Elastic studies of Glass Materials Studied by Ultrasonic Technique


Outline• Glass in General• Ultrasonic Waves• Physical Properties of Glass

– Preparation– Density and Molar Volume

• Elastic Properties– Compositional Dependence – Temperature Dependence– Hydrostatic Pressure Dependence

• Conclusion

Glass prism


Glass – An IntroductionWhat is a glass?

Glass - hard, brittle solid material that is normally lustrous and transparent in appearance and shows great durability under exposure to the natural elements.

The latin term glesum, probably originated from a Germanic word for a transparent, lustrous substance.

The term glass developed in the late Roman Empire.

Natural heat-producing processes like volcanoes and lightning strikes are responsible for creating various forms of natural glass.

Obsidian - super-heated sand or rock

that rapidly cooled.

Moldavite formed by meteorite impact (Besednice, Bohemia)


General Introduction

• These three properties—lustre, transparency, and durability

— make glass a favoured material for such household objects as windowpanes, bottles, and lightbulbs.

Clear glass for Incandescent light bulb

ThermalInsulationGlass


Glass ApplicationSpecial properties of glass make it

suitable for folllowing applications– flat glass– Tempered glass– Annealed glass– Laminated glass– container glass– optics and optoelectronics material– laboratory equipment– thermal insulator (glass wool)– reinforcement fiber (glass-

reinforced plastic, glass fiber reinforced concrete)

– and art.


Glass Technology

Glass building

Energy saving mirror


Glass Arts

Decoration Glass- Venetian millefiori.Chinese ring

Glass beads are made with silica (usually from sand). Cross section of Korean broken beads


Glass Art

Roman Cage Cup from the 4th century A.D.

Roman glass A 16th-century stained glass window.

A vase being created at the Reijmyre glassworks, Sweden

Stained glass


Next-generation large-scale panels

Glass substrates for LCDs

To form various functional films on glass substrates.

ASAHI Glass, Japan


Specific Potential Application of Glassy Materials

CD memory device

Optical switching device

Non-linear optical devices

Electrochemical devices

Laser host

Infra-Red Fiber Optics

Optical waveguides


Outline• Glass in General

– Definition

• Ultrasonic Waves• Physical Properties of Glass



• Conclusion

Glass prism


.

Glass Definition♦ Glass is an inorganic product of fusion that has cooled to a rigid condition without crystallization (American Society for Testing Materials ,1945) True for most commercial materials (e.g., soda-lime-silica) but ignores organic, metallic, H-bonded materials, ignores alternate processing routes (sol gel, CVD, electron or neutron -bombardment, etc.)

♦ Glass is an X-ray amorphous material which exhibits a glass transition. (Wong and Angell, 1976)

Not all amorphous solids are glasses; wood, cement, a-Si, thin film oxides, etc. are amorphous but do not exhibit the glass transition.

♦Glass is an undercooled liquid." Problems: glasses have 'solid' properties (e.g., elastic material)No flow at room temperature

♦ Glass as any isotropic material, whether inorganic or organic, which lacks three dimensional atomic periodicity and has a viscosity greater than about 1014 poise (Mackenzie, 1960)


Zachariasen’s model

According to Zachariasen, in order for a given oxide AmOn to form a glassy solid, it must meet the following criteria:

(1) the oxygen should be linked to no more than two atoms of A,

(2) the coordination number of the oxygen about A should be small, on the order of 3 or 4,

(3) the cation polyhedra must share corners only, and

(4) at least three corners must be shared.

In 1932, physicist W.H. Zachariasen defined glass is an extended, three-dimensional network of atoms that form a solid which lacks the long-range periodicity (or repeated, orderly arrangement) typical of crystalline materials.


Definition

• These criteria are useful guidelines for the forming of conventional oxide glasses, but unsuitable for nonoxide glasses.

• Chalcogenide glasses

– for instance, are chains of random lengths and random orientation formed by the bonding of the chalcogen elements sulfur, selenium, or tellurium.

– Ions of these elements have a 2-coordination requirement, and the chains are cross-linked by 3- or 4-coordinated elements such as arsenic, antimony, or germanium.

