# A Simple Method for Estimation of the Scattering Exponent of Nanostructured Glasses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**The first (direct) method**to achieve this is to measure the optical densities (${D}_{1\mathrm{m}}\left(\lambda \right)$, ${D}_{2\mathrm{m}}\left(\lambda \right)$) of two samples of different thicknesses (${h}_{1}$, ${h}_{2}$) and use the equation (see, for example, Equation (3) in Ref. [34]):

**The second method**to determine the internal optical density $D\left(\lambda \right)$ of the sample is to calculate the reflection losses using the measured value of the refractive index $n\left(\lambda \right)$ of the material and to subtract the optical density ${D}_{\mathrm{refl}}\left(\lambda \right)$ of reflection losses from the measured optical density ${D}_{\mathrm{m}}\left(\lambda \right)$ of the sample (e.g., see Equations (15)–(18) in Ref. [34]):

**The third method**can be applied if the variation of the refractive index of the material during phase transformation (phase separation and crystallization) is insignificant. This means that the optical density ${D}_{\mathrm{refl}}\left(\lambda \right)$ of reflection losses does not change in the course of phase transformation. In this case, one can measure the optical density ${D}_{\mathrm{ig},\mathrm{m}}\left(\lambda \right)$ of a sample of the initial glass and the optical density ${D}_{\mathrm{htg},\mathrm{m}}\left(\lambda \right)$ of this sample after its heat treatment. These optical densities can be presented as

^{3+}ion absorption peaks and the appearance of light scattering due to the precipitation of ZnO nanocrystals, and to determine the values of the scattering exponent. This method seems to be the simplest of the three.

## 2. A Simple Method for Estimation of the Scattering Exponent and Its Testing

#### 2.1. The Method

#### 2.2. Application of the Novel Method and Comparison to the Results with Those Obtained Using the First Method

#### 2.2.1. Phase-Separated Sodium Borosilicate Glass

_{2}O·36.0B

_{2}O

_{3}·50.1SiO

_{2}(mol% by analysis) with the weight of 500 kg was melted in an industrial furnace at a temperature of 1250 °C, cooled in the crucible from 1250 to 500 °C for 70 h, and held at that temperature for 8 h, after which it was allowed to cool to room temperature at a rate of 4 °C per h. The phase-separated glass G2 was prepared from the initial glass by heat treatment for 10 h at 610 °C. The wavelength step of absorption spectrum recording was $\u2206\lambda =2\mathrm{nm}$. It was found that for $\lambda >360\mathrm{nm}$, the spectral dependence of the extinction coefficient is described by Equation (1) with the value of the scattering exponent:

#### 2.2.2. Gahnite-Based Zinc Aluminosilicate Glass-Ceramic Studied in Reference [34] and Denoted as GC1

_{2}O

_{3}·50SiO

_{2}(mol% by synthesis) doped by a mixture of 5TiO

_{2}and 5ZrO

_{2}as nucleating agents [37] was subjected to two-step heat treatment (750 °C, 6 h + 1000 °C, 6 h). As a result, a glass-ceramic was formed consisting of ZnAl

_{2}O

_{4}, ZrO

_{2}, and ZrTiO

_{4}nanocrystals with diameters d = 14, 12 and 24 nm, respectively, distributed in amorphous matrix whose composition is close to that of silica glass [38].

#### 2.2.3. Gahnite-Based Zinc Aluminosilicate Glass-Ceramic Studied in Ref. [34] and Denoted as GC3

#### 2.2.4. Spinel-Based Magnesium Aluminosilicate Glass-Ceramic

_{2}O

_{3}·60SiO

_{2}(mol% by synthesis) doped by 10TiO

_{2}as a nucleating agent (MAS composition). Initial glass 300 g in weight was melted in a laboratory electric furnace with SiC heating elements in crucibles made of quartz ceramics at 1560 °C for 8 h with stirring, quenched by pouring onto a cold metal plate and annealed at 660 °C. The glass-ceramic was prepared from the initial MAS glass by two-step heat treatment (750 °C, 6 h + 1000 °C, 6 h). It will be denoted as MAS-GC. This glass-ceramic contained crystals of magnesium aluminate spinel with the average size of ≈8 nm and of magnesium aluminotitanate solid solution with the average size of ≈15 nm distributed in the highly siliceous residual glass.

