CHAPTER 6

ULTRASONIC STUDIES ON Ca0-B203-A1203-Na20

AND Ca0-B203-AI2O3-Fe203 GLASS SYSTEMS

CHAPTER VI

ULTRASONIC STUDIES ON Ca0-B203-A1203-Na20 and

CaO-B 0 -A1203-Fe203 GLASS SYSTEMS 2 3

6.1. Introduction

There is an ever increasing interest in the

measurement of elastic properties of solids using

ultrasonic methods, due to their non-destructive nature.

Elastic and acoustical properties of glasses are

si9nificant from the point of view of their application in

special devicesli]. The main reason for extensive

ultrasonic investigations of solids is the need for

elastic properties of materials like crystals, alloys,

plastics, ceramics, glasses and so on in a variety of

applications. The development of electronic circuits has

resulted in a variety of techniques, ranging in precision

from a per cent to a hundredth of per cent under various

conditions of temperature and pressure. The older static

and dynamic methods of measuring elastic constants of

large samples have gained wide acceptance due to their

simplicity. Among the various newer techniques pulse echo

methods are useful where measurements of highest precision

are needed.

An ultrasonic investigation of solids will help to

understand various solid state phenomena such as grain and

domain boundary effects in metals, ferromagnetic and

ferroelectric materials, the diffusion of atoms, molecules

and vacancies through a solid, the motion of imperfection

such as dislocations as well as the interaction between

the lattice sound vibrations and free electrons in metals

at low temperatures. All these effects are studied by

measuring elastic properties, internal friction properties

and their change with temperature, frequency and applied

electric field[2-41.

The measurement of elastic constants of solids is of

considerable interest and significance to both science and

technology. This measurement yields valuable information

reqarding the forces operative between the atoms or ions

in a solid. Since the elastic properties describe the

mechanical behaviour of materials, this information is of

fundamental importance in interpreting and understanding

the nature of bonding in the solid state. When a material

is subjected to a stress it will get strained and within

the elastic limit stress applied on a material is directly

proportional to strain (Hooke's law). The proportionality

constant relating the stress and strain is the modulus of

elasticity or the elastic constant. Commonly there are

three types of elastic constants[5]. They are (i) Young's

modulus (Y) (ii) Bulk modulus ( K ) and (iii) Rigidity or

shear modulus ( G ) . The Young's modulus relates a

unidirectional stress to the resultant strain. It also

represents the resistance to traction along the axis of a

thin bar or rod. The Bulk modulus ( K ) provides a good

link between the macroscopic elasticity theory and the

atomistic view points such as lattice dynamics. Basically

'l- it relates the pressure with volume st,ain. The shear

modulus (G) shows the relation between shear stress and

shear strain. In addition to the above elastic constants

there is a longitudinal modulus (L ) determined from the

velocity of propagation of longitudinal waves through a

solid. The kinds and number of elastic constants for

non-isotropic solids like crystals have been discussed by

various workers like Huntinqton[6], Nye[7It

Bhagavantam[B], Hearmon[9,10], Federov[ll], Musgrave[l2]

and others, and the use of physical acoustics to study the

properties of solids has been discussed by Mason[13-151.

Amorphous materials like glasses exhibit some unique

properties which are not usually found in other

engineering materials. These materials lack the long-

ranye periodicity in the arrangement of atoms. The study

of the propagation and attenuation of waves in

c,lasses[16,17] is of special and vital significance due to

the observation of anomalous specific heat[l8] and thermal

conductivity at low temperature[l9]. The ever increasing

study of glasses is also due to their anomalous physical

properties apart from practical applications[2]. Inspite

of the immense use of ultrasonic techniques in

understanding the structure and properties of glasses only

a limited number of reports have appeared on such studies.

Ultrasonic studies on binary alkali oxide and other oxide

glasses have been reported. The studies of ternary

glasses are sparse, while on quarternary glasses are

almost totally lacking. A brief review of the latest

ultrasonic studies in binary and ternary glasses is given

in the following section.

