Investigations of the Elastic Moduli of Er2O3NPs NPs Doped TeO2 − B2O3 − SiO2 Glasses using Theoretical Models

Elastic moduli of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y glasses with y = 0.01, 0.02, 0.03, 0.04, 0.05 were studied in this work using the theoretical elastic models. The Makishima & Mackenzie, Rocherulle and bond compression models were employed for the study. In the Makishima and Mackenzie model, the packing density was calculated from the bulk glass molar weight and the bulk glass density whereas in Rocherulle model it is determined as the individual oxides. Young, shear and bulk moduli as well as the Poisson ratio were calculated for the glasses in the Makishima and Rocherulle models, while longitudinal, was calculated in addition to young, bulk and shear moduli using the bond compression model. Bond per unit volume number (nb), bulk modulus, bulk modulus ratio (Kbc/Ke), atomic ring size (`) and stretching force constantwere also calculated and presented. The values of the Young, bulk and shear moduli obtained from Makishima model increased from 52.854 to 55.335 GPa, 35.754 to 39.862GPa and 21.080 to 21.809GPa respectively with Er2O3NPs composition increase from 1% to 5%..The Rocherulle model presented increasing values for Young, bulk and shear moduli as 56.910 to 58.432GPa, 41.452 to 44.450 GPa and 22.385 to 22.809 GPa respectively with Er2O3NPs composition increase from 1% to 5%. The bond compression model presented much higher values of the elastic moduli compared to the experimentally obtained values and showed an increasing trend as the Er2O3NPs concentration increases. In the glass network, the atomic ring size value decreased from 0.5698 to 0.5091 nm indicating an increase in the close packing of atoms. Based on the elastic moduli values presented by all the models, Makishima and Mackenzie model presented a more reliable data and hence represents the best model for the studied glass system. DOI:10.46481/jnsps.2022.222


Introduction
Tellurite based glasses doped with rare earth ions have been used in amorphous silicon solar cells for the improvement of the cell efficiency [1]. These glasses are developed widely in various optical applications in glass sensors, optoelectronics, fibre optics and light emitting diode (LED) [2,3,4,5]. In most cases glasses requiring mechanical strength for special applications are fabricated in combination with SiO 2 as silica combination is known to provide thermal stability, mechanical strength and chemical stability [6,7,8,9]. Silica incorporation in glasses improves their optical transparency in both lasing and excitation wavelengths [10]. B 2 O 3 is considered an excellent raw material in combination with other TeO 2 based glasses for quality improvement in terms of hardness, optical transparency, easy glass fabrication (as it lowers the glass's melting temperature) and rare earth ion solubility [11].
Various researchers in the area of glass science, technology and engineering have been engaged in the study of different glass properties for applications in the areas of communications, optical, laser and other [12,13,14]. Glass's application in any technology is affected by either its mechanical strength, optical behaviours, elastic properties, thermal stability or chemical stability [15,16].Because of mechanical strength importance in optical disc, optical fibres medical and dental implants, electronic displays and radiation shielding applications, SiO 2 are used as substrates to provide such mechanical strengths [17,18,19].
Elastic moduli of glasses of different compositions have been studied by various researchers over the years. Makishima and Mackenzie developed a model for the calculation of the elastic moduli of glasses theoretically, in consideration to its importance in the manufacture of strong optical fibres [20,21]. Rocherulle et al. [22] proposed a model that improved the Makishima and Mackenzie model through the extension of the dissociation energy and packing factors. The bond compression model is based on the consideration of the oxides' atomic geometry which include their coordination number as well bond length [23].
In this work, the elastic moduli of the Er 2 O 3 doped TeO 2 -B 2 O 3 -SiO 2 glasses were determined using Mackishima and Mackenzie, Rocherulle and bond compression models. The study of the elastic moduli is important for the definition of its applicability in optical fibre, laser and other optoelectronic applications. The objectives of the study are as follows; 1. To determine the elastic moduli of the glasses under study using theoretical models. 2. To compare the values of the elastic moduli obtained from each model with the others.
This work is important considering the importance of the elastic moduli to the application in different areas of technology and the fact that there is no such work carried out on the studied glasses composition.

Theoretical Models
This section presents the four theoretical models adopted in this work for the study of the elastic properties of the Er 2 O 3 and Er 2 O 3 NPs doped rice husk silicate borotellurite glass systems. The models used include the Makishima-Mackenzie model, Rocherulle model, bond compression model and the ring deformation model.

