Journal of Minerals & Materials Characterization & Engineering, Vol. 11, No.3, pp.267283, 2012 jmmce.org Printed in the USA. All rights reserved 267 Elastic Prope rties of Clin opyroxene Ba sed Glasses along D iop side (CaMg Si2O6)Jadeite (NaAlSi2O6) Join Rinkel Jindal, a, b,* Widiya Jatmiko, b Indra Vir Singh, c R. Jayaganthan, a a Department of Metallurgical and Materials Engineering & Centre of Nanotechnology, Indian Institute of Technology Roorkee, Roorkee 247667, India b Department of Glass and Ceramic Composites, Institute of Mineral Engineering, RWTH Aachen, Mauerstr. 5, 52064 Aachen, Germany c Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India * Corresponding Author: rinkeljindal@gmail.com (Rinkel Jindal) ABSTRACT The elastic properties of glasses along Diopside (CaMgSi2O6)Jadeite (NaAlSi2O6) join (Dix  Jd1x where x=2 0 , 40, 60, 80, 100 mole %), were obtained by the ultrasonic echography technique, at room temperature. The correlation of elastic moduli with the atomic packing density of these glasses w as discussed. The derived experimental values of Young’s modulus, bulk modulus, shear modulus and Poisson’s ratio for in vestigated glasses were compared with those theoretically calculated values in terms of the Makishima–Mackenzie model and the modified model presented by Rocherulle. Keywor d s: Glass, Diopside, Mechanical properties.
268 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 1. INTRODUCTION Glass is cha racterized as brittle and easily broken, unlike metals and plastics, when subjected to thermal or mechanical stresses. The thermal shock resistance and fracture toughness of glasses are very important properties, because they estimate the resistance to these stresses. These parameters are directly related to Young’s modulus, which, in turn, is influenced predominantly by the glass chemical composition thus, the estimation of the Young’s modulus based on glass composition is very useful for the development of glass materials [1, 2]. Moreover, the strength of materials increases with their elastic moduli; it is therefore possible to assess strength indirectly from their elastic properties. Studies of the elastic moduli of the glassy materials give considerable information about the structure of noncrystalline solids, since they are directly related to the interatomic forces and potentials [311]. It is useful to predict the elastic properties of polycomponent oxide systems solely from knowledge of the system composition, density and well known tabulated physical properties. Makishima and Mackenzie [3, 4] correlated the elastic moduli of oxide glasses to both packing density and the average strength of chemical bonds in the glass. Rocherulle [8] extended the analysis of Makishima and Mackenzie [3, 4] to oxynitride glasses. They introduced a thermodynamic factor, which results from the substitution of oxygen by nitrogen within the vitreous network. Their results showed that the calculated values of elastic moduli are in good agreem en t with the experimental values. The purpose of the present work is to calculate theoretically the elastic moduli and Poisson's ratio from the chemical composition and density data on the basis of Makishima and Mackenzie's [3, 4] model, Rocherulle [8] model and to compare with t he experimental values of the investigated glasses. Furthermore, a correlation between the predicted and experimental values of elastic moduli and Poisson’s ratio is stu died to verify the applicability of these models for the studied glass system. The solid solution between diopside (CaMgSi2O6; hereafter referred as Di) and jadeite (NaAlSi2O6; hereafter referred as Jd) is a subject of relevance from petrologic as well as
