J. Nig. Soc. Phys. Sci. 4 (2022) 9–15 Journal of the Nigerian Society of Physical Sciences Investigations of the Elastic Moduli of Er2O3NPs NPs Doped TeO2 − B2O3 − SiO2 Glasses using Theoretical Models U. S. Aliyua,∗, I. G. Geidamb, M. S. Ottoa, M. Hussainic a Department of Physics, Faculty of Science, Federal University Lafia, Nasarawa Sate, Nigeria bDepartment of Physics, Faculty of Sciences, Yobe State University Damaturu, Nigeria cDepartment of Physics, Faculty of Sciences, Umar Suleiman College of Education Gashua, Yobe State, Nigeria Abstract 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 Keywords: Erbium oxide, Tellurite Glass, Elastic Moduli, Poisson Ratio, Theoretical Models Article History : Received: 05 May 2021 Received in revised form: 04 December 2021 Accepted for publication: 05 December 2021 Published: 28 February 2022 c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: B. J. Falaye 1. 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 vari- ous optical applications in glass sensors, optoelectronics, fibre ∗Corresponding author tel. no: +2348039364898 Email address: usaltilde@yahoo.com (U. S. Aliyu ) 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 SiO2as silica combination is known to provide thermal stability, mechanical strength and chemical stability [6, 7, 8, 9]. Silica incorporation in glasses im- proves their optical transparency in both lasing and excitation wavelengths [10]. B2O3 is considered an excellent raw mate- rial in combination with other TeO2 based glasses for quality 9 Aliyu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 9–15 10 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, technol- ogy and engineering have been engaged in the study of differ- ent glass properties for applications in the areas of communi- cations, optical, laser and other [12, 13, 14]. Glass’s appli- cation 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 applica- tions, SiO2 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 Mak- ishima and Mackenzie model through the extension of the dis- sociation energy and packing factors. The bond compression model is based on the consideration of the oxides’ atomic ge- ometry which include their coordination number as well bond length [23]. In this work, the elastic moduli of the Er2O3 doped TeO2 – B2O3 – SiO2 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. 2. Theoretical Models This section presents the four theoretical models adopted in this work for the study of the elastic properties of the Er2O3 and Er2O3 NPs doped rice husk silicate borotellurite glass systems. The models used include the Makishima-Mackenzie model, Rocherulle model, bond compression model and the ring de- formation model. 2.1. Makishima and Mackenzie model The Makishima and Mackenzie model proposed a theoret- ical approach to determine the elastic moduli of oxide glasses with consideration to to the chemical composition (xi) of the constituting oxide, individual oxides’ packing densities (Vi), and their corresponding dissociation energies (Gi) [24]. The glass Young modulus is expressed in terms of the packing den- sity (V t), and the dissociation energy (Gt) as; Em = 2Vt ∑ i Gi xi = 2VtGt (1) From the oxides’ packing density, Vtcan be determined as by Vt = ( ρ M ) ∑ i Vi xi (2) where M = glass molecular weight, ρ = glass density, xi =ithcomponent’s molar fraction (i), and V i is calculated for an oxide (Ax Oy) as: Vi = NA (4π/3) ( xR3A + yR 3 O ) , (3) where RO and RA respectively represent the ionic radii of oxy- gen and cation respectively [23]. According to Makishima and Mackenzie, bulk modulus (Km), Shear modulus (Gm)and Pois- son ratio (σm) for oxide glasses on any component are calcu- lated as follows: Km = 1.2Vt E (4) Gm = (3EK/9K − E) (5) σm = (E/2Gm − 1) (6) 2.2. Rocherulle Model A modified expressions of the Makishima and and Macken- zie model was proposed by Rocherulle at al. (1989) [22]. The packing density, Vi in the Makishima-Mackenzie model is re- placed with Ci which is expressed as follows: Ci = NA (4πρ/3M) ( xR3A + yR 3 O ) (7) For glasses of polycomponent nature, the Ct factor is therefore expressed as follows: Ct = ∑ i Ci xi (8) Ci = ∑ i ρi Mi Vi xi (9) The Young Modulus (Er ), bulk modulus (Kr ), shear modulus (Gr ) 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 ox- ides’ density and molecular weights [22]. 10 Aliyu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 9–15 11 2.3. Bond Compression and Ring Deformation Models The theoretical model of bond compression takes into con- sideration the atomic networking in a material and the bond stretching force constants between them to theoretically esti- mate the elastic characteristics of the material [25]. For single oxide glass systems, the bulk modulus is obtained as; Kbc= nb Fr2 9 (10) The expression for a multicomponent oxide glasses is given as: Kbc = (ρNA/9M) ∑ i (xn f Fr 2)i, (11) where F = the average stretching force constant, and nb= the bond number per unit volume and r is the cation-anion bond length. The bond number per unit volume is calculated as: nb = ( n f ρNA/M ) (12) n f is the number of bonds per unit glass formula, ρ is the glass density,NA 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: F= ∑ i ( xn f f ) i∑ i ( xn f ) i (13) The calculations of Poisson’s ratio (σbc) and the corresponding average cross link density (nc) of the glasses are deduced as in equations (13) and (14) respectively. σbc = 0.28(nc) −0.25 (14) nc = 1 η ∑ i (xnc Nc)i, (15) where nc= oxide (i) cross-link number per cation and Nc= cation number per unit glass formula. The total number of cations per unit glass formula for a mul- ticomponent glass system (η) is obtained as: η = ∑ i (xNc)i (16) Kbc and σbc are used to calculate the Young, shear and longi- tudinal moduli. The ring deformation model theoretically esti- mates the atomic ring size in the glass network structure. The model uses the experimental bulk modulus (Ke) values and the bending force constant (Fb) values to estimate the atomic ring size [28]. In the approximation process, the average stretching force constant is used in place of Fb. The following formula is used in the determination of the atomic ring size: Ke = 0.0106Fb(l) −3.84 (17) 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]. 