Makishima and Mackenzie model
The Makishima and Mackenzie model proposed a theoretical approach to determine the elastic moduli of oxide glasses with consideration to to the chemical composition (x i ) of the constituting oxide, individual oxides' packing densities (V i ), and their corresponding dissociation energies (G i ) [24]. The glass Young modulus is expressed in terms of the packing density (V t ), and the dissociation energy (G t ) as; From the oxides' packing density, V t can be determined as by where M = glass molecular weight, ρ = glass density, x i =i th component's molar fraction (i), and V i is calculated for an oxide (A x O y ) as: where R O and R A respectively represent the ionic radii of oxygen and cation respectively [23]. According to Makishima and Mackenzie, bulk modulus (K m ), Shear modulus (G m )and Poisson ratio (σ m ) for oxide glasses on any component are calculated as follows:

Rocherulle Model
A modified expressions of the Makishima and and Mackenzie model was proposed by Rocherulle at al. (1989) [22]. The packing density, V i in the Makishima-Mackenzie model is replaced with C i which is expressed as follows: For glasses of polycomponent nature, the C t factor is therefore expressed as follows: The Young Modulus (E r ), bulk modulus (K r ), shear modulus (G r ) and the Poisson ratio are calculated as in equations (1), (4), (5), and (6) respectively. The basic difference between the two models is that the Makishima and Mackenzie model take into consideration the bulk density and molecular weight of the glass, while the Rocherulle model considers the individual oxides' density and molecular weights [22].

Bond Compression and Ring Deformation Models
The theoretical model of bond compression takes into consideration the atomic networking in a material and the bond stretching force constants between them to theoretically estimate the elastic characteristics of the material [25]. For single oxide glass systems, the bulk modulus is obtained as; The expression for a multicomponent oxide glasses is given as: where F = the average stretching force constant, and n b = the bond number per unit volume and r is the cation-anion bond length. The bond number per unit volume is calculated as: n f is the number of bonds per unit glass formula, ρ is the glass density,N A is Avogadro's number and M is the glass molecular weight [26]. The stretching force constant ( f ) is deduced for multi-component glass using the expression as reported by [27] as: The calculations of Poisson's ratio (σ bc ) and the corresponding average cross link density (n c ) of the glasses are deduced as in equations (13) and (14) respectively.
where n c = oxide (i) cross-link number per cation and N c = cation number per unit glass formula. The total number of cations per unit glass formula for a multicomponent glass system (η) is obtained as: K bc and σ bc are used to calculate the Young, shear and longitudinal moduli. The ring deformation model theoretically estimates the atomic ring size in the glass network structure. The model uses the experimental bulk modulus (K e ) values and the bending force constant (F b ) values to estimate the atomic ring size [28]. In the approximation process, the average stretching force constant is used in place of F b .
The following formula is used in the determination of the atomic ring size: The value l represents the atomic ring size and is defined as the diameter of the external ring. The ring perimeter can be determined using l as bond number x bond length divide by π [29].

Makashima Model
Using the Makishima and Mackenzie model, the results of the theoretical elastic properties data obtained is presented in this section. Figure 1 illustrates the packing density and dissociation energy variation of the {[(TeO 2 ) 0.7 (B 2 O 3 ) 0.3 ] 0.8 (SiO 2 ) 0.2 } 1−y (Er 2 O 3 NPs) y glass system with increasing concentration of Er 2 O 3 NPs. The packing density value increased from 0.5637 to 0.6003 cm −3 . The value increases maybe associated to the increment in the density of the material. This increment is connected to increase in the material compactness and rigidity. It can also be said to be due to increase in the conversion of TeO 3 to TeO 4 structural form [30].
The dissociation energy decreased from 46.878 to 46.088 kJ/cm 3 with the increased Er 2 O 3 concentration from 1% to 5%. The decrease in the value maybe due to the introduction of more low dissociation energy Er 2 O 3 into the system. It may also relate to the observed increase in the molar volume of the glasses [31]. Figure 2 and Table 1 presents the elastic moduli of the glass  Table 1 increased from 0.2536 to 0.2686 with the increase in the Er 2 O 3 concentration from 1% to 5%. The Poisson ratio increases normally with increase in the crosslink density of a material [33]. According Makishima and Mackenzie, Poisson ratio of a silicate glass should be around 0.25 when the work done after glass deformation involves the compression or stretching of the Si-O-Si links [21]. concentration was increased from 1% to 5%. The increase in the packing density might be due to the decrease in the interatomic spaces resulting from the formation of more bridging oxygens (BOs) with the formation of more TeO 4 structural units in the glass network [34]. This can also be due to the formation of more BO 3 units and decreasing concentration of BO 2 O nature containing one (1) non-bridging oxygen (NBO) and two (2) bridging oxygen atoms attacked to the boron atom [35].   The Young and shear moduli increased from 56.9097 to 58.4324 GPa and 22.3445 to 22.8090 GPa, respectively with an increase in the Er 2 O 3 molar fraction from 0.01 to 0.05. The decrease in the values may be associated with decrease in the dissociation energy which overrides the observed increase in the packing density. The bulk modulus was found to have increased from 41.4525 to 44.4498GPa. The bulk modulus increase might be due to the increase observed in the packing density of the glasses. The bulk modulus dependence to the packing density is more than its dependence on the dissociation energy [22]. This leads to the overriding effect of the packing density on the bulk modulus value than the dissociation energyon the bulk modulus value [36]. The Poisson ratio value for the glass system of {[(TeO 2 ) 0.7 (B 2 O 3 ) 0.3 ] 0.8 (SiO 2 ) 0.2 } 1−y (Er 2 O 3 NPs) y composition as shown in Table 2 increased from 0.2712 to 0.2809 as the Er 2 O 3 concentration increased from 1% to 5%. The Poisson ratio is a rigidity dependent parameter, as such increase in the Poisson ratio maybe due to an increase in the rigidity of the glasses. This is due to the closure of the interstices between atoms in the glass network This allows large structural relaxation with the introduction of the sound waves, which makes the lateral strain to grow larger compared to the longitudinal strain [31]. The increase can also be attributed to the decrease in the iconicity caused by the introduction of more polar Er 3+ ions in the network [37].