♦ Glass is a solid that possesses no long range atomic order and, upon heating, gradually softens to the molten state. Non-crystalline structure Glass transformation behavior


Outline• Glass in General• Basic Ultrasonics• Physical Properties of Glass



• Conclusion

Glass prism


Basic Ultrasonics

• “Ultrasonic" refers to sound that above the frequencies of audible sound, (beyond 20 kHz)

• Ultrasonic can be produced by transducers – piezoelectric effect

– magnetostrictive effect

• Piezoelectricity is the ability of some crystals (quartz) and certain ceramics materials to generate an electric potential in response to applied mechanical stress.

• Magnetostrictive transducers use magnetic strength to produce high intensity ultrasonic sound in the 20-40 kHz range for the ultrasonic cleaning and also other mechanical applications


Ultrasonic wave are able to propagate in medium by method such as:• Reflection • Refraction • Propagation• Transmission• Dispersion• etc

Advantages of ultrasonic wave

• ability to form coherent wave in which amplitude, frequency, • direction of propagation can be controlled

Advantages of ultrasonic wave


Type of ultrasonic wave

2 type of ultrasonic wave traveling in solid such as glass: longitudinal wave

shear wave

Longitudinal wave also known as compressional waves•the oscillation of the particle is forward and backward, compressing, and depressing.

Shear wave also known as transverse wave•the oscillation of particle in medium is at right angles to the direction of propagation.•Shear wave can only propagate through solid and cannot propagate in liquid and gas.

N

N

U

U


Outline• Glass in General• Ultrasonic Waves• Physical Properties of

Glass– Glass Preparation– Density and Molar Volume


• Conclusion

Glass prism


2 methods of preparing glass samples

Glass Preparation

Conventional Method Cooling from the liquid

state/ Melt quenching technique

Condensation from the vapor

Pressure quenching Solution hydrolysis

Unconventional Method

Unconventional meltingSolution methodsDeposition methodsSolid-state transformations


Glass FormationNormally, glass is formed upon the cooling of

a molten liquid in such a manner that the ordering of atoms into a crystalline formation is prevented.

Instead of the abrupt change in structure that takes place in a crystalline material such as metal as it is cooled below its melting point

in the cooling of a glass-forming liquid there is a continuous stiffening of the fluid until the atoms are virtually frozen into a more or less random arrangement similar to the arrangement that they had in the fluid state.


Crystal vs Glass

Glasses•lack of long-range order results in larger volumes (lower density), higher energies; atoms could rearrange to form denser structures if given enough thermal energy and time.

•thermodynamically metastable phase

Crystalsordered atomic structures mean smaller volumes (high density) & lower energies

•thermodynamically stable phase


Glass StructureGlass • amorphous • isotropic macroscopic physical properties • no grain structure when viewed under an optical microscope.

Glass structure relates to various physical properties such as density, thermal expansion, viscosity, surface tension and also miscellaneous mechanical (including elastic), chemical and electrical properties of glass.


Glass Structure

Two-dimensional representation of (a) an oxide crystal and (b) a glass of the same chemical composition (A2O3) due to Zachariasen (1932)

Schematic two-dimensional representation of the microscopic structure of binary oxide glass; (a) composed of basic glass former and glass former; (b) showing the effect of network

modifying cations on the network of the glass former.


Structure of Binary Borate Glass

Some structural groupings in borate glasses as indicated from nuclear magnetic resonance experiments (Bray 1985). Small solid circles represent boron atoms, open circles oxygen atoms and an open circle with negative sign indicates non-bridging oxygen.





• Conclusion

Glass prism

Quartz sand (silica) as main raw material for commercial glass production.


Substance is weight First Furnace400 C for 30 min

Second Furnace750-800 C for 60 min

Pour melt into mould Placed molten and mould in

First Furnace and annealed at 400 C

Removed the mould after the melt

was hard enough

Cut and polished sample

GLASS PREPARATION PROCESS


Preparing Commercialized Glass• Pure silica (SiO2) melts at a viscosity

of 10 Pa s (100 P)— of over 2300 °C (4200 °F).

• Sodium carbonate (Na2CO3) lowers the melting point to about 1500 °C (2700 °F) in soda-lime glass

• Soda makes the glass water soluble, which is usually undesirable, so lime (calcium oxide (CaO), some magnesium oxide (MgO) and aluminium oxide are added to provide for a better chemical durability.

