Strain and grain size of CeO2 and TiO2 nanoparticles: Comparing structural and morphological methods

Various crystallite size estimation methods were used to analyze X-ray diffractograms of spherical cerium dioxide and donut-like titanium dioxide anatase nanoparticles aiming to evaluate their reliability and limitations. The microstructural parameters were estimated from Scherrer, Monshi, Williamson-Hall, and their variants: i) uniform deformation model, ii) uniform strain deformation model, and iii) uniform deformation energy density model, and also size-strain plot, and Halder-Wagner method. For that, and improved systematic Matlab code was developed to estimate the crystallite sizes and strain, and the linear regression analysis was used to compare all the models based on the coefficient of determination, where the Halder Wagner method gave the highest value (close to 1). Therefore, being the best candidate to fit the X-ray Diffraction data of metal-oxide nanoparticles. Advanced Rietveld was introduced for comparison purposes. Refined microstructural parameters were obtained from a nanostructured 40.5 nm Lanthanum hexaboride nanoparticles and correlated with the above estimation methods and transmission electron microscopy images. In addition, electron density modelling was also studied for final refined nanostructures, and μ-Raman spectra were recorded for each material estimating the mean crystallite size and comparing by means of a phonon confinement model.


Introduction
Nanotechnology is the branch of Science in which materials are obtained and studied on a nanometric scale, named as nanoparticles (NPs), where novel and outstanding physical and chemical effects are found between 1 and 100 nm [1]. NPs can be obtained through various physical, chemical or biological synthesis methods. The NPs manufacturing process is essential since it can affect the photocatalytic, adsorptive, thermal, and optical properties of metal oxides such as cerium oxide (CeO2) and titanium oxide (TiO2), which depend on the particle size, shape, and crystal morphology [2]. Hence, the tuning of structural and morphological properties is highly relevant due to their possible usefulness in different applications. For example, CeO2 NPs are widely used due to their wide range of application in electrochemistry, such as electrode materials in supercapacitors, and medicine, due to their antibacterial properties [3]. On the other hand, TiO2 NPs are highly used in photocatalysis, solar cells, biomedicine, chemical sensors, and lithium storage [2],and have been also explored for heavy metal water cleaning purposes [4]. between 2θ =20°-100° for CeO2 and TiO2 , with 0.02° and 5s per step. The crystallographic phases were identified using Match version 3 software obtaining crystallographic cards [96-434-3162] and [96-500-022] with the crystallographic information file (CIF) #9009008 and #5000223 for the CeO2 and TiO2 anatase phases, respectively. The Origin pro 9.0 software was used to estimate the FWHM using a pseudo-Voigt fitting model. By making use of the instrumental width correction (βinst = 0.01°); the microstructual parameters were systematically ontained by using a code in Matlab version R2017b which is given in the Supplementary material. The values for microstructural parameters estimated from the code (including corrected instrumental resolution) are close in value to the obtained by the linear fit for each crystallite estimation model. Therefore, we only reported these values in Table 1.
For the RM of the diffractograms, the software Fullprof was employed, the CeO2 and TiO2 crystallographic information files (CIF) obtained from Match v3 software were used as initial parameters, which crystallographic data for CeO2 are: cubic crystalline structure, space group Fm-3m, and cell parameters a=5.4110 Å. And, for TiO2 anatase they were: tetragonal crystalline structure, space group I 41/amd, cell parameters a=3.78435 Å, c=9.50374 Å for both cases the Caglioti's initial parameters were U=0.004133, V=-0.007618, and W= 0.006255 and refined using the pseudo-Voigt function. Finally, the average crystallite size was determined in the Fullprof program. To do that, we first characterized the standard Lanthanum hexaboride reference material (nano LaB6), its measurement on the diffractometer was performed between 2θ =10°-80° with a step of 0.02°. For the LaB6 refinement, the CIF #1000055 was inserted as the initial model and the Thompson-Cox-Hastings (TCH) pseudo-Voigt Axial divergence asymmetry function was used to obtain the instrumental parameters of the equipment which is added to the instrumental resolution file (IRF), and later used to determine the average crystallite sizes of the CeO2 and TiO2..