Bond Compression Model and Ring Deformation Model
The results of the elastic moduli and the Poisson ratio calculated using the bond compression model for the{[(TeO 2 ) 0.7 (B 2 O 3 ) 0.3 ] 0.8 (SiO 2 ) 0.2 } 1−y (Er 2 O 3 NPs) y glassesare presented in this section. Also discussed in this section is the ring deformation model results for the glass system of {[(TeO 2 ) 0.7 (B 2 O 3 ) 0.3 ] 0.8 (SiO 2 ) 0.2 } 1−y (Er 2 O 3 NPs) y composition. Figure 4 and Table 3 [31]. It can also be attributed with observed increase in cross-link density [36] as shown in Table  3. The decrease in the value maybe also associated with the increase in the glasses' molar volume resulting from increased interstitial spacing between constituting glass atoms [34].
The value of the average stretching force constant for the studied glasses decreased from 362.5248 to 349.8995 Nm −1 as the Er 2 O 3 NPs concentration increased from 1% to 5%. The decrease in the first order average stretching force constant might be associated with the substitution of oxides of higher stretching force constant; SiO 2 [25], B 2 O 3 [38] and TeO 2 [39] with Er 2 O 3 [14,15] having much lower stretching force constant value.
The elastic moduli for glass system of {[(TeO 2 ) 0.7 (B 2 O 3 ) 0.3 ] 0.8 (SiO 2 ) 0.2 } 1−y (Er 2 O 3 NPs) y composition is presented in Figure 5 and Table 4. The values of the bulk, shear, longitudinal and Young moduli ranged from 79.56544 to 81.35993 GPa, 53.76034 to 55.97828 GPa, 151.2532 to 155.9976GPa and 131.636 to 1236.6052GPa respectively with increase in the Er 2 O 3 concentration from 1% to 5%. Generally, Young modulus is related to the bond stretching force constant whereas the shear modulus is related to the bond bending force constants of the constituting oxides of a glass [37].The decrease in the values of the moduli maybe due the introduction of Er 2 O 3 with larger cation-anion length (r= 0.225 nm) compared to the substituted TeO 2 (with r = 199 nm), B 2 O 3 (r=0.138 nm) and SiO 2 (r= 161) [23,25,40]. The decrease reflects the dependence of the elastic moduli on the average stretching force constant which decrease due the introduction of more Er 2 O 3 lower stretching force constant [41]. Figure 6 presents the variation of the atomic ring size of the glass systemof {[(TeO 2 ) 0.7 (B 2 O 3 ) 0.3 ] 0.8 (SiO 2 ) 0.2 } 1−y (Er 2 O 3 NPs) y composition with Er 2 O 3 molar fraction. The atomic ring size value decreased from 0.5996 to 0.5786 nm when the Er 2 O 3 concentration was increased from 1% to 5%. The decreased in the value maybe attributed to the increase in density, crosslink density and rigidity of the glasses [37]. This implies the close packing of the material in the glass network [42].       values of the elastic moduli from the elastic models showed an increasing trend just as obtained from the experimental data but demonstrating comparatively very high values from the bond compression model. Data obtained from the Rocherulle model showed a very much closer relation to the data obtained from the experimental assessment compared to the bond compression model. The increase in the elastic moduli is generally associated with increased rigidity and network connectivity [43,44].

Conclusions
Makishima and Mackenzie, Rocherulle and bond compression models were used in this work for the theoretical determination of the elastic moduli of Er 3+ ions doped silica borotellurite glass system with empirical formula {[(TeO 2 ) 0.7 (B 2 O 3 ) 0.3 ] 0.8 (SiO 2 ) 0.2 } 1−y (Er 2 O 3 NPs) y with y = 0.01, 0.02, 0.03, 0.04, 0.05. The elastic moduli (Young, shear and bulk) increased with increasing concentration of Er 2 O 3 in the Makishima and Mackenzie model. The values of the elastic moduli calculated presented a decreasing value of Young and shear elastic moduli with increasing pattern of the corresponding bulk modulus value as the Er 2 O 3 concentration was increased. The bond compression model presented rather much higher values of the elastic moduli compared to the other two models with an increasing value variation against the Er 2 O 3 molar concentration. This might be due to the fact that, unlike the Makishima & Mackenzie and Rocherulle models which commonly considers densities and dissociation energies, the bond compression model considers the bond strength and numbers. Among the three models considered in this work, the Makishima and Mackenzie model presented a more reasonable values of the elastic moduli and thus can be regarded as the best model for the studied glass system.