• The resulting glass contains about 70 to 74 percent silica by weight and is called a soda-lime glass. Soda-lime glasses account for about 90 percent of manufactured glass.

Lead glass, such as lead crystal or flint glass, is more 'brilliant' because the increased refractive index causes noticeably more "sparkles",

while boron may be added to change the thermal and electrical properties, as in Pyrex.

Marlinda Daud, Pusat Minerologi Ipoh


Special Glass

• Adding barium also increases the refractive index.

• Thorium oxide gives high refractive index and low dispersion, (for high-quality lenses) but due to its radioactivity has been replaced by lanthanum oxide in modern eye glasses.

• Large amounts of iron are used in glass that absorbs infrared energy, such as heat absorbing filters for movie projectors,

• cerium(IV) oxide for absorbs UV wavelengths (biologically damaging ionizing radiation).

Finally, fining agents such as sodium sulfate, sodium chloride, or antimony oxide are added to reduce the bubble content in the glass.


Other Types of GlassBesides common silica-

based glasses, many other inorganic and organic materials may also form glasses, including – plastics (e.g., acrylic glass) – phosphates, – borates, – chalcogenides, – fluorides, – germanates (glasses based

on GeO2),

•tellurites (glasses based on TeO2), • antimonates (glasses based on Sb2O3), • arsenates (glasses based on As2O3), • titanates (glasses based on TiO2), • tantalates (glasses based on Ta2O5), • nitrates, carbonates and many other substances.


Colored Glass• Color in glass may be obtained by addition

of electrically charged ions and by precipitation of finely dispersed particles (such as in photochromic glasses).[

• Ordinary soda-lime glass appears colorless• iron(II) oxide (FeO) impurities of up to 0.1

wt% produce a green tint

• Further FeO and Cr2O3 additions may be used for the production of green bottles.

• Sulfur, together with carbon and iron salts, is used to form iron polysulfides and produce amber glass ranging from yellowish to almost black.

• Manganese dioxide can be added in small amounts to remove the green tint given by iron(II) oxide.


Colored Glass

IonSilicate-based

GlassPhosphate-based Glass

Fe+2 deep blue-green slight greenish blue

Fe+3 yellowish-brown slightly brownish

The color of a glass may depend upon the nature of the glass as well as the coloration ion.

For example, iron ions have the following color influences:

SEM Photos XRD Pattern of Starting Materials

TeO2 powder

TeO2 glass

ZnO Powder TeO2-ZnO glass

0200400600800

1000120014001600180020002200240026002800300032003400

10 20 30 40 502 theta

Inte

nsity

(a.

u)

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

10 20 30 40 502 Theta

Inte

nsity

(a.

u)

0

5000

10000

15000

20000

25000

30000

35000

10 20 30 40 502 Theta

Inte

nsity

(a.

u) TeO2-ZnO-AlF3 glassAlF3 (97.0%)

Powder


XRD patterns

100

600

1100

1600

2100

10 20 30 40 502 theta

Inte

nsity

(a.u

)

TZ7

TZ6

TZ5

TZ4

TZ3

TZ2

TZ1

TZ0

• no discrete or continuous sharp peaks • but broad halo at around 2 260 - 300, which reflects the characteristic of amorphous materials. • absence of long range atomic arrangement and the periodicity of the 3D network in the quenched material

400

600

800

1000

1200

1400

1600

1800

10 20 30 40 502 theta

Inte

nsity

(a.u

)

S5

S4

S3

S2

S1

TeO2)1-x (ZnO)x (x = 0.1 to 0.4 in 0.05) (TeO2)90(AlF3)10-x(ZnO)x (x = 1 to 9)

binary ternary


Glass Forming Region


Density & Molar Volume

M

V

Molar volumes

acaca

as ww

w

Density Measurement (Archimedes Method)



Variation of density and molar volume with mol% Bi2 O3 in Bi2 O3–B2 O3 glass systems.

The increase of the density of the glasses accompanying the addition of Bi2 O3 is probably attributable to a change in cross-link density and coordination numbers of Bi3+ ions.

