Sidekaziz glass by Adzrif Effendy

Elastic Properties of Glass Materials Studied by Ultrasonic Technique Sidek Ab Aziz Universiti Putra Malaysia

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

Glass prism

– Preparation – Density and Molar Volume

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

Glass â&#x20AC;&#x201C; An Introduction What 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.

Obsidian - super-heated sand or rock

that rapidly cooled.

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. Moldavite formed by meteorite impact (Besednice, Bohemia) Glass in General | Ultrasonic Waves | Physical Properties | Preparation | Elastic Properties | Conclusion

General Introduction • These three properties— lustre, transparency, and durability — make glass a favoured material for such household objects as windowpanes, bottles, and lightbulbs. Thermal Insulation Glass

Clear glass for Incandescent light bulb

Glass Application Special 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 (glassreinforced plastic, glass fiber reinforced concrete) and art.

Glass Technology

Glass building

Glass Arts

Decoration Glass- Venetian millefiori.

Glass beads are made with silica (usually from sand).

Chinese ring

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.

Next-generation large-scale panels

Specific Potential Application of Glassy Materials

Optical waveguides

CD memory device Non-linear optical devices

Laser host

Outline • Glass in General – Definition

Glass prism

• Ultrasonic Waves • Physical Properties of Glass – Preparation – Density and Molar Volume

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

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)

♦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)

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

Zachariasenâ&#x20AC;&#x2122;s model In 1932, physicist W.H. Zachariasen defined glass is an extended, threedimensional network of atoms that form a solid which lacks the long-range periodicity (or repeated, orderly arrangement) typical of crystalline materials.

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.

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 2coordination 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

Glass prism

– Preparation – Density and Molar Volume

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

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

Advantages of ultrasonic wave 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

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

 shear wave

U N U

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.

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

Glass prism

– Glass Preparation – Density and Molar Volume

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

Glass Preparation 2 methods of preparing glass samples Conventional Method  Cooling from the liquid state/ Melt quenching technique  Condensation from the vapor  Pressure quenching  Solution hydrolysis

Unconventional Method Unconventional melting Solution methods Deposition methods Solid-state transformations

Glass Formation Normally, 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

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

â&#x20AC;˘thermodynamically stable phase

â&#x20AC;˘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.

â&#x20AC;˘thermodynamically metastable phase

Glass Structure Glass • 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 nonbridging oxygen.

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

Glass prism

– Preparation – Density and Molar Volume

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

• Conclusion

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

GLASS PREPARATION PROCESS Substance is weight

Pour melt into mould

First Furnace 400 C for 30 min

Placed molten and mould in First Furnace and annealed at 400 C

Second Furnace 750-800 C for 60 min

Cut and polished sample

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.

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 Glass Besides common silicabased 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 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:

Ion

Silicate-based Glass

Phosphate-based Glass

Fe+2

deep blue-green

slight greenish blue

Fe+3

yellowish-brown

slightly brownish

SEM Photos XRD Pattern of Starting Materials Intensity (a.u)

3400 3200 3000 2800 2600 2400 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0

TeO2 powder

TeO2 glass

30 2 theta

50000 45000 40000

Intensity (a.u)

35000 30000

TeO2-ZnO glass

25000 20000

ZnO Powder

15000 10000 5000 0

30 2 Theta

35000 30000

Intensity (a.u)

25000

TeO2-ZnO-AlF3 glass

20000 15000

AlF3 (97.0%) Powder

10000 5000 0

2 Theta

XRD patterns 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)

2100 1800 Intensity (a.u)

1600

TZ6 TZ5

1100

600

1600

Intensity (a.u)

TZ7

1400

TZ4

1200

TZ3

1000

TZ2

S5 S4 S3

800

TZ1 TZ0

100

30 2 theta

600

400

binary

30 2 theta

ternary

• 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 Glass in General | Ultrasonic Waves | Physical Properties | Preparation | Elastic Properties | Conclusion

Glass Forming Region

Density & Molar Volume Density Measurement (Archimedes Method)  s   



wa  ac wa  wac 

Molar volumes

V 



5000

28.5

4950 4900

4850 27.5

4800

4750

26.5

Density (kg m -3)

Molar volume(cm 3 mol -1)

Density & Molar Volume

4700

26 0.55

0.6

0.65

0.7

0.75

4650 0.85

0.8

Mole fraction of TeO2

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.

30.5

5300

5200

29.5

5100

5000

28.5

4900

28 0.05

0.10

0.15

0.20

0.25

0.30

4800 0.35

Pecahan Mol Ag 2O

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

Ketumpatan (kg/m 3)

Variation of density and molar volume with mol% Bi2 O3 in Bi2 O3â&#x20AC;&#x201C;B2 O3 glass systems.

