Glass Transformation- - Glass Transformation Range Behavior Range - - PowerPoint PPT Presentation

▶
glass transformation glass transformation range behavior
SMART_READER_LITE
LIVE PREVIEW

Glass Transformation- - Glass Transformation Range Behavior Range - - PowerPoint PPT Presentation

Jun 08, 2023 199 likes •717 views

Web Course Web Course Physical Properties of Glass Physical Properties of Glass Glass Transformation- - Glass Transformation Range Behavior Range Behavior Richard K. Brow Missouri University of Science & Technology Department of

slide-1
SLIDE 1

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-1

Web Course Web Course Physical Properties of Glass Physical Properties of Glass

Glass Transformation Glass Transformation-

  • Range Behavior

Range Behavior

Richard K. Brow Missouri University of Science & Technology Department of Materials Science & Engineering

slide-2
SLIDE 2

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-2

Glass Transformation-Range Behavior

  • Structural relaxation and the glass

transition

  • Rheology and configurational entropy
  • Thermal history effects on glass properties
slide-3
SLIDE 3

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-3

Supplementary References on Glass transformation-range behavior

  • Chapter 13 in ‘the good book’*
  • Structure, Dynamics and Properties of Silicate Melts,

Reviews in Mineralogy, Vol. 32 (1995), ed. JF Stebbins, PF McMillan and DB Dingwell (Mineralogical Society of America)

– CT Moynihan, Chap. 1, Structural relaxation and the glass transition

  • CT Moynihan et al, “Dependence of the Fictive Temperature of

Glass on Cooling Rate,” J. Amer. Ceram. Soc. 59 12 (1976)

– DB Dingwell, Chap. 2, Relaxation in silicate melts – P Richet and Y. Bottinga, Chap. 3, Rheology and configurational entropy of silicate melts

  • GW Scherer, Glass formation and Relaxation, Chap. 3 in

Materials Science and Technology, Vol. 9, ed. J. Zarzycki, VCH, 1991.

*AK Varshneya, Fundamentals of Inorganic Glasses, 2nd Ed (2006)

slide-4
SLIDE 4

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-4

Why should we Why should we care? care?

  • Glass properties depend
  • n thermal history
  • The nature of the glass

transition is ‘the deepest and most interesting unsolved problem in solid state theory’

  • Philip W. Anderson, 1995
slide-5
SLIDE 5

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-5

Class Exercise- Consider the question ‘What is the glass transition?’

slide-6
SLIDE 6

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-6

Structural relaxation and the Structural relaxation and the glass transition glass transition

slide-7
SLIDE 7

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-7

Glass transition: Glass transition:

  • Changes in the

structure of a super- cooled liquid ‘fall out

  • f equilibrium’ as

the liquid is cooled

  • Relaxation time is

long compared with the observational time

  • Properties (including

Tg) depend on thermal history; viz., quench rate through the glass transition region

slide-8
SLIDE 8

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-8

Structural relaxation

  • Average structure specifies thermo-dynamic state of

the liquid (T,P,V specified) – At equilibrium, average structure is time- independent

  • Relaxation involves breaking/remaking network bonds

– Dynamic equilibrium – Viscous characteristics of super-cooled liquid – Contribute to ΔH, ΔS, ΔV of the liquid as f(T,P)

  • Structural relaxation rate decreases with decreasing

temperature

slide-9
SLIDE 9

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-9

Time Strain (ε) ε0 Time Stress (σ) σ0=G0ε0 σ(t)=G(t)ε0 σ∞=0

Stress Relaxation- Maxwell Model

ε0 Mathematical form:

σt = σ0 exp(-Gt/η)

G = shear modulus (Pa) (η/G) = τ: time for stress to decay to (1/e) σ0 = (0.367 σ0) = Relaxation Time

σt = σ0 exp(-t/τ)

Exponential relaxation curve

slide-10
SLIDE 10

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-10

Voigt-Kelvin Element- Anelastic behavior

slide-11
SLIDE 11

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-11

Burger element- permanent deformation Note: viscoelastic substances may be modeled by a distribution of mechanical elements representing a distribution of structural features

slide-12
SLIDE 12

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-12

From Moynihan, 1995

Consider isothermal relaxation after ΔT

Fictive Temp.