26

26.5

27

27.5

28

28.5

29

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Mole fraction of TeO2

Mo

lar

volu

me(

cm3 m

ol-1

)

4650

4700

4750

4800

4850

4900

4950

5000

Den

sity

(kg

m-3

)

Density and molar volume of TeO2.B2O3 glasses

28

28.5

29

29.5

30

30.5

0.05 0.10 0.15 0.20 0.25 0.30 0.35

Pecahan Mol Ag2O

Isip

adu

mo

lar

(cm

3)

4800

4900

5000

5100

5200

5300

Ket

um

pat

an (

kg/m

3)

Density and molar volume of [(TeO2)x (B2O3)1-x)]1-y [Ag2O]y


Density and Molar Volume

3500

4500

5500

6500

7500

0 20 40 60 80Bismuth Oxide (mol%)

Dens

ity (k

gm-3

)

Dependence of density on the composition of bismuth oxide glass systems as measured by El-Adawy and Moustafa (1999) (5 - 45 mol%), Wright et al (1977) (20 – 42.5 mol%) and present works (40 – 70 mol%).


4700

4800

4900

5000

5100

5200

5300

5400

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4

Mole fraction of ZnO

De

ns

ity

(k

g/m

3 )

22

24

26

28

30

32

34

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4


Mol

ar v

olum

e (1

0-6

m3 m

ol-1

)

•The increase in density indicates zinc ions enter the glassy network

•The decreases in the molar volume was due to the decrease in the bond length or inter-atomic spacing between the atoms

• The stretching force constant (216 N/m – 217.5 N/m) of the bonds increase resulting in a more compact and dense glass.

• Atomic Radius (Shelby, 2005).

•R(Zn2+)(0.074 nm) << R(Te2+)(0.097 nm)

•there is no anomalous structural change (non-linear behaviour)




• Elastic Properties– Theory– Compositional Dependence – Temperature Dependence– Hydrostatic Pressure Dependence

• Conclusion

Glass prism

A modern greenhouse in Wisley Garden, England, made from float glass.


Elastic Properties of MaterialsApplication of external forces to a

solid body produces complex internal forces which cause

– motion of the body, – in the form of linear translation,

rotation and deformation.

The body is in a condition of stress and any changes in the shape or volume are then referred to as strain

Elastic body - after removal of the external force, the material returns to its original unstressed condition.

To study of elasticity we consider infinitesimal elastic deformations, where stress is linearly proportional to strain, as stated in Hooke's law.


Elastic Properties of Materials

Elastic constants connect stress and strain

Elastic constant can be determined by propagating ultrasonic elastic waves travel through a medium.

The velocity of these waves and density of the sample can be used to determine the values of these elastic constants.

The concept of elastic continuum is applied to explain quantitatively the elastic behavior of a solid body, under external stresses (temperature and hydrostatic pressure)

Materials are assumed to behave like a homogeneous continuous medium.

This approximation valid for elastic waves of wavelengths λ longer than 10-6 cm, i.e. frequencies below 1012 Hz, and ultrasonic waves fulfill this criterion.


Theory of Elasticity

Theory of elasticity and the anharmonicity of solids, both crystal and noncrystalline (glass) systems

The propagation of elastic waves in crystals, including the thermodynamic definition of elastic constants

The concept of anharmonicity is also outlined.

The effect of hydrostatic pressure and temperature on the elastic


Elasticity and Hooke's Law• For sufficiently small deformations of

the solid body, each component of stress is linearly related to each component of strain by Hooke's law

σij=Cijklϵ ( i,j, k,l= 1,2,3)

6 components of stress

6 components of strain

σij=Cijklϵ ( i,j, k,l= 1,2,3)


Thermodynamic Definition of Elastic Constants

Elastic constants can be also defined thermodynamically i.e. with respect to thermodynamic parameters (Brugger, 1964).

4 main thermodynamic potentials can be involved,

U - internal energy

F - Helmholtz free energy

H - enthalpy and

G - Gibbs free energy

Each of these to be a function of entropy S, temperature T, thermodynamic tension components tij and reduced Lagrangian strain components ηij/ρo where ρo, represents the density of the unstrained solid.

ijijdηtρ

1TdsU

d

ijijdηtρ

1SdTF

d

ijijdtρ

1TdsH

d

ijijdtρ

1SdTG

d

0',

ijkls

...

