**3. Results and discussion**

### **3.1 Structural properties**

X-ray diffraction (XRD) patterns of prepared Mg2TiO4 nanoparticles milled for different hours via high energy ball milling was taken and is illustrated in **Figure 3**. XRD patterns indicates that for the samples milled for 5 hrs exhibited the peaks corresponding to initial compounds MgO and TiO2 only. When the milling time increases, the intensities of the parent oxide peaks appear to be depressed gradually and the formation of associated MgTi2O5 phase was observed. When the milling time increased to 30 hrs, all the starting oxides peaks are disappeared completely. At the same time high intense diffraction peaks of pure- Mg2TiO4 phase are observed with small significance of MgTiO3 and MgTi2O5 phases. However, when the milling time reached to 35 hrs, the

#### **Figure 3.**

*XRD patterns of the MgO and TiO2 oxides milled for 5, 20, 30 and 35 hrs (adapted with permission from Bhuyan et al., 2020,* @ Springer *[19]).*

sample showed more intensified Mg2TiO4 phases along with small MgTiO3 peak. This signifies that crystallite nature of pure Mg2TiO4 sample enhances and is confirmed from ICSD – PDF # 06–5792. The presence of associated phases such as MgTi2O5 and MgTiO3 in the MgO - TiO2 system is mostly due to the difference in the degree of the incipient mechanical reaction. This can be explained as follows: at the time of milling, the mechanical energy of the grinding media transforms into the given oxide particles that causes structural destruction followed by reduction in particle size [26].

#### **3.2 Williamson - Hall (W - H) method**

The crystallite size of nanoparticles can be determined with several techniques that rely upon the peak width of the X - ray diffraction patterns. In the present study, Williamson-Hall (W-H) plot method as well as Scherrer formula have been chosen in order to understand the origin of the broadening in the XRD peak.

The broadening of XRD peaks is due to crystallite size and strain contributions. The average crystallite size was calculated from XRD peak width based on Debye– Scherrer's equation,

$$D = \frac{k\mathcal{X}}{\beta\_{\text{hkl}}\cos\theta} \tag{1}$$

where *β*hkl is the full width half maximum, *D* is the crystallite size, *k* is the shape factor which is taken as 0.9 for spherical particles, *λ* is the wavelength of incident X-ray radiation (λ = 0.154 nm for Cu-Kα) and *θ* is the Bragg angle of the analyzed peaks.

According to Williamson and Hall, the strain-induced broadening in nanocrystalline powders due to crystal imperfection and distortion was calculated using the formula [35],

$$\varepsilon = \frac{\beta\_{\text{hkl}}}{4 \tan \theta} \tag{2}$$

Here, *ε* is the effective strain associated with mechanical alloying. Now, the total peak broadening is defined as the sum of the contributions of crystallite size and strain present in the material and can be expressed as [36],

$$
\beta\_{\text{hkl}} = \beta\_{\text{D}} + \beta\_{\text{s}} \tag{3}
$$

where *β*D is due to the contribution of crystallite size, *β*ε is due to strain-induced broadening and *β*hkl is the width of the half-maximum intensity of instrumental corrected broadening. This *β*hkl can be calculated by using the relation,

$$\mathcal{J}\_{hkl} = \left[ \left( \boldsymbol{\beta}^{\mathrm{2}}\_{\ \mathrm{\,\,kdl}} \right) \mathrm{Measured} - \left( \boldsymbol{\beta}^{\mathrm{2}}\_{\ \mathrm{\,\,kdl}} \right) \mathrm{Instru} \,\mathrm{mental} \right]^{\mathrm{\,\,k}} \tag{4}$$

If we consider the particle size and strain contributions to line broadening are independent to each other and both have a Cauchy-like profile, then the observed line breadth is the sum of Eqs. (1) and (2) and is given by [35],

$$\mathcal{J}\_{\text{hkl}} = \left[\frac{k\mathcal{X}}{D\cos\theta}\right] + 4\varepsilon\sin\theta \tag{5}$$

By rearranging the above equation, we get,

$$
\beta\_{\text{hll}} \cos \theta = \left[\frac{k\mathcal{A}}{D}\right] + 4\varepsilon \sin \theta \tag{6}
$$

*Synthesis of Nano-Composites Mg2TiO4 Powders via Mechanical Alloying Method… DOI: http://dx.doi.org/10.5772/intechopen.94275*