6.2. Ultrasonic Investigations in Oxide Glasses - A Brief Review

The ever increasing interest in the investigation of

5lasses is motivated by their widespread practical

application and the fact that they exhibit a number of

anomalous physical properties, which suggest specific

structural singularities that differentiate the glassy

state of matter from the crystalline as well as the

ordinary amourphous state[21. So far, however, a unified

theory of the glassy state of matter has failed to emerge,

and so the specific structure of glasses continue to

be less than fully understood. These specific attributes

are extremely pronounced, in particular, in the acoustical

properties of glasses, primarily in the composition and

temperature dependence of the velocity and absorption of

ultrasonic waves[2,20]. For that reason a great many

publications have been devoted to the investigation of

glasses by ultrasonic methods. A brief review of the

earlier works on ultrasonic studies of inorganic glasses

is given in this section.

Reports on ultrasonics investigations on glasses up

to 1976 have been reviewed by Kul'bitskaya et a1.[20]

In 1985 Kodama[21] reported the elastic properties

of barium borate glasses. By making use of the ultrasonic

pulse echo overlap method, ultrasonic velocities in barium

borate glasses were measured at 298 K over the single

phase composition range. The results of the elastic

constant measurements of the glasses as a function of

composition were discussed with the help of the relation

wv2 2 2 = ( a Vm/ and ) which was derived from the

Sm finite elastic strain theory, where M is the molar mass, V

the velocity of the longitudinal or transverse wave, Urn per

the internal energy unit mole, nH the Lagranqean strain

component specifying the sound wave, and Sm the molar

entropy. Based on this relation, elastic internal

energies per unit mole of the glasses are determined as

functions of composition in relation to the behaviour of

N4r the fraction of boron atoms in tetrahedral

coordination.

Elastic constants and structure of the glass system

Co 0 -P 0 had been determined by Higazy et a1.[21] by the 3 4 2 5

ultrasonic techniques at 15 MHz. They found that Young's,

bulk, shear and longitudinal moduli and the Poisson ratio

are sensitive to the composition of the glass. From the

ultrasonic data obtained, it was found that the glass

system could be divided into three "compositional

regions". This behaviour had been qualitatively

interpreted in terms of the cobalt coordination, crosslink

densities, interatomic force constants and atomic ring

sizes. They also presented a full discussion of effects

of annealing on elastic properties of the cobalt phoskhab

glasses.

Ultrasonic sound velocities behaviour in silver

borate glasses were investigated by Carimi et a1.[23].

They studied the sound velocity of 5 MHz longitudinal and

transverse waves in silver iodide - silver borate glasses

and observed in the 77-430 K temperature range the

presence of dispersive effects, whose contribution

increased with the AgI content. These effects arise from

+ the thermally activated jumps of Ag ions, between nearly

equivalent positions available in the glassy network. The

whole behaviour was explained by the overlap of two

different mechanisms, the relaxational one and the one

coming out from the anharmonicity of the system. This

last effect implies, in a quasi-harmonic approximation, a

linear temperature dependence of the elastic constants in

all the explored ranges.

The velocity and absorption coefficient o f

longitudinal ultrasonic waves of frequency 5 and 10 MHz in

molten glassy Na 0-SiO K 0-SiO and PbO-SiO and molten 2 2' 2 2 2'

Na 0-B 0 and PbO-B 0 were measured by means of the 2 2 3 2 3

pulse-echo method at 300 to 1600 K by Kazuhira

Nagata et a1.[24]. They observed that the velocity of

sound decrease with increasing ternprature and decreased

rapidly near the transition temperature of the glass

system. The mean free path of phonons was also estimated

from the velocity of ultrasonic sound, thermal

conductivity, and specific heat capacity.

The temperature dependence of 15 MHz ultrasonic bulk

wave velocity in the range 4 to 600 K in Moo3-P205 glass

system was reported by Bridge et a1.[25] in 1987. They

concluded that a complete understanding of temperature

gradients of elastic moduli in glasses generally requires

the measurement of both acoustic wave velocity and wave

absorption as a function of temperature, so that the

relaxational contribution to the gradients can be computed

and substrated from the experimental gradients.

Damodaran et al.[26] reported the elastic properties

of lead containing MOO -P 0 glasses using ultrasonic 3 2 5

velocity measurements at 10 MHz. They observed that the

composition dependence of elastic moduli, Poisson's ratio

and the Debye temperature were consistant with a

structural model proposed by Selvaraj et a1.1271.