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 269 technological point of view. AboMosallam et al. [12] studied the structure and crystallization behavior of glasses and glasscerami cs by MASNMR along (CaMgSi 2O6)1x  (NaAlSi2O6) x  (Ca5 (PO4)3F) y where 0 ≤ x ≤ 30 mole % and y = 7 mole %. He has reported that with increasing Jd content in the glasses, the polymerization in silicate glass network shifts from Q2 to Q3 (Qn: degree of polymerization; n: number of bridging oxygen’s) and Al exists predominantly as Al (IV) species. The glass compositions under investigation have been designed along diopside (CaMgSi2O6)jadeite (NaAlSi 2O6) join (Dix  Jd1x where x =20, 40, 60, 80, 100 mole %) with varying diopside/jadeite molar ratio as shown in the Table 1. Th e pa rtial s ubstituti on of B2O3 for SiO2 has been made in all compositions in accordance with substitution scheme 0.3 Si4+↔ 0.4 B3+ so as to decrease the melting point of the glass batch. 2. EXPERIMENTAL High purity chemical powders SiO 2 (Sigma Aldrich, German y, purity >99.7%), CaCO 3 (Merck, Germany >99.8%), Al2O3 (Merck, Germany, ≥98%), H3BO3 (Merck, Germany, 99.8%), MgO (Merck, Germany >99.7%), Na2CO 3 (Merck, Germany, 99.9%) were used for glass melting. For each glass composition as shown in Table 1 the batch of 200 grams was taken and thoroughly mixed by using an agate ball mill. The mixed powder was taken in a platinum crucible and heated in the electric furnace to 1500 °C for one hour and one hour dwell time to obtain bubble free and homogeneous glass. Glasses in the bulk form were produced by pouring the melts on the preheated graphite moulds followed by annealing at 550 °C for one hour. The amorphous nature of glasses was confirmed by powder Xray diffraction (XRD) (Philips PW 3710). The density of glass was determined by Archimedes’ method in which the sample was weight both in air and immersed in liquid. The liquid used in the present study for density measurement was Ethylene Glycol of known density (1.1132 g/cm3). The accuracy of the measurement was about ±0.002 g/cm3.
270 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 Table 1 Batch composition of glasses Glass MgO CaO Na2O SiO2 B2O3 Al2O3 Di0 wt.%   15.65 51.58 7.03 25.74 NaAlSi1.7B0.4O6 mole %   17.27 58.57 6.89 17.27 Di20 wt.% 4.01 5.58 12.34 50.84 6.93 20.3 Ca0.2Mg0.2Na 0.8Si1.7Al0.8B0.4O6 mole % 6.42 6.42 12.92 54.90 6.42 12.92 Di40 wt.% 7.91 11.01 9.12 50.12 6.83 15.01 Ca0.4Mg0.4Na 0.6Si1.7Al0.6B0.4O6 mole % 12.11 12.11 9.09 51.44 6.06 9.09 Di60 wt.% 11.7 16.28 6.0 49.42 6.74 9.87 Ca0.6Mg0.6Na 0.4Si1.7Al0.4B0.4O6 mole % 17.12 17.12 5.73 48.57 5.73 5.73 Di80 wt.% 15.39 21.41 2.96 48.74 6.65 4.87 Ca0.8Mg0.8Na 0.2Si1.7Al0.2B0.4O6 mole % 21.60 21.60 2.72 45.93 5.43 2.72 Di100 wt.% 18.97 26.40  48.08 6.55  CaMgSi1.7B0.4O6 mole % 25.65 25.65  43.57 5.13  The elastic constants such as Young’s modulus (E), shear Modulus (G), bulk modulus (K) and Poison's ratio (σ) of glasses were determined b y ult rasonic echograph y at room t emperature. Fo r this purpose, velocities of longitudinal (10 MHz) and transverse (4 MHz) ultrasonic waves in the investigated glass specimens were measured using piezoelectric transducers and associated electronics (ultrasonic flaw detector USD15, Krautkramer GmbH & Co., Huerth, German y). The overall uncertainty in calculated value is estimated to be ±2% due to several influential effects, such as multiple internal reflections within the transducer, sample thickness and the acoustic impedance mismatch between glass sample and transducer.