2.2.μ-Raman experimental details
Structural and vibrational features of the nano powders were analyzed by Raman spectroscopy using a confocal μ-Raman microscope inVia TM by Renishaw. The spectrometer was configurated with a 1200 grooves/mm diffraction grating and a ×50 objective with N.A. 0.75 and working distance of 0.37 mm. The excitation wavelength was set to 785 nm from a laser diode. Laser power was set to ~ 1 mW. After identifying the main Raman peaks, we use a phonon confinement model (PCM) for the estimation of nanocrystals size [18][19].

p-XRD analysis
p-XRD diffractograms for CeO2 and TiO2 NPs are shown in Figure S1(a-b). For CeO2, it indicates a monophasic phase and could be indexed to a cubic structure. The commercial TiO2 powder presents two characteristic crystal phases: rutile and anatase [20]. Figure S1 (b) shows the TiO2 diffractogram which only detected the Bragg peaks of anatase phase as there is no diffraction lines that have a rutile-phase TiO2.

Scherrer method
Scherrer obtained his equation for the ideal condition of an ideal parallel, infinitely narrow, and monochromatic X-ray beam diffracting on a monodisperse powder of cubicshaped crystallites [21]. The broadening of the diffracted Bragg peak in the nanocrystals is due to the crystallite size and the intrinsic strain effects. This broadening often consists of one physical and instrumental broadening parts, this last one can be corrected with the following relationship [22]: where is the corrected peak broadening. The instrumental broadening and physical broadening of the sample have been measured as FWHM. So, with the Scherrer method we can calculate the average particle size ignoring the contribution of the strain, the average crystallite size is calculated by the following equation: where is the morphological parameter or shape factor for spherical particles equal to 0.94 nm -1 , the wavelength (λ) of the radiograph is 1.54056 Å for radiation, the Bragg diffraction angle (θ) and the FWHM is rewritten as and expressed in radians. The plot 1/ vs shown in Figure 1a gave R 2 values higher for the CeO2 NPs than TiO2 NPs (Figure 1b). To have accurate results it is important to highlight that the Scherrer equation can only be used in: i) for average sizes up to 100-200 nm, ii) sample and signal/noise ratio, because the broadening of the XRD peak decreases as the crystallite size increases and it is difficult to separate the broadening from the peak [21].

Monshi method
Monshi [6] introduced some modifications to the Scherrer equation. Scherrer's equation has been seen to show an increment in the nanocrystalline size values as the dhkl (diffracted planes distance) values decrease and the 2θ values increase, as βcosθ cannot be kept constant. The modified Scherrer equation depends on the fact that the crystallite size is obtained during each main peak and the error in crystallite size assessment is reduced, due the advantage of reducing the sum of absolute values of errors ∑(±∆ ) [23].
Plot vs , we can observe a straight line with a slope of around one and an intercept of from which it was calculated the average crystallite size, see Figures 1 (c) and (d). As we can see in Figure 1 (d), the value of R 2 increases with respect to the Scherrer equation. This is because if there are different N peaks in the range of 2ϴ =20°-100° it is assumed that all these peaks should represent equal values for the crystallite size. But as seen in the development of this research for each peak a different value was obtained and there is a systematic error in the results for each peak, this correction gave us a decrease in the average crystallite sizes obtaining 24.1 nm for CeO2 and 12.7 nm for TiO2 NPs, respectively.

W-H method
In comparison to the Scherrer formula, the W-H method considers the effect of straininduced in the XRD peak broadening and can be used for the calculation of intrinsic strain separated from crystallite size. As already mentioned, the broadening of the physical line of the XRD peak occurs due to the size and micro-deformation of the nanocrystals. Therefore, the total broadening can be written as [22]: where, is the broadening due to size and the broadening due to strain. In the next section we will analyze the crystallite size and micro-deformation using the modified W-H equation as UDM, USDM, and UDEDM.

UDM method
The UDM assumes that the deformation is uniform along the crystallographic direction, so this model considers the deformation, which is isotropic nature and is also known as isotropic deformation model (ISM). The intrinsic deformation affects the physical broadening of the XRD profile and the broadening βs is related to the effective stress and Bragg angle by the equation [24]: where the deformation ε can be calculated from the expression = . Therefore, the total broadening representing the FWHM of a diffracted peak due to the contribution of the lattice strain and the size of the crystallites in a particular peak that can be expressed as: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 June 2021 doi:10.20944/preprints202106.0156.v1 Eq. (8) can be mathematically represented by: From the slope of the straight line between 4sinθ and cosθ the strain can be estimated and the average crystallite size can be estimated by extrapolation of the Yintercept Eq. (10); see Figures 2 (a) and (b) .
By taking the βinst = 0.01°, we obtained the average crystallite size of the y-intercept from the linear fit. Values of 28.9 nm for the CeO2 and 16.3 for TiO2 NPs were obtained, respectively. In principle, this method is not realistic at all due to the consideration of isotropic nature.

