Tellurite glasses Part 1. Elastic properties

Tellurite glasses Part 1. Elastic properties

ELSEVIER IMaterials Chemistry and Physics 53 ( 1998) 93-120 Review Tellurite glasses Part 1. Elastic properties Raouf El-Mallawany Phpsicb Departme...

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IMaterials Chemistry and Physics 53 ( 1998) 93-120


Tellurite glasses Part 1. Elastic properties Raouf El-Mallawany Phpsicb Department,


of Sciemz



* Shebwn



Received 21 January 1997; received in revised form 14 October 1997; accepted 14 October 1997

Abstract Tellurite glasses (pure, binary, ternary and quaternary systems containing transition metal or rare earth oxides) are now a new type of noncrystalline solids with a lot of applications over a wide range of compositions, temperatures and frequencies. Elastic properties provide much information about the structures of solids and they are directly related to the interatomic potentials. Glasses are isotropic and have only two independent elastic constant, Cl 1 and C43. These two parameters have been collected from the longitudinal and shear sound velocities and density of the glass. The rest of the elastic constants (bulk, Young’s modulus, Poisson’s ratio) could be deduced. The hydrostatic and uniaxial pressure dependences of ultrasonic waves in these glasses at room temperature have been collected. The data proved both second and third order elastic constants of the glass and in consequence the shear and the longitudinal acoustic mode Gruneisen parameters have both been collected. The estimated bulk modulus and Poisson’s ratio have been calculated using the bond compression model according to the cationanion bond of each oxide present in the glass. Information about the structure of the glass can be deduced after calculating the number of network bonds per unit volume, the value of the average stretching force constant, the average ring size, the structure sensitivity factor and the mean cross-link density. Comparisons between the calculated end experimental elastic moduli and Poisson’s ratio have been carried out. Also, the role of halogen inside the glass network has been discussed. 0 1998 Elsevier Science S.A. All rights reserved Ke~~pords: Telluritr glasses; Elastic properties: Halogen

Contents I. 2. 3.


5. 6.

Introduction ............................................................................................................. 1.1. What are tellurite glasses’?...................................................................................... 1.2. Glass forming ranges of tellurite glasses and bonding nature in tellurite glasses .............................. Elastic properties of glass .............................................................................................. 2.1. Semi-empirical formulae for calculating elastic constants ..................................................... Elastic properties of TrO? glass under uniaxial and hydrostatic pressure.............................................. 3.1. Glass preparation ............................................................................................... 3.2. General physical characteristics of tellurite glass .............................................................. 3.3. Experimental techniques for measuring ultrasonic velocities and results ...................................... 3.4. Physical significance of the second order elastic constants (SOEC) .......................................... 3.5. Hydrostatic pressure derivatives of the second order elastic constants, third order elastic constants (TOEC) and vibrational anharmonicity .................................................................................. Elastic constants of binary transition metal tellurite glasses ........................................................... 4.1. Tr02-WOA andTeO,-ZnC1, glasses........................................................................... 4.7. TeO?-Moo, glasses ............................................................................................ 4.3. TeO,-ZnO glasses.............................................................................................. 4.4. TeO,-V,OS glasses ............................................................................................. Elastic constant5 of binary, ternary and quatemary rare earth tellurite glasses ........................................ Quantitative analysis of the elastic constants of rare earth tellurite glasses............................................

References. ................................................................................................................... * Fax: i-20-202-777-620; e-ma\\: sigsr-a\[email protected] 0X4-0584/98/$19.00 PUSO254-0584(97)02011-5

0 1998 Elsevier Science S.A. All rights reserved

94 94 94 95 97 98 98 98 98 99 100 103 103 106 107 109 112 114 118


R. El-Mulian~an~/Maierials


1. Introduction 1.1. What are teliwite glasses? Considerable international progress has been made in the discovery of new tellurite glasses, and in knowledge of the optical and physical properties, structural and bonding nature of these glasses during the 1990s [l-121] and late 1980s. Although the vibrational spectra of crystalline solids are now understood in essence, this is not the case for amorphous materials. The considerable theoretical difficulties experienced for amorphous solids are increased by a lack of precise experimental information on the vibrational spectra. To fill this gap, the pressure dependence of ultrasonic wave velocity, the second and third order elastic constants ( (SOEC) and (TOEC) ) of amorphous tellurite glasses have been measured. The SOEC are of central importance in any study of the vibrational properties of a solid because they determine the shape of the dispersion curves at the long wavelength limit, and their pressure dependences provide information on the shift of these vibrational energies with compression. The TOEC are of interest because they characterize the anharmanic properties. that is the non-linearity of the atomic forces with respect to atomic displacements. Tellurite glasses have been studied for over 150 years [ 1221 but it is only recently that TeO, glass ofpurites exceeding 98.5 mol% have been made [ 1231. Tellurite glasses are of technical interest on account of their low melting points and absence of hygroscopic properties, which limit the application of phosphate and borate glasses. Also, they have high densities and low transformation temperatures [ 124,125 ] , They have large refractive indices, and hence high dielectric constants, and are good IR transmitters for wavelengths up to 5 p.m. The first measurements of the static dielectric constant have been measured previously [ 1261 for pure tellurite glass and binary transition metal tellurite BTMT glasses. The static dielectric constant of vitreous TeO, at room temperature was 20.1, which was of the same order as the mean value for crystalline TeO, ( paratellurite) : the molarpolarizabilities of the glass (2.2 X 10W6 m3) andcrystalline (2.8 X 10m6 m3) forms are very similar. However, the pressure dependence of the static dielectric constant of pure tellurite and binary TMTG at an elevated pressure of O-70 kbdr and a temperature of 295-380 K were positive dependencies [ 1271. This was in contrast with the behavior of crystalline insulators. Using complex admittance techniques, the first measurements of electrical conductivity of TeO, and BTMT glasses have been measured [ 1281 throughout the temperature range 90-430 K. The conductivities were analyzed using the smail polaron theory to establish the thermal and disorder activation energies for the carriers in both glasses [ 791. Also, a comparison between the experimental and calculated elastic moduli has been carried out on pure and binary tellurite glasses[ 1091. The numberof binary systemsdependsupon the kind of glass former used. In phosphateor tellurite, the number of binary

and Phpics

53 (1998)


systemsis generally largerthan in borate, silicate,germanate, etc. It seemsthat the above difference dependsupon the electronegativity of the glass-formerions; at high temperaturesthe acidity of the oxide of the larger electronegative&E is stronger, and thus the ionic bond-forming range becomes wider. The linear and non-linearoptical propertieshavebeen measured by the third harmonic generation methods2 [5 1,86,92,93]. The largest2 obtainedforNb,O,-TeO,glass was50 times larger thanthat of purefusedsilica glasses[ 921. Because in recent years magnetic recording devices have becomesmallerin size and their memory density hasbecome greater, tellurite glasshas been used as a bonding glassin magnetic heads[ 1011. 1.2. Glassfuming rangesof tellurite glassesand bonding nature in tellurite glasses TeO, crystallizes in three different structures at atmospheric pressure:an orthorhombic structure with DZhissymmetry and two tetragonalstructuresof symmetry D,,15 (rutiie structure) and D,“. The form with DA4is paratellurite and undergesa secondorder pressure-inducedphasetransition at about 9 kbar (9 X lo* Pa) at room temperature [ 1291 to an orthorhombic structure of D2’ symmetry. Shearwaves propagating along a ( 110) crystal direction and polarized along a ( I 10) direction travel with anextremely smallphasevelocity, so that l/2 (C, ,-C,2) is small. Mode softening of thi’ shearwave drives this strain-inducedelastic transition from the D4’ to the D,’ phase.Previously it has been reported whether mode softening occurs in the glassy form of TeOz or not [ 1231. TeO, hasbeen regardedas an oxide that can form a vitreous network only when mixed with substantial amounts of other oxides. For example, in 1978 [ 1301 the glassformation range of a large number of binary tellurite glassescontaining transition oxides and lower limits for the addedoxides. varied from 2.5 mol% (for Fe, 0,) to 30 molg (for CuzO). The possibility of additional SiOZimpurity being imparted by the quartz crucible was not considered.Moreover, their datarepresentthe starting compositionsratherthan the analyzed values, and in view of the relatively low boiling point of TeO, comparedwith the other constituents,the probability is that the proportions of the modifier in theseglasses is somewhathigher than the quotedvalues. For thebereasons the glasspreparedasdescribedin Section 3 of this article, in which no deliberately addedmodifier was involved, is by far the purest specimenof vitreous TeO, yet reported. In 1962 tellurite glasseswere prepared I 13I] for Raman scattering and IR absorption by melting 99.5% purity telluna oxrde powder in an alumina crucible, but on analysisthe specimenwas found to contain up to 6 mol% A1203. A glassusedfor an X-ray study of vitreous TeOz [ 1321was found to contain 7.5 mol% of L&O after microanalysis.In 1957 [ 1321 it was reported that some10mol% of oxide modifier must be added to TeOz before there is any tendency of glassformation. The interpretation of the glassstructurein light of this result was

based on the assumption that in the cystalline form TeO, octahedra share three edges with their neighbouring ones so it is possible for glass formation. It is not sufficient simply for bond angles to be distorted, as in the formation of vitreous silica, for example, for this would conflict with the Zachariasen model of glass formation that only comers of structural groups can share. So the model [ 1321 proposed that group edges were broked in the glass and that for each broken edge oxygen must be supplied by the modifying oxide to complete the octets of the two cations involved. However, for all edges to be broken, in accordance with the Zacharisen model, the minimum mole ratio of Li,O to TeO, required would be 3/4 rather than the observed I / IO. So the model proposed that TeO, glass contains crystallites of about the size of the unit cell, which comprises eight TeO, units. Then about (3/4) X ( l/8) m 10 mole ratio of modifier oxides are required to maintain the broken edges between the crystallites. The starting point of the model [ 1321 argument is unsound. After that, in 1978 and 1979 [ 133-1351 a model of the tellurite glasses was illustrated by using neutron diffraction. In both the orthorhombic form (the mineral tellurite) and the tetragonal paratellurite there is fourfold coordiantion of the tellurim. the neighbouring atoms being arranged at four of the vertices of a trigonal bypyramid [ 136,137] (Fig. 1). In this coordination polyhedron there are two equatorial (Te-O,, = 1.9 A) and two axial (Te-O,, = 2.08 A) bonds. In paratellurite all the vertices of the TeO, groups are shared in a three-dimensional configuration of 4:2 coordination in which the oxygen bond angle is 140” [ 1331. Tellurite has a layer structure comprised of a Te,O, ring formed by comer sharing [ 1341. Thus the previous arguments presented for the need of a modifier to form shared comers in vitreous TeOz is no longer valid. After the atomic radial distribution functions for tellurite glasses have been determined from neutron diffraction [ 136,137], a number of possible structural models were dis-

- -:

Fig. 1. Basic coordination polyhedron in vitreous Te02 and in crystalline tellurite and paratellurite ( after Neov et al. [ 136.137 ] 1. The upper and lower atoms (large circles) are described as brin, 0 ‘axial’. the centre two as being ‘equatorial’. Tellurium is denoted by the smaller circle. In all cases each oxygen atom is bonded in the equatorial position to one tellurium atom and in the axial position to another. In paratellurite all polyhedron vertices are shared to form a three-dimensional structure of Te,O, rings.

cussed in some detail. However, in general terms, the results for TeOz-rich glasses are consistent with the structure being based upon trigonal bypyramidal TeOz units in comer sharing configurations. Therefore, the structure is an open one; a coordination number for the Te atoms being close to 4. As a result of comer sharing, the formation of the glassy form of TeO, is possible simply by a variation in the Te,,-O,,-Te angles. Nevertheless, following Refs. [ 1321 and [ 136,137], TeO, cannot form a glass by itself, the Te-0 bond being considered to be too strongly covalent to permit the requisite amount of distortion-an argument open to doubt. The production of vitreous TeOz has been assumed to require the introduction of a quantity of modifier which could effect randomization of the structure by helping to distort the Te,,O,,-Te bond and/or simply by the breaking up of the TeO, chains. This description is equivalent to a crystallite model of vitreous TeO, (although the authors did not describe it as such). In this respect the recent view of Refs. [ 136,137] is substantially the same as that of the earlier Ref. [ 1321, although in fact their structural work varies from Ref. [ 1321 suggestions. Figs. 1 and 2 illustrate the mechanism of structural recombination as reported in Refs. [ 136,137].

