Journal of Minerals & Materials Characterization & Engineering, Vol. 11, No.3, pp.267-283, 2012
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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 Nano-technology, 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 -
Jd1-x 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 non-crystalline solids, since they are directly
related to the interatomic forces and potentials [3-11].
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. Abo-Mosallam et al. [12] studied the structure and crystallization
behavior of glasses and glass-cerami cs by MAS-NMR along (CaMgSi 2O6)1-x - (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 - Jd1-x 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 X-ray 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
Di-0 wt.% - - 15.65 51.58 7.03 25.74
NaAlSi1.7B0.4O6 mole % - - 17.27 58.57 6.89 17.27
Di-20 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
Di-40 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
Di-60 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
Di-80 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
Di-100 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)
CaO
MgO
Na
2
O
SiO
2
B
2
O
3
Al
2
O
3
M (g/mol) 56.077 40.304 61.979 60.084 69.62 101.961
G
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 Di-100 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:- X-ray diffractograms of glass-powder
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)
Longitudinal
Velocity Vl (m/s)
Vt (m/s)
Di-0 2.462 5776.580 3519.403
Di-20 2.510 5933.704 3488.286
Di-40 2.607 6205.345 3599.386
Di-60 2.655 6277.778 3604.466
Di-80 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 two-dimensional layer structures, and d
= 1 for one dimensional chain s, respe cti vel y. The cal culated d-val 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 (
σ
) and fractal bond connectivity (d) of
investigated glasses
Glass C11
(GPa) C44
(GPa) C12
(GPa) E (GPa) K (GPa)
σ
d = 4C44/K
Di-0 82.118 30.470 21.178 73.423 41.461 0.229 2.940
Di-20 88.223 30.493 27.247 75.376 47.576 0.243 2.564
Di-40 100.309 33.749 32.811 84.135 55.310 0.248 2.441
Di-60 104.595 34.480 35.635 86.486 58.620 0.252 2.353
Di-80 119.941 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 low-valency ions by high-valency 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 d-value 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 diopside-jadeite-fluorapatite glasses by Abo-Mosallam
et al. [12].
Table 5 Theoretical calculated Packing density (Vt), Young’s modulus (Ecal), Shear
Modulus (Scal), Bulk Modulus (Kcal) and Poisson’s ratio (
σ
cal) of investigated glasses from
the model of Makishima and Mackenzie [3, 4]
Glass Vt Ecal ( GPa) Scal (GP a) Kcal (GPa)
σ
cal
Di-0
0.523
75.925
32.874
47.498
0.427
Di-20
0.525
75.543
32.680
47.441
0.427
Di-40
0.539
76.900
33.070
49.580
0.425
Vol.11, No .3 Elastic Properties of Clinopyroxene Based Glasses 277
Di-60
0.547
77.589
33.258
50.767
0.424
Di-80
0.569
80.210
34.093
54.593
0.411
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 (
σ
cal*) of investigated glasses
from the model of Rocherulle [8 ]
Glass Ct Ecal* (GPa) Scal* (GPa) Kcal* (GPa)
σ
cal*
Di-0
0.62886
91.292
38.050
68.672
0.41266
Di-20
0.62608
90.088
37.579
67.476
0.41304
Di-40
0.62308
88.896
37.115
66.255
0.41346
Di-60
0.62159
88.169
37.256
65.556
0.41367
Di-80
0.61962
87.346
36.506
64.738
0.41394
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 2-20% and 5-40% 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
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