Note: bonds break & reform ‘elastic’ ‘viscous’

slide-13
SLIDE 13

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-13

Fictive Temperature Fictive Temperature

  • Introduced by Tool, NBS

(1946)

  • Describes the contribution of

structural relaxation to a property, expressed in units of temperature

  • In equilibrium: Tfic=T and

dTfic/dT=1

  • For glass with ‘frozen

structure’: Tfic=const. and dTfic/dT=0

  • Not a fundamental property,

but a conceptual view

time

Temperature, T, Tfic

ΔT

Tfic

T

Fictive: feigned, sham (from fiction)

slide-14
SLIDE 14

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-14

Relaxation Time

Assumption 1: the rate of structural relaxation is described by a characteristic relaxation time, τ Assumption 2: rate at which volume approaches equilibrium defined by first

  • rder rate constant (k=1/t) and depends
  • n deviation of volume from the new

equilibrium value (at T2):

( ) ( ) ( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − = Φ − − ≡ − − = Φ − = − τ t kT t T T T T V V V V t V V k dt V V d

fic e e e e

exp exp ) ( ) (

2 1 2

Φ(t) is a relaxation function Φ(t) = 1 at t=0, Φ(t) = 0 at t=∞

time Volume Temp

T1 T2 V1 Ve

V0 Fast (glassy) response Slower (viscous) response

slide-15
SLIDE 15

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-15

Assumption 3: for small departures from equilibrium and small ΔT, the temperature dependence of τ is described by the Arrhenius relationship:

( )

RT H * exp Δ =τ τ

where ΔH* is the activation enthalpy ΔH*>0, so as T decreases, τ increases and the rate of structural relaxation decreases

slide-16
SLIDE 16

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-16

Relaxation during heating and cooling Relaxation during heating and cooling

Equilibrium property Deviation between experiment and equilibrium

τ>>t, He is reached τ<<t, no relaxation (glassy behavior)

From Moynihan, 1995

slide-17
SLIDE 17

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-17

Some observations: On cooling, H>He after Δt On heating, H<He after Δt Note on heating: H initially decreases when approaching He, then increases at greater temperatures cooling/heating rates are defined by q=dT/dt; series of ‘isothermal’ holds for Δt=ΔT/q Information about relaxation time can be obtained by measuring property changes at different ‘q’…. From Moynihan, 1995

slide-18
SLIDE 18

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-18

Note hysteresis in properties after cooling and reheating τ>>Δt τ>>Δt→∞ Glass transition occurs when τ≈Δt, entirely kinetic in origin!

From Moynihan, 1995

slide-19
SLIDE 19

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-19

Recall Heat Capacity Cp=dH/dT Tg is ‘observed’ when t≈Δt=ΔT/q Note: sigmoidal shape of Cp(T) is a consequence of the hysteresis in the ΔH/ΔT due to the relaxation kinetics Fast cooling rate: ‘fall out of equilibrium’ at greater temperature (shorter relaxation time), greater limiting fictive temperature (Tf’)- the intersection of the H-values for the glass and liquid

From Moynihan, 1995

slide-20
SLIDE 20

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-20

Measuring the ‘limiting fictive temperature’- one measure of Tg

( ) ( )

∫ ∫

<< >> >>

− = −

g g f g

T T T T gl p p T T T gl p e p

C C dT C C

) ( ' ) ( ) (

Area I Area II

slide-21
SLIDE 21

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-21

The glass transition temperature depends on the thermal history

CT Moynihan, et al., JACerS, 59 12 (1976)