3. Results and Discussions 4. 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 {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y glass system with increasing concentration of Er2O3NPs. 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 con- nected to increase in the material compactness and rigidity. It can also be said to be due to increase in the conversion of TeO3 to TeO4 structural form [30]. The dissociation energy decreased from 46.878 to 46.088 kJ/cm3 with the increased Er2O3 concentration from 1% to 5%. The decrease in the value maybe due to the introduction of more low dissociation energy Er2O3 into the system. It may also re- late to the observed increase in the molar volume of the glasses [31]. Figure 2 and Table 1 presents the elastic moduli of the glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y composition. The Young, bulk and shear moduli increased from 52.8537 to 55.3347 GPa, 35.7544 to 39.8619 GPa and 21.0804 to 21.8087 GPa respectively. The increase in the elastic moduli increase with increase in the Er2O3 concentration may be due to increase in material compactness associated with increase in the TeO4 units’ concentration. The elastic moduli increase is generally associated with rigidity increase [15, 32]. The Poisson ratio value for the {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y presented in Table 1 increased from 0.2536 to 0.2686 with the increase in the Er2O3 concentration from 1% to 5%. The Poisson ratio increases normally with increase in the crosslink density of a material [33]. Accord- ing Makishima and Mackenzie, Poisson ratio of a silicate glass should be around 0.25 when the work done after glass deforma- tion involves the compression or stretching of the Si-O-Si links [21]. 5. Rocherulle’s Model Table 2 presents the packing density, elastic moduli, and the Poisson ratio values for the glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)ycomposition. The packing density values increased from 0.6008 to 0.6094 when the Er2O3 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 TeO4 struc- tural units in the glass network [34]. This can also be due to the formation of more BO3 units and decreasing concentration of BO2O nature containing one (1) non-bridging oxygen (NBO) and two (2) bridging oxygen atoms attacked to the boron atom [35]. 11 Aliyu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 9–15 12 Table 1: Packing Density (Vt ), Dissociation Energy (Gt ), Elastic Moduli (Em, Km and Gm) and Poisson Ratio (σm) for glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)ycomposition. y (mol) Vt( cm3 mol−1 ) Gt (KJ/cm3) Em (GPa) Km (GPa) Gm (GPa) σm 0.01 0.5637 46.8784 52.8538 35.7544 21.0804 0.2536 0.02 0.5710 46.6809 53.3064 36.5233 21.2081 0.2567 0.03 0.5755 46.4833 53.4977 36.9424 21.2521 0.2586 0.04 0.5879 46.2857 54.4262 38.3990 21.5333 0.2638 0.05 0.6003 46.0882 55.3347 39.8619 21.8087 0.2686 Figure 1: Packing density and dissociation energy variation with the molar frac- tion of Er2O3NPs in {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y glass system Figure 2: The variation of the elastic moduli with molar fraction of Er2O3 in {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y glass system 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 Er2O3 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 in- creased from 41.4525 to 44.4498GPa. The bulk modulus in- crease 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 Table 2: Packing density (Ct ), elastic moduli (ER, KR, and GR), Poisson ratio (σR) for glass systemof {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)ycomposition. y (mol) Ct( cm3 mol−1 ) ER (GPa) KR (GPa) GR (GPa) σR 0.01 0.6070 56.910 41.452 22.385 0.2712 0.02 0.6137 57.298 42.198 22.493 0.2737 0.03 0.6205 57.682 42.947 22.600 0.2762 0.04 0.6272 58.060 43.697 22.705 0.2786 0.05 0.6339 58.432 44.450 22.809 0.2809 Figure 3: Elastic moduli variation with molar fraction of Er2O3NPs for glass systemof {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y composition. energy [22]. This leads to the overriding effect of the packing density on the bulk modulus value than the dissociation ener- gyon the bulk modulus value [36]. The Poisson ratio value for the glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)ycomposition as shown in Table 2 increased from 0.2712 to 0.2809 as the Er2O3concentration increased from 1% to 5%. The Poisson ra- tio 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 relax- ation 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 Er3+ 12 Aliyu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 9–15 13 ions in the network [37]. 