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 271 3. THEORY Makishima and Mackenzie [3, 4] presented a theoretical model to calculate the elastic modulus of oxide glasses in terms of chemical composition, packing density and dissociation energy of the oxide constituents. They derived the following relations: Young’s Modulus, Bulk Modulus, Shear Modulus, Poisson’s Ratio where Vt is the packing density of the glass sample which is calculated by using the equation: where M is effective molecular weight, ρ is the density, Xi is the molar fraction of component i and Vi is a packing factor obtained from the following equation for an oxide AXOY: where RA and RO are the respective ionic radius of metal and oxygen (In the present study, Pauling’s ionic radii are used). The dissociation energy per unit volume (Gi), the effective molecul ar weight (M) and packing densit y (Vi) of each oxide component present in the Diopside (CaMgSi2O6)jadeite (NaAlSi2O6) are given in Table 2. Rocherulle [8] introduced some
272 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 modifications in the expression of the packing factor and expressed it for an oxide AmOn as:where ρi and Mi are the density and molecular weight of ith oxide component respectively. The elastic moduli and Poisson's ratio of a multicomponent glass are given by Rocherulle [8] as: Young’s Modulus, Bulk Modulus, Shear Modulus, Poisson’s Ratio, Packin g Facto r The values of packing factor (Ci) of various oxides used in Diopside (CaMgSi2O6)jadeite (NaAlSi2O6) system are given in Table 2. Table 2 Effective molecular weight (M), Dissociation energy per unit volume (Gi), Packing density (Vi) and Packing factor (Ci) 2 2 2 3 2 3 M (g/mol) 56.077 40.304 61.979 60.084 69.62 101.961 i (Kcal/cm 3) 15.50 20.00 8.90 15.40 18.60 32.00 Vi (cm3) 9.4 7.6 11.2 14.0 20.8 21.4
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 273 Ci 0.5530 0.6750 0.4102 0.6174 0.7619 0.8333 4. RESULTS The investigated glass compositions were readily castable after 1 h of soaking time at 1500 ºC resulting in homogeneous and transparent glass. The amorphous natures of the quenched glasses were confirmed by XRD analysis as shown in Fig. 1. On the contrary Di100 crystallized immediately after pouring the melt on the graphite mould as expected (due to direct contact to air and it prone to cracking during annealing so no ex periment was done on this sample) but further increase in Jd/Di ratio in glasses led to the formation of stable, transparent and monolithic bulk glasses. Figure 1: Xray diffractograms of glasspowder In an amorphous solid such as glass, the elastic strain produced by a small stress can be described by two independent elastic constants, C11 and C44 [13]. The Cauchy relation 2C44 = C11  C12 allows to determine C12, and for pure longitudinal waves and for pure transverse waves where Vl and Vt are the longitudinal and transverse velocities,
274 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 respectively. The sound velocities also allow the determination of Young’s modulus (E), bulk modulus (K), shear modulus (S) and Poisson’s ratio ( σ ) by the following equations: The Density, longitudinal velocity (Vl) and transverse (Vt) sound velocities of diopside (CaMgSi2O6) – jadeite (NaAlSi2O6) glasses are given in Table 3. Table 3 Experimental values of Density (ρ), Longitudinal (Vl) and Transverse velocity (Vt) (g/cm3) Velocity Vl (m/s) Vt (m/s) Di0 2.462 5776.580 3519.403 Di20 2.510 5933.704 3488.286 Di40 2.607 6205.345 3599.386 Di60 2.655 6277.778 3604.466 Di80 2.766 6560.816 3730.021 Table 4 gives the calculated elastic constants (C11, C44 and C12), Young’s Modulus (E), Bulk Modulus (K) and Poisson ratio ( σ ) from experimental sound velocities for diopside (CaMgSi2O6)jadeite (NaAl Si2O6) glasses. The overall u ncertain ty for ab ove calcu lated v alue is estimated to be ±2%. The expression of d = 4C44/ K, which was derived by Bergman and Kantor [14] for an inhomogeneous random mixture of fluid and a solid backbone near the percolation limit, gives