UC

Sklij

n

o

0',

ijklT

...

FC

Tklij

n

o

0',

ijklS

...

HS

tSklij

n

o tt

0',

ijklT

...

GS

tTklij

n

o tt


Elastic Constant

.........6

1

2

1)(

5

5o

mnkljijklmn

klijlijk

iC

CU

Energy density of the solid may then be written as

0'

ijkls

...

UC

klij

n

o

0'

3

ijklmnsC

mnklij

o

U

Second Order Elastic Constant

Third Order Elastic Constant

Since the form of this expansion is similar to the expansion of potential energy in terms of interatomic displacement,

It is possible to relate the elastic stiffness constants to the interatomic forces in a solid.


Propagation of Elastic Waves In Solid

One way to determine the elastic properties of any solid is through a dynamic test in which elastic waves are propagated in the solid.

Assumption

• Propagated waves behave adiabatically: • Entropy is conserved; • Wavelengths are much greater than the interatomic spacing • Small displacement of the atoms

To ensure that the deformation is elastic and Hooke's law is obeyed.

j

ijio xt

u

2

2

i

j

j

iijkl

k

io dx

u

x

uC

xt

u

2

12

2

kj

iijkl

io xx

uC

t

u

2

2

2

02 oljkolijklilo UNNUCv

Christoffel's equation

For a specific combination of N and U, the equation of motion will provide only three solutions of wave velocities, one of which resembles a longitudinal and two shear waves.

N - Propagation DirectionU - Polarization direction


Elastic Contants – Wave Velocity

Mode NoPropagation Direction

N

Polarization direction

U(ρv2)

1 [100] [100] C11 (L)

2 [100] In [100] plane C11

3 [110] [110] (C11+C12+2C44)/2

4 [110] [001] C44 (G)

5 [110] [1 0] C’=(C11-C12)/2

6 [111] [111] (C11+2C12+4C44)/3

7 [111] In [111]plane (C11+C44-C12)/3

1

Ultrasonic wave velocity and elastic stiffness constant relationships for a cubic crystal


Elastic constants of the glasses

Longitudinal modulus

Shear modulus

Bulk modulus

Poisson’s ratio

Young’s modulus

Debye Temperature

2lVL

2sVG

22

3

4sl VVK

22

22

2

2

sl

sl

VV

VV

22

222 43

sl

sls

VV

VVVE

mDt VM

Np

k

h 3

1

4

9

3

1

33

12

lS

mVV

V


Ultrasonic SystemSchematic representation of (a) simple pulse ultrasonic system. (b) Envelope of pulse echo train and (c) detail of each echo as seen on oscilloscope display


Ultrasonic Pulse Echo Overlap System

Pulse echo overlap systemPulse echo overlap waveforms

Block diagram of the experimental set up – ultrasonic wave velocity and attenuation measurement (Mepco Engineering College, INDIA)


Ultrasonic System

Ultrasonic – MBS 8000 Ultrasonic Data Acq. System


Interatomic Potential Energy

In a simple lattice dynamical model, the restoring forces between atoms and hence their potential energy are generally considered to be a function of the atomic displacement from equilibrium positions.

In the harmonic and anharmonic lattice vibration models for a solid

TOEC play an important role in accounting for the anharmonic and nonlinear properties of solids in the long wavelength acoustic modes.


Wave Velocity

Compositional dependence of the velocity of longitudinal and shear acoustic waves in Bi2 O3–B2 O3 glass systems.

Both increase at first with increasing Bi2 O3 mol% up to a maximum at 25 mol% Bi2 O3 and then decrease as the Bi2 O3 mol% increases further.

1000

1500

2000

2500

3000

3500

4000

0.05 0.10 0.15 0.20 0.25 0.30 0.35

Pecahan mol of Ag2O

Halaj

u ul

traso

nik (

m/s)

Compositional dependence of the velocity of longitudinal and shear acoustic waves in [(TeO2)x (B2O3)1-x)]1-y [Ag2O]y glass

1500

2000

2500

3000

3500

4000

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4


Velo

city

(m/s

)

Longitudinal

Longitudinal

Shear

Shear

Compositional dependence of the velocity of longitudinal and shear acoustic waves in [(ZnO)(TeO2) glass


Lead Magnesium Chloride Phosphate Glass


Elastic Modulus

Dependence of longitudinal modulus on the composition of Bi2 O3–B2 O3 glass systems.

