Viscosity of glass-forming liquids Yuanzheng Yue Aalborg - - PowerPoint PPT Presentation

Viscosity of glass-forming liquids

Yuanzheng Yue

Aalborg University, Denmark Wuhan University of Technology, China

Joint ICTP-IAEA Workshop, Trieste, Italy, Nov. 6-10, 2017

Outline

• Background and motivation
• Viscosity models
• Iso-structural viscosity
• Non-Newtonian flow
• Fragile-to-strong transition

Background and motivation

Confucius (孔夫子) stood by a river: "Everyting flows like this, without ceasing, day and night”. Deborah: "Everything flows if you wait long enough, even the mountains”.

About flow

Haraclitus: "everything is in a state of flux".

Flow is everywhere!

Flow is remarkable, but sometimes dangerous!

In philipin In Hawaii

How to judge whether a substance is liquid

• r solid?

t e D  

If  < t, a substance is a liquid, otherwise, a solid! Time of relaxation Time of observation

A fu fundamental number of f rh rheology: Deborah number (D (De)

Some liquids flow easily, some not. How to quantify this? Measure Viscosity by vis iscometers:

• Concentric Cylinder
• Parallel-Plate Compression
• Capillary
• Beam Bending
• Fiber Elongation
• Sphere penetration
• Melt containerless levitation
• ……..

Vis iscosit ity is is a crucia ial quantit ity of gla lass technology.

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

• 2

2 4 6 8 10 12 14

SiO2 Basalt Anorthite fining window annealing glass blowing

log (Pa s) Tg/T (K/K)

fiber drawing window

Viscosity determines

• Melting conditions
• Fining behaviour
• Working ranges
• Annealing range
• Upper temperature of use
• Devitrification rate
• Glass forming window
• Glass fiber drawing window
• Every step of industrial glass formation depends critically on the

viscosity.

• The glass product relaxation depends on the nonequilibrium viscosity of

the glass, which is a function of composition, temperature, and thermal history.

Viscosity is a key quantity of glass science.

It provides information on

• Glass dynamics
• Transport properties
• Glass structure
• Liquid fragility
• Thermodynamics
• Geology
• Crystallization
• ........

Angell plot

Viscosity of a melt varies with

• Temperature
• Time
• Deformation rate
• Pressure
• Composition
• Hydroxyl
• Crystallization
• Phase separation
• Inclusions
• .......

The non-Arrhenian behavior of liquids is described by liquid fragility.

0.0 0.2 0.4 0.6 0.8 1.0

• 4
• 2

2 4 6 8 10 12

SiO2 (Infrasil) DGG NCS (16Na2O10CaO74SiO2) Basalt Seltso Basalt Komso Diabase Obersheld) Diabase Karshamn Anorthosite Diopside 25Na2O25Li2O50P2O5 CaP2O6

Tg/T Log ( in Pa s)

m (slope at Tg)

g T T g T

T d d m



 ) / ( log

It is quantified by the kinetic liquid fragility index m.

• It is defined as the rate of

the viscosity or relaxation liquid at Tg upon cooling.

• It is an important dynamic

parameter of glass-forming liquids.

Connection between fragility index (m) and heat capacity jump (Cp) in glass

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.8 1.0 1.2 1.4 1.6

Cp (Jg

• 1K
• 1)

Ca(PO3)2 0.37 NaO-CaO-SiO2 0.27

T/Tg (K/K) Cp/Cpg Tg

Cp 0.4 0.6 0.8 1.0

• 2

2 4 6 8 10 12

high m

Na2O-CaO-SiO2 (F1/2=0.46) Ca(PO3)2 (F1/2=0.81)

Tg/T Log0 (Pa s)

low m

) 1 (

0 

 

m m g T A p

C

Smedskjaer, et al. J. Phys. Chem. B, 2011

Viscosity models

Vogel-Fulcher-Tamman (VFT) Model

) exp(

' T T A  

 

where ∞ is the high temperature limit of viscosity, and A and T0 are constants. Or

log log

T T A   

  

Vogel, Phys. Zeit. 22 (1921) 645; Fulcher, J. Am. Ceram. Soc. 8 (1925) 339 Tammann, Hesse, Z. Anorg. Allg. Chem. 156 (1926) 245

T0 = Tk? This is a debating problem.