2. Elastic properties of glass Elastic properties differentiate a solid from a liquid. The application of a shearing force to solid is met with considerable resistance. The magnitude of the resulting deformation is proportional to the force and remains constant with time if the force is constant. The deformation is instantaneously recovered upon the removal of the force. On the contrary, liquids flow or deform at a constant rate over time if aconstant shear force is applied, or the shear force relaxes over time if a constant strain is applied. The terms ‘instantaneous’ and ‘deform over time’, and hence the terms ‘solid’ and ‘liquid’, need to be recognized in their proper perspectives. Elastic properties of glass are important because most glass products critically depend upon the solid-like behavior of glass. At temperatures well below the glass transition range, glass may be considered a linear elastic solid obeying Hooke’s law when the applied stress magnitudes are low relative to the fracture strength. This means that upon application of a stress (force per unit area). glass undergoes instantaneous deformation such that the ratio of the stress to the resulting strain (change in length per unit length) is a constant called the ‘modulus of elasticity’ (Pa = N m-‘) which is independent of the magnitude of the strain. An example of non-Hooken elastic behavior which is commonly observed is the stretching of a rubber band, where the modules of elasticity varies significantly with the deformation. Non-Hooken behavior such as the non-constancy of the elastic moduli in glass is often observed when the applied stresses are very high, perhaps about two-thirds of the fracture strength. This situation is reached, forinstance, when glass is loaded by an indenter in a microhardness test, which is a measure of the abrasion resistance. Because of the


Fig. 2. Diagram showing chains by a modifier.

the structural


model in teilurite

glasses: (a) ol-TeO,,

small area of contact, the applied load translates to very large local stresses that cause yielding and plastic deformation of the glass network. Other examples of plastic behavior, such as irreversible compaction under hydrostatic compression, and elastic-viscous (viscoelastic) behavior such as delayed elasticity, also exist in glass. The magnitude of such nonlinearity or non-elastic behavior generally increases with the magnitude of applied stresses and with the temperature approaching that of the glass transition range. The SOEC are of central importance in any study of vibrational properties of a solid because they determine the slope of the dispersion curves at the long-wavelength limit; their pressure dependences provide information on the shift of these vibrational energies with compression. Although the vibrational spectra of crystalline solids are now understood in essence, this is not the case for amorphous materials. To fill this gap at least in the long-wavelength limit, measurements were made of the

(b) TeO, chains,

(c) deformation

and breaking

of the TeOT

pressure and temperature dependences of ultrasonic wave velocities to determine the SOEC and TOW for the semiconducting tellurite glasses. The TOEC are of interest because they characterize the anharmonic properties, that is, the non-linearity of the atomic displacements. The higher order elasticity of glasses is known [ 138,139 J to fall into two quite distinct regimes. On the one hand, the non-linear elastic properties of glasses based on silica [ 140-1421 and BeFi [ 1411 are quite different from most materials in that the

pressure derivatives of the bulk and shear moduli are negative while their temperature derivatives are positive and the TOEC are anomalously positive. On the other hand, the elasticbehavior of many glasses including amorphous arsenic [ 1381, amorphous As,O, [ 1391, chalcogenide glasses [ 1435 and a flurozirconate glass [ 1441 resembles that of crystalline solids (those which do not exhibit some form of lattice instability which includes acoustic mode softening).

R. El-~Mullununy/

2.1. Sftni-errlpiricnlfij~ll~l~l~e cOnStcLnt.~

for calcdnting







The force betweenions is JU,li)i-, and so the stressg is a=( llr’){av,/ar)



but this isjust Ed6 where the strain de= drlr,. Thus E =da/de

-( llr;,)(~*U,lilr’)


=2ae’lr’ Eq. (5) showsthat the Young’s modulusof ionic crystals is inversely proportional to the fourth power of atomic spacings i;, and this relationship was confirmed by many ionic and covalent crystals. This relation has not been evaluated for glass.Eq. (5) can be written asfollows: dr”)(e’/r,)





where A4 is the effective molecular weight, p is the density, X, is the mole fraction of component I and V, is a packing factor obtained from the following equation for an oxide A,rO,.: V,=6.023x10’3{4z-/3(xR~+~R~)}

( 10)

where R, and R, are the respective ionic radiusof metal and oxygen (Pauling’s ionic radii [ 1471). Thus, the Young’s modulusof glassis theoretically given by [ 1481 V, I$ G,X,


This expressiongives E in units of kbar if units of Gi are in kcal cm-?. The simple division of the pure vitreous oxides into two distinct categoriesaccording to their valuesof Poisson’s ratio g, for P,05, BzOj and As203, az0.3; for GeOz and SiO,, ~~0.15, which suggestsa straightforward theoretical interpretation [ 1491,asfollows: accordingto the three hypothetical chain networks of Fig. 3 identical except for having cross-link densities(defined asthe numberof bridging bonds per cation, lesstwo) of 0, 1 and 2, respectively. Cross-link density also seemsto provide the key to an understandingof the radically different melting points and melt viscositiesof vitreous oxides (whose bond strengthsare relatively constant). (4)

Tensile Stress



Zi GiXi



Then the change of stressfor a change in r is doldr, and therefore da=(drlr’)(i,‘U,,,/i,r’}

53 (1998)

The packing density V, is defined by

In order to account for the many interactions between ions within a crystal, this is multiplied by the Madelung constant cy,giving the Madelung energy u,=ff



According to Gilman [ 145a,145b,145c], the Young’s modulus (E) of ionic crystals can be approximately derived asfollows. For apair of ionsof oppositesignswith the spacing ro, the electrostatic energy of attraction U is equal to U=-e211;,


= 0


[email protected]+-c-++


=2ff u,lr-i From Eq. (6 j and Eq. (2), the Young’s modulusistwo times the binding energy U, per cubic volume of r3. The single bond strength of oxides hasbeen determinedby Sun [ 1461 from the ratio of the dissociationenergy and the coordination number. If the A-O bond energy in one molecule of oxide A.,O,. is similar for the crystal and the glass,providing the coordination number is the same,then it is reasonableto apply the above treatement to oxide glasses.However, becauseof the disordered structure of the glass, it is difficult to adopt a meaningful Madelung constant as for crystalline oxides. In place of U, per cubic volume I’, we consider that a more appropriate binding energy of glass (U’,) is given by the production of the dissociation energy per unit volume (G) and the packing density of ions V,. For example, in a single one-componentglasssuch assilica E=2V,G For polycomponent glasses


Fig. 3. Diagram showing the variation strain/inorganic strain) with cross-link applied parallel to oriented chains.

of Poisson’s ratio [ 1491 (lateral density (n,) for tensile stresses

3. Elastic properties of TeO, glass under uniaxial and hydrostatic pressure 3.1. Glnss prepcrmtiof7 White crystalline tetragonal TeO, powder (BDH grade, 99%)) melting point733”C, was placed in an electric furnace, preheated to a temperature of SOO”C, and kept at this temperature for 30 min, after which complete fusion had taken place. The melt was then cooled at a rate of 20°C min - ’ to a temperature of 700°C and the melt, which had a high viscosity, was cut into a cube shaped split-mould of mild steel, which had been preheated to 400°C. The glass so produced, of dimensions 1.5 cm3, was annealed in a second furnace at 300°C for 1 h, after which time the furnace was switched off and the glass allowed to cool in situ for 24 h [ 1231. A large number of alternative thermal cycles were tried which faiIed to produce glass. For example, when TeO, melt was cast from a temperature of 800°C straight into a mould preheated to 400°C a white polycrystalline porcelain-like substance was obtained. through which no ultrasonic echoes would propagate. The same result was obtained by casting melts from various temperatures between 800°C and the melting of tetragonai TeO,. The interim stage of cooling from 800°C to about 700°C is evidently crucial to glass formation. We interpret this qualitatively as follows: to form a glass rather than a polycrystal the melt must be very viscous, which requires casting (i.e. supercooling) from as low a temperature as possible, consistent with achieving the necessary flow. However, the starting material must initially be heated to well above its melting point so that any residua1 crystallinity in the melt has been removed. The six glass faces were ground roughly on a lapping machine with 600 grade Sic powder. Then they were polished with 0.3 ,LLalumina powder using a precision polishing machine (Metals Research Multipol II). Opposite faces were finished optically flat and parallel to within 1 or 2 S or arc with the aid of a purpose builr apparatus by Metals Research (MK II polishing jig in conjunction with an autocollimator). As both the preparation and annealing furnaces hadcapacities greatly exceeding the volume of the crucible, the temperature gradients across the glass at any time during melting and annealing were negligible. Therefore, the glass was expected to be homogeneous, under uniaxial and hydrostatic pressure, as well as analysis of the pulse-echo decay patterns described in the following section. 3.2. General physical clzarcrcter-istics of tellurite glass The glass is semi-transparent with a very pale lime green color. An X-ray exposure using a Debye-Scherrer powder camera yielded the diffuse haloes typical of the vitreous state. Electron microscopy using the plastic replica technique presented no evidence of phase separation. Thermal analysis gave a transformation temperature of 320°C and a linear expansion coefficient (over the temperature range of O-

Table 1 Second order elastic constanta [ 1231

SOEC of TcOz glass at room temperature

Longitudinal ultrasonic wave velocity Shear ultrasonic wave velocity Second order elastic stiffness constants Adiabatic Adiabatic Adiabatic Density

bulk modulus Young’s modulus Poisson’s ratio

v V, C’, , C’I4 K’ E’ u\ P

3403 2007 59.1 20.6 31.7 50.7 0.233 --5105

ms-’ ms-’ GPa GPa GPa GPa kgmzZ

320°C) of 15.5 X lop6 “C-‘-a figure which we discuss in Sections 3 and 4. It is perhaps not well known that possibly the most rapid method of espablishing the vitreous nature of cast inorganic oxide melts is the use of ultrasonic wave prop: agation. NIegacycle frequency ultrasonic echoes cannot be obtained from polycrystalline castings, whereas echoes arereadily obtained from glasses. This result can be obtained immediately after grinding two roughly parallel faces, whereas the X-ray method requires lengthy exposure times, For this reason it was the production of successful ultrasonic propagation that gave the first indication that a high purity TeO, glass had been prepared. Furthermore, the degree oft homogeneity of glass samples can readily be estimated ultrasonically: if a velocity gradient is present in the wave propagation direction, interference maxima and minima will appear in the exponential decay envelope of successive pulse echoes, even when all other causes of such interference patterns have been removed. If the fractional velocity change over a distance of the order of diameter of the transducer is (VC/C), the envelope modulation is given by the function 211 (2 kn/z VCIC) /(2 knit VC’IC), where tzis the order of the echo, k is the wave number, CI is the transducer radius, and J is the first order Bessel function. Typically, with good bonds, we found the glass gave about 30 echoes above noise level (using conventional narrow band instrumentation), at 15 MHz, using a transducer of 5 mm radius. Thus, from the preceding equation ( taking the value of C from Table I ), the fractionai velocity change over a distance of 1 cm could not have exceeded 0.05% and might have been much less. ~~

The velocities of longitudinal V, and shear V, ultrasonic waves propagated through the TeO, were determined at room temperature by the pulse echo overlap technique [ 123 j. Ultrasonic pluses at a frequency of 10 MHz were generated and received by X- and Y-cut quartz transduceres. The meas:ured velocities and C, ,% ( =pV,‘) and CjA” ( =pVS”) are given in Table 1 together with the technical elastic moduli, the bulk modulus p { = p( 3V,‘-- 4V,‘) /3}, Young’s modulus E: { =pV’ (3V,‘-4V>‘)/( VIZ-vs2j}, Poisson’s ratio cTb( =(v,~-2v,~)/2(vI~-v,~)}.

