Thermodynamics of viscous flow and elasticityof glass forming liquids in the glass transition range

Tanguy ROUXELLARMAUR, ERL CNRS 6274, University Rennes 1, France

RISC-E octobre 2011

Elastic moduli, Poisson’s ratio and the short to mediumrange order in glass

High temperature elastic behavior and thedepolymerization of the atomic network

The viscous flow process and the non-Arrhenian behavior

Elastic moduli, Poisson’s ratio and the short to mediumrange order in glass

There is no direct correlation between E and Tg

Elastic moduli are expressed in Pascals, i.e. in J/m3 , and are thusgoverned by the volume density of energy

J. Am. Ceram. Soc. 90 [10] 3019-3039 (2007)

Chemical-Physics

Mechanics of

Materials

(constititive laws)

Numerical

Modelling and

Simulation (FEM,

MDS) Mechanical Engin. -

Structural Design

Silicate glasses: Glass formers (Si, Al, B, Zr) Modifyers and charge compensators (Li, Na, K, Ca, Ba) Anions: O, N or C

Cation substitution: modifyers ! Uo ; formers ! CgIntermediate elements occupying former or interstitial sites: Hf, Be,Zr, Ti, Li and Th.Electronegativities: 1.25 to 1.75.Emax=145 GPa: magnesium aluminates + 25 mol.% de BeO.

Si Si

Si Si

Si Si

Si Si

Si Si

Si Si

O

O

O

O N C

O

O

Anionic substitutions: more efficient but Tg

However: UoSiC(447) kJ/mol)~UoSi-N(437 kJ/mol)<UoSi-O(800 kJ/mol)

E oxycarbides and E oxynitrides >> E oxides

This is more the architecture (reticulation) of the network than the individual bondstiffnes that governs the glass elasticity

Elastic moduli and structural changes are correlated

!cd<!bcc<!fcc,hcp

For a given crystalline structureand valency, Poisson’s ratiomostly increases with atomicnumber (Z) (Lead (!Pb=0.44) andthallium (!Tl=0.45) with high Zhave remarkably high Poisson’sratios)

High melting points favor lowPoisson's ratio (beryllium,combining a small Z and amelting point (Tm=1560 K) muchhigher than those of the otherelements in the same column,exhibits a remarkably smallPoisson’s ratio of 0.032)

General tendencies:

SeSe

SeSe

SeSe

SeSe

Ge

Ge

GeSeSe

SeSeSe

Se

SeSe

SeSe

SeSeSe

Se Ge

OSiO

OO

SiO

OO

SiOO

O

SiO

OO

Si

OOO

Si OO

O

N

CSi

OOO Si

OOO

SiOO

O

SiOO

O

Si

OOO

Si OO

O

Substitution of 3O2- for 2N3-

Substitution of 2O2- for 1C4-

1D: Chains and rings

2D: Layers

3D: 3D units andcrosslinking

0D: Clusters, litlle short to medium range ordering

Molecular scale:

Poisson’s ratio and atomic network dimensionality

"L

"D

D

L# = -$t/$l = - L/Dx"D/"L

"V/V=Trace $=(1-2#)%/E

%

#& 0.153D

#& 0.2862D

#& 0.3231D

#& 0.370D?

a-SiO2 GeSe4 a-Se Zr55Cu30Al10Ni5

Poisson’s ratio and dimensionality

High temperature elastic behavior and thedepolymerization of the atomic network

High temperature measurementsMeasurements at ambient temperature

Measurement of elastic moduli by ultrasonic echography

Generator

Receiver

Signal analyser

Magnetostrictive Transducer (100 – 400kHz)

Ar-7%H2

Furnace Sample

Tungsten guide

Refractory cement

Thermocouple

(Fourier's analysis)

E = ' Vl2

L

Long beam bar mode : e<<L and e << ( (10 to 100 mm)

e

Oscilloscope

Sample

Transmitter/receiver

Piezoelectrictransducer (5 to 20MHz)

Signal

Time synchronization

Elastic wave in semi-infinite medium : (<<L and ( << )

E = ' (3Vl2-4Vt

2)/((Vl/Vt)2-1)G= ' Vt

2

#=E/(2G)-1

Where: ': specific massVl: longitudinal wave velocityVt: transverse wave velocity

HL

Small size specimens: Acoustic microscopyWhen (L,H)<(, regular piezoelectric transducers are

unable to efficiently promote the propagation of shear wavesthrough the specimen. Focused piezoelectric transducers canbe used to propagate surface-type waves, also calledRayleigh waves, which velocity is given by: VR=*Vt, where* is a function of Poisson´s ratio, or of the Vl/Vt ratio. VRand Vl are measured and Vt is optimised to satisfy thefollowing equation:

)/V(V-0.750))/V(V-(0.715V

Vlt2ltt2

R =

0

20

40

60

80

100

120

140

160

200 400 600 800 1000 1200 1400 1600

T (K)

Youn

g's

mod

ulus

(GPa

)

SiOC Window glassSe Ge10Se90Ge15Se85 Ge25Se75Ge30Se70 YSiAlOYSiAlON(7.5 at% N) YSiAlON(11.2 at% N)YSiAlON(Si/Al=1.5) YSiAlON(Si/Al=1.95)YSiAlON(Si/Al=3.75) YMgSiAlONZrCuAlNi GlycerolGe22As20Se58 a-SiO2Pd42.5Ni71Cu30P20 ZBLANDiopside Grossulara-B2O3

Soda-lime-silica (window)

SiOC

Chalcogenides

Oxynitride

Bulk metallic Basaltic

a-SiO2

Tg

For T>Tg, µ/µ(Tg)=(Tg/T)+

where + ranges between 0.07 (a-SiO2) and 10 (a-Se)

The viscous flow process and the non-Arrhenian behavior

Log stress

Log ,

So-called equilibrium viscosity!