USDM method
This model takes into account the uniform deformation stress for a more realistic crystalline system where the anisotropic nature is considered. In addition to the uniform deformation energy density, the anisotropic nature of the Young's modulus of the crystal is more realistic [2].
Remembering Hooke's law, within the elastic limit, there is a linear proportionality relationship between strain (ε) and stress (σ), the constant of proportionality being the modulus of elasticity or simply Young's modulus and is given by = , where the constant of proportionality is the modulus of elasticity or Young's modulus, denoted by . Isolating the strain ε=σ/y and substituting in Eq. (9) we have: In Eq. (11), depends on the crystallographic direction perpendicular to the set of planes (hkl) or the Miller indices. Next, the expressions for the cubic and tetragonal crystal system that are in relation to the elastic compliances constants ( ) and stiffness constants ( ) are presented. i. Cubic crystal For a cubic crystal, Young's modulus is calculated using the following equation: where: The elasticity constants , , for cubic crystals are 71, 380, 275 GPa respectively. Replacing this data in Eqs. (13), (14) and (15)  Young's modulus ( ) is in the direction perpendicular to the crystalline lattice plane set (hkl), and for a cubic crystal it is represented by the following equation: = 1 (19) where the value of elasticity stiffness constants , , for cubic CeO2 is 455.06 × 10 ,188.7, 81.48 × 10 respectively [26]. Using these values of elasticity constants, we can calculate the elastic compliances values as 2.904 × 10 , -8.513× 10 and 1.227× 10 , respectively. Therefore, the value of Young's modulus for each peak was calculated taking as average 262.9 GPa, this value is higher than the value calculated in Eq. (12) because the (hkl) planes are taken into account.
ii. Tetragonal crystal Young's modulus is given by the following relation [2]: where , , , , are elastic strain for TiO2 anatase, their values are 5.1 × 10 , − 0.8 × 10 , −3.3 × 10 , 10.7 × 10 , 18.5 × 10 , and 16.7 × 10 N/m , respectively [27]. Using these elastic strains, the value of Young's modulus for each peak was calculated by taking as average 127 GPa. Figure S2 (a-b) shows a plot between 4 / vs , a linear fit was done where the slope represents the strain, and the average crystallite size was calculated of the intersection with the axis obtaining a value of 27.5 nm for CeO2 and 13.4 nm for TiO2. As mentioned, in contrast to the above method, this model uses the corresponding Young's modulus.

UDEDM method
It has been seen that the UDM model assumes an homogeneous crystal which is isotropic in nature. This homogeneity and isotropy is no longer justified for a real crystal. Since a crystal is anisotropic, the W-H equation must be modified by anisotropic terms [8]. This modified model is the USDM model which assumes a linear relationship between stress and strain, according to Hooke's law. But, in real crystals, the isotropic nature and linear proportionality between stress and strain cannot be considered, because there are several defects, such as dislocations and agglomerations that create imperfections in almost all crystals.
Thus, we have the UDEDM which considers the deformation of crystals, the uniform anisotropic deformation of the lattice in all crystallographic directions, and the cause of that uniform anisotropic deformation of the lattice is the deformation energy density (u). Therefore, the proportionality constants associated with the stress-strain relationship left to be independent. The strain energy (energy per unit volume) as a function of strain is given by Hooke's law as: where stress and strain are related as = × , so the intrinsic strain can be written as a function of energy density.
The W-H equation is modified in the UDEDM by: Plot of Eq (23), whit the term 4 / along X-axis and along Y-axis corresponding to each diffraction peak where the density of energy is obtained from the slope and the average crystallite size is got from the y-intercept of the linear fitting. Figure  S2 (

SSP method
This method has a better result for isotropic broadening, since at higher diffacted angles , the XRD data are of lower resolution and the peaks overlap [28]. Figure 3  The SSP is one of the methods that considers the XRD peak profile to be a combination of the Lorentzian and the Gaussian functions. In this assumption, the strain profile is shown by the Gaussian function and the size of the crystallites by the Lorentz function.
= + (24) where and are the broadening peak due to the Lorentz and Gauss functions, respectively. The SSP equation is presented below: In this particular method, less value is given to high angle diffraction data as an advantage it gives less weight to these, where the precision is usually lower [11]. This is because, at higher angles, XRD data are of lower quality and peaks are generally highly overlapped at higher diffracting angles [22]. It was clearly observed that the average crystallite size obtained for both CeO2 and TiO2 NPs are smaller compared to Scherrer's method, this difference can be given by the deformation. Therefore, when using a method that does not consider stress, it can give us inaccurate results [29]. It seems that the reason why the crystalline size decreases with this method is due to the deformation of the compounds, that is uniform and hence the peak broadening is reduced.