Adam-Gibbs (AG) Model (Entropy model )

) exp(

) ( ' T TS B

c



 

where ∞ is the high temperature limit of viscosity, B is constant, and Sc(T) is the configurational entropy as a function of temperature:

Adam and Gibbs, J. Chem. Phys. 43 (1965)139

       

K

T T C T S

p c

ln ) (

This is a problem too.

Avramov-Milchev (AM) Model

F T Tg

AM

B ) ( log log  



 

where ∞ is the high temperature limit of viscosity, BAM constant, and Tg the glass transition temperature, and F is a measure of liquid fragility. F=m/BAM, where m is the Angell fragility index

Avramov and Milchev, J. Non-Cryst. Solids 104 (1988) 253

Angell-Rao (AR) model

1000 1500 2000 2500

• 2

2 4 6 8 10 12 14 16

data Angell-Rao model

log  (Pa s) T (K)

log  = -3.45+0.434exp(3931/T-(-0.09))

Anorthite

This 4-parameters model with fits the data excellently and bears physical meaning. Angell and Rao, JCP (1972)

𝑚𝑝𝑕𝜃 = 𝑚𝑝𝑕 𝜃∞ + 𝐵𝑓𝑦𝑞(𝐶 𝑈 − 𝐷)

Other models

• Free volume model
• Doremus model
• Shoving model
• Sanditov model
• Parabolic model
• …….

See recent reviews:

• M. I. Ojovan, Adv. Condensed Mat. Phys., 2008,

S.V. Nemilov, J. Non-Cryst. Solids, 2011

• Q. Zheng, J.C. Mauro, J. Am. Ceram. Soc., 2017.

Derivation of our new model (MYEGA)

 

kT H

f   exp 3

  ln fNk Sc

c

TS B3

log log  



 

 

T C T K exp

log log  



 

Topological degrees of freedom A simple two-state system The configurational entropy Adam-Gibbs expression

Mauro, Yue, Ellison, Gupta, Allan, PNAS 106 (2009) 19780

The viscosity-temperature relation for most liquids can be described by VFT and AM models, even better by MYEGA:

฀ log  log  B T exp C T      

0.0 0.2 0.4 0.6 0.8 1.0

• 4
• 2

2 4 6 8 10 12 SiO2 Window glass Corning aluminosilicate Basalt Anorthite Glycerol Propylene carbonate Triphenylethe O-terphenyl 4Ca(NO3)2-6KNO3

log (Pa s) Tg/T (K/K)

fragility

 

                           

  

1 1 log 12 exp log 12 log log T T m T T

g g

   

  T relation for oxide, ionic and molecular liquids

The new model is physically reasonable. (Fitting results based on 1000 glasses)

20 40 60 80 100 120 140 160

• 6
• 5
• 4
• 3
• 2
• 1

Log[ (Pa-s)] Count



VFT AM Current Model

high T low T

New model:

• Sc converges at T=
• Sc = 0 at T=0
• log: the narrowest distribution
• log=-3: A universal value?

log=-3

The new model is practically useful.

2 4 6 8 10 12 0.5 0.6 0.7 0.8 0.9 1.0 1000/[T (K)] Log[Viscosity (Pa-s)] Measured VFT Avramov New Model

Used for Fitting Predicted Isokom

• 0.5 K
• 5.6 K

9.4 K

• 12
• 8
• 4

4 8 12 16 Average Error (K)

(c)

VFT AM Current Model 5.6 K 8.7 K 12.1 K 4 8 12 16 VFT AM Current Model RMS Error (K)

(b)

The new model shows stronger ability to predict low T viscosity data from high T viscosity data than the other 3-parameter models.

Is there a universal log η∞ value?

Results on 946 Corning compositions

It is about -3!

Zheng, et al. Phys. Rev. B 2011

Tg,vis (from viscosity) and Tg,DSC (from DSC)

• Y. Z. Yue, J. Non-Cryst. Solids 2008, 2009

600 800 1000 1200 1400

0.8 1.0 1.2 1.4 1.6 1.8

qh=qc=10 K/min

upscan downscan

Tg

Cp (Jg-1K-1) T(K)

2 4 6 8 10 12 14 data AM fit

Ca(PO3)2 melt

log  (Pa s)

Tg,10K/min = Tlog=12

200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400

NaPoLi

water?

measurements linear fit

Tg,DSC (K) Tg,vis (K) silicate Silica?