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 275 an interesting information on effective dimensionality of the materials [15, 16]. Bogue and Sladek [15] called this new par ameter ‘‘d’’ the fractal bond connectivity, where d = 3 for three dimensional tetrahedral coordination polyhedra, d = 2 for twodimensional layer structures, and d = 1 for one dimensional chain s, respe cti vel y. The cal culated dval ues fo r inv estigated gl asses are given in Table 4. Table 4 Experimental calculated values of elastic constants (C11, C44 and C12), Young’s Modulus (E), Bulk Modulus (K), Poisson ratio ( ) and fractal bond connectivity (d) of investigated glasses Glass C11 (GPa) C44 (GPa) C12 (GPa) E (GPa) K (GPa) σ d = 4C44/K Di0 82.118 30.470 21.178 73.423 41.461 0.229 2.940 Di20 88.223 30.493 27.247 75.376 47.576 0.243 2.564 Di40 100.309 33.749 32.811 84.135 55.310 0.248 2.441 Di60 104.595 34.480 35.635 86.486 58.620 0.252 2.353 Di80 119.941 38.623 42.696 97.422 67.994 0.256 2.272 5. DISCUSSION The calculated values of longitudinal and transverse elastic constants (C11, C44), Young’s modulus (E) and bulk modulus (K) decreases as the mole % of Jd increases as shown in the Table 4. This implies that glass containing more percentage of Di have a rigid structure in the investigated glasses. The values of Young’s modulus are increasing as mole % of Di increases because Di contains Mg+2 and Ca+2 ions so as we the increase Di, the Mg+2 and Ca+2 ions increases and the substitution of lowvalency ions by highvalency ions enhance elastic moduli because the internal energy is proportional to the effective charge of cations and anions. Thus, the glasses containing an alkaline earth show high elastic modulus [17].
276 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 Both interatomic energies (U0) and atomic packing densities have been taken into account to interpret the elasticity data. Here the substitution of Na+ and Al 3+ by Ca2+ and Mg2+ as the mole % of Di increases and U0 Mg (7.646), U0 Ca (6.113) is greater than U0 Na (5.139), U0 Al (5.986) so modulus of elasticity increases [18]. Simultaneously the atomic packing density decreases with increase of Jd as shown in Table 5 so the elastic modulus decreases [18]. Poisson’s Ratio decreases as J d i ncreases be cau se as we substitutes the Na+ and Al3+ by Ca2+ and Mg2+ as the mole % of Di increases so the amount of alkali content decreases and packing density is also increasing as shown in Table 5 so Poi sson’s ratio increases [18]. In that glas ses as the amount of aluminum increases Al coordination changes from 6 (small Al quantities) to 4 (Al is network forming). As the network former Al decreas es t he atom ic p acki ng densi t y, where as in comparison packing density is enhanced b y sixfold network modifying Al atoms. Consequentl y, Poisson’s ratio exhibits a slight increase at low Al contents, However when we increases the mole % of Jd the n etw or k co nn ecti vi ty is increase d [12] so the poison ratio decrease, because the higher connectivity of network decrease the Poisson’s ratio. The fractal bond connectivity data, which shows the dvalue of these, glasses around 2.1 to 2.7 as shown in Table 3 implies that as we increase the Di the connectivity of the structure decreases. Similar results have also been reported on diopsidejadeitefluorapatite glasses by AboMosallam et al. [12]. Table 5 Theoretical calculated Packing density (Vt), Young’s modulus (Ecal), Shear Modulus (Scal), Bulk Modulus (Kcal) and Poisson’s ratio ( cal) of investigated glasses from the model of Makishima and Mackenzie [3, 4] Glass Vt Ecal ( GPa) Scal (GP a) Kcal (GPa) σ cal
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 277 For the studied glasses, Table 5 gives th e th eo ret i c ally calculat ed p a cki n g dens ity (Vt), the elastic moduli (Ecal, Scal and Bcal) and the Poisson’s ratio based on the model of Makishima and Mackenz ie. The theoretical values for the packing density (Ct), Young’s modulus, shear modulus, bulk modulus, and Poisson ratio based on the Rocherulle model for the studied glass samples are given in Table 6. Table 6 Theoretical calculated Packing factor (Ct), Young’s modulus (Ecal*), Shear Modulus (Scal*), Bulk Modulus (Kcal*) and Poisson’s ratio ( cal*) of investigated glasses from the model of Rocherulle [8 ] Glass Ct Ecal* (GPa) Scal* (GPa) Kcal* (GPa) σ cal* The cor relation b etween t he experiment al values of Young’s modulus and those calculated from the theory of Makishima and Mackenzie [3, 4] is shown in Fig. 2 (a). This fi gure shows th at the cal culat ed values ar e less than the observed values and the correlat ion is not satisfactory and Fig. 2 (b) shows the correlation between the observed and theoretically calculated values of Young's modulus on the basis of the Rocherulle [8] model. This figure clearly shows that this model is also not good in predicting most of the observed values of Young's modulus. The calculated values are more than that of experimental values while same