6. Bond Compression Model and Ring Deformation Model The results of the elastic moduli and the Poisson ratio cal- culated using the bond compression model for the{[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)yglassesare presented in this section. Also discussed in this section is the ring de- formation model results for the glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y composition. Figure 4 and Table 3 presents the variation of the bond per unit volume number and the average stretching force constant with molar fraction of Er2O3 composition for the glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y compo- sition. The bond number per unit volume increased, decreased and then increased with the increase in the Er2O3 concentra- tion. The increase in the value might be due to the increase in the number of TeO4 and BO3 structures with decreasing num- ber of TeO3 and BO2O structures, indicating an increase in the number of bridging oxygen [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 Er2O3 NPs concentration increased from 1% to 5%. The de- crease in the first order average stretching force constant might be associated with the substitution of oxides of higher stretch- ing force constant; SiO2 [25], B2O3 [38] and TeO2 [39] with Er2O3 [14, 15] having much lower stretching force constant value. The elastic moduli for glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y composition is pre- sented 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 Er2O3 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 Er2O3 with larger cation-anion length (r= 0.225 nm) compared to the substituted TeO2 (with r = 199 nm), B2O3 (r=0.138 nm) and SiO2 (r= 161) [23, 25, 40]. The decrease reflects the depen- dence of the elastic moduli on the average stretching force con- stant which decrease due the introduction of more Er2O3 lower stretching force constant [41]. Figure 6 presents the variation of the atomic ring size of the glass systemof {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)ycomposition with Er2O3 molar fraction. The atomic ring size value decreased from 0.5996 to 0.5786 nm when the Er2O3 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]. Figure 4: Bond per unit volume and stretching force constant variation with molar fraction of Er2O3NPs for glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)ycomposition. Figure 5: Elastic moduli variation with molar fraction of Er2O3 for glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y composition. Figure 6: Atomic ring size variation with molar fraction of Er2O3 for glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y composition Table 5 presents the elastic moduli (Young, bulk and shear moduli) obtained from experiment and theoretical models of Makishima & Mackenzie, Rocherulle and Bond compression. The data from Makishima & Mackenzie model presented a much closer elastic moduli values to the experimental values compare to the Rocherulle and bond compression models. The 13 Aliyu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 9–15 14 Table 3: Bond per unit volume number (nb), bulk modulus, bulk modulus ratio (Kbc/Ke), atomic ring size (`) and stretching force constant (F) for {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y glass system. y nb (×1028 m−3) Kbc (GPa) Ke (GPa) Kbc/Ke l (nm) F (N/m) nc 0.01 8.4014 79.5728 30.6560 2.5957 0.5828 362.525 2.4289 0.02 8.4360 79.7584 31.1124 2.5636 0.5792 359.322 2.4704 0.03 8.4304 79.5654 33.3348 2.3869 0.5680 357.029 2.5112 0.04 8.5419 80.4765 41.1798 1.9543 0.5360 353.009 2.5516 0.05 8.6507 81.3599 43.1171 1.8870 0.5284 349.900 2.6161 Table 4: Elastic moduli and Poisson ratio for glass system of {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 NPs)y composition. y Kbc (GPa) Gbc(GPa) Lbc(GPa) Ebc(GPa) σbc 0.01 79.5738 53.7603 151.253 131.636 0.22429 0.02 79.7584 54.1123 151.908 132.396 0.22334 0.03 79.5654 54.2003 151.833 132.512 0.22243 0.04 80.4765 55.0357 153.858 134.457 0.22154 0.05 81.3599 55.9783 155.998 136.605 0.22016 Table 5: Comparative presentation of Bulk, Young and Shear moduli obtained from non-destructive ultrasonic spectroscopic analysis (experimental) and Makishima & Mackenzie, Rocherulle and Bond compression models y Bulk Modulus Young Modulus Shear Modulus Ke Km Kr Kbc Ee Em Er Ebc Ge Gm Gr Gbc 0.01 30.656 35.754 41.452 79.573 31.801 52.854 56.910 131.64 20.546 21.080 22.385 53.760 0.02 31.112 36.523 42.198 79.758 33.951 53.306 57.298 132.40 21.685 21.208 22.493 54.112 0.03 33.335 36.942 42.947 79.565 34.770 53.498 57.682 132.51 22.436 21.252 22.600 54.200 0.04 41.180 38.399 43.697 80.477 35.412 54.426 58.060 134.46 23.841 21.533 22.705 55.036 0.05 43.117 39.862 44.450 81.360 37.720 55.335 58.432 136.61 25.302 21.809 22.809 55.978 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 com- pression model. The increase in the elastic moduli is gener- ally associated with increased rigidity and network connectivity [43, 44]. 7. Conclusions Makishima and Mackenzie, Rocherulle and bond compres- sion models were used in this work for the theoretical de- termination of the elastic moduli of Er3+ ions doped silica borotellurite glass system with empirical formula {[(TeO2)0.7 (B2O3)0.3]0.8 (SiO2)0.2}1−y (Er2O3 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 Er2O3 in the Mak- ishima and Mackenzie model. The values of the elastic mod- uli calculated presented a decreasing value of Young and shear elastic moduli with increasing pattern of the corresponding bulk modulus value as the Er2O3 concentration was increased. 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