mA, and it was registered at a scanning rate of 2.5 degrees/minute and a step size of 0.02 degrees. The XRD peaks were interpreted by High Score X'Pert software (4.9.0) analysis to get the output ASC type files. The pulse-echo technique was applied for the determination of sound velocity for both transverse and longitudinal ultrasonic signals. For ultrasonic measurement, the model of pulser receiver and oscilloscope were Panametrics Co. (waltham, MA, USA) and Iwatsu (Tokyo, Japan) (100 MHz), respectively. For powder pressing, the model of mechanical machines was CD04-Z and CIP (CP 360). Additionally, the specific surface area of the sample was investigated by desorption isotherms of nitrogen (N2) gas

via using a Brunauer-Emmett-Teller (BET) apparatus Gemini V analyzer, micrometrics GmbH (Tehran, Iran). Moreover, transmission electron microscopy (TEM) CM 10-Philips (Tehran, Iran) with acceleration voltage from 50 to 80 KV was utilized.

### *2.3. Methods*

## 2.3.1. Synthesis of Nano-Powder CaTiO3

Calcium titanate (CaTiO3) was synthesized by solvothermal method. A simple procedure, namely the solvothermal method, was performed for the synthesis of CaTiO3 (Figure S1). In the first step, (1) calcium chloride dehydrate was stirred with ethanol and deionized water. (2) Titanium (IV) butoxide and ethanol were added to the system drop by drop, under stirring for around 10 min (750 rpm). The molar ratio of ingredients was achieved to calcium chloride dehydrate = 1, ethanol = 5, Titanium (IV) butoxide = 1 and deionized water = 100 respectively. (3) To create pH = 14, sodium hydroxide solution was utilized. (4) The produced solution was placed into the autoclave and the temperature was ~250 ◦C for 5 h. (5) Afterward, the product was under the drying conditions involved at 110 ◦C and 0.76 bar, respectively. (6) After a day, the mixture was washed, (7) filtered and dried (110 ◦C for 4 h), respectively. This method was used in previous studies [18,19].

### 2.3.2. X-ray Diffraction of CaTiO3 and Planar Density Calculations

Combining X-ray diffraction of crystalline CaTiO3 and calculation of planar density values of each diffracted plane was performed. In our study, the atomic density of each plane was considered as the planar density, which was determined as the area of atoms with the center positioned at the plane divided by the total area of the plane, and it is a determinant factor for mechanical properties of each plane. Planar density is a unitless parameter, and its value is less than 1 in each cell. Furthermore, the values of planar density are related to the positions and situations of atoms in the planes. For determination of atomic area, the Crystal Maker, Version 10.2.2 software was performed. First of all, the three-dimensional (3D) geometry of crystal structures was designed, and then, from the intersection area of each diffracted plane with atoms located at the plane, the atomic area was calculated. When an atom with diameter D was involved completely, the atomic area will be A = π D 2 2 ; otherwise, it will be a percentage of this amount.