Borosilicate crown glass Tg (and Tfic) increases with increasing quench rates (qc)

slide-22
SLIDE 22

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-22

Note: Over small temperature intervals, centered on Tr, the activation energies for enthalpy relaxation are equivalent to those from shear-viscosity measurements (solid line) and from volume relaxation measurements (Ritland)

d(ln qc)/d(1/Tf’) = -ΔH*/R

CT Moynihan, et al., JACerS, 59 12 (1976)

slide-23
SLIDE 23

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-23

Shear viscosity relaxation (closed symbols) and enthalpy Shear viscosity relaxation (closed symbols) and enthalpy relaxation (open symbols) processes have the same relaxation (open symbols) processes have the same activation energies activation energies

Dingwell, 1995

Volcanic obsidian glasses

slide-24
SLIDE 24

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-24

From Moynihan, 1995

Enthalpy Relaxation vs. Viscous Flow

slide-25
SLIDE 25

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-25

Example: Geospeedometry

Volcanic obsidian ‘natural cooling rate’

Dingwell, 1995

slide-26
SLIDE 26

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-26

Example: Geospeedometry

1º/minute 1º/hour 1º/day 1º/week Calibration range

Natural extrapolated cooling rates

slide-27
SLIDE 27

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-27

Kinetics of structural relaxation

  • Qualitative ‘first order’ kinetic model
  • Quantitative model must account for

– Non-linear character of the relaxation function – Non-exponential character of the relaxation function

( )

RT H t t * exp exp ) ( Δ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Φ τ τ τ

slide-28
SLIDE 28

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-28

Non-linear isothermal relaxation

From Moynihan, 1995

Note that the relaxation time depends on the instantaneous structure (fictive temp)

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Δ − + Δ =

f

RT H x RT H x * ) 1 ( * exp τ τ

Tool-Narayanaswamy (TN) Equation x is the nonlinear parameter (0<x<1)

slide-29
SLIDE 29

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-29

Relaxation is non-exponential

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2

Normalized time Normalized property

beta=1.0 beta=0.8 beta=0.6 beta=0.4

β

τ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Φ t t exp ) (

Stretched exponential function (KWW)- β is the non- exponentiality parameter (0<β<1)

slide-30
SLIDE 30

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-30

Distribution of relaxation times Distribution of relaxation times

∑

=

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− ≈ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −

N k k k p

t w t

1

exp exp τ τ

β

Broader distributions are associated with smaller values of β

Scherer, 1991

slide-31
SLIDE 31

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-31

Is there a physical source for non Is there a physical source for non-

  • exponentiality

exponentiality? ?

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Δ − + Δ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− = Φ

∫ ∑

f i t i i i

RT H x RT H x dt g t * ) 1 ( * exp / ' exp ) ( τ τ τ

Consider a distribution of relaxation times (τi) incorporated into the relaxation function with a weighting coefficient (gi, where Σgi=1), and each relaxation time described by the T-N form: Microscopic interpretation:

  • Relaxation involves coupled

responses of a series of processes with different ‘reaction rates’- bond 1 breaks, then bond 2…..

  • Different regions within liquid

relax at different rates because

  • f structural differences

(differences in configurational entropy from μ-region to μ- region)

slide-32
SLIDE 32

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-32

Structural relaxation models

  • Four adjustable parameters: τ0, ΔH*, x, β

2 1 2

) ( ) ( ) ( T T T t T P P P t P t

fic e e

− − ≡ − − = Φ

From Moynihan, 1995

slide-33
SLIDE 33

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-33

From Moynihan, 1995

Cooling rates

0.62K/min 2.5K/min 10K/min 40K/min

B2O3 Glass- reheated at 10K/min

slide-34
SLIDE 34

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-34

Structural Relaxation Summary Structural Relaxation Summary

  • The glass transition is a

kinetic phenomena

  • Thermal history

dependence

  • Thermal history effects
  • n glass properties

described using the ‘fictive temperature’ concept.