The hydrostatic pressure dependences of the ultrasonic wave transit times (T,,) were measured in a piston-and-cylinderapparatus using castor oil as the liquid pressure medium. The hydrostatic pressure was found by measuring the change in electrical resistivity of a manganin wire coil within the pressure chamber. To account for pressure induced changes in crystal dimensions and density, the ‘natural velocity’ II/ (I,, T,,, where I,, is the path length at atmospheric pressure) technique [ 1501 was used, the experimental data for the change in ultrasonic wave transit time with pressure being converted to change of natural velocity { ( W/W,) - I }. The pressure dependenccs of the relative changes in natural velocity of the longitudinal and shear wave velocities were found to be linear up to the maximum pressure applied ( 1.4 k bar). The values of the present derivatives (d(pW) ldP},,=,, obtained from this data were 8.03 ( F 1..5)% and 1.46 ( + 1.5) % for longitudinal and shear modes, respectively. To obtain sufficient information to determine all the TOEC, the change in one of the ultrasonic wave velocities with uniaxial pressure must be measured in addition to the hydrostatic pressure data. Thus the sample was loaded under uniaxial composition in a screw press, the uniaxial stress being measured by a calibrated proving ring. Changes in ultrasonic longitudinal wave velocity were measured using an automatic frequency controlled, gated-carrier pulse superposition apparatus [ 15 11. The uniaxial pressure dependence was linear up to the maximum pressure applied (50 bar), the pressure derivative (d( pW) /dP],,,,I for the longitudinal mode being - 1.05 (&2)%. 3.4. Plzysical sig~~ijcurnce of the seco~ld order elastic COnStmltS (SOEC) Recently it has been pointed out [ 1491 that the pure vitreous inorganic oxides SiOl, GeOz, P,05, AslO and B,03 can be divided into two groups according to the magnitude of their cross-link densities and Poisson’s ratio. Thus SiO, and GeOz with two cross-links per cation have Poisson’s ratios of 0.15 and 0.19, respectively, while for P,Os, As,O, and B203, all having one cross-link per cation, the ratios are 0.29, 0.30 and 0.32, respectively. It will be observed that vitreous TeO, continues this pattern, with its cross-link densities of two per cation and Poisson’s ratio of 0.233, it belongs to the first group of oxides. It seems that bond bending distortions do not play a significant role in the elastic behavior of this material. A useful guide to the structure of simple covalently bonded glasses and compounds containing only one type of bond can be obtained by calculating the bulk modulus on the assumption that isotropic compression of the structure results in uniform reductions in bond lengths without any changes in bond angles [ 149,152]. The bulk modulus obtained on this bond compression model is given by Kbc=il,,



where PZ~is the number of network bonds per unit volume, r is the bond length and f is the first order stretching force

constant; higher order force constants and bond-bond interactions are neglected. For structures which approximate closely to microscopic isotropy (diamond for example) Eq. ( 12) gives a result of the right order-typically within about 30% of the experimental value K,. Usually, however, Kbc will be found to be greater than the experimental modulus (typically by a factor of 3 to 10) and the ratio K,,/K, forms a rough measure of the degree to which bond bending processes are involved in isotropic elastic deformation of the structure. It has been argued further that, at least for the pure inorganic oxides, f&,/K, will increase systematically with atomic ring size (i.e. the shortest closed circuits of network bonds), provided the ratio of bending to stretching force constants for the different glasses is the same. For TeO, we take the bond length as that (2.08 A) of Te-O,, [ 1331. The value of ~(2.1X10’NmW2 ) has been obtained by extrapolating a curve of known force constant versus bond lengths for a number of simple oxides [ 1521 and assuming fourfold coordination for tellurium, IZ~= 7.74 X IO’” rnp3. Substitution of thisdataintoEq.(12)givesK,,as7.15XlO’”Nm-’.Comparison of this with the experimental bulk modulus of 3.17 X 10” N m-* shows that the ratio K,,IK, is about 2.3, which is considerably smaller than that obtained [ 1491 for all the other pure inorganic oxide glass formers. This suggests that isotropic deformation results in much less bond bending in the TeO, network compared with (for example) SiO, (K,,IK, = 3) or GeO, (K,,IK, =4.4) so plausibly TeOz can be considered as a three-dimensional network comprised of (TeO)jz rings, where the average number of 212cations and anions in a ring is somewhat smaller than that in silica glass (in which n averages 6). Alternatively the result could indicate that the ratio of first order bending and stretching force constants for the Te-0 bond is substantially larger than the ratio occuring for the Si-0 and Ge-0 bonds. Similar conclusions can also be reached in the following manner. It has been proposed that the average ring sizes in simple oxide glasses can be related to bulk modulus by an empirical relation of the form [ 149,152] K,,=constant

F, II”


where 1is an ‘extremal ring diameter’ defined as a ring perimeter (i.e. number of bonds in ring X bond length) divided by r, Fh is a bending force constant, II is a high positive power (typically of the order of 4), and Krd is called the bulk modulus according to the ‘ring deformation model’. Assuming F, to be proportional to the stretching force constantf, the above equation was fitted to assumed values of K, f and 1 for the pure inorganic oxide glasses (SiOz, GeO,, As,O,, B203, P,O,) and for diamond, by means of linear regression which yielded [72] (with a correlation coefficient of 99%) Eq. (14) Cl491 K,d=0.0106fXI-3,X”



where 1 is in nm, and f is in N m- ‘. Applying this result to TeO, we find a ring diameter of 0.5 nm, which (round off to

the nearest whole number) corresponds to Te,Oj ( &atom) rings. This result suggests that TeO, glass is a disordered version of paratellurite which is indeed a three-dimensional network of Te,O, rings [ 1231. On the contrary, if the glass is to be envisaged as a disordered, three-dimensional version of tellurite with (on average) Te,O, rings, then the Te-0 bending force constants must be unusally strong (i.e. compared with these constants in the other vitreous oxides) which would make the constant used in Eq. ( 14) to obtain our ring size estimate in error. This conclusion is certainly consistent with the difficulties evident for producing the vitreous TeO, network by deformation of the Te-0 Te angles.

These three independent elastic constants have been obtained from the measurements of the hydrostatic pressure derivatives {d(p W,‘)IdP},,=,accordingto Ref. [ 1511

IWW?W’l,,=, =-l-(1/3




c,12=c223=c133=ct22=c233; C 1.55 =c

244- -c344=c~~~=c26(1=cj55

However, taken as

only three of these are independent. If these are

C,23=~~, and C,44=112 and C456=~i3


then the others are given by the linear combinations



lf2+4 ~1~) ( 19j

= 1.46 i 1.5%



The values obtained for I{,, lt2, zf3 and hence of C,,, are given in comparison with those of other glasses in Table 2. The hydrostatic pressure derivatives of the bulk modulus ([email protected]),,,=-(Cl,,+6

C 144- -c 255---cm;



3.5. Hydrostatic pressure derivckses of the second order elastic conslants, third order elmtic constants (TOEC) and ~Gv-ationnl anhamorzicity The third order elastic constants of an isotropic material are




~~_~ (w

and shear modulus are also given in Table 2. Glasses fall into two distinct categories so far as the temperature and pressure dependence of their elastic properties are concerned. For Te02 glass the pressure derivatives of the bulk (aB/P),,,, and shear (d,u/dP),,, moduli are positive and the third order elastic constants are negative in sign. The physical principles underlying the TOEC of a material can be understood by considering the force acting between pair3 of atoms vibrating longitudinally in a linear chain, i.e. F=-ax-I-bx’+cx”


I!~ and C,ss=zf2+2

C,,,=zii+611,+8 Table 2 Comparison

uj and



of the TOEC


of TeO, glass with thuae of other amorphous


TeO, glass v231




(GPd) (GPd) cc-1 (GPJ) (GPd) (GPd)

-732 - 120 - 186 +33 - 153 - 93

(dB‘/ilP),,,, (i)pLlt/P)p=o

Ref. Cl,, c,,, c,z3 Cl,, C,,, c,,,

Yi Yl kh,liC B-‘(WaPIpz, p-‘(cQ.L/dP) ct (lo-““c-‘)

(IO-“‘Pa-‘) P = Cl(lo-‘[‘Pa-‘)


where CI,I? and c are all positive and x is the displacement of the interatomic separation about the equilibrium value. The at room temperature