Isipadu molar (cm 3)

Density and molar volume of TeO2.B2O3 glasses

Density and Molar Volume Density (kgm-3)

7500 6500 5500 4500 3500 0

40 60 Bismuth Oxide (mol%)

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 â&#x20AC;&#x201C; 42.5 mol%) and present works (40 â&#x20AC;&#x201C; 70 mol%).

Density & Molar Volume •The increase in density indicates zinc ions enter the glassy network

5400 5300

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

Density (kg/m )

5200 5100 5000 4900 4800 4700 0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

Mole fraction of ZnO

• Atomic Radius (Shelby, 2005). •R(Zn2+)(0.074 nm) << R(Te2+)(0.097 nm)

-6

-1

Molar volume (10 m mol )

0,4

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

28 26

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

24 22 0

0,05

0,1

0,15 0,2 0,25 Mole fraction of ZnO

0,3

0,35

0,4

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

Glass prism

– Preparation – Density and Molar Volume

• Elastic Properties – – – –

Theory Compositional Dependence Temperature Dependence Hydrostatic Pressure Dependence

• Conclusion

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

Elastic Properties of Materials Application of external forces to a solid body produces complex internal forces which cause â&#x20AC;&#x201C; motion of the body, â&#x20AC;&#x201C; 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 位 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

σij=Cijklϵ

6 components of strain

( 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

1 dU  Tds    t ij dηij  ρ  1 dF  SdT    t ij dηij  ρ  1 dH  Tds   ij dt ij  ρ  1 dG  SdT   ij dt ij  ρ 

 nU   Cs ijkl  o     ...   ij kl  S , '  0  nF   CT ijkl  o     ...   ij kl T , '  0  nH   SSijkl   o   t t ...   ij kl  S ,t '  0   nG   ST ijkl   o   t t ...   ij kl T ,t '  0

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.

Elastic Constant Energy density of the solid may then be written as 1 2

oU ( )   C 5ijk lij kl 1   C 5ijklmni j kl mn  ......... 6

 nU   Cs ijkl  o     ...   ij kl  '  0

Cs ijklmn  o

 3U ij kl mn

 ' 0

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.

Second Order Elastic Constant

Third Order Elastic Constant

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.

 2ui  ij o 2  t x j o

 u u    2ui 1    Cijkl  i  j   2  x  t 2 xk   j dxi  

 2ui  2ui o 2  Cijkl t x j xk Christoffel's equation

 v  2



 CijklU ol N k N j U ol  0

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 Direction U - Polarization direction

Elastic Contants – Wave Velocity Ultrasonic wave velocity and elastic stiffness constant relationships for a cubic crystal

Propagation Direction

Polarization direction

[100]

C11 (L)

[100]

In [100] plane

C11

[110]

(C11+C12+2C44)/2

[110]

[001]

C44 (G)

[110]

[1 0]

C’=(C11-C12)/2

[111]

(C11+2C12+4C44)/3

[111]

In [111]plane

(C11+C44-C12)/3

Mode No

(ρv2)

Elastic constants of the glasses Longitudinal modulus Shear modulus Bulk modulus Poisson’s ratio

L  Vl 2

G  Vs2  2 4 2 K   Vl  Vs  3  

V  2V   2V  V  2

Young’s modulus

E Debye Temperature



Vs 2 3Vl 2  4Vs 2 Vl  Vs 2

1 3

h  9 Np   Vm k  4M 

t   D  

  2 1  Vm   3  3  Vl   VS



1 3

Ultrasonic System Schematic 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 system Pulse echo overlap waveforms

Block diagram of the experimental set up â&#x20AC;&#x201C; ultrasonic wave velocity and attenuation measurement (Mepco Engineering College, INDIA)

Ultrasonic 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 Halaju ultrasonik (m/s)

4000 3500

Longitudinal

3000 2500 2000 1500

Shear

1000 0.05

0.10

0.15

0.20

0.25

0.30

0.35

Pecahan mol of Ag 2O

Compositional dependence of the velocity of longitudinal and shear acoustic waves in [(TeO2)x (B2O3)1-x)]1-y [Ag2O]y glass 4000

Compositional dependence of the velocity of longitudinal and shear acoustic waves in Bi2 O3â&#x20AC;&#x201C;B2 O3 glass systems.

Velocity (m/s)

3500 3000

Longitudinal

2500 2000

Shear

1500

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

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

Mole fraction of ZnO

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

Lead Magnesium Chloride Phosphate Glass

Elastic Modulus Modulus kenyal (GPa)

70 60

30 20

10 0.05

0.10

0.15

0.20

0.25

0.30

Pecahan mol Ag 2O

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

Dependence of longitudinal modulus on the composition of Bi2 O3â&#x20AC;&#x201C;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.