According to this model lead acts both as a network former

and as a network modifier in different composition

regimes. They suggested that the incorporation of lead

into the network is accompanied by the conversion of

three-connected tetrahedra into four-connected tetrahedra

in the network. Longitudinal and shear velocities were

found to decrease gradually as the concentration of PbO

increased. The results were interpreted with the help of

the structural model proposed by Selvaraj et a1.[27].

Ultrasonic studies and calculation of elastic and

thermodynamic properties of alkaline earth containing

silicate glasses were investigated by Batti et a1.[28].

They made an effort to test the model proposed by

Makishima and Mackenzie[29,30] for the direct calculation

of the Young's modulus of silicate glasses of different

compositions. Batti et a1.[31] also studied the

softening temperature and Debye temperature for the

alkaline earth silicate glasses.

They also reported[32] the attenuation and velocity

measurements of ultrasonic waves in strontium borate

glasses and their elastic properties. They observed a

variation of velocity, attenuation, longitudinal modulus

and coefficient of thermal expansion of the glasses with

the frequency of the ultrasonic waves.

Ultrasonic velocities in Vanadium-barium-borate

glasses were measured at 298 K by making use of the

ultrasonic pulse-echo technique at three frequencies by

Anand Pal Singh et a1.[33]. They calculated the molecular

weight, packing density, mean atomic volume and effective

number of atoms in these glass samples. They also

calculated the longitudinal modulus of elasticity,

internal friction and thermal expansion coefficient with

the help of the ultrasonic propagation velocity. They

observed that the values of ultrasonic velocity and the

dynamic modulus of elasticity exhibit considerable

variation at each frequency due to variation in structure

and composition of the glass. Values of longitudinal

modulus were found to increase with the B203 content and

with the frequency of the ultrasonic waves. The results

of ultrasonic, X-ray and infrared measurements on xBaO-

(0.9-x)B203-0.10Fe203 glasses have been reported recently

by Anand Pal Singh et a1.[34]. They have concluded that

introduction of Fe203 in the matrix of BaO-B 0 softens 2 3

the material and that Fe203 do not enter the boron-oxygen

network but, after dissociation into Fe3+ and 02-, sit in

cavities inside the structure.

Ultrasonic studies in sodium borate glasses were

reported by Sidkey et a1.[35] in 1990. They observed

that ultrasonic velocity increased as the sodium oxide

concentration was increased upto 27.2 mol%. A similar

trend was observed in the case of Young's, bulk and shear

moduli. The increase in velocity was attributed to the

increase in packing density due to a decrease of B203, and

therefore an increase in the B04 groups and consequently

occupation of the intersticies by the alkali ions. They

compared the experimental results with those calculated

theoretically from equation derived by Makishima and

Mackenzie[29,30]. They also studied the boron anomaly and

the results showed that this anomaly should appear at

concentrations of sodium oxide above 28 mol%.

Padake et a1.[36] investigated ultrasonic velocity,

and absorption in ZnO - B203 glasses at 2 MHz frequency

for different temperatures. They observed a peak in the

value of attenuation for all glasses and the velocity was

found to be decreasing with increase of temperature.

Experimental results were explained on the basis of

tunneling defect atom and the structural mechanism which

is totally responsible for the strong absorption in

glasses. Ultrasonic studies in binary zinc borate glasses

xZn0-(1-x) B203 were also reported by Singh et al. in

1992[371. They had calculated the elastic moduli of the

glasses and compared the results with those predicted by

P,lakishima-Machenzie mode1[29,30].

Temperature dependence of velocity of longitudinal

and transverse ultrasonic waves in V 0 -P 0 glass 2 5 2 5

system was investigated by Mukherjee et a1.[381. The

experimental results showed that unlike most of the

glasses having tetrahedrally coordinated structures, '2'5-

'2'5 glasses which contain both tetrahedral and octahedral

structures[39] do not indicate any minimum in the

variation of sound velocity with temperature but instead

show a steady decrease of velocity with a small negative

temperature coefficient.

Recently Kodama[40] reported ultrasonic velocity in

potassium borate glasses as a function of concentration of

K20. They observed a strong dependence of the ultrasonic

velocity on the concentration of K20.

The elastic properties of these glasses were analysed

in terms of the three structural units, on the assumption

that these structural units have their respective elastic

constants. They have shown that the elastic constants of

these structural units are defined on the basis of the

elastic internal energy due to deformation.