[ 1231 Pyrex


(Fe203)tl.38 I P,Os) U.M glass [ 1531

[ 1511

Fused silica [1551




[ 1561

[ 1571

[ 1581

-465 -33 - 162 f64 - 108 -86

- 267 -78 -26 -26 -47 -11

-450 - 200 -160 - 18 -62 -22

400 30 260 -120 90 105

530 240 50 90 70 - 10

+ 6.4 + 1.7 +2.14 + 1.11 + 1.45

+ t t t +

6.42 1.73 2.34 1.45 1.75

t 6.52 t 1.87 +2.61 t 2.49 +2.53

t 4.73 -0.16 + 1.1 -0.3 + 0.7

-4.72 - 2.39 - 1.74 - 1.50 - 1.58

- 6.3 -4.1 -2.8 - 2.36 -2.5

t 0.28

- 0.m


f2.0 t 0.83

t 3.67 t 1.87

$5.11 t 2.92

+ 1.03 - 0.057

- 1.23 -0.87

- 1.69 - 1.04

t 3.88 t 0.52

+ 153

t 8 - 10.3


t 7.5 - 14

+ 3.2

t 0.45

-I- 13.5 - 15.5


~~~ -603 -0.61 i-7.5

- 1.67_ - 1.52~ +7.5

first term is the harmonic one in the energy (F= - au/ax) and is the Hooke’s law approximation. The second term is asymmetric in the potential; its effect is to increase the force less rapidly (than expected from Hooke’s law) as the displacement .Y is increased in the positive direction but to increase the force less rapidly as s is made more negative, reflecting the fact that interatomic repulsive forces have a shorter range than the interatomic attractive forces. The fourth term is symmetric with respect to s so far as the potential is concerned and causes F to increase less rapidly with x at a large vibrational amplitude - an effect which has a pronounced influence during phonon mode softening in materials which show incipient acoustic mode instabilities. Now consider a steady uniaxial pressure applied to the chain causing the mean value of s to become X (the latter being negative). The pressure can be written

( 1/p) ( 1ld,u/rlP),,=, and Poisson’sratio u of a substantial number of glasseshave been reported [ 1441. TeO, fits into this correlation, which for the glassesin Table 2 yields a correlation coefficient of 86%. A linear regressionpreformed on ( 1lB) ( dBIJP),=o againstg gives an even better correlation (96%). The effect of hydrostatic pressureon the modevibrational frequencies can be quantified by considerationof the Gruneisenmode gammaswhich expressthe volume (or strain) dependenceof the normal modefrequency w, yt=-d

In w,ld In V


For an isotropic solid there are only two componentsof the Gruneisen parameter for acoustic modesat the long wavelength limit, namely y, and y, which refer to longitudinal and shearelastic waves, respectively. Thesetwo parameterscan be obtained from the secondand third order elasticconstants using [ 1.591

Hence the effective elastic modulus for a wave motion along the chain of amplitude I A.YI -K X, is

y,,<=-( 1/6~1,,,,)(3BT+2~~,.,+k,,,)


where \t’, = C, ,. it’, = Cjli, k,=CI,,+2CIIz and k,=0.5 ( C, , , - C,,,). It can be shown that for an isotropic solid












where the Lame constant ,Uis C,, and =(dMldlXI)(





The meanlong wavelength acousticmodeGruneisenparameter ye, is given by

For X+ 0, IX] --) Plcr hence dM/dP=l/M(2b-(6cPlci)J

C,,+3 C,z+C,,,-C,23)


Hence a positive pressuregradient for the elastic modulus M usually arisesfrom the third order constant b, although at a sufficiently high pressureP, a negative gradient can arise through the fourth order constant c. However, this simple approach predicts that (dM/dP),),, is always positive, wherasfor somematerials,in particular vitreous silica (Table 2), the pressurederivatives of the elastic moduli for p -+ 0 are negative. For real materialsthe physical interpretation of TOEC is of coursefar morecomplex becausethe interatomic forces are functions of angle as well as atomic separation, and in generalthe application of uniaxial or pressureinvolves changesof both bond anglesand bond lengths. In summary. when a material is subjectedto pressure,collapse is resisted by the interatomic repulsive forces. which have a shorter rangethan the attractive forces and tend to dominatethe third order elastic constant. Thus the third order elastic constants are negative and the hydrostatic pressurederivatives of the elastic constantsare positive in a material which showsthe normal effect of stiffening under the influence of stressas TeO, glassdoes. A good correlation between values of



The longitudinal and shear mode Gruneisen parameters, which measurethe shift of the long wavelength mode frequencieswith volume change,and ye,of TeO, glassare compared with those of other glassesin Table 2. The positive signsof y, and y, found for TeO, glassshow that the application of hydrostatic pressurecausesthe long wavelength acousticmode frequenciesto increase.This is normal behavior in that the energiesassociatedwith those modesarise if the glassis subjectedto volumetric strain: the acousticmodes stiffen. In contrast, in silica and Pyrex both the longitudinal and shearacoustic modeGruneisenparametersare negative; the acousticmodessoften under pressureastheir frequencies and energiesdecrease.The linear thermal expansioncoefficient cymeasuredat room temperaturefor TeO, is largecompared with thoseobtained in vitreous material which exhibit anomalousnegative pressurederivatives of the elastic constantsand positive TOEC. In fact, for Te02 LYis somewhat larger than that of B,O,. anoxide glassknown to shownormal elastic behavior under pressure(Table 2) and temperature. In addition to determining the elastic behavior of a material

under pressure, vibrational anharmonicity is responsible for thermal expansion, although that of a glass has not been described quantitativiely at a microscopic level in the absence of sufficiently detailed knowledge of the vibrational density of states. Experimentally, an average thermal Gruneisen parameter 3/th can be evaluated from yth=3nV yth=3cr

3,/C, V&/C,


=z c~y,Iccj where Ci is the mode contribution to specific heat ( C, = J$Ci) and the mode gamma ‘/I is given by Eq. (24). Since long wavelength acoustic modes must contribute substantially to the collective summations C over their modes, there should be a strong correlation between ‘/th and ye,, which in principle would enable an assessment of tht: relative contributions to yh of the long wavelength acoustic phonons and the shorter wavelength excitations, which do not have a well-defined wavevector k. However, thermodynamic data is sparse on glasses whose elastic behavior under pressure is known, so a comparison has been made between the room temperature thermal expansion and ycyel(Fig. 4) for the glasses listed in Table 2. It can be seen that there is a systematic trend between these two properties, which are determined by vibrational anharmonicity for this wide variety of glassy materials. At one end of the scale vitreous silica has anomalously negative elastic constant pressure derivatives, and hence (Eqs. (25)(29)) a negative Y~,; the thermal expansion coefficient is small because of the existence of individual mode 7%with both negative and positive signs, depending upon the nature of the atomic displacements associated with the mode [ 1601. TeOz is positioned in the range of glasses showing the more usual positive value of yeyricoupled with a moderate thermal expansion. Eqs. (3)-( 12) show that negative values of the pressure derivatives of the elastic modulus (p -+O) for a wave motion

Fig. 4. Corrdation between the mean long wavelength acoustic mode Gruneisen parameter y<, and rhc linear coefficient LYof thermal expansion for a number of glasses [ 1231,

along a linear chain and negative values of the corresponding Gruneisen mode gamma, cannot arise from longitudinal atomic vibrations alone. It is therefore natural to consider whether transverse components of atomic motions are: responsible for TOEC anomalies. Intuitively it is easy td understand that pure transverse vibrations can lead to small or even negative 01 and therefore, according to Eq. (30)(which is derived solely from thermodynamical arguments), to negative mode gammas. However, as to the mechanisms by which transverse vibrational components give rise to these effects, several qualitative models have been given in the literature, for example Refs. [ 1601 and [ 16 11. The following discussion differs from these accounts in that it predicts neg, ative mode gammas from a consideration of pure transverse vibrations alone on central force theory, rather than from an analysis of the simultaneous effects of transverse and compressional vibrations of directional bonds. Consider a chain of atoms with equilibrium separation r;, reducing to r on application of uniaxial pressure along the chain. If an atom vibrates transversely with instantaneous displacement dfrom the axis, then assuming central forces the atom will move in a double-well potential having minima at positions defined by the equation d’+ r’ = rU2, and a central %aximum on the= chain axis [ 1521. For a sufficiently small pressure the total energy of the atom E (even in the ground state) will exceed= the barrier height V so that the atom may be considered (classically) to vibrate across the barrier with kinetic energy E-V. As the applied pressure P increases from zero V will increase and so for a given E, the kinetic energy and correspondingly the vibrational frequency 13both decrease. Thus dp/d\, is negative making it plausible that negative Gruneisen mode gammas can be associated with transverse components of atomic motions. The above arguments predict TOECanomalies in crystals. In glasses the same mechanism takes place but is complicated by the fact that applied pressure now perturbs two well systems which are already present because of the spread of interatomic spacing present in the amorphous state. So it is not surprising if a given material, which is available as both a crystal and a glass, exhibits different TOEC behavior in the two forms. In general, the vibrational anharmonicity of the acoustic modes of TeO, glass does not exhibit the unusual features of silica-based glasses which can be ascribed to their low coordination number and relative ease of bending vibrations [162]. The anomalous decreases of the bulk and shearmoduli induced by hydrostatic pressure in silica and Pyrex are not a necessary consequence of the glassy state but irem part of systematic behavior which is controlled by the nature of the anion, the coordination number and the structure. Silica-type glasses are comprised of fourfold coordinate cations. The negative pressure dependences (and the corresponding positive temperature dependences 11611) of the elastic moduli stem from this open structure with the low coordination number based upon SiOl tetrahedrd which allows bending vibrations of the bridging oxygen ions corresponding to transverse motion against small force constants. Although the coordi-

nation number of the tellurium atoms in TeO, is also small (being close to 4 [ 136,137] ), the elastic behavior under the pressure of TeO, differs entirely from those of silica-based glasses, resembling rather the chalcogenide glasses and amorphous arsenic [ 1381 and arsenic trisulfide [ 1391 (Table 2). Clearly the low coordination alone is not sufficient to produce the anomalous elastic behavior. However, there are structural differences between TeO, and SiO, glasses which account plausibly for their different elastic responses to applied pressures. The neutron diffraction studies [ 136.1371 of TeO, glasses show that there arc well defined peaks at 2.00 A in the radial distribution function. There are closely corresponding Te-0 and O-O distances in crystalline paratellurite and tellurite and both are characterized by fourfold coordination of the tellurium atom. The good correlation between the Te0 and O-O distances with the two tctragonal crystallineforms points to the basic coordination polyhedra being the trigonal bypyramids (Fig. 1) [ 1371. which are connected vertex to vertex so that the equatorial-axial bonding of oxygen is preserved in the glass. A certain freedom of rotation of these TeO, units is permitted in the glass. However, the anomalous elastic behavior of silica glasses under pressure arises from the motion of the bridging oxygen atoms which are free to move individually. Inspection of the structure of the TeO, glass shows that a similar motion of individual oxygen atoms is not permitted because they are bound rigidly in the TeO, trigonal bypyramid. More explicitly. this rigidity stems from the fact that each oxygen is bound in an axial position to one Te atom and in an equatorial position to another. Thus a transverse motion of a Te-O,,, bond involves a fairly direct compression of a Te-O,, bond. and vice versa: and so a pure transverse vibrational mode is only possible if the entire structural grouping is broken up, which involves a large amount of energy compared with that available in elastic waves. Plausibly the improbability of pure transverse vibrations could be linked with the low value of Kb,/Kc which has been attributed to the relatively small amount of bond bending taking place under isotropic compression. Hart [ 1621 has measured the elastic moduli of tellurite glasses. He found that the bulk and shear moduli decrease with temperature and increase with pressure in the normal way. This normal elastic behavior of these heavily modified tellurite glasses might well have been construed as arising from the inhibition of bending vibrations by the modifying ions-as happens in the silica-based glasses. Thus Sato and Anderson [ 1611 found for a soda-lime silica glass ( SiO?, 70.5 wt.%: CaO, 11.6 wt.%: Na,O, 8.7 wt.%‘; K70,77 wt.%) and for a lead-silica glass (SiO?, 46.0 wt.%: PbO, 45.32 wt.%; KZO, 5.62 wt.%) that while the longitudinal ultrasonic velocity increased under pressure, that of the shear wave decreased. The pressure derivatives of the bulk moduli for these two glasses were found to be + 2.6 and -I- 2.80, respectively, and those of the shear moduli were -0.235 and +0.0057, respectively. The elastic behaviors of these two silica based glasses represent a middle position between that of silica itself and those of the glasses, such as arsenic and

arsenic trisulfide. which are normal. In the silica-based glasses the pressure of alkali metal ions in the silicainterstices inhibits the transverse vibrations responsible for the anomalous elastic behavior. A similar situation occurs for the iron phosphate glass of composition (Fe,0,)0.38(P205)0.62 for which i dB’lJP),=o is strongly positive while (d~‘/~P),=, is just negative [ 1531. However, as the third order elastic constants of pure PzO, glass have not yet been measured, the role of the modifier cannot be decided upon. Nevertheless, the results obtained on the parent glass allow a definite statement to be made concerning TeO,-PbO and TeO,-LazOX; unlike silica glasses there is no need to postulate that the modifier inhibits bending vibrations and produces normal elastic behavior. since pure TeO, glass itself does not show the anomalous effects found in vitreous silica.

4. Elastic constants of binary transition glasses

metal tellurite

4. I. TeO,- WO.? and Te02-ZlzC12 glasses Insofar as their elastic behavior depends upon temperature and hydrostatic pressure, glasses divided into two distinct sorts [ 1381. Vitreous silica and silica-based glasses have anomalously negative pressures and positive temperature dependences of the bulk B and shear k moduli [ 163.1641. However. most other glasses behave more normally in that the pressure derivatives of these moduli are positive and their temperature dependences are negative. To establish where tellurite glasses fit into these categories, the elastic moduli of two ternary tellurite glasses has been measured and they have been found to have normal elastic behavior [ 1621. This was confirmed when it was shown that the third order elastic constants of ternary tellurite glasses have the negative signs expected in this instance [ 1651. Until recently there was doubt about whether TeO, itself could exist in a true state. However, vitreous TeO, was eventually made. Its elastic stiffnesses was found to increase under pressure [ 1231. This finding negated the possibility that the ternary glasses might have contained enough modifier to inhibit bending vibrations and thus produce normal elastic behavior. No previous measurements on binary tellurite glasses have been reportedbefore those of Ref. [ 1661. Our objective in Ref. [ 1661 was to study the compositional dependence of the elastic properties of such materials. Two glass series have been selected having modifiers which were expected to produce quite different changes in physical properties. One modifier. the transition metal oxide WO, has previously attracted some interest. measurements having been made of capacitance and dielectric loss [ 1261 molecular refraction, molar volume, glass-transition temperature, thermal expansion and infrared spectra [ 95,100,167,168]. Also, a.c. conductivity and pressure dependence of the dielectric constant have been measured [ 1271 with the optical absorp-

tion edge [ 1691. The other modifier, whose effects on elastic

io4 Table 3 SOEC and TOEC WO, content Density,

of tungsten


glasses at room temperature


Q (kg m- ‘)

Ultrasonic wave velocity (m s- ‘) Longitudinal mode, vL Shear mode. v, Adiabatic Young’s modulus, p (GPa) Adiabatic bulk modulus, B” ( CPa) Adiabatic Poisson’s ratio, us Second-order C! I Cl2 C -1.3

elastic stiffness


[d(@i”)/dP],,_,~: mode 1 mode 2 mode 3 mode 4 mode 5 Third-order c III C! I? c 123 c UJ C,,, C ah (aBiaP)p,(> i il,ulW,=,, YL YT hl.lW

[ 1661 0








3403 2007 50.7 31.7 0.233

3532 2031 54.3 36.6 0.253

3561 2080 57.9 37.0 0.241

3555 2098 61.9 38.6 0.23

59.10 18.00 X,60

65.50 22.18 21.66

68.35 21.71 23.32

8.03 1 .-I6 - 1.05

7.07 1.48

8.16 1.34

( GPa)

0.42 1.20 elastic constants


72.04 25.09 25.09 x2x 1.39 0.75 1.14

(GPa) - 732 - 120 - 186 -33 - 153 -94

-685 - 166 -39 -63 - 130 -33

6.4 1.7 2.14 1.11 I .15

5.42 1.68 1.98 1.30 1.52

-770 -224 - 130 -47 -136 -45 6.9 1.6 2.28 1.09 1.49

6.14 1.60 2.22 1.02 1.42 -~