  • Tfic represents the

contribution of structural relaxation to the property of interest, expressed in temperature units

MA Debolt et al., 1976

slide-35
SLIDE 35

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-35

Structural Relaxation Summary Structural Relaxation Summary

The relaxation function Φ(t)

  • 1. Is non-linear
  • Up-quench ≠ down-quench relaxation rates
  • Φ(t) depends on instantaneous structure (Tfic)
  • Tool-Narayanaswamy ‘non-linearity’ parameter ‘x’
  • 2. Is non-exponential
  • ‘Stretched exponential’ function (KWW)- β
  • Modeled by a distribution of relaxation times
  • Is their a ‘microscopic’ explanation?

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Δ − + Δ =

f

RT H x RT H x * ) 1 ( * exp τ τ

β

τ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Φ t t exp ) (

slide-36
SLIDE 36

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-36

How is enthalpy relaxation How is enthalpy relaxation connected to viscosity? connected to viscosity?

From C. A. Angell, Science, 267, (1995), 1924. Fragile Strong Tg/T

Log (viscosity in poise)

g g

RT H T T d d 3 . 2 * ) / ( ) (log

η

η Δ =

slide-37
SLIDE 37

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-37

Moynihan, 1995

How is enthalpy relaxation How is enthalpy relaxation connected to viscosity? connected to viscosity?

xNa2O (1-x)B2O3 glasses Fragile melt behavior (greater Eη/Tg) correlated with larger ΔCp at Tg… Related to greater structural changes as glass is heated through transition range?

g g

RT E T T d d 3 . 2 ) / ( ) (log

η

η =

slide-38
SLIDE 38

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-38

From Moynihan, 1995

Enthalpy Relaxation and Viscous Flow have similar activation energies- similar ‘structural’ mechanisms? Relaxation time at Tg≈ 400-600 sec Angell defines Tg at 1012 Pa·

slide-39
SLIDE 39

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-39

Glass Transition Theories

  • Free Volume-Viscosity
  • Adam-Gibbs Cooperative Relaxations

– Configurational Entropy

slide-40
SLIDE 40

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-40

Free Volume Theory Free Volume Theory

  • Turnbull, Cohen, J Chem Phys 52 3038 (1970); Cohen, Grest Phys Rev

B, 20 1077 (1979)

  • Consider ‘ideal’ close-packed structure representing a

thermodynamic minimum volume, V0

  • Flow occurs by movement of molecules into voids or holes larger

than a critical size, Vh – Thermal/density fluctuations open up voids – Increase temperature, increase specific volume (V) of melt – Free volume (Vf) within a structure becomes available to accommodate viscous flow

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ,...) , ( exp P T V V

f

η η

hole

slide-41
SLIDE 41

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-41

Free volume depends melt properties Free volume depends melt properties

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − Δ =

∫ ∫

, , , ,

) , (

P T P T P T P T T P f

dP K dT V P T V α

( ) ( )(

)

T T dT V V

glass liq T T glass liq f

− − ≈ − ≈ ∫ α α α α

αP=isobaric expansion coefficient KT=isothermal compressibility For constant pressure, Substitute into the Arrhenian η equation to get the VFT eq.:

log T T B A − + = η

“T0” represents the temperature at which free volume disappears

slide-42
SLIDE 42

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-42

Configurational Entropy Model Configurational Entropy Model

  • G. Adam, JH Gibbs, J. Chem. Phys., 43 139 (1965); GW Scherer, J. Am

Ceram Soc, 67 504 (1984)

  • Fluidity of a system depends on the rate of disappearance of the

configurational entropy

  • A system at the ideal glass transition temperature (T2) has no more

configurational entropy to lose

– System is ‘frozen’ into ‘ground state’ of amorphous packing

  • Adam-Gibbs model assumes that a liquid consists of a number of

regions that can cooperatively rearrange

– Each region consists of ‘Z’ molecules that can rearrange independently in response to an enthalpy fluctuation

  • As a liquid is supercooled, configurational entropy of the system is

reduced and the size of the cooperatively rearranging subsystems grows larger

– Increased coupling between neighboring molecules with decreasing temp.

slide-43
SLIDE 43

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-43

EA DiMarzio, J. Res. Natl. Inst. Stand. Technol. 102, 135 (1997) Lattice model version: glass regions imbedded in sea of liquid- glass transition occurs (on cooling) when the percolation limit for imbedded regions is reached

Visualizing configurational entropy Visualizing configurational entropy… …

slide-44
SLIDE 44

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-44

Adam Adam-

  • Gibbs (cont.)