Ultrasonic velocity and elastic properties of the

ternary glass system Sr0-Ba0-B203 were reported very

recently by Anand Pal Singh et a1.[41]. They observed that

ultrasonic velocity and acoustic impedance in these

glasses increased with the concentration of strontium

oxide. The role of SrO and BaO (modifier) was shown to be

diametrically opposite to their role in silicate glasses.

The elastic moduli of these glasses were obtained making

use of Makishima and Mackenzie mode1[29,30].

6.3. Theory

The ultrasonic velocity in solids yields the

appropriate elastic modulus of the mode being propagated.

The relation can be expressed as

Where P is the density of the solid and M is the

apgropriate combination of the elastic moduli of the

solid. The combination depends on the mode of

propagation, and the mode in turn depends on the

interaction of the wave with the boundaries of the solid.

Since solids can sustain shearing strains elastically,

they will support the propagation of waves with transverse

as well as longitudinal particle motion. The moduli of

materials are influenced by many physical phenomena which

may in turn be studied by measuring the ultrasonic wave

velocities.

Within the elastic limit, majority of solids obey

Hooke's law which states that stress is directly

proportional to strain. Then,

Where p is the normal (tensile) stress and is the

strain. E is the moduli of elasticity. Similarly the

shear stress 1 is directly proportional to the shear

strain.

where G is the modulus of elasticity in shear. When a

sample is extended in tension, there is an accompanying

decrease in thickness; the ratio of the thickness

decrease to the length increase in the Poisson's ratio 6

where A d and ~l are the change in thickness and length,

and d and 1 are original thickness and length

respectively.

Poisson's ratio relates the Young's modulus and shear

modulus by the following equation.

This relationship is only applicable to an isotropic

body in which there is only one value for the elastic

constant independent of direction. Generally this

equation is a good approximation for glasses and for most

polycrystalline ceramic materials.

Under conditions of isotropic pressure the applied

pressure P is equivalent to a stress of -P in each

principal directions. In each principal direction, we

have a relative strain.

The relative volume change is given by

The Bulk modulus K defined as the isotropic pressure

divided by the relative volume change is given by

The elastic constants of the solids are calculated

from the measured densities and the velocities of

longitudinal (VL) and transverse (Vs) ultrasonic waves 5 ,

using the following expressions[$2].

Longitudinal modulus L = 2 "L ..... (6.9)

-~

Shear modulus G = P V s 2 ..... (6.10) Bulk modulus K = L - (4/3) G ..... (6.11)

1-2 (VS/VL) 2 Poisson's ratio 6 = -------------

2 ..... (6.12) 2-2 (VS/VL)

Young's modulus E = (1 +6 ) 2G ..... (6.13)

6.4. Work Undertaken in the Present Study

In the present study two systems of quarternary

glasses CaO-B 0 - A1203-~a 0 and CaO-E 0 -A1 0 -Fe 0 2 3 2 2 3 2 3 2 3

containing different concentrations of Ma20 and Fe203

respectively were prepared. Longitudinal and transverse

ultrasonic velocity in these glasses were determined using

ultrasonic pulse echo overlap technique. The elastic

moduli and Poisson's ratio with concentration of Na20 and

Fe 0 are discussed. 2 3

6.5. Experimental Details

Two systems of ylass samples 10Ca0-(75-x) B 0 -15 2 3

A1 0 -xNa 0, x varying from 15 to 24 mol% and 20 2 3 2

CaO-(70-y) B 0 - 10 A1203 - y Fe203, y varying from 2 to 2 3

8 mol% were prepared as described in Section 3620f

Chapter 3. Glass samples of thickness about 10 mm and

with smooth and parallel end faces were obtained.

Velocity of longitudinal and transverse ultrasonic waves

in the glass samples were determined using Matec 7700

ultrasonic velocity system and using respectively x cut

and y cut quartz transducers each of frequency 3 MHz. The

block diagram of the experimental set up (figure 2 . 5 ) and

the procedure for the measurement of the ultrasonic

velocity are described in detail in Section 2 . 5 of

Chapter 2 . The path length of the ultrasonic waves in the

glass samples were determined by measuring the thickness

of the glass samples using a micrometer. Longitudinal and

transverse ultrasonic velocity in the glass samples

containing different concentrations of Na 0 and Fe203 were 2

determined. The density of the glass samples were

measured making use of Archimede's principle and using

water as the immersion liquid.