One reason for this difference may come from the volume effect, in that C44 expresses the resistance of the body to deformation where no change in volume is involved, while C11 expresses the resistance where compressions and expansions are involved.

10

20

30

40

50

60

70

0.05 0.10 0.15 0.20 0.25 0.30 0.35

Pecahan mol Ag2O

Mod

ulus

ken

yal (

GPa

) L

E

K

G

Compositional dependence of the longitudinal and shear modulus of [(TeO2)x (B2O3)1-x)]1-y [Ag2O]y glass


15

20

25

30

35

40

45

50

55

60

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4Mole fraction of ZnO

Elasti

c Mod

uli (G

Pa)

Longitudinal Modulus, L

Young’s Modulus, E

Bulk Modulus, K

Shear Modulus, G


Elastic Properties

Mole fraction, x 0.3 0.4 0.45 0.5 0.6

Elastic stiffness (GPa)

C11

C44

C12

48.9

18.0

12.9

48.8

18.0

12.7

47.5

17.4

12.7

47.3

17.5

12.3

47.3

17.2

13.0

Young's modulus, E

(GPa)

43.5 43.5 42.2 42.2 41.8

Bulk modulus, B (GPa) 24.9 24.7 24.3 24.0 24.4

Poisson's ratio, 0.208 0.207 0.211 0.207 0.215

Fractal dimension 2.90 2.92 2.87 2.92 2.82

Molar volume, V

(cm3/mole)

34.2 33.8 34.2 33.9 33.3

Number of atoms per

volume (x1028

atoms/m3)

9.67 8.90 8.37 8.00 7.24

Debye Temperature (K) 291 275 263 255 238

The room temperature elastic properties of (PbO)x(P2O5)1-x glasses

Mole fraction, y 0.04 0.06 0.07 0.1

Elastic stiffness (GPa)

longitudinal, c11

shear, c44

c12

50.4

17.1

16.3

44.3

16.0

12.3

43.0

15.9

11.2

35.7

14.8

6.03

Young's modulus, E

(GPa)

42.4 39.0 38.4 33.9

Bulk modulus, B (GPa) 27.6 23.0 21.8 15.9

Poisson's ratio, 0.244 0.217 0.206 0.145

Fractal dimension 2.47 2.79 2.92 3.73

Molar volume, V

(cm3/mole)

33.5 33.5 33.3 33.4

Number of atoms per

volume (x1028 atoms/m3)

9.60 9.65 9.72 9.78

Debye Temperature (K) 276 266 264 251

Room temperature elastic properties of (PbCl2)y(PbO.2P2O5)1-y glasses


Elastic Properties





• Conclusion

Glass prism


Temperature Dependence

Low temperature dewar system

Sample holder


Temperature Variations of the SOECThe dependence of elastic constants upon temperature is another consequence of anharmonicity of the interatomic potential energy in solids.

The normal behaviour of the thermal variations of the SOEC of most crystals,, is characterized by two general features; •a linear increase with decreasing temperature and

•a zero slope in the region where the temperature approaches zero Kelvin.

The linear dependence of elastic constants on temperature, especially above the Debye temperature θD ,is due to the anharmonic nature of the lattice vibrations.

The linearity of this dependence may break down in the vicinity of a phase transition.

Typical curve of second order elastic constant versus temperature

))/(1( DoJI TLFCC

T

DD

D

dxxxTTF/

0

34 1)exp()/(3)/(

3

2 v

o

TIJ

JI

dT

df

fC

dT

dC



Variation of velocity and elastic moduli with temperature of Lead Bismuth Tellurite (BTP) Glasses



At sufficiently low temperatures it is expected that in most crystalline solids the slope of dCIJ/dT would decrease i.e. dCIJ/dT →0 as T →0 which is a direct consequence of the third law of thermodynamics.

However in certain materials like glasses, this particular feature is not always observed; instead of showing a zero slope of dCIJ/dT, the elastic constant increases to a maximum value at low temperature (~1K).

This behaviour has been ascribed to interactions with two-level systems (Anderson et al. 1972, Phillips 1972).