#### 2.3. Application of the Novel Method and Comparison of the Results with Those Obtained in Ref. [35] for Glass-Ceramics Based on Sr^{II}Nb^{IV}O_{3} Crystals Using the Second Method

_{2}—18.25K

_{2}O—9Bi

_{2}O

_{3}—9SrO—9Nb

_{2}O

_{5}—0.5CeO

_{2}—0.5Eu

_{2}O

_{3}(mol%) was heat-treated at 450 °C for 10 h and then at 500 °C for 20 and 40 h to prepare the glass-ceramics GC20 and GC40, respectively [41].

^{II}Nb

^{IV}O

_{3}crystals (JCPDS file card No. 79-0625) with sizes of 20–25 nm [41] and are characterized by strong light scattering in visible and IR ranges.

#### 2.4. Application of the Novel Method and Comparison of the Results with Those Obtained Using the Third Method

#### 2.4.1. Er-Doped Potassium Zinc Aluminosilicate Glass-Ceramic Containing ZnO Nanocrystals

_{2}O—32ZnO—14Al

_{2}O

_{3}—40SiO

_{2}+ 1.5Er

_{2}O

_{3}(mol%) were studied in Ref. [36].

#### 2.4.2. Yb-Doped Potassium Zinc Aluminosilicate Glass-Ceramic Containing ZnO Nanocrystals

_{2}O·32ZnO·14Al

_{2}O

_{3}·40SiO

_{2}+1.5Yb

_{2}O

_{3}(mol%) and prepared the glass-ceramic (ZnOYb-GC) by heat treatment at 750 °C for 2 h [42]. This ZnOYb-GC contains ZnO nanosized crystals.

## 3. Discussion

#### 3.1. Comparison of the Scattering Exponent Values Obtained by the New and Traditional Methods

^{II}Nb

^{IV}O

_{3}crystals (compare Equations (23) and (24)). Let us try to find the reasons for this discrepancy. In Section 2.3, we applied the novel method in the wavelength ranges from 552 to 652 nm (GC20, Figure 6a) and from 517 to 663 nm (GC40, Figure 6b). Since the ranges are narrow and there are strong oscillations of the curves, we could not find non-linear behavior of the curves in these ranges. However, Figure 4c,d in Ref. [35] show that Equation (1) is not satisfied in these ranges, and thus, portions of straight lines cannot be found on curves in Figure 6a,b. We will illustrate this conclusion below.

#### 3.2. On the Choice of the Wavelength Step in the Spectrum of the Measured Optical Density

#### 3.3. On the Choice of the Sample Thickness

#### 3.4. Applicability of the New Method to the Cases of Wavelength-Dependent Reflection Losses

- The internal optical densities of the thick and thin samples calculated by Equation (36) differ only by a constant factor: ${D}_{1}\left(\lambda \right)=({h}_{1}/{h}_{2}){D}_{2}\left(\lambda \right)$. The similar equation also applies to derivatives: $d{D}_{1}\left(\lambda \right)/d\lambda =({h}_{1}/{h}_{2})d{D}_{2}\left(\lambda \right)/d\lambda $. Thus, in the logarithmic representation (Figure 15), the curve of $d{D}_{1}\left(\lambda \right)/d\lambda $ (curve 1′) can be obtained by shifting the curve of $d{D}_{2}\left(\lambda \right)/d\lambda $ (curve 2′) along the y-axis. Figure 15 confirms this conclusion. A slight difference in slopes ${S}_{\mathrm{lp}}$ of curves 1′ and 2′ (see the text preceding Equation (37)) is due to different wavelength ranges of the linear approximation, shown by arrows in Figure 15.
- If the thicknesses of the thin and thick samples are significantly different (as in the case shown in Figure 15), the internal optical densities ${D}_{1}\left(\lambda \right)$ and ${D}_{2}\left(\lambda \right)$ of both samples (see Equation (36)) are determined to a greater extent by the measured optical density of the thick sample, the relative measurement error for which is smaller than for the thin one. Therefore, the curve 1′ is smoother than the curve 1, obtained from measurements made for the thin sample.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A log–log plot of the dependence of $(-d{D}_{\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ constructed for the measured optical density ${D}_{\mathrm{m}}\left(\lambda \right)$ of the glass G2 sample with the thickness ${h}_{2}=10.00\mathrm{mm}$ (red curve). The green straight line presents the linear least squares approximation of the curve in the spectral range of $360-500\mathrm{nm}$.