properties have been examined, was ZnCl?. It was observed that while WO, increases, the glass density increases as in Table 3. ZnCl, has the opposite influence as shown in Table 4. It was anticipated that changes by these modifiers correlated with their different effects on density. An increase of ZnCl, content decreases ultrasonic wave velocity; the elastic stiffness is reduced. In contrast, addition of WO, stiffens the glass elastically. Hence, both sets of glasses follow the usual trends between density and elastic stiffnessacrossglasssystems. The ultrasonic velocity measurementswere repeated on Te02 in a similar manner to that studied in earlier work [ 1231 with three years difference in the time the specimen had been exposed to the atmosphere.The new resultswere indistinguishablefrom the previous data. This agreementhas somesignificance.The unchangedelasticpropertiesanddensity establish that vitreous TeO,, whose very existance had been doubted, is stableat room temperatureand also that it cannot have absorbedany significant amountof water vapor. To study how the acousticmodeanharmonicity varieswith composition in these two binary tellurite-glass systems, measurements have beenmadeof the hydrostatic anduniaxial stressdependencesof ultrasonic wave velocities so as to determinethe hydrostatic pressuredependencesof the elastic moduli and alsothe third-order elastic constantsasin Tables

3 and 4 with the third-order elastic constantsand the hydrastatic pressurederivatives (aB/#),,=, and ( ~~/??P),,=Oof the bulk modulusand shearmoduli. The negative signsof the third-order elasticconstantsand thepositive hydrostaticpressure derivatives of the second-orderelastic constantsshow that thesebinary tellurite glasses,like vitreous TeO, itself, behave normally in that they stiffen under the influence of external stress.The increaseof ZnCl, reducesUleglassdensity and stiffness extends to the pressurederivative of the bulk modulus (aBIJP),=O. The addition of either modifier has little influence on the pressurederivative ( 8$;)P)p=0. Knowledge of the compressionV(P) /V, (the ratio of the volume V(P) at a pressureP to that V, at atmosphericpressure) is useful in both experimental and theoretical studies now underway of the behavior of thesematerialsunderpressure.This quantity hasbeenmadeby the Murnahan equation of state [ 1701which is expressedin logarithmic form as ln(V,IV(P)=(IIB~,‘T)ln{B,‘T(PIB;f)+l}


and describeswell the compressionof many solids. Ultrasonic measurementsgive adiabatic (S) moduli, so it is necessaryto transformthe datato isothermal(T) quantitiesusing Eq. (32) [ 1711, BoT=Bd( 1SayT) (325~


Table 4 SOEC and TOEC ZnClz

of zinc chloride

content (mol4



glasses at room temperature


@ (kg m “)


wave velocity

elastic stiffness constants




5 105.0




3403 xl07 50.7 31.7 0.233

3362 1879 44.9 32.9 0.273

3324 1812 41.2 32.5 0.289

3312 1807 39.0 30.6 0.289

59.1 18.0 20.6

56.5 21.2 17.7

53.x 21.8 15.9

50.8 20.6 15.1

t GPa)

Cl, Cl2 GA [[email protected]’)/dP],,,,: mode I mode 2 mode 3 mode 4 mode 5 Third-order c /II c I I? c I23 C 144 C,,, C 4%


t m s- ’ )

Longitudinal mode, uL Shear mode, vs Adiabatic Young’s modulus, Es (GPa) Adiabatic bulk modulus, B” ( GPa) Adiabatic Poisson’s ratio, (7’ Second-order

[ 1661

8.03 I .46 - I.05

elastic constants

7.85 I.41

X.27 I so

7.44 1.38 - 1.15

- 0.48 I .23

- 0.29 1.29

-732 - 120 - 186 -33 - 153 -93

- 655 - 166 -90 -31 - I22 -46

- 614 - 1x0 -92 -46 - 115 -35

-616 - 129 - 121.0 -3.x - 122 -59

6.4 1.7 2.14 1.11 I .45

6.28 1.59 2.29 1.33 1.85

6.52 1.67 2.5 1.5 1.8

5.94 I .54 2.24 1.38 1.67


(aBIaP),=, (apLIiJP)P=c YL YT Y&WC



Table 5 The compression

of vitreous


glasses [ 1661

X( l-(2/aB;f)-2B,,“) The volume thermal expansion coefficient cy of TeO, is 46.5 X 10-hoC-’ [ 1231; while CY(TeO,--33 mol% WOj) being about 37 X IO-“‘C‘. The temperature dependences of the bulk moduli (r?B/ilT),,,, has been obtained and equal to - 7 X 10’ Pa K- ‘. By replacing the Gruneisen parameter Y by ~e~nrt,c: Te02, B;f=31.1 Gpa & B,,‘T=6.0 TeO,-33





Gpa & BclrT=6.3

BIf=3.0 Gpa, & B,‘T=5.S

These isothermal quantities have been inserted into the equation-of-state (Eq. (3 1) ) to obtain the compressions for these glasses as shown in Table 5. The reduction of stiffness according to modifying the glass by ZnCl, has the concomittent effect of reducing V(P) /V, from that in Too, itself, while W03 does the opposite. Insight into the anharmonicity of the long wavelength acoustic modes can be gained by considering the Gruneisen gammas which quantify the volume dependence ( FJIn w,/i)

In V) of the normal mode frequencies w,. The longitudinal

Pressure ( IO” Pa)


I 2 3 4 5 6 7 8 9 10


TrO, - ZnCl> (33 mol%)

TeOz - WO, (33 mol%)

0.97 I 0.947 0.927 0.909 0.893 0.880 0.867 0.856 0.845 0.836

0.970 0.945 0.923 0.905 0.888 0.874 0.861 0.849 0.838 0.837

0.976 0.955 0.938 0.922 0.908 0.896 0.884 0.872 0.865 0.856

y,, and transverse tively, by yl=-(


V(P) / V,,

y, Gruneisen parameters given, respec-

3 B’+2C,,+C,,,+2



(34) (35)

are given for these glasses in Tables 3 and 4. The positive

signs show that the application of hydrostatic pressure to

R. El-Mc~llnlr,fln~,/~inreriols



tellurite glasses leads to an increase in the frequencies of the long wavelength acoustic modes. This is normal behavior, corresponding to an increase in vibrational energy when the glasses are subjected to volumetric strain. The elastic behavior of tellurite glasses is in accord with their structure. The basic coordination polyhedra in TeO, glasses comprise trigonal bipyramids joined vertex to vertex [ 136,137]. In silica-based glasses, the coordination number is closed to 4. Now the elastic behavior of tellurite glasses under pressure indicates that this small coordination number does not, in itself, lead to anomalous behavior. In silica glasses the anomalous elastic and thermal properties are associated with motion of bridging oxygen atoms. However, a similar bending motion is not permitted in the tellurite glasses because the oxygen atoms are rigidly bound in the TeO, trigonal bipyramids. The binary TeO,-WO, and TeO,-&Cl, glasses, like those of the parent material TeO,, follow normal trends in their elastic behavior under pressure. Fig. 5 represents the proposed structure from the neutron diffraction measurements [ 1721. Break

Fig. 5. Model showing the manner of bonding of the nearest coordination polyhedra in binary tellurite tungstate glasses [ i 721. Table 6 Elastic constants

of molybdenum


rind Physics 53 (19981 93-120

4.2. TeO,-MOO3 glasses Since 1990 [ Ill] molybdenum tellurite glasses of various compositions have been reported as semiconductors over the temperature range 100-500 K. Multicomponent oxide catalyst based on TeO, and MOO, are also of interest because of their selective and highly oxidizing catalytic properties [ 173: 1761, The phase diagram [ 177,178] as well as the crystalline structures of the tellurium molybdate [ I79- 18 I] are knownData have been published about a tendency toward glassformation and ihe properties of the glasses in binary MOO,TeO, [ 1X2], Melts of binary MoO1-TeOz systems are eas~ily fixed in a glassy state at normal cooling rates. The glass formation limits in the system are in the range of 12.5 to 58.5 mol% MOO, [ 1301. Structural studies of the molybdenum tellurite glasses have been reported [ 183,184]. The tellurium molybdenum oxide glass structure is not sufficiently studied. There are only X-ray diffraction data published [ 1831 which barely reflect the distribution of oxygen atomsThe large O’- ions from the anion network of the glass and exhibit a determination influence on the cation locations of the glass-former or modifier. The short-range atomic order iti molybdenum tellurite glasses has also been studied by neutron diffraction [ 1841. Previously [77] we prepared the binary molybdenum tellurite glasses of various compositions. Densities of the glasses~ decrease with the substitution of TeOz by MOO,. The molar volume of these glasses has an opposite behavior. The observed variations in the molybdenum-substituted glasses are consistent with the differences in the molecular weight of the TeOl and Moo3 oxides. Sound velocity (both longitudinal and shear) decreases with increasing MOO, concentraL tion in these glasses. The measured elastic properties of different molybdenum tellurite glasses are-listed in Table 6, Among the various elastic moduli we will first consider the composition dependence of the bulk modulus. The following observation can be made about the composition dependence of the bulk modulus; with increasing Moo3 concentration in the glass, the bulk modulus decreases from 3 1.7 to 25.6 Gpa. The quantitative interpretation of this experimental result is as follows. 1. Due to the larger volume of the glass unit cell, the average number of bonds per unit volume decreases while the

glasses [77]











0.0 20.0 30.0 45.0 50.0


159.61 156.5 154.9 153.3 151.77

31.29 31.23 31.61 32.2 32.9

3403 3272 3190 3147 3137

2007 1870 1823 1798 1793

59.1 53.6 49.9 47.0 45.3

20.6 17.5 16.3 15.4 13.8

50.6 44.0

31.7 30.28 28.15 26.56 25.55

5.01 4.9 3.75 4.6

“p=Density (gmcm-“), M,=glass molecular weight, K= longitudinal, shear, Young’s and bulk moduli (GPa),

V=molar volume o, 8, I=Poisson’s

41.0 38.7 37.2

(cm”), V,, I/,=shear and longitudinal ratio, ring diameter (nm).




-0233 -0.257 0.258 0.257 0.257 velocities

53~~- z. 5.5



(m s- ‘), L, G. E,

R. Ei-A4olla~~art~



000 \I/MO




/ Materink

(b) ci%o

0 Go I o- Te-0-MO--o-Te-0 I I\ Cc) 0 0 0

0 I I 0

Fig. 6. Schematic two-dimensional representation of (a) crystalline TeOZ [ 1331, (b) crystallineM00~ [ 1851 and (cl binaryTeO,-MOO, [77].

modifying oxide is 6-coordination for MOO, in a medium of 4-coordination for TeOz. 2. The value of the stretching force constant are nearly equal as calculated using the crystal structure shown in Fig. 6 [77,133,185], i.e. the equal connectivity in the binary Moo,-TeO, glass has a small effect on the decrease of the bulk modulus. The previous quantitative interpretation of the experimental elastic behavior was based on the ring deformation model of the bulk modulus [ 1491. By using Eq. ( 14) we have estimated the ring sizes for the modified tellurite glasses. The value for pure TeO? glass was 5.3 A which increases to 5.7 A for 50-50 TeO,-Moo, glass. Also, it is clear from Fig. 5 that the ring size increases due to some edge sharing of Moo,. The interpretation of the behavior of the other elastic moduli is the same. The composition dependence of Poisson’s ratio is clear in Table 6. Poisson’s ratio increased from 0.233 for TeO, [ 1231 to 0.257 for binary glass with 50% MOO,. This slight increase in Poisson’s ratio could be attributed to the breaking of network linkages. 4.3. TeO,-ZrzO glnssts In this section the longitudinal elastic moduli of binary zinc oxide tellurite glasses will be discussed without any external effect and the effect of halogen substitution in the tellurite glass network will be investigated. The zinc-tellurium glasses are very stable and are considered as a choice for the superheavy optical flint glasses [ 1861. The analysis will depend upon the most crucial quantities for the moduli Table 7 Density, molar volume, Glass

TeO, [I231 Te02-ZnO



ultrasonic Modifier



[ 1661

53 (1998)



of the very importent systems TeO*-ZnO and TeO,-ZnCl,. Both glass systems have been reported as a basis for multicomponent optical glass synthesis and as a useful medium for ultra-low loss ( 1 dB 1000 m- ‘) optical fibers for wavelengths in the 3.54 pm region [ l&7]. Recently we analyzed the effect of halogen substitution on the optical properties of tellurite glasses [ 951. Halogen substitution lowered the dielectric constant and density. So the reduced number of polarizable ions per unit volume was primarily responsible for the reduction in the dielectric constant. Variations of densities and molar volume with ZnO concentration are shown in Table 7 [91]. The densities of all glasses increase with the substitution of TeOl by ZnO, while molar volume decreases. From Table 7 it is clear that the behavior of ZnO is the opposite to that of ZnCl, in tellurite glasses. Variations in the longitudinal sound velocity and elastic constants with the composition of prepared binary tellurite glasses are shown in Fig. 7 (a) and (b). Among the structure variations of these binar systems we will consider the composition dependence of the longitudinal modulus. Firstly, the addition of ZnO (coordination number = 6) results in a higher network rigidity, and this results is an increase of the longitudinal modulus. Secondly, the strength of the modifying bonds and their disposition will be considered. By disposition we mean the type of ring structure formed by the bonds. Thus, bond strength and greater connectivity give rise to higher values of the moduli. The values of the stretching force constants were F(Zn-0) = 219 N m -’ while F(Zn-Cl) = 185 N m- ‘. This behavior was opposite to that of binary zinc chloride tellurite glasses [ 1661. These two opposite behaviors are interesting and were discussed quantitatively according to the structural changes from which the role of the halogen could be determined. The quantitative analysis of the above experimental results was based on the bond compression model [ 1881 for the multicomponent glass of the formula xA,,O,, - ( 1 -x)Gn2 0,,12, which depends on the crystal structure of each oxide in the glass as shown in Table 8 [ 133,189]. The parameters are the bond length r (nm), the coordination number ~zr,and the force constant which has been calculated according to the relation F= 17/t3 [ 1881. The structure of these two series of tellurite glasses has been carried out after calculating the number of bonds per unit volume !T~,

and elastic constants for binary


[ 9 11

and Physics

zinc oxide and zinc chloride

glasses [91]

P (g cm-‘)





(m s-‘)


31.27 27.8 25.6

3403 3468 3775






5.101 5.46 5.47


128.366 157.2 152.5

5.5 5.00 4.87

23.33 31.44 31.3

3819 3362 3324





10.0 20.0 33.0

65.7 77.9 80.2 56.5 53.8 50.8


R. El-Mdlawnry/



rind Ph,ysirs 53 (I 998) 93-120


80 Lb)







(123) A

Fig. 7. Variation Table 8 Crystal structure




TeO, ZnO ZnCl,





of the longitudinal

I1331 I1331 11891

IA 30

(a) sound velocity



0.199 0.1988 0.232

216 219 185

1 6 4


where Nr is the number of formula units per unit volume xpNAIMG, NA is the Avogadro’s number, x is the mol% of the oxide, p is the density, and M, is the molecular weight of the glass. The average cross-links per cation in the glass have been calculated according to E,=( l/S){(ncN~)l+(n,N,)z}


where Eis the total number of cations per glass formula unit. n, is the number of cross-links per cation, and N, is the number of cations per glass formula unit. The values of both quantities for each glass sample are shown in Table 9. Table 9 provides the calculated number of bonds per unit volume and the calculated number of cross-links per cation which are the two main and effective factors to interpret the Table 9 Number of network bonds per unit volume and average cross-links zinc oxide and zinc chloride tellurite glasses


of binary


TeOz [ 1231 TeO,-ZnO [ 9 1]

[ 9 I]

10.0 25.0 40.0 10.0 20.0 33.0


7.74 9.11 10.06 12.28 7.65 7.51 7.34



L * I




and (b) elastic constants for binary

F (N m-‘)



.", F 2




40 MoOg%




. TeO2



2.0 2.2 2.5 2.8 2.0 2.0 2.0


iA 30

1 40


and ZnCl,-TeO,

glasses [ 9 11.

modulus. Previously, we reviewed the available information on TeO&ch tellurite glass and gave the_first report on-t& pure tellurite glass [ 1231. On the basis of the elastic data we consider glassy TeO, to be disordered paratellurite with a mean ring diameter of eight atoms (TeO, rings). Possible structuresfor pure Z&l, and TeO,-ZnCiz glassesbasedon the elastic data have been discussed[ 190,191], and alsoby meansof the neutron diffraction methodin Ref. [ 1921.Since our experimental data suggestedthe complete absenceof phaseseparation [ 981, we need to consider how TeO, and ZnO can combine into a single homogeneousstructure w$h the role of chlorine or oxygen insidethe network. The atomic arrangementof zinc-tellurite glasseshas been both experimentally and theoretically invesitgated by using netron d/ffraction patterns [ 1921asshown in Fig. 8. The study of the elastic moduli of the mono- or polycoti ponent three-dimensionalvitreous telluritenetwork hasdetermined the following points. I. The elastic modulusincreaseslinearly with the force constant for constantmeanring size. 2. The elastic modulus increaseslinearly with the value of cross-linksfor fixed force constant, i.e. inversely proportional with mean ring diameter which equals the mean number of atoms in a ring times the mean bond lengh divided by II. In this light, there areno coordination changesfor Cl atorn? to be substitutedfor 0 atomsin the TeO,ZnCl, glasses.T&e structureparametersin Table 9 confirm thisbecausethe averagecross-link is 2.0 and the numberof bondsper unit volume changesonly from 7.74 X 10” crnp3 for pure TeO, glassto 7.34X 1022cm-j for Te02-ZnO glasses;the number of bondsbecomes 12.28X 10” crne3 for 40 mol% ZnO. This is due to the 6 coordination number of ZnO, which in turn increasesthe average cross-links from 2.0 of TeO, glassto 2.8 for 40 mol% ZnO. The change in the force constant obtainedby changing one Te-0 bond into a Te-Cl bond~and one Zn-Cl bond into a Zn-0 bond is 10%. The most like5 causeof the higher modulus is therefore a reduction in thk

Fig. 8. Model showing the nature of the atomic arrangement tellurite glasses [ 19’1.

in binary


size of the rings. Since we have proposed rings of eight atoms for the basic TeO, glass, this implies the introduction of some rings with six atoms at the most. 4.4. Td-V,O,


The first switching phenomena in tellurite-vanadate glasseswas measuredin 1973 [ 1931, and the bipolar threshold switching [ 1941and memory switching of theseglasses [ 1951 have alsobeen measured.Very recently we measured the variation of ultrasonic wave velocities (shear and longitudinal) and the density with a different percentageof the vanadium oxide [ 31. The changein density (Fig. 9) accom-

"E < sE x .5 E 2