Practical use of the MYEGA

       



T C T B exp log log  

 

                             

  

1 1 log log exp log log log log T T m T T

g Tg g Tg

     

                           1 1 15 exp 15 3 log T T m T T

g g



For inorganic systems η∞ ≈ 10-3 Pa s For inorganic systems ηTg ≈ 1012 Pa s

Tg T T Tg d d

m





) / ( log

Now, only two parameters, m and Tg, remain. Meaning: the entire log~T relation can be estimated just by DSC!

Be careful with the difference between mvis and mDSC

• mvis > mDSC
• mvis - mDSC due to Arrhenian approximation
• f non-Arrhenius behavior
• mvis – mDSC increases as fragility increases

     

1 m m f m m m m

DSC DSC vis

    

A model:

28

mvis = 1.289(mDSC-m0)+m0

Zheng, Mauro, Yue, J. Non-Cryst. Solids. 2017

The entir ire vis iscosit ity-temperature curve can be be determin ined by y DSC!

Advantages of the DSC method:

• It is simpler.
• Takes much less time than

viscometry technique.

• Uses smaller samples.
• Measure both good and poor

glass forming systems.

Example

Based on the facts:

• Tg and mDSC are measurable by DSC
• mDSC can be converted to mvis.
• log10=-3
• Tg corresponds to 1012 Pa s

29

                           1 1 15 exp 15 3 log T T m T T

g g



Derivation of VFT from MYEGA

• In the high temperature limit, -K/T can be expanded in a Taylor series:

M.M. Smedskjaer, J.C. Mauro, Y.Z. Yue, J. Chem. Phys. 131, 244514 (2009).

Divergent at a finite T ?

Zhao, Simon, McKenna, Nature Comm, (2013)

Using 20-million year-old amber, Zhao, et al. provided an implication against the existence of the divergence at a finite T.

MYEGA VFT Parobolic form

See also Hecksher, et al., Nature Phys. (2008)

Iso-structural viscosity or non- equilibrium viscosity

Comparison between the measured iso data and the iso data calculated from models

6 8 10 12 14 16 4 8 12 16 measured iso measured eq linear fit of iso MYEGA fit of eq AM AGS VFT

log  ( in Pa s) 1/T (10

• 4 K
• 1)

window glass

MYEGA

Tg=824 K

Data from Mazurin (1982)

11 12 13 14 15 8 10 12 14 16 18 MYEGA AM AGS Tf=0.96Tg

log  ( in Pa s) 1/T (10

• 4 K
• 1)

window glass

Tg VFT

Non-Newtonian flow

(Shear rate dependence of viscosity) (Shear thinning)

0.0 0.2 0.4 0.6 0.8 200 400 600 800

Newtonian flow

NaPoLi

NCS

 (MPa)

Non-Newtonian shear flow of glass-forming liquids (soda lime silicate vs Li-Na metaphosphate)

0.01 0.1 1 0.1 1

NaPoLi NCS

/0

in s-1

    in s-1

)) exp( (1 ) η (η η

g g

            

)] exp( [1 ) (1

g

                  

  g

Y.Z. Yue and R. Brückner, J. Non-Cryst. Solids (1994)

0=109 Pa s It is attributed to orientation of structural units.

Fragile-to-strong transition

(An abnormal liquid dynamic behaviour)

A normal liq liquid id – a a win indow gla lass!

6 8 10 12 14 4 8 12 16

log (Pa s) 1/T (10-4 K-1) window glass

A normal liquid – a window glass!

6 8 10 12 14 4 8 12 16

log (Pa s) 1/T (10-4 K-1) window glass

                           1 1 15 exp 15 3 log T T m T T

g g



An abnormal case – a metallic liquid!

0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 4
• 2

2 4 6 8 10 12

La55Al25Ni15Cu5 log (Pa s) Tg/T

An abnormal case – a metallic liquid! Its dynamics cannot be decribed by a 3-parameters model.

0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 4
• 2

2 4 6 8 10 12

La55Al25Ni15Cu5

SD: 0.181 loginf = -13.2

log (Pa s) Tg/T

SD: 0.48 loginf = -7.5

                           1 1 15 exp 15 3 log T T m T T

g g



We recall a famous liquid – water, which shows an abnormal dynamic behaviour to – fragile-to-strong transition

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 4
• 2

2 4 6 8 10 12 Water viscosity data

from Hallett (1963) from Angell (2002)

MYEGA fit water F-S

log(Pa s) Tg/T

SiO2 Tg=162 K

Ito, Moynihan, Angell, Nature 1999

More metallic liquids similar to water, which exhibits Fragile-to-Strong (F-S) Transition

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 4
• 2

2 4 6 8 10 12 water OTP

Gd55Al25Co20 Gd55Al25Co10Ni10 Pr55Ni25Al20 water

log(Pa s) Tg/T

SiO2

The data of these liquids cannot be described by a single model.