H-W method
In the above method, the XRD peak profile size extension has been assumed as a Lorentzian function, while strain broadening, as a Gaussian function. But actually, the XRD peak is neither Lorentzian function nor Gaussian function, as XRD peak region matches well with the Gauss function, whereas its tail falls off too fast matched and; on the other hand, the profile tails fit quite well with the Lorentz function but does not full the total area of the Bragg diffracted peak [22]. That is why H-W method is proposed, which is based on the assumption that peak broadening is a symmetric Voigt function which is a convolution of Lorentz and Gauss's function. Hence, for Voigt function, the full width at half maximum of the physical profile can be written by H-W method as: where, and are the full width at half maximum of the Lorentzian and Gaussian function. This method gives more weight to Bragg peaks in the low angle range and middle angle, where overlap of diffractant peaks is low. Now, the relationship between the size of the crystallite and the lattice strain according to the H-W method is given by [22]:  By comparing the results of the crystallite size and lattice strain, shown in the Table 1, we can see that the H-W method shows a decrease in crystallite size as well as lattice strain, a common feature between the W-H and H-W method is that the dispersion of data points increases with increased lattice strain, which would indicate that lattice strain is anisotropic [30], but in our case you see a decrease in the dispersion of the points and also a decrease in deformation so you could say that lattice strain is isotropic in nature.  The R² values are important to differentiate among all of the studied linear methods (see Table 1). We obtained only positive values of R² for all of the crystallographic phases, one method is more accurate if the R² is near 1 or in other words, data points of x-y are more touching the fitting line [31]. The H-W method is more accurate, suggesting that this model better fits the XRD diffractograms presented in this study. Figures 5 (a) and (b) shows the refined diffractogram of CeO2 and TiO2 NPs using the TCH profile functions, and the refined parameters are displayed in Table 2. For CeO2 NPs, the RM confirms the cubic structure of the CeO2 NPs where the characteristic peaks are very close to the fluorite-structured CeO2 crystal and no peak of any other phase was detected, indicating the high purity of the sample. For TiO2, the corresponding tetragonal structure of the TiO2 NPs shows the presence of anatase phase only. The weighted profile residual (Rwp) and the profile residual factor (Rp) were taken into account to follow the progress and as indicator of the refinement improvement. The goodness of refinement, χ² (chi-squared), indicates the statistical error. In both samples, small values of 1.94 and 2.25 were obtained using the TCH profile for TiO2 and CeO2 NPs, respectively.. For the estimation of the mean crystallite size the TCH profile was used, for which an IRF file is yielded by fitting the XRD diffractogram of the LaB6 nano-standard, which subtracts the instrumental broadening. The mean apparent size was 12 and 14.1 nm, for the CeO2 and TiO2 nanocrystallites, respectively, and of 40.5 nm for the LaB6 standard (see Figure S3 (ad) and Table S1).  Table 2.

Electron density modeling
Electronic density measures the probability of finding an electron in a certain region of the atom. The higher the concentration of electrons at a given point, the higher the electronic density. Therefore, depending of the material will exhibit certain characteristics. It can be seen on a scale composed of rainbow colors, red to indicate the region of highest electronic density and blue for that region where there are fewer electrons. Figures 6a and  6b show that there is no formation of ionic bonds using as an indicator the red color that are the covalent bonds. The Vesta program was used with a resolution of 0.6 Å for both samples.

μ-Raman analysis
For spherical particles with a diameter D and no vibration mode degeneration, the Raman intensity can be written as shown in Eq. (28) Here, is the phonon wave vector ranging in the BZ from − / to / , with a the lattice parameter. The volume differential under the spherical symmetry approximation is written as According to previous reports, for particle diameters greater than D ≥ 10 nm [17,18,32], a Gaussian confinement weighting function is a good approximation, and thus the Fourier coefficients are written as depicted in Eq. (29). The confinement factor can vary from 1 to 2 for the Ritchter confinement model [33] and Campbell model, respectively [34]. Here, for the sake of simplicity we fix = 1. Previous reports suggest that the dispersion relation for anatase TiO2 NPs near the Eg band can be modeled as ( ) = 0 + (1 − ( ) ) with = 20 . For the case of CeO2 NPs, we simply approximate the dispersion relation with a polynomial function of order 3, ( ) = 0 + 1 + 2 2 + 3 3 . Distinct reports put for the F2g mode between 464 and 466 cm -1 . Here the best fitted value is obtained for =464.4 cm -1 in agreement with Spainer et al [32]. However, our analysis did not include strain effects.
In order to fit Eq. (28) we normalize the Raman intensity to the peak area. We do this with the experimental data and with Eq. (28). The fitting procedure is based on the minimization of the unbiased squared error with free parameters , and . For the TiO2 NPs the Eg and B1g bands are identified. The A1g band cannot be resolved. The analysis is performed with the Eg band centered at 144 cm -1 . The characteristic F2g band at 464 cm -1 of the CeO2 particles is observed and used for the analysis. Both peaks were fitted using Eq. (29) after normalization. The estimated NPs size are 11.5 nm and 14.5 nm for TiO2 and CeO2, respectively with a fitting error close to 0.01 nm. Additional error source could be attributed to a particle size distribution not considered in this analysis.