paning the addition of V20, is related to the change in the atomic massand atomic volume of the constituent elements. The atomic massof Te and V are 127.6 and 50.94, respectively, and their respective atomic radii are 1.6 and 1.34 A. This explains the observedlinear decreasein the density with increasingV20j. Fig. 10 shows a plot of molar volume V and the oxygen ion molar volume V, versus mol% V20, concentration. It is clear that the oxygen ion molar volume of the tellurite vanadateglassesdecreases rapidly in the rangefrom 0 to 20 mol%. Above 20 mol% V205, the oxygen ion molar volume decreasesgradually with V,O, content up to 50 mol%. On the contrary, the variation of the molar volume with V,O, concentration is not sensitive up to 10mol% but it increases linearly and rapidly with increasingV205 in the range from 10to 50 mol%. The variation of longitudinal and shear wave velocities with V,O, concentration is given in Fig. 11. The addition of V,O, up to 20 mol%, to vitreous TeO,, decreasesboth longitudinal and shearwave velocities, while with further addition of V,O, modifier above 20 mol% up to 50 mol% a noticable increasein wave velocities (longitudinal andshear) is observed. From the compositional dependenceof ultrasonic velocities, the studiedglasssystemcan be divided into two composition regions 0


0123 . 4.0





. . .


i 0

1 10


1 30

/ 40 WS

Fig. 9. Dependence

V205 glass system.

of density

on V205 molR

I 50


/ 10

I 20

/ 30

/ 40 ho5




in the TeO-

/ 50



Fig. 10. Variation of the molar volume and oxygen ion molar volume as a function of VzOs mol% concentration in the Te02-V,O, glass system.

R. El-Mullan~anv/Mnterials

1 IO


md Physics

53 (1998)



? E


x 5 9 B .2 5




123 .

l .



l .




? 5



. 2000


s+ A



d A




A 1000












Fig. 11. Dependence of ultrasonic wave velocities (longitudinal on V,OS 17101% concentration in the Te02-V105 glass system.

and shear)



Fig. 13. Dependence of bulk modulus on V,OS rnol% Te02-Vz05 glass system.


in the



50 2 2. : 2


. .

-; 2 >




i l



. .














40 WA




Fig. 12. Variation of the elastic moduli (longitudinal and shear) as a function of Vz05 mollo concentration in the TeO,-VZOS glass system.

to pure TeO, decreases both ultrasonic velocity and density values and this in turn decreases the elastic moduli. When V205 is increased above 20 mol%, the density decreases linearly while the velocity increases rapidly and therefore all elastic moduli values increase gradually. Fig, 15 shows the variation of Poisson’s ratio with composition for the investigated tellurite-vanadate glasses. It can be seen that the behavior of the variation is nearly opposite to that of the elastic moduli variation, i.e. Poisson’s ratio increases in the first composition region and then decreases variably in the second composition region. Poisson’s ratio is defined as the ratio between the lateral and longitudinal strain produced when tensile force is applied. For tensile stresses applied parallel to the chains, the longitudinal strain produced will be the same and is unaffected by the cross-link density while lateral strain is greatly decreased with the cross-link density (number of bridging bonds per cation minus two).









Fig. 14. Variation of Young’s modulus as a fur&on tration in the TeO,-V,O, glass system.


of V205 mol% concen-

1 l

l l









s .;



2 I







30 W5

Fig. 15. Compositional glass system.


of Poisson’s




50 [email protected])

ratio for the TeO,-V&

These cross-links generate a strong covalent force resisting the lateral contraction as shown previously. Values of mean cross-link per cation in pure TeO, and TeO1--50 mol% V,O, are 2.0 and 2.67, respectively [ 31. Moreover, Poisson’s ratio increases with increasing the atomic ring size and in the first composition region, the elastic moduli and Poisson’s ratio are more affected by the increase of the atomic ring size. In the second composition region, increasing the V,O, content increases the average cross-link density and the average atomic ring size slightly decreases with a subsequent decrease in Poisson’s ratio. In pure TeO, glasses. the basic coordination polyhedra are trigonal bipyramids [ 1361 which are joined vertex to vertex. The replacement of TeO, by V,O, causes a change in the atomic ring size of the network. The average atomic ring size is 0.535 nm for 20 mol% V,05 while for all other mol% V205, the average atomic ring size is slightly less. For TeO, glasses, the average atomic ring size is reported to be equal to 0.500 nm. In the first composition region ( V,05 5 20 mol%) the three-dimensional tellurite network is partly broken by the formation of TeO, trigonal pyramids. This in turn leads to the reduction of the glass rigidity as evidenced by the increase in ring size and the decrease in elastic moduli, Debye temperature, microhardness and ultrasonic velocity. In the second compositional region ( V,05 > 20 mol%) the glass structure changes gradually from the continuous tellurite network (which is soft and easily deformed by stresses) to the continuous vanadate network with higher stretching force constant and cross-link density. Increasing VZ05 raises the resistance of the network to deformation (stiffens the glass elasticity) and decreases the average atomic ring size. Moreover, for each glass we compute a theoretical bulk modulus Khc according to the bond compression model. The bulk modulus is given by an expression in the form:

(38) Where x and ( 1 -x) are the mole fractions of the two component oxides, II~ is the number of formula units per unit volume, I’ is the bond length, f is the first order stretching force constant. N, is Avogadro’s number, p is the density and A4 is the molecular weight of the glass. After substitution, the above equation can be rewritten as K,,=10”[3104~+2289.76(1-x)]p/hl


It is clear from this equation that the bulk modulus depends on mole fractions of component oxides, the density of the glass and the molecular weight of the glass. As the content of VzO, is increased, the glass molecular weight increases and the density decreases so that the value of p/M varies. Therefore, the number of network bonds per unit volume increases from 7.74X 10” mm3 for pure TeO, to 7.96 X 10” m-3 for glasses containing 10 mol% V,Oj.

Above 10mall’ V205, the number of network bonds per

1 * ’


e l

x *














Fig. 16. Dependence of bulk modulus centration in TcO?-V,05.

(KbL. and &)






50 V205 mol% on V,Oj mol% con-

unit volume decreases steadily up to 50 mol% V,05 concentration. Fig. 16 represents the dependence of the calculated and experimental bulk modulus on VzO, content in binary V205TeO, glass. It is clear from the figure that Kbc and Kexp have the same trend up to 20 mol% V,05 and above this concentration Kbc continues decreasing while Kexp increases gradually. The important parameter in experimental bulk modulus calculation is the measured ultrasonic velocity and density, while for theoretically calculated bulk modulus, the parameters are density and molecular weight. The observed agreement between the behavior of Kbc and Kexp through the first composition region can be explained by the fact that both density and ultrasonic velocity decreases with an increase of V,05 content up to 20 mol%. Above 20 mol% V,O,, the ultrasonic velocity increases rapidly and therefore an increase in Kenpwill result. At the same time, the continuous decrease in density will influence calculations of Kbc and its value consequently decreased. From Table 10 it is clear that the ratio between the calculated and experimental bulk modulus is 2.3 for pure TeO, glass and ranges between 2.7 and 1.6 for the other glasses. These values are considerably smaller than those for other pure inorganic oxide glass formers K,,,/K,,, = 3.08 for P,05, 4.39 for GeO,, and 10.1 for Bz03. The relation between (KJK,) and the calculated atomic ring size 1 is illustrated in Fig. 17. Table 10 Calculated

bulk modulus

and Poisson’s

ratio [ 31

.Y (mol%)




( X 10bm-“)

( X 10” m-j)


00 [6.8] IO 20 25 30 35 40 35 50

0.031 0.032 0.030 0.028 0.027 0.026 0.025 0.024 0.023

7.74 7.96 7.56 7.16 7.11 6.78 6.65 6.48 6.31

71.5 76.4 73.3 69.7 69.6 66.6 65.6 64.2 63.1




2.31 2.2 2.7 2.3 2.4 2.0 1.8 1.7 1.6

2.00 2.18 2.33 2.40 2.46 2.52 2.57 2.62 2.67

0.235 0.230 0.227 0.225 0.224 0.222 0.221 0.220 0.219






I B2°3

' i




0.4 e hm)

Fig. 17. Variation of the estimated of KJK,,,. After [ 1481.

average atomic ring size with the value

A value of KbclKexp. 2 1 indicates a relatively open threedimensional network with ring size tending to increase with ( KbolKcxp) . The relatively high value of ( Kb,IK,,,) = 2.7 of tellurite glasses containing 20 mol% V,05 is attributed to the very open three-dimensional structure (0.535 nm) at that composition. Poisson’s ratio can be calculated with the knowledge of the average cross-link density fi,, according to the formula gca,=( 0.28)/( n,.)0.2”

(40) Where 17, is the number of cross-links per cation. The average cross-link density and the calculated Poisson’s ratio are given in Table 10. It is clear from Table 10 that the calculated Poisson’s ratio decreases steadily withthe increase of V,O, mol% concentration and this steady decrease is mainly due to the fact that the mean cross-link density in these glasses increases from 2.0 for pure TeO, to 2.67 for TeO, 50 mol% concentration. 5. Elastic constants of binary, ternary and quaternary rare earth tellurite glasses Glasses containing rare earth ions in high concentrations are potentially useful for optical data transmission or in laser 11 Composition,

systems. Earlier studies of the effect of incorporating rare earth cations like lanthanum, neodynium, samarium, europium or gadolinium on the structure and physical properties of tellurite glasses have been reported [ 197-202 ] . An interest in erbium containing tellurite glasses developed because the photochromic properties of erbium-doped tellurite glasses arr substantially more pronounced than those of silicate glasses containing an equivalent erbium-ion concentration [ 2061. Rare earth tellurite glasses have been prepared as in Table 11 in order to measure their elastic properties I20 I]. Data for the densities, room temperature ultrasonic wave velocities, the second-order elastic stiffness constants, the adiabatic bulk and Young’s moduli and Poisson’s ratio for rare earth tellurite glasses are shown in Table 12 with those of pure TeO, and also rare earth phosphate glasses. Inspection of the elastic constants data for the binary glasses shows that inclusion of 10 mol% La?O,, CeO, or Sm,O, in tellurite glasses increases the elastic stiffness: both shear and bulk moduli are larger in rare earth tellurite than in vitreous TeO, itself. The introduction of an ionic binding component due to the presence of the rare earth ion stiffens the structure. It is interesting to note that while the rare earth phosphates are much less dense, they have elastic stiffnesses which are much the same as those of the tellurite glasses. Prdusibly this arises from substanially greater ionic contributions to the binding in the phosphates than that in the tellurites. The elastic properties of the vitreous samarium phosphates in relation to structure and binding have been discussed elsewhere [ 203,204]. The velocities of both longitudinal and shear ultrasonic waves in the binary, ternary and quatemary tellurite glasses are closely similar to those of vitreous TeOz itself. This is an unusual feature; ulrasound velocities in other glass systems, such as silicates, borates or phosphates are usually quite sensitive to the inciusion of other oxides whether as formers or modifiers. It would appear that the tellurite matrix determines the long wavelength acoustic mode velcities (V, or V,). Because the elastic moduli are defined by the appropriate value pl/‘, they become larger as the density of the oxide additional to the tellurite former is increased (Table 12). For samarium phosphate glasses it hab been found that both the bulk and shear moduli decrease with the application of hydrostatic pressure: these materials show the extraordi-


Glass formula


and color of binary,


and quaternary

rare earth trllurite

glasses [201]



A B c D E F G H




5706 5782

6713 6018 6027

6110 5781

(kg cm-‘)




pale lime-green (T) dark reddish-b&n (0) yeilow (0) pink CT) lime-yellow (Tj lime-yellow (T) yellow (T) very dark reddish-brown pink (T)

0 = opaque7


R. El-Mulluwony

/ Marerids




53 (1998)



nary property of becoming easier to compress as the pressure on them is increased [ 203,204]. The similarity of the Raman spectra of these samarium glasses to those of other phosphate glasses, including lanthanum, suggests that the samarium glasses have similar structural features to those of otherphosphate glasses. Hence, the unusual elastic behavior under pressure could be due to the variable valence of the samarium ion. To test this, it is useful to measure the elastic properties under the pressure of samarium ions in a glass based on a different former; tellurium dioxide is well-studied for this purpose. A lanthanum tellurite glass has also been studied. In contrast to samarium, whose ions can be 2+ or 3 +, lanthanum ions can only be 3+. To seek other possible valence effects, cerium, which can have 3 + or 4 + ions, has also been included in the study. The pressure dependences of the relative changes in natural velocity of the longitudinal and shear wave velocities for the rare earth tellurite glasses were found to increase linearly up to the maximum pressure (3 k bar). This is a good indication that the glasses are homogeneous on a microscopic scale. The hydrostatic pressure derivatives (aC, , /dP),,,,. ( aB/8P),,o and (dC,/dP),,, of the moduli obtained from the experimental measurements are given for each glass in Table 13. The positive values obtained for the hydrostatic pressure derivatives of the second-order elastic constants show that these rare earth tellurite glasses, like vitreous TeO, itself, behave normally in that they stiffen under the influence of external stress. The hydrostatic pressure derivatives of the SOEC are combinations of the TOEC and hence correspond to cubic terms in the Hamiltonian with respect to strain. They measure the anharmonicity of the long wavelength vibrational modes, and thus relate to the non-linearity of the atomic forces with respect to atomic displacements. Insight into the mode anharmonicity can be gained by considering the acoustic mode Gruneisen gammas, yi, which represent in the quasiharmonic approximation the volume (V) dependent ( -r3 In w,/J 1nV) on the normal mode frequency w,. The longitudinal ( y,) and transverse ( yt) acoustic mode Gruneisen parameters in the long wavelength limit are given in Table 13. Their positive signs show that the application of hydrostatic pressure to rare earth tellurite glasses leads to an increase in the frequencies of long wavelength acoustic modes. This is normal behavior, corresponding to an increase in vibrational energy of the acoustic modes when the glass is subjected to volumeteric strain. In contrast, when a hydrostatic pressure is applied to samarium phosphate glasses their moduli have the anomalous property of decreasing [ 203,204]. Although uncommon, such acoustic mode softening behavior is also known in glass based on silica [ 1.541, which has been attributed to the open fourfold coordination structure that enables bending vibrations of the bridging oxygen ions, corresponding to transverse motion against a small force constant [ 16 1] . If P-0bl-idgine-P bending were to be the cause of the anomalous elastic behavior of samarium phosphate glasses under pressure, then other phosphate glasses

would be expectedto have a negative value of dBldP and

114 Table 13 Hydrostatic



of the elastic constants of rare earth tellurite

Giass composition

TeO, [ 1231 iLa203),l ,iTeOZh, iCe02),, ,(Te02),,, iSm20,),, ,(Te02hL) (SmGh iP,O,h, (Sm,O,),,,,(PiO,),,,, (La,O),, IiPzO,h, (ErzO&, I~W03MTe02)o, (Y202)1,.I,~(WOI),I,2(TeO~),,,7 (LazO~l),,,,,(WO~),,2(Te0,),,,,, (SmzOi)l,,irs(W03)1,2(TeOl),,is

(CeO,),,,,,(PbOi,,,,,(TeO,),,,, IEr,0,),,,,,lW0,),,29(PbO),~.L(Te02),,,,

Hydrostatic constants

glasspa [201]

pressure derivatives

of elastic stiffness

Long wavelength xoustic Griincisen paramerers






t 8.03 +9.29 +%I9 + 8.52 - 0.88 - 1.09 + 2.22 + Il.64 + 10.19 + 10.10 +7.41 + 6.67 -I- 9.99

+ 1.7a + 1.79 + 1.87 + 1.71 - 0.69 - 0.37 +0.11 + 1.77 + 1.06 -I- 2.46 t 1.83 + 1.95 + 3.38 -

+ 6.40 + 6.80 +5.70 + 6.24 + 0.045 - 0.77 + 2.07 + 9.28 + 8.78 + 6.83 + 4.97 -l-4.07 + 5.62

t 2.14 +2.15 + 1.88 + 1.89 - 0.40 - 0.45 + 0.48 + 2.83 +X6 +2.58 + 1.83 t 1.62 + 2.4.l

+ 1.11 t 1.03 t 1.07 + 0.89 - O-70 - 0.70 - 0.06 +1.09 + 0.67 + 1.79 t L26 + 1.33 +2.21

dp/dP. However, this is not so for molybdenum [20.51,lanthanum [ 203,204] or iron phosphateglasses[ 2061. On that comparative basisthe anomalousnegative pressurederivatives seem to be characteristic of the samariumion of the phosphatenetwork [ 202,203]. The valence transition from 2 + towards 3 + involves size collapesof the samariumion {&I-?+ (3P) is about 20% smaller than Sm”+ (4p) } and could lead to the observed reduction under pressureof the bulk and shearmoduli. One objective of the study [201] was to find out whether samariumin a glassytellurite matrix glass alsoshowsthis extraordinary behavior. It doesnot. Either the pressureeffects on the valence state or the way in which samariumion is bound would seemto be different in the tellurite than in the phosphateglasses.Theseopening studies of sucha fundamentalproperty asthe elastic stiffnessand its behavior underpressureshowthat thereis muchto learnabout the nature of the way in which rare earth ions in high concentrationsarebound in a glassymatrix, andthe way in which they behave under an applied stress.

6. Quantitative analysis of the elastic constants of rare earth tellurite glasses The estimatedbulk modulusand Poisson’sratio of binary, ternary and quaternary rare earth tellurite glasseshave been calculated using the bond compressionmodel [ 1881according to the cation-anion bond of each oxide present in the glass. Information about the structure of the glass can be deduced after calculating the number of network bonds per unit volume, the value of the average stretching force constant.the averagering size, the structuresensitivity factorand the mean cross-link density. Comparisonsbetween the calcuiated and the experimental elastic moduli and Poisson’s ratio have beencarried out. Also, the longitudinal and shear elastic stiffness of binary rare earth tellurite glassesarecom-