6.6. Results and Discussions

Longitudinal ( V L ) and transverse ( V ) velocities of T *

ultrasonic waves of frequency 3 MHz in quarteAnary glass

systems CaO-B 0 -A1203-Ba20 and 2 3 CaO-B 0 -A1 0 -Fe203 2 3 2 3

containing different concentrations of Na20 and Fe203,

respectively, are given in table 6.1. The density of the

glass samples was found to increase with increase in the

concentration of Na 0 and Fe203. 2 It is seen from

figure 6.1 and 6.2 that both VL and VT increase almost

Table 6.1

Variation of ultrasonic velocities, Poisson's ratio and elastic moduli in CaO-B 0 -Al 0 -Na 0 (SS) with varying concentration of Na 0 and in CaO-B 0 -Al 0 -Fe 0 ?F$)

2 3 2 2

with varying concentration of P$ 3 2 3 2 3 2 3

Sample Longitudinal Transverse Dens'ty Poisson's Longitudinal Shear Bulk Young's f Name velocity velocity kg/m ratio modulus modulus modulus modulus

m/sec m/sc K bar K bar K bar K bar

Figure 6.1 Variation of longitudinal and transverse velocities in CaO-B 0 -A1 0 -Na 0 with varying concentrations20? N~;O! 2

Figure 6.2 Variation of longitudinal and transverse velocities in CaO-B 0 -A1 0 -Fe203 with varying concentration$ df ~ 6 ~ d ~ .

regularly with the concentration of Na20 or Fe203. But

the rate of increase of V is greater than that of VT for L

both the glass systems investigated. The values of the

three elastic constants and the ~oisson's ratio evalu ated

usins expressions 6.9 to 6.13 are given in tables 6.1.

It is seen that for both the glass systems the modulii of

elasticity show almost a regular increase over the entire

variation of concentration of Na 0 and Fe 0 [figure 6.3 2 2 3

and 6.4). But the Poisson's ratio exhibit a reverse trend

(figure 6.5 and 6.6).

From X-ray diffraction studies by Biscoe and

Warren[43] had pointed out that as an alkali oxide is

added to B203, the coordination of boron which is 3 in

B203 changes to 4. It is known for some time that the

physical properties of binary borate - glasses display

unusual trends with change in their composition. This

behaviour known as "boron oxide anomaly", has been

investigated by many workers. Internal friction studies

of sodium borate glasses[44,45] showed that this anomaly

occurs at 15 mo18 of alkali oxide, while Abe screening

theoryL461 suggested the saturation of BO to occur at 4

16 mol%. But ultrasonic studies of Gladkov and

Tarasov[47] showed this anomaly to occur at 35 mol% Na20.

Figure 6.3 Variation of elastic constants in CaO-8 0 - iil 2 o 3 -Na20 with varying concentrations2 a t Na20.

Figure 6 . 4 Variation of elastic constants in CaO-B 0 - A1203-Fe203 with varying concentrations2 af Fe 0

2 3'

Figure 6.5 Variation of Poisson's ratio in CaO-B 0 - A 1 2 0 3 -Na20, with varying concentrations2 a f Na203.

Figure 6.6 Variation of Poisson's ratio in CaO-B 0 - A1203-Fe 0 with varying concentrations2 df Fe203. 2 3

Recent ultrasonic studies by Sidkey et a1.1351 on sodium

borate glasses have showed that both longitudinal and

transverse ultrasonic velocity in sodium borate glasses

and the elastic constants increased with concentration of

Na20 upto 27 mol%. The increase in ultrasonic velocity

has been attributed to an increase in packing density due

to the transformation of coordination of boron from 3 to 4

and consequent occupation of the intersticies by the

alkali ions. But once BO groups get saturated, non- 4

bridging oxygens start appearing producing a loose

structure. This phenomenon was not observed by

Sidkey et a1.[35] upto a concentration of 28 mol% of Na20.

In these studies Poisson's ratio was found to increase

with increase in Na 0 concentration. They pointed out 2

that addition of Na 0 changes the coordination of boron 2

from three to four making the glass strong and rendering

maximum rigidity. KodamaI401 have measured the elastic

properties of potassium borate glasses as a function of

concentration of K 0 and analysed the elastic properties 2

in terms of the three structural units represented by Bg3,

+ X+B g26 and K B o4 , where P) represents a bridging

- oxysen and 0 a non-bridging oxygen, on the assumption

that the three structural units have their respective

elastic constants. It was shown numerically that the

structural unit B B 4 increases the rigidity of the glass

+ whereas the unit K ~(3~0- decreases it.