This is the Williamson- Hall equation, which represents the uniform deformation model. The average crystallite size is estimated for selected peaks of nanocrystalline MTO powders milled for different hours by using Eqs. (1) and (6). The variation of average crystallite size as a function of milling time calculated by both the method is plotted and is depicted in **Figure 4(a)**. It is clear from **Figure 4(a)** that up to 30 hrs of milling the average crystallite size decreases sharply and then attains a constant value. The average crystallite size of the parent sample was found to be nearly 2.5 μm. But for 20 hrs of milling, the crystallite size reduced to 100-120 nm and for 35 hrs of milling it becomes 40-60 nm, as calculated by W-H method. From Scherer formula the average crystallite size for MTO powder are found to be 28 nm and 17 nm, respectively after 20 and 35 hrs of milling. Thus the crystallite size calculated from the Scherer equation is smaller than that of the W-H method. This is due to the fact that the Scherer's equation does not account for the lattice strain effect that causes line broadening.

Mg2TiO4 has an inverse spinel structure and having structural formula *Mg O* [MgTi] <sup>4</sup> and belonging to the cubic space group of Fd3m (227) [15]. The Ti and Mg atoms occupy the tetrahedral (8a) and octahedral (16d) sites and the oxygen atoms are in (32e) site symmetry position [20]. According to Bragg's law [37],

$$2d\sin\theta = n\lambda\tag{7}$$

where, *n* is the order of diffraction and it is usually taken as *n* = 1, λ is the wavelength of incident X-ray and *d* is the spacing between parallel planes of given miller indices *h, k and l*. Since, Mg2TiO4 has cubic structure, so the lattice constants are *a = b = c*. The *d*-spacing is related to the lattice constant *a*, and the miller indices *h, k and l*, by the following relation [37],

$$d = \frac{a}{\sqrt{h^2 + k^2 + l^2}}\tag{8}$$

By using Eq. (7), the lattice constant of selected planes is calculated by following relation,

$$a = \frac{\lambda}{2\sin\theta}\sqrt{h^2 + k^2 + l^2} \tag{9}$$

The variation of lattice parameter as a function of different milling time is plotted and is illustrated as inset **Figure 4(a)**. The results showed that the lattice constant decreases with increase of milling time from 8.436 Å to a stable value of 8.412 Å. This difference in lattice constant stipulates the occurrence of atomic disorder due to the milling process. That means the grinding of the powders via high energy ball milling techniques not only reduces the crystallite size into nanoscale range (< 100 nm) but also causes in the enhancement of lattice strain. Thus, the net X-ray line broadening is due to decrease of crystallite size, development of lattice strains and also due to the instrumental effects. Normally, crystallite size is a measure of the size of a coherently diffracting domain. So, when the crystallites of the materials are <100 nm, they have very less number of parallel diffraction planes that causes broadened diffraction peaks. Similarly, the non-uniform strains arises out of heavy plastic deformation during the course of high energy mechanical milling process that causes broadening of the diffraction peaks [35].

The milling dependence of internal microstrain (*e*) of mechanically derived nanocrystalline -MTO powders was evaluated. The graph between 4sin*θ*/*λ* (taken along *x*-axis) and *β*hkl cos*θ*/*λ* (taken along *y*-axis) for selected diffraction

**Figure 4.**

*(a) Variations of average crystallite size and lattice parameter (inset) with milling time and (b) variation of lattice strain with milling time. Inset: W-H plot for 35 hrs milled powders. (adapted with permission from Bhuyan et al., 2020,* @ Springer *[19]).*

peaks for 35 hrs milled MTO nano-powders, is plotted and is depicted in inset **Figure 4(b)**. In the present case, the crystal is considered as isotropic in nature and it is assumed that the properties of material do not depend on the direction along which it is measured. From inset **Figure 4(b)**, (called W-H plot), it shows that

*Synthesis of Nano-Composites Mg2TiO4 Powders via Mechanical Alloying Method… DOI: http://dx.doi.org/10.5772/intechopen.94275*

the data points are not much deviated from the straight line suggesting isotropic nature of the strain. From the linear fit of the data, the average crystallite, *D* was estimated from the y-intercept and the microstrain (*ε*) from the slope of the fit (inset **Figure 4(b)**). For 35 hrs milled powders, the crystallite size is found to be approximately 38 nm and microstrain is around 9.01× 10−3 respectively. Thus from the W-H analysis it is clear that the broadening of the X-ray peaks is due to the contribution of smaller crystallite size and the induction of strain. Further, it was noticed that with increase in milling duration the internal microstrain increases and it attains a constant value after a particular milling period.