~~~ ~~ 9’

q -

+ 1.45 + 1 .-NT+ 1.34 i-L23 -0.60Lo - OAX21 + ox -I- 1.67Y t 1.31 i 2.02 + 1.451~ t 1.43 t 2.29

pared and analyzed with those of other binary rare earth glasses. The bulk modulusof a polycomponent oxide glassof tie formula.uA,,,O,, -J~B,,,~O~~- (1 -s-y)%,O, (where G is the glassforming cation, A and B are the addedcation ands and y are mole fractions) hasbeen calculated from the formula suggestedin Ref. [ 188]. The bulk tnodulus on this compressionmodel is given by Kc,=SItKbc where Khc=nb y’s/9 and

where Kbc is the calculated bulk modulus according to the bond compressionmodel, S is the stucture factor, K,.] is the final estimatedbulk modulus, u,, is the number of network bondsper unit volume, r is the bond length, andfis the-first order stretching force constant. Finally, 12,.is the numer of network bondsper formula unit, NA is Avogadro’s number, p is the density, and M is the molecular weight of the glass, respectively. The Poisson’sratio is calculatedafterobtainiqg the averagecross-link density, which was suggestedin Ref. [ 1491 according to formulae (36) and (37) ,where II, is theaveragecross-linksper cation in the glass,(n,) i is the number of cross-linksper cation, (N,), is the number of cations pear glassfromula unit, and v is the total numberof cations per glassformula unit. The other calculatedelasticmoduli follow readily by combining the calculated bulk modulusandPoisson’sratio for eachglasssystemasfollows: Shear modulus,G,,=1.5 K,,((l-2a,,)l(l+~~,)) Longitudinal modulus,L,,=K,,+1.33 Young’s modulus,E,,=2G,,( It-g,,)