In the present ultrasonic investigations both the

longitudinal and transverse ultrasonic velocities were

found to increase with concentration of Na 0 and Fe203. 2

Also the elastic constants showed almost a regular

increase with concentration of Na20 or Fe203. These

results may be explained by making use of the results of

ultrasonic investigations on binary borate glasses

reported in the literature[35,40] as is done in the case

of laser Raman spectra where results from the Raman

studies of binary glasses are made use of in the

interpretation of spectra of ternary and quarternary

glasses[48,49]. Results of laser Raman studies (Chapter 5

of this thesis) of the ternary glass CaO-B 0 -A1203 showed 2 3

that the structure of the glass consists of mainly boroxol

rings containing only bridging oxygen. When Na20 is added

to this glass system (so that the resultant glass is CaO-

B 0 -A1203- Na20) the structure was found to consist 2 3

mainly of tetraborate groups and at high concentration of

Na 0 pentaborate groups were formed (Chapter 5 of this 2

thesis). A few percentage of diborate-pentaborate and

other groups having bridging oxygens were also detected

in this structure. The main structural units in

quarternary glass CaO-B 0 -A1 0 -Fe203 were found to be 2 3 2 3

boroxol rings for low concentration of Fe 0 where as at 2 3

high concentration boroxol rings transform into other

9roups all having only bridging oxygens (Chapter 5).

In these transformations boron undergo a change from three

coordinated to four coordinate @"and it is reported that

the presence of B a 4 increases the rigidity of the

glass[40]. It has also been reported that in the case of

binary sodium borate glasses as the concentration of Na20

is increased the packing density increased due a

transformation of coordination of boron from 3 to 4 and

consequent occupation of the intensities by the alkali

ions[35]. The increase in ultrasonic velocity and the

elastic moduli in the present study may also be attributed

to the increase in packing density and rigidity of the

c,lass samples as the concentration of Na20 or Fe20j is

increased. The laser Raman spectra indicated the presence

of a few loose diborate and loose B04 groups. Their

concentration should be small since a large concentration

of these groups should Lead to a decrease in the rigidity

of the glass resulting in the decrease of ultrasonic

velocity and the elastic constants, whereas an increase in

these quantities were observed. Poisson's ratio had been

reported to be increasing with alkali oxide concentration

in binary oxide glasses[35], while it had been observed

to decrease upto a certain concentration of ZnO and then

increase in the case of zinc oxide glasses[37]. In the

present study, Poisson's ratio showed a regular decrease

with increase in concentration of both Na 0 and Fe203. 2

The regular variation of ultrasonic vel(-~ities and the

elastic constants of the two systems of quarternary

glasses investigated in the present study show that the

transformation of the structural groups in these glasses

to other groups is systematic and does not cause a

disruption of the structure which is also supported by the

Raman scattering results (Chapter 5 of thls thesis) that

the Ranan peak characteristic of a continuous random

network was prominently present in the spectra of all the

$lass samples investigated. The ultrasonic velocity or

the elastic constants do not show a decreasing trend in

any of the glasses. This may be attributed to the reason

that within the variation in Na 0 and Fe 0 studied, B 0 4 2 2 3

groups do not get saturated and show a trend for the

fornation of nonbridging oxygens leading to a loose

structure.

6.7. Conclusion

Ultrasonic velocity of longitudinal and transverse

waves of frequency 3 ElHz has been determined in two

quarternary glass systems. The elastic constants and

Poisson's ratio have been evaluated. The increase in the

values of ultrasonic velocity and elastic constants has

been attributed to an increase in the packing density and

rigidity of the glass samples as a result of a

transformation of the coordination of boron from 3 to 4

when the concentration of Na 0 and Fe 0 respectively in 2 2 3

the two systems of glasses is increased. It is also

concluded that the transformation of the groups

constituting the structure of the glass into other groups

on increasing the concentration of Na 0 or Fe 0 does not 2 2 3

affect the rigidity of the glass so that the random

continuous network of glass is maintained.

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