278 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 behavior is observed with the bulk modulus, experimental values are not in satisfactory agreement with the calculated values from both of the model as shown in Fig. 3 (a) and (b). 020406080100 120 140 0 20 40 60 80 100 120 140 Eexp (GPa) Ecal (GPa) (a) 020406080100120140 0 20 40 60 80 100 120 140 Eexp(GPa) Ecal* (GPa) (b)
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 279 Figure 2: Agreement between observed and theoretical calculated values of Young’s modulus in the present study according to (a) Makishima and Mackenzie’s [3, 4] model; (b) Rocherulle [8] model
280 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 020406080100 120 140 0 20 40 60 80 100 120 140 Bexp (GPa) Bcal (GPa) (a) 020406080100 120 140 0 20 40 60 80 100 120 140 Bexp (GPa) Bcal * (GPa) (b) Figure 3: Agreement between observed and theoretical calculated values of Bulk modulus in the present study according to (a) Makishima and Mackenzie’s [3, 4] model; (b) Rocherulle [8] model
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 281 020406080100 120 140 0 20 40 60 80 100 120 140 Sexp (GPa) Scal (GPa) (a) 020406080100 120 140 0 20 40 60 80 100 120 140 Sexp (GPa) Scal* (GPa) (b) Figure 4: Agreement between observed and theoretical calculated values of Shear modulus in the present study according to (a) Makishima and Mackenzie’s [3, 4] model; (b) Rocherulle [8] mo del
282 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 The disagreement between the experimental and the theoretically calculated results are in the range of 0.2–18% and 0.3–20% for Young’s modulus and for bulk modulus, respectively by Makishima and Mackenzie's theory and this disagreement between the experimental and the theoretically calculated results are in the range of 220% and 540% for Young’s modulus and for bulk modulus, respectively by Rocherulle model. The calculated values of Poisson's ratio from the theory of Makishim a and Mackenzie [3, 4] and Rocherulle [8] are not satisfactory with the experimental values. Fig. 4 (a) and (b) shows the agreemen t between the observed and theoret ically cal culated v alues of shear modulus from the Makishima and Mackenzie model [3, 4] and the Rocherulle [8] model, respectively. Fig. 4 (a) shows that the calculated values from the Makishima and Mackenz ie model [3, 4] is in satisfacto ry agreement (between 88 and 98% i n Fig. 4 (a)) with the experimental values, So this model is valid for the studied glass system taking into account the uncertai nt y of exper iment al data. Fig. 4 (b) shows t hat the calcu lated valu es from the Rocherulle [8] model are not in the satisfactory agreement (between 80 and 95% in Fig. 4 (b)) with the experimental values. 6. CONCLUSION Elastic moduli and Poisson’s ratio decreases with the increase of Jd content in glasses along diopside (CaMgSi2O6)jadeite (NaAlSi2O6) join. Comparison of theoretical and experimental values of elastic moduli and Poisson's ratio of the diopside (CaMgSi2O6)jadeite (NaAlSi2O6) glass system leads to the conclusions that: 1. The correlation between the observed and calculated values of Young’s modulus from Makishima and Mackenzie [3, 4] model as well as Rocherulle [8] model is not satisfa ctory . 2. The satisfactory agreement betw een the obser ved and theor eticall y calcul ated values of s hear modulus is valid only for Makishima and Mackenzie [3, 4] model and values from Rocherulle [8] model are not satisfactory. 3. The calculated values of bulk modulus and Poisson's ratio from the theory of Makishima and Mackenzie [3, 4] and R ocherulle [8] are not satisfactory.