**Figure 2.**Log–log plots of the dependences of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (curves) constructed for the measured optical densities ${D}_{i\mathrm{m}}\left(\lambda \right)(i=1,2,3)$ of three samples of the glass-ceramic GC1. The thicknesses of samples are: ${h}_{1}=0.27\mathrm{mm}$ (green curve), ${h}_{2}=0.77\mathrm{mm}$ (cyan curve), and ${h}_{3}=3.02\mathrm{mm}$ (orange curve). The portion of each curve limited by arrows is approximated by a dashed straight line using the least squares method, and the slope ${S}_{\mathrm{lp}}$ of the straight line is indicated in the figure.

**Figure 3.**Log–log plots of the dependences of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (curves) constructed for the measured optical densities ${D}_{i\mathrm{m}}\left(\lambda \right)(i=1,2)$ of two samples of the glass-ceramic GC3. The thicknesses of the samples are: ${h}_{1}=2.11\mathrm{mm}$ (magenta curve) and ${h}_{2}=3.02\mathrm{mm}$ (red curve). The portion of each curve limited by arrows is approximated by a dashed straight line using the least squares method, and the slope ${S}_{\mathrm{lp}}$ of the straight line is indicated in the figure.

**Figure 4.**(

**a**): Log–log plots of the measured optical densities for two samples of the MAS-GC with thicknesses indicated in the figure. (

**b**): A log–log plot of wavelength dependence of the extinction coefficient $\alpha \left(\lambda \right)$ determined by Equation (5) on the basis of the data presented in Figure 4a (the first method).

**Figure 5.**Log–log plots of the dependences of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (curves) constructed for the measured optical densities ${D}_{i\mathrm{m}}\left(\lambda \right)(i=1,2)$ of two samples of the MAS glass-ceramic studied in the present work. The thicknesses of the samples are: ${h}_{1}=0.54\mathrm{mm}$ (magenta curve) and ${h}_{2}=3.06\mathrm{mm}$ (red curve). The portion of each curve limited by arrows is approximated by a straight dashed line using the least squares method, and the slope ${S}_{\mathrm{lp}}$ of the straight line is indicated in the figure.

**Figure 6.**Log–log plots of the dependences of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (red curves) constructed for the measured optical densities ${D}_{\mathrm{m}}\left(\lambda \right)$ of the samples of the glass-ceramics GC20 (

**a**) and GC40 (

**b**). The samples have the same thickness $h=2.00\mathrm{mm}$. The portion of each curve limited by arrows is approximated by blue straight line using the least squares method, and the slope ${S}_{\mathrm{lp}}$ of the straight line is indicated in the figure.

**Figure 7.**A log–log plot of the dependence of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (blue curve) constructed for the measured optical densities ${D}_{\mathrm{m}}\left(\lambda \right)$ of sample of the glass-ceramic ZnOEr-GC. The sample thickness $h=1.06\mathrm{mm}$. The portion of the curve limited by arrows is approximated by a red straight line using the least squares method, and the slope ${S}_{\mathrm{lp}}$ of the straight line is indicated in the figure.

**Figure 8.**The measured optical densities ${D}_{\mathrm{ig},\mathrm{m}}\left(\lambda \right)$ of the sample of the initial glass (with thickness $h=3.00\mathrm{mm}$) (black curve) and ${D}_{\mathrm{htg},\mathrm{m}}\left(\lambda \right)$ of the sample of the ZnOYb-GC (with thickness $h=3.02\mathrm{mm}$) (blue curve), see Equation (7), and their difference $\u2206D\left(\lambda \right)$ (red curve) (Equation (8)) (log–log plot). The slope of the linear portion of $\u2206D\left(\lambda \right)$ curve was determined by the least squares method in the spectral range shown by arrows, and is indicated in the figure.