At very low temperatures where only ground states have to be considered, the groups of atoms are still able to tunnel through a barrier. This gives rise to an energy splitting of the ground state for this two-level system given by

20

2 E

)exp(00

2/2 mVd

where

The parameter Δ designates the asymmetry of two-level potential, Δo represents the tunneling energy and λ is a tunnelling parameter describing the overlap of the wavefunctions of two states in a quantum mechanical theory (Phillips 1981). The parameter ωo is the frequency of oscillation in an individual well.

Schematic representation of a double well potential, sometimes known as the two level system, characterized by a barrier V, asymmetry energy LI and distance d.


Debye Temperature

3

1

33

32

31

3

1

1 111

4

9

d

vvvk

h

V

NeD

The Debye temperature θD is useful in describing the thermal behaviour of solids and it plays an important role in the theory of lattice vibrations.

θD measure of the separation of the low temperature quantum mechanical region, where the vibrational modes begin to be "frozen out", from the high temperature region where all modes begin to be excited according to classical theory.

θD can be obtained directly from heat capacity measurements, and it can be also derived from a set of the elastic constants.

3

1

33

21

3

1

SL VVVm

VL (= (C11/ρ) 1/2) and VS= (=C44/ρ) 1/2

meD V

k

h

V

N

3

1

1

4

9





• Conclusion

Glass prism


Pressure Dependence

To compare between the measured values of (ρW2)'p=0 and the pressure derivatives of the effective SOEC (ρV2)', the following relations has to be employed,

dP

WVdW

dP

dW

dP

dWW

dP

Vd oo

ooooo

)/()/(2)(

2222

2

kmiimk

T

o

IJIJ SNN

dP

df

fC

dP

dC2

2

Here CoIJ represents the SOEC at ambient

condition, df/dP is the gradient of measured frequency in the pulse echo overlap experiment (see section 3.5) versus pressure, fo is the overlap frequency and βT and Skmii are the isothermal volume compressibility and the elastic compliances respectively.

Hydrostatic pressure cell and sample holder


Mode Grüneisen Parameters

The mode Grüneisen parameter is an important tool for investigation of anharmonic effect in solid.

One way to derive the Grünisen parameter is by considering the entropy of the system in the quasiharmonic form (Barron 1995)

T

TVSS i

ii

),(

1

ln

ln1

ln)(

TVTC ii

ii

Vd

d ii ln

ln

11

)(

TTVTTC

V

S i

V

i

T

i

ii

V

VT

th

C

V

V

S

VTVP

SV CVCV //

α is the coefficient of volume thermal expansion, V is the volume, and is isothermal and adiabatic compressibility respectively and Cv and Cp are specific heats at constant volume and constant pressure respectively

dP

vdTl

1ln1

3

1

dP

vdTs

1ln1

3

1

3

21 sel


Relation between SOEC and TOEC

Even though we have not measured a set of individual TOEC, the TOEC can be obtained from the pressure derivatives of the second order elastic constants

)2(

)22(

1211

16614444121144

CC

CCCCC

P

C

)2(2

)233(

1211

1231111211'

CC

CCCC

P

C

)2(3

)26(

1211

123112111

CC

CCC

P

B





• Conclusion

Glass prism


Conclusion• The general discussion on basic

glass and its preparation, as well as the fundamental of elastic properties have been discussed

• Experimental shows the physical and elastic properties of glasses were found slightly affected by the changes in the glass composition.

• The densities of most glasses increases as the glass modifier content was added to substitute the glass former content while their molar volume increases or decreases due to the composition.

•The experimental setup has been discussed for evaluating the ultrasonic and elastic properties of materials at low temperatures and high pressure.

•There are possibility in evaluating the elastic properties of any glass based materials at elevated temperatures and pressure in order to obtain their elastic behaviour.


Acknowledgement

Glass Research Group• Dr Halimah Ahmad Kamari

• Prof Madya Dr Zainal Abidin Talib

• Prof Madya Dr Wan Daud Wan Yusof

• Dr Khamirul Amin Matori

• Prof Dr Abdul Halim Shaari

• Our postgraduate students

Special thanks to MOSTI and UPM for financial support.


Thank you for your attention

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Elastic studies of Glass Materials Studied by Ultrasonic Technique

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