G,, (4Q

R. El-Mullarcany

Table 14 Parmeters adopted from the crystal Oxide Reference [ 1331 Coordination number Average bond length (nm) Bond force constant ( N m- ’ )


/ Materinls


ccnd Physics

53 11998)



of each oxide used

TCOz (2) 4 0.199 216

PbO (b) 4 0.230 139

0 I -0-Te-0-Te-0 I (4 0

LazO, Cc) 7 0.253 105

WO3 (a) 6 0.187 261

ceoz Cd) 8 0.248 112

Sm303 Cc) 7 0.249 110

Y203 (b) 7 0.228 143

ErA (b) 7 0.225 149

0 I I 0

000 \I/ ( ,050



Fig. 18. Schematic

two- and three-dimensional


of the crystalline

Table 14 and Fig. 18 give all the parameters obtained from the crystal structure of each oxide to calculate both bulk modulus and Poisson’s ratio. The parameters are average bond length r (nm), stretching force constant f (N m- ‘). and the coordination number of each cation. Table 15 gives the complete set of variables needed to calculate the bulk modulus and Poisson’s ratio using Eqs. (36)) (37) and (41) The important variable for the bulk modulus calculation is the number of network bonds per unit formula unit IZ,, which equals 4 for TeO,. By introducing a modifier with a higher value of I+, the structure will be more linked. For example, the number of network bonds per formula unit equals 6 for WO,, 7 for La,O,, Sm,O,, Y,03 and Er,O, and 8 for CeOz. By introducing PbO which has an rrr=4, no significant change in the structure occurs. Thus a more linked structure occurs by introducing a modifier with a higher number of network bonds per formula unit. The value of average cross-links per cation in the pure TeO, glass is 2.0, which changes to 2.55, 2.4 and 2.54 for different binary systems and reaches the value of 3.09 and 3.3 for the ternary and quaternary systems, respectively. Consequently, after increasing the number of cross-links per cation of the glass, the number of network bonds per unit volume will increase. After calculating these two main parameters, we conclude that the number of network bonds per unit volume, lzb has changed from 7.74 X 10” m- ’ for pure TeO, to 8.25 X 10’” m- ’ for binary glass A, 9.1 X 10” m- ’ for ternary system F,




(d) ‘.‘_____ :-e’ TeO,. PbO, W03, LazO,,



Er,OA and GO,.

and 9.64 X 10’” m- ’ for the quaternary system I. An increase in 12~increases the bulk moduli to values of 76.9, 87.1 and 9 1.1 Gpa for glasses A, F and I respectively. The high K, (Gpa) =25.3 (P,O,), 23.9 (GeO,), 12.1 (B,03) 36.1 ( SiOZ) moduli for pure or multicomponent glasses are due to the high number of network bonds per unit volume in the glass-forming structure. Previously [ 1231, we proved that the ratio between the calculated and the experimental bulk modulus is 2.3. This value is considerably smaller than that for other pure inorganic oxide glass formers [ 1491, I&,/ K,=3.08 (P205), 4.39 (GeO,), 10.1 (B,O,) 3.05 (SiO,). From Table 15 the ratio of K,,,/K, is in the range of 2.3 + 6% for all systems. This suggests that the elastic properties of tellurite glasses are mainly due to the Te-0 bond rather than the modifier bond. The relation between the ratio of K,,IK, and the calculated atomic ring size (K,=O.O106 F lm3.*‘, where 1 is in nm and F is in N rn-‘) is illustrated with the systematic relationship in Fi g. 17. The value of the ring size is in the same range as for pure TeO, glass which has a ring size of 0.5 nm. The structure sensitive factor, S has been found to be 0.43 + 10% for this family of tellurite glasses. Finally, the calculated Poisson’s ratio decreases steadily for this family of tellurite glasses. The steady decrease is interesting because it occurs due to the fact that the mean cross-link density per cation increases from 2.0 for pure TeO, to reach 3.3 for glass I.

Table 16 Comparison Glass


between Calculated (GPa)


and experimental

elastic moduli

elastic moduli

Experimental (GPa)

[ 1091

elastic moduli








EC :

32.6 37.0 33.2 39.6 37.7 37.0 36.3 34.3 38.7

24.5 24.9 23.3 28.3 26.1 25.6 24.7 19.3 19.9

65.2 70.1 64. I 77.0 72.3 71.0 69.8 60.0 65.1

58.7 61.0 56.6 68.5 63.6 62.4 60,2 47.5 48.0

33.1 33.5 33. I 43.6 39.4 39.7 40.7 36.0 45.1

24.9 25.2 26.7 30.7 24.8 25.0 26.1 23.3 31.1

66.3 67.1 67.6 84.6 72.5 73.0 75.5 67.2 86.7

59.8 60.5 ~ dzL 7476L.h 58 61.5 ~~ 72.6 75.9-

Table 17 Comparison between elastic moduli phosphate glasses [ 1091

of rare earth tellurite

and rare earth -

Glass composition

TeO, [ 1231 (Te0,hv(Sm203),,

Density (kg m-“)

, [2011

UeOd,,dLa2W,, , [2011 pzos I1881 JPLOS)r~.sFfSm,03),, tP,O,),,,,,(L~*OS)o.I,

,j 12031 PO31

5101 5782 5685 2520 3280 3413

Experimental (GPa) L


59.1 68.7 66.3 41.4 66.4 67.6

20.6 26.7 24.9 12.1 23.6 23.1



31.7 33.1 13.1 25.3 34.9 36.9

50.7 59.8 -605 31.4 57.8 57.2


With the calculated values of the bulk modulus and Poisson’s ratio, it is possible to do the following. ( 1) Compute the rest of the elastic moduli (longitudinal, shear and Young’s) using Eq. (42) (Table 16). The calculated moduli are in the range of the experimental values which have been measured before [ 2011. (2) Compare tellurite with other glass formers with the same modifier (Table 17). We have used binary tellurite and binary phosphate with samarium and lanthanum rare earth oxides. Also, the data for pure TeO, and Pz05 are included in Table 17. Comparison of these elastic properties reveals a rather surprising fact: tellurite glasses when modified with different rare earth oxides undergo a small change in elastic moduli in contrast to the phosphate glasses which undergo a large change in the elastic moduli when modified with different rare earth oxides. As an example, the measured bulk modulus of telluiite glass containing 10 mol% Sm,O, is 33.1 Gpa while that of pure TeO, glass is 3 1.7 Gpa. In binary rare earth phosphate glasses the situation is opposite; the addition of 15 mol% Sm203 causes a change in the buJk modulus from 25.3 to 34.9 Gpa. In phosphate glasses the difficulty in in: terpreting the trends of elastic moduli is substitution of a phosphate atom by another rare earth which produces simul-

117 Table IS Different factors which arc necessary Glass formula

for the calculation ,,I

Pure TeOz La,O,),, ,TeOz),,,, Ce0,1,, ,TeOz)c,c, Sm203),, lTcOZ),,v Er,O?),,,.~110~),,,TeO,),,, Y?OI),I,,~WOI),,?T~O~),,~, LGh),,,~ WOj),, ?TeO, ),I 7, Sm~OJ),,,,sWO~),,2Tr02),,,~ CeOL),,,,~W03),,2,T~02)1,,,, ErA 1o 1j2WO,~),,~,,~~O),I~T~O~),,I~

159.61 176.23 160.86 176.53 203.58 176.02 179.04 176.30 173.99 197.77

of Young’s

P (g cm-‘)


5.101 5,685 5.607 5.782 5.713 6.018 6.027 6.1 IO 5.78 I 6.813

1.5961 I .7623 I .6086 I .7653 2.0358 I .7602 I .7904 I .7630 I .7399 1.9777

modulus V, im’X 41.08 39.80 38.70 39.60 33.60 36.70 36.80 36.40 35.80 29.10

Table I9 Comparison het\veen the cxperimcntal and calculated elastic moduli bond compression and Mahishima and Mackenzie’s models) Glass formula

La20,h ,TeO,j,,,, Ce02),, ITeOz),,c, Sm,W,, iTeOzLlc, Er2031cl IWO~)cl.J~O~)l,h Y?W,,i,iWOi),~ zTQ),, La,Od,,,,, W03h2Te02h,, Smz0,),,,,sWO~),,~TeOz),7, CeOZ)~,,,SW0,),,21Te01)l17, Er203h,,2 WO,),,,<,pbO),,




II (kJ mol-‘X 146.85 142.817 144.584 143.972 158.983 156.132 155.137 154.890 155.628 152.282

G, ( X IO”)

V, ( X 10”)

G! ( X IO”)

c, ( X IO-?)

E (GPa)

1.69 4.61 5.13 1.72 1.46 5.34 5.22 5.37 5.17 5.25

0.2095 0.2262 0.1958 0.2289 0.1916 0.2209 0.2218 0.2224 0.2069 0.1983

7.48 8.12 8.25 8.33 9.08 9.40 9.35 9.47 9.00 10.38

65.570 70.14 62.25 69.90 68.40 64.60 65.90 64.20 62.28 57.55

37.15 47.60 42.90 48.70 51.90 50.80 51.50 50.83 46.90 49.90



E <,1/



[ 1071


L’s’ [71

58.7 61.0 56.6 68.5 63.6 62.3 60.2 47.5 48.0

59.8 60.5 63.1 74.7 61.6 58.8 64.5 72.6 75.9

47.6 42.9 48.7 51.9 50.8 51.5 50.83 46.9 49.9

-I 20 80






Fig. 20. Agreement and that calculated

between the experimental values of Young’s using Makishima and Mackenzie’s model.


taneous changes in coordination number, force constant, number of P=O bonds replaced by bridging bonds and pro-

glasses the calculated variables affecting the moduli. i.e. coordination number and average force constant are nearly the same. By using the previous equations in Section 2, the author’s group recently analyzed the elastic moduli of rare earth tellurite glasses [ 71. Tables 18 and 19 show that the value of the occupied volume of the glass in pure TeO,=41 X lo-’ changed to 38.7 X IO-’ m3 for binary cerium tellurite glass. The values of the occupied volume decreased for the tricomponent glass and also for the tetracomponent systems. Secondly. the dissociation energy of the present glasses increased more for multi-systems than that of pure tellurite glass. as in Table 18. The important variable of the dissociation energy is the number of network bonds per formula unit. Thirdly, the packing density of the glass depends upon the kind of modifier (the ionic radius of the modifier). The elastic moduli calculated by using the bond compression model and Makisima and Mackenzie’s model shows good agreement (Figs.

portions of the pyro- andmeraphosphatestructure.In tellurite


20 40 60 80 CALCULATED YOUNG'S MODULUS BY BOND COMPRESSION MODEL Fig. 19. Agrermcnt between the experimental values of Young’s modulub and that calculated using the bond compressional model.


R. El-Mdlwmy/



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