**Figure 9.**A log–log plot of the dependence of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (red curve) constructed for the measured optical density ${D}_{\mathrm{m}}\left(\lambda \right)$ of sample of the glass-ceramic ZnOYb-GC with the thickness $h=3.02\mathrm{mm}$. The portion of the curve limited by arrows is approximated by a black dashed straight line using the least squares method, and the slope ${S}_{\mathrm{lp}}$ of the straight line is indicated in the figure.

**Figure 10.**Log–log plots of the internal optical density $D\left(\lambda \right)$ (red curves) for samples of the glass-ceramics GC20 (

**a**) and GC40 (

**b**). The samples have the same thickness $h=2.00\mathrm{mm}$. The portion of each curve limited by arrows is approximated by red dashed straight line using the least squares method, and the slope of the straight line is indicated in the figure.

**Figure 11.**Log–log plots of the dependences of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (red curves) constructed for the measured optical densities ${D}_{\mathrm{m}}\left(\lambda \right)$ of samples of the glass-ceramics GC20 (

**a**) and GC40 (

**b**). Notice that the spectral range is wider than in Figure 6 for the same glass-ceramics; the wavelength step $\Delta \lambda =20\mathrm{nm}$ is greater than the step $\Delta \lambda =1\mathrm{nm}$ used to draw the curves in Figure 6.

**Figure 12.**A log–log plot of the dependence of $(-d{D}_{i\mathrm{m}}(\lambda )/d\lambda )$ on $\lambda $ (green curve) constructed for the measured optical density ${D}_{\mathrm{m}}\left(\lambda \right)$ of sample of the glass-ceramic ZnOYb-GC with the thickness $h=3.02\mathrm{mm}$. The portion of the curve limited by arrows is approximated by blue dashed straight line using the least squares method, and the slope ${S}_{\mathrm{lp}}$ of the straight line is indicated in the figure. The wavelength step $\Delta \lambda =5\mathrm{nm}$ was used.

**Figure 13.**Optical densities of reflection losses, ${D}_{\mathrm{refl}}\left(\lambda \right)$, calculated for the glass-ceramics GC20, GC40 and MAS-GC (log–log plots). For the glass-ceramics GC20 and GC40, Equation (6) and analytical expressions for refractive index $n\left(\lambda \right)$ were used. In the case of the glass-ceramic MAS-GC, calculation was carried out by Equation (34) using the measured optical densities of two samples.

**Figure 14.**The ratio $[d{D}_{\mathrm{refl}}\left(\lambda \right)/d\lambda ]/\left[d{D}_{\mathrm{m}}\right(\lambda )/d\lambda ]$ for the glass-ceramics GC20 and GC40. The wavelength step is $\Delta \lambda =20\mathrm{nm}$.

**Figure 15.**The plots of $\left[{\mathrm{log}}_{10}\right(-d{D}_{i\mathrm{m}}\left(\lambda \right)/d\lambda \left)\right]-\left[{\mathrm{log}}_{10}\left(\lambda \right)\right]$ (curves 1 and 2) and ${[\mathrm{log}}_{10}(-d{D}_{i}(\lambda )/d\lambda )]-{[\mathrm{log}}_{10}\left(\lambda \right)]$ (curves 1′ and 2′) for the thin ($i=1$, curves 1 and 1′) and thick ($i=2$, curves 2 and 2′) samples of MAS-GC (Section 2.2.4).

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**MDPI and ACS Style**

Shepilov, M.; Dymshits, O.; Zhilin, A.
A Simple Method for Estimation of the Scattering Exponent of Nanostructured Glasses. *Materials* **2023**, *16*, 2630.
https://doi.org/10.3390/ma16072630

**AMA Style**

Shepilov M, Dymshits O, Zhilin A.
A Simple Method for Estimation of the Scattering Exponent of Nanostructured Glasses. *Materials*. 2023; 16(7):2630.
https://doi.org/10.3390/ma16072630

**Chicago/Turabian Style**

Shepilov, Michael, Olga Dymshits, and Aleksandr Zhilin.
2023. "A Simple Method for Estimation of the Scattering Exponent of Nanostructured Glasses" *Materials* 16, no. 7: 2630.
https://doi.org/10.3390/ma16072630