Zhang, Hu, Yue, Mauro, J. Chem. Phys. (2010) Way, Wadhwa, Busch, ACTA Mater. (2007)

f = m'/m

0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 4
• 2

2 4 6 8 10 12

m

La55Al25Ni20 La55Al25Ni15Cu5 La55Al25Ni5Cu15 Al87Co8Ce5 Ce55Al45

b

Tg/T

m'

• 4
• 2

2 4 6 8 10 12

Sm55Al25Co10Ni10 Sm50Al30Co20 Sm55Al25Co10Cu10

log(Pa s)

a

The extent of the F-S transition can be determined by: f > 1: F-S transition f = 1: no F-S transition f < 1: never seen (unphysical?)

More….

Zhang, Hu, Yue and Mauro, JCP (2010)

The factor f confirms the existence of the F-S transition in the investigated MGFLs.

The calculated f values for different MGFLs

Is there a model that can describe the abnormal liquid dynamic behaviour? Question:

Yes! But, to do so, the MYEGA has been generalized to the form:

0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 4
• 2

2 4 6 8 10 12

La55Al25Ni20 La55Al25Ni15Cu5

log(Pa s) Tg/T

                    



T C W T C W T

2 2 1 1

exp exp 1 log log  

C1 and C2: two constraint onsets . W1 and W2: normalized weighting factors. If C1 = C2, the equation reduces to that for normal liquids.

Fragile term Strong term

Zhang, Hu, Yue, Mauro, JCP (2010)

Two “phases” co-exists in the F-S crossover regime: Strong and fragile phases

0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 4
• 2

2 4 6 8 10 12

data

• Eq. (7)

fragile term strong term

log(Pa s) Tg/T

b

La55Al25Ni15Cu5

• 4
• 2

2 4 6 8 10 12

fragile phase

data

• Eq. (7)

fragile term strong term

a

La55Al25Ni20

strong phase

Fragile phase (LDA):

• higher Tg
• higher activation enthalpy
• higher entropy
• lower density

Strong phase (HDA):

• lower Tg, i.e., actual Tg of the

mixed liquid

• lower activation enthalpy
• lower entropy
• higher density

The fragile phase is cooled, the F-S transition intervenes, mitigating the sharp increase in viscosity with decreasing T.

Non-montonic structural response to sub-Tg annealing measured by x-ray scattering

Annealing dependence

• f the structural unit size

Annealing dependence

• f the correlation length

Critical temperature for the dramatic decreases in Rc: Tc ~ around 1.3Tg

Total structural factors PDF

Schematic scenario of the structural evolution during fragile-to-strong transition

Zhou, et al. J. Chem. Phys. (2015)

Containerless aerodynamic levitation (ADL) melting to avoid heterogeneous nucleation

2180 2200 2220 2240 2260 2280 2300

0.2 0.4 0.6

Melting of Al2O3

η = 3𝑁 20𝜌𝑆 Γ

Forced oscillation and decay

• Extend the supercooled region
• Measure viscosity
• In-situ structural characterization

Fragile-to-strong transition in aluminates

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 2

2 4 6 8 10 12

log (Pa s) Tg/T (K/K)

m=45 m=81 log0= -2 (Pa s) 3CaO-Al2O3 MYEGA fitting Tg=1075.2 K

                         1 1 15 exp 15 3 log T T m T T

g g



Fitted with

(MYEGA)

The data can be described by the generalized MYEGA

Parameter Value log0 (Pa s)

• 2.039

W1 0.018 C1 7324 W2 1.68E-4 C2 1407

𝑚𝑝𝑕𝜃 = 𝑚𝑝𝑕𝜃0 + 1 𝑈[𝑋

1 exp −𝐷1

𝑈 + 𝑋

2exp(−𝐷2

𝑈 )] 𝑈

𝑔−𝑡 =

𝐷1 − 𝐷2 𝑚𝑜𝑋

1 − 𝑚𝑜𝑋 2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

• 2

2 4 6 8 10 12

log (Pa s) Tg/T (K/K)

Tf-s=1263 K 3CaO-Al2O3

Strong term Fragile term Generalized MYEGA fit Viscosity data

I would like to all my co-authors and collaborators.

Thank you for your attention!