TEM analysis and comparison
Commercial nanoceria NPs have average TEM particle sizes in the range of 15-25 nm [35]. The nanoceria are agglomerated and present a particle size distribution (PSD). While the crystallite estimation methods including Rietveld refinement, gave values between 10 and 20 nm, which is reasonable agreement with TEM determination and hence suggesting that the Scherer, W-H, SSP, and H-W methods are accurate suggesting the presence of PSD. On the other hand, donut-like TiO2 NPs shown in Figures 8(a) and (b) shows a particle size of ca. 20 nm with mean pore sizes of 5-7 nm. By comparing with the size estimation methods, the crystallite sizes are in the interval of 5.2 to 16.3 nm. Then the anatase TiO2 NPs is composed of these individual crystallites. It can be concluded that in the case of PSD, the NPs are formed by nanocrystallites with anisotropic size behavior, then a crystallite size distribution is expected By comparing all the methods, it can be said that the crystallite sizes obtained by the W-H method are more accurate, but an overestimation of 35% with respect to Scherrer method is obtained, as shown in TiO2 NPs, where W-H crystallite sizes are on average 3-5 nm times higher than the sizes of the Scherrer equation [36]. Moreover, the H-W method underestimate the crystallite size obtained from Rietveld method in ca. 5 nm. This suggest that the strain contribution must be considered in the sample with exotic morphologies as compared with nanoceria.   In the case of the H-W method, it shows a decrease in the crystallite size, which confirms the nanostructured character presented in the sample. On the other hand, the RM is consistent with the size strain method. This provides reliability in the method and allows its use to obtain average crystallite sizes, being one of the most important part in the characterization of nanomaterials. Table 2. Rietveld refinement parameters of CeO2 and TiO2 samples using the Fullprof program: cell parameters, cell volume and agreement factors.
(%) and (%) are the profile residual and the weighted profile residual factors, respectively, used to verify the Rietveld refinement quality. The goodness of fit, chi-square (χ²).

Conclusion
In this work, an exhaustive analysis of XRD data for CeO2 and TiO2 NPs are presented using crystallite size estimation methods, among them: the Scherrer method, the Monshi method, the W-H model, the UDM, UDEDM, SSP, and H-W method, where all of them suggested an important isotropic broadening contribution, assuming Lorentzian and Gaussian profile contributions to estimate the crystallite size and microdeformation physical parameters. However, the method of Scherrer and W-H have less precision for the determination of the crystallite size for these metallic nanooxides. Furthermore, the crystallite size was calculated using the RM. Fort that, the IRF function was considered and obtained from the refinement of the standard (nano LaB6). By employing the RM, it was possible to carry out the refinement of CeO2 and TiO2 nanopowders corroborating the phases of cubic nanoCeO2 and TiO2 anatase using the TCH profile and hence allowing calculation of microstructural parameters. After comparing all the presented models, it was found that the average crystallite sizes determined by the SSP and H-W methods are close to the results obtained in RM, in case of the CeO2, but for the TiO2 the Monshi method was the closest model. Then, we can say that the average crystallite size of CeO2 and TiO2 NPs are in a range of 10-15 nm as also corroborated by TEM analysis. Thus, suggesting that each NP is made up of two or at least three crystallites. Crystallite size determined by Raman analysis are in agreement with this particle crystallite distribution. Therefore, we have presented in detail XRD characterization that strongly correlates with Raman and TEM analysis. This perspective can be used in future works in order to analyze and estimate accurately the cystallite size distribution presented in NPs as prepared by different physical and chemical methods as well.
Supplementary Materials: Figure S1. Pure p-XRD pattern of CeO2 (a) and TiO2 NPs (b). Figure S2. The modified W-H analysis of CeO2 (a) and TiO2 NPs (b), assuming USDM. The modified W-H analysis of CeO2 (c) and TiO2 NPs (d), assuming UDEDM. Figure  Data Availability Statement: The simulated data of the present research can be provided upon reasonable request at the email juan.ramos5@unmsm.edu.pe.