0.35

60 55

Longitudinal Modulus, L

Elastic Moduli (GPa)

50 Youngâ&#x20AC;&#x2122;s Modulus, E

45 40 35

Bulk Modulus, K

30 25

Shear Modulus, G

20 15 0

0,05

0,1

0,15 0,2 0,25 Mole fraction of ZnO

0,3

0,35

0,4

Elastic Properties The room temperature elastic properties of (PbO)x(P2O5)1-x glasses Mole fraction, x

0.3

0.4

0.45

Room temperature elastic properties of (PbCl2)y(PbO.2P2O5)1-y glasses 0.5

0.6

Mole fraction, y

0.04

0.06

0.07

0.1

Elastic stiffness (GPa)

Elastic stiffness (GPa) C11

48.9

48.8

47.5

47.3

longitudinal, c11

50.4

44.3

43.0

35.7

C44

18.0

17.4

17.5

17.2

shear, c44

17.1

16.0

15.9

14.8

C12

12.9

12.7

12.3

13.0

c12

16.3

12.3

11.2

6.03

Young's modulus, E

43.5

42.2

41.8

42.4

39.0

38.4

33.9

Bulk modulus, B (GPa)

27.6

23.0

21.8

15.9

Poisson's ratio, ď ł

0.244

0.217

0.206

0.145

Fractal dimension

2.47

2.79

2.92

3.73

Molar volume, V

33.5

33.3

33.4

9.60

9.65

9.72

9.78

276

266

264

251

(GPa)

(GPa) Bulk modulus, B (GPa)

24.9

24.7

24.3

24.0

24.4

Poisson's ratio, ď ł

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

34.2

33.8

34.2

33.9

33.3

9.67

8.90

8.37

8.00

7.24

(cm3/mole) Number of atoms per

Young's modulus, E

(cm3/mole) Number of atoms per

volume (x1028

atoms/m3)

Debye Temperature (K)

291

275

263

255

238

Debye Temperature (K)

Elastic Properties

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

Glass prism

– Preparation – Density and Molar Volume

• Elastic Properties – – – –

Theory Compositional Dependence Temperature Dependence Hydrostatic Pressure Dependence

Temperature Dependence

Sample holder

Low temperature dewar system

Temperature Variations of the SOEC The 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

CI J  Co (1  LF (T /  D )) F (T /  D )  3(T /  D )

D / T 4

 x



exp( x)  1 dx

•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.

Typical curve of second order elastic constant versus temperature dCI J 2  df   v  CIJT   dT f o  dT  3

Temperature Dependence

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

Temperature Dependence At sufficiently low temperatures it is expected that in most crystalline solids the slope of dCIJ/dT would decrease i.e. dCIJ/dT â&#x2020;&#x2019;0 as T â&#x2020;&#x2019;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). Glass in General | Ultrasonic Waves | Physical Properties | Preparation | Elastic Properties | Conclusion

Temperature Dependence 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 twolevel system given by

E  2  20

where

 0   0 exp( )   d 2mV /  2 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 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.

1 3

 9 N   h    1 1 1  d     3  3 3   4V   k    v1 v2 v3   

 De1  

1  1 2  Vm    3  3   3  VL VS 



1 3

VL (= (C11/ρ) 1/2) and VS= (=C44/ρ) 1/2 1 3

 9N   h    Vm  4V   k 

 De1  



1 3

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

Glass prism

– Preparation – Density and Molar Volume

• Elastic Properties – – – –

Theory Compositional Dependence Temperature Dependence Hydrostatic Pressure Dependence

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, V 2

2 2 dW 2 d (  / o ) 2 d (V / W ) d( )  2 oWo  oWo  oWo dP dP dP dP

Hydrostatic pressure cell and sample holder

 2 df  dCIJ  C  IJ    T  2 N k N m Skmii  dP  f o dP  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. Glass in General | Ultrasonic Waves | Physical Properties | Preparation | Elastic Properties | Conclusion

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

 th 

S V

1 i i   Ci (T ) T V T T i

 ln i  ln i   Ci (T ) 1 V  ln T i

i  

dln i d ln V

 V

i T

V CV

 V / VS CP  V / VT CV

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

One way to derive the Grünisen parameter is by considering the entropy of the system in the quasiharmonic form (Barron 1995) i (V , T ) S   Si

S V

1

1 3

l  

1

s 

1 d ln v1  T dP

1 1 d ln v1  3  T dP el

 

 1  2 s  3

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

C44 (C11  2C12  C44  C144  2C166 )  P (C11  2C12 )

C ' (3C11  3C12  C111  2C123 )  P 2(C11  2C12 )

B

(C111  6C112  2C123 )  P 3(C11  2C12 )

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

Glass prism

– Preparation – Density and Molar Volume

• Elastic Properties – – – –

Theory Compositional Dependence Temperature Dependence Hydrostatic Pressure Dependence

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