Stress or strain-rate dependence

Boltzmann-Arrhenius approach of thermally activated processes:

- = -o exp["Ga /(RT)] -: the characteristic relaxation time-o: a constantT: temperatureR: perfect gas constant"Ga: free activation enthalpy of the flow process

Assuming a simple Maxwell relaxation model,= µ -o exp["Ga /(RT)] ,

where µ is the shear elastic modulus.

, = ,o exp["Ga /(RT)]Fragile

Strong

Log ,/,(Tg)

Tg/T

Observations: the heat for flow is temperature dependent, especially in the case ofshort glasses VFT, WLF, AM, MYEGA… numerous empirical expressions

Adam and Gibb's (1965) approach of the viscous flow process:

- = -o exp[z"g /(RT)] -: the characteristic relaxation time; -o: a constant; T: temperature; R: perfect gas constant; z: number ofmolecules or monomeric segments; "g: height of the potential energy hindering the cooperativerearrangement of the elementary structural units involved in the process

Assuming a simple Maxwell relaxation model,= µ -o exp[z"g /(RT)] ,

where µ is the shear elastic modulus.z is inversely proportional to the configurational entropy Sc and "g is taken as temperature independent

, = ,o exp["G /TSc]

The configurational entropy is a temperature dependent parameter (especially for T>Tg) which iscommonly assimilated to the excess entropy

!

Sc(T ) = Sc(Tg ) +Cp

Liq "CpGlass

TTg

T# dT

Good fit of experimental data in the few cases where actual calorimetric data are available(Richet, Botinga)

But fundamentals uncertainties remain (« excess » versus « configurational » entropy;« Configurational » versus « vibrational entropy »)

Present analysis:

i) As in the case of plasticity in crystalline materials, the energy barrier for viscous flowchiefly depends on the shear elastic modulus. Therefore µ comes into play both in theprefactor and implicitly in "Ga

ii) Glassy materials differ from crystalline ones by a rapid softening in the supercooled liquirange (i.e. for T>Tg), so that µ is mostly very sensitive to temperature. Consequently, "Gadepends much on temperature and this can be analyzed within the classical framework ofthermodynamics, recalling that the entropy change writes

!

"S = # $"G$T

%

,= µ -o exp["Ga /(RT)]

The problem of concern reduces to the determination of the temperature dependence of !Ga, !Saand the heat for flow !Ha

ACTIVATION ENTROPY OF THE VISCOUS FLOW PROCESS

!

˙ " # exp $%Gb $ %Wa( )

RT&

' (

)

* +

!

˙ " # exp $%Ga( )RT

&

' (

)

* +

"Gb = "Ga + "Wa

!

" = # / ˙ $

!

R " ln#

"1T $ , stru

= %Ga + 1T

"%Ga

"1T $ , stru

!

" #$Ga

#T % , stru

= $Sa

!

R " ln#

"1T $ , stru

= %Ga + T%Sa = %Ha

!

"Sa = # $"Gb

$T % , stru

+%$Va$T % , stru

!

"#Gb

"T $ , stru

=#Gb

µ"µ"T $ , stru

!

"Sa = # $T

("Ga +%Va ) + % &Va&T

!

" = Tµ#µ#T

!

"Saµ

!

"Sa#

For -=10 MPa and Va=50 cm3.mol-1, -Va=500J.mol-1, to be compared with the order ofactivation energies, 100 to 1000 kJ.mol-1

!

"Ha = R # ln$

#1T % , stru

!

"Sa = # $(1# $ )T

"Ha

!

"Ga = 1(1# $ )

"Ha

(1)

(2)

(3)

(4)

(2)+(5)

(5)

("Wa = - Va)

(6)

+(6)

!

"Ha = R# ln$#(1/T ) % , stru

"#$%&,"#$%&...

This is where entropy comes into play…

"Ga = "Ga(Tg)(Tg/T)+

!

" = "o exp#Ga(Tg )(Tg /T )$

RT

%

& '

(

) *

!

"Ga(Tg ) = "Ha(Tg )

1 #Tg

µ(Tg )$µ$T Tg

%

& ' '

(

) * *

,o = µ(T),(Tg)/µ(Tg)

with

!

"#Ga

"T $ , stru

= #Ga

µ"µ"T $ , stru

µ/µ(Tg)=(Tg/T)+

Equivalent to Avramov-Milchev equation

with: µ(T)=µ.exp(Eµ/T)Equivalent to MYEGAequation