Gibbs (cont.)

  • Probability for a single cooperative transition:

where A is a frequency factor and δμ is the energy barrier (per molecule) to rearrangement. The average transition probability depends on the lower limit to the sizes of the cooperative regions (z*):

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− = T k z A T p

B

δμ exp ) (

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− ≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− = ∑

∞ =

T k z A T k z A T p

B z z B

δμ δμ * exp exp ) (

*

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− =

c c B A c

TS C A TS k N s A T p exp exp ) (

*

δμ

Entropy of the entire system (Sc) depends on the number (n) of rearranging units

  • f size z and the entropy contribution of each unit: Sc=nsc. For one mole of

molecules, there are n=NA/z independent regions, so z*=sc*NA/Sc and sc*≈kBln2 is the configurational entropy of a minimally sized region.

slide-45
SLIDE 45

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-45

Adam Adam-

  • Gibbs (cont.)

Gibbs (cont.)

Since fluidity (1/η) is proportional to the transition probability, then

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

c

TS C exp η η

Note that η→∞ as Sc decreases (Δs→0); viz., fewer configurations are accessible, more molecules must cooperate to permit flow. At some temp (TK), Z→∞ and flow no longer occurs.

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ≈ = Δ Δ ≈ ∫ T T D S T D C T C S

K c p T T p c

K

1 1 , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ≈

K

T T B exp η η

VFT-form

Angell: Fragile liquids have small values for ‘D’, strong liquids have large values.

slide-46
SLIDE 46

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-46

A A-

  • G Model

G Model fits viscosity fits viscosity data very data very well. well.

Richet and Bottinga, 1995

slide-47
SLIDE 47

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-47

A A-

  • G Model

G Model fits viscosity fits viscosity data very data very well. well.

Richet and Bottinga, 1995

slide-48
SLIDE 48

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-48

Kauzmann Temperature (TK)

Does the extension of the supercooled liquid entropy below Tg by slower cooling lead to the condition where this entropy is equal to that of the crystal?

  • Kinetic barrier to a

thermodynamic catastrophe

  • “Frozen in transition”

without any specific thermodynamic order*

  • TK provides a

thermodynamic basis for the VFT relationship

*Varshneya (p.328)

slide-49
SLIDE 49

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-49

Potential energy landscapes represent configuration distributions in a system From Adam-Gibbs, each transition probability depends

  • n δμ, the energy barrier (per

molecule) to rearrangement

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− = T k z A T p

B

δμ exp ) (

Fragile liquids are characterized by many different configurations that are accessible at greater temps: greater Sc, lower η. Strong liquids have few local minima- single barrier, Arrhenius dependence?

slide-50
SLIDE 50

Adv Vitreous State/Glass Properties FS08 Richard K. Brow/Missouri S&T brow@mst.edu Web-based Course Glass Transformation-50

Summary Summary-

  • Glass Transition

Glass Transition

  • Upon cooling a liquid through the supercooled region, viscosity rapidly

rises

– Glass transition range: 108-1015 Pa-s – Glass transition temperature sometimes defined at 1012 Pa-s

  • Rapid rise in viscosity implies a rapid reduction in free volume and a

loss of configurational entropy

  • Fictive temperature

– Divergence of the temperature dependent properties of a supercooled liquid and glass; e.g., V-T curve – structure of a glass corresponding to the structure of the liquid at Tfic

  • A change in temperature in the transition range produces a nonlinear

approach to equilibrium for glass properties, like viscosity

– Properties depend on changing experimental temperature and changing fictive temperature – Fictive temperature history determines the physical properties of a glass