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 283 REFERENCES [1] D.P.H. Hasselman, Thermal stress resistance parameters for brittle refractory ceramics: A compendium, Am. Ceram. Soc. Bull. 49 (1970) 1033–37. [2] K. Hirao, M. Yoshimoto, N. Soga, and K. Tanaka, Densification of magnesium and calcium metaphosphate glasses, J. NonCryst. Solids 130 (1991) 78–84. [3] A. Makishima and J. D. Mackenzie, Direct calculation of Young’s modulus of glass, J. NonCryst. Solids 13 (1973) 35–45. [4] A. Makishima and J. D. Mackenzie, Calculation of Bulk modulus, Shear modulus and Poisson's ratio of glass, J. Non Crystalline Solids 17 (1975) 14757. [5] A. Makishima and J. D. Mackenzie, Calculation of Bulk modulus, Shear modulus and Poisson's ratio of glass, J. Non Crystalline Solids 17 (1975) 14757. [6] A. Makishima, Y. Tamura, T. Sakaino, Elastic moduli and refractive indices of aluminosilicate glasses containing Y2O3, La2O3 and TiO2, J. Am. Ceram. Soc. 61 (1978) 247249. [7] B. Bridge, N.D. Patel, D.N. Waters, On the elastic constants and structure of the pure inorganic oxide glass es, Phys. St at. Sol. 77 (1983) 655668. [8] J. Rocherulle, C. Ecolivet, M. Poulain, P. Verdier, Y. Laurent, Elastic moduli of oxynitride glasses: Extension of Makishima and Mackenzie's theory, J. NonCryst. Solids 108 (1989) 187193. [9] A. Elshafie, M aterials science communication room temperature ultrasonic wave velocity and attenuation in Ge10Se80−xSbxTe10 bulk glassy samples, Mater. Chem. Phys. 51 (1997) 182185. [10] A. Abd ElMoneim, I.M. Yousssof, M.M. Shoaib, Elastic moduli prediction and correlation in SiO2based glasses, Mat er. Chem. Phys. 52 (1998) 258262. [11] R. E lMallawany, Tellurite glasses Part 1. Elastic properties, Mater. Chem. Phys. 53 (1998) 93120. [12] H.A. AboMosallam, R.G. Hill, N. Karpukhina, R.V. Law, MASNMR studies of glasses and glassceramics based on clinopyroxenefluorapatite system, J. Mater. Chem. 20 (2010) 790797.
284 Rinkel Jindal, Widiya Jatmiko, Indra Vir Singh, Vol.11, No.3 [13] J. Schroeder, in: M. Tomozawa, R.H. Doremus (Eds.), Treatise on Material Science and Technology, vol. 12, Academic Press, New York (1977) 157–222. [14] D.J. Bergman, Y. Kantor, Critical properties of an elastic fractal, Phys. Rev. Lett. 53 (1984) 511514. [15] R. Bogue, R.J. Sladek, Elasticity and thermal expansivity of (AgI)x(AgPo3)1x glasses, Phy. Rev. B 42 (1990) 52805288. [16] G.A. Saunders, T. Brennan, M. Acet, M. Cankurtaran, H.B. Senin, H.A.A. Sidek, M. Federico, Elastic and nonlinear acoustic properties and thermal expansion of cerium metapho sp hat e glass es, J. NonCryst. Solids 282 (2001) 291305. [17] N. Soga, Elastic moduli and fracture toughness of glass, J. Noncryst. Solids 73 (1985) 305313. [18] T. Rouxel, Elastic Properties and Shortto MediumRange Order in Glasses, J. Am. Ceram. Soc. 90 (10) (2007) 3019–3039.
