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

About flow Haraclitus : " everything is in a state of flux ". 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 ”.

Flow is everywhere!

Flow is remarkable, but sometimes dangerous! In philipin In Hawaii

How to judge whether a substance is liquid or solid? A fu fundamental number of f rh rheology: Deborah number (D (D e )  Time of relaxation  D e t Time of observation If  < t, a substance is a liquid, otherwise, a solid!

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. 14 SiO2 Basalt 12 annealing Anorthite 10 log  (Pa s) 8 glass blowing 6 4 2 fiber drawing window 0 fining window -2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 T g / T (K/K)

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. Angell plot It provides information on • Glass dynamics • Transport properties • Glass structure • Liquid fragility • Thermodynamics • Geology • Crystallization • ........

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. It is quantified by the kinetic liquid fragility index m . m (slope at T g ) log  12 d  SiO 2 (Infrasil) m 10 DGG d ( T g T / )  NCS (16Na 2 O10CaO74SiO 2 ) T T g Basalt Seltso 8 Basalt Komso Log  (  in Pa s) Diabase Obersheld) 6 Diabase Karshamn Anorthosite 4 • It is defined as the rate of Diopside 25Na 2 O25Li 2 O50P 2 O 5 2 the viscosity or relaxation CaP 2 O 6 liquid at T g upon cooling. 0 • It is an important dynamic -2 parameter of glass-forming -4 liquids. 0.0 0.2 0.4 0.6 0.8 1.0 T g / T

Connection between fragility index ( m ) and heat capacity jump (  C p ) in glass 12 1.6 Na 2 O-CaO-SiO 2 (F 1/2 =0.46) -1 K -1 )  C p (Jg 10 Ca(PO 3 ) 2 (F 1/2 =0.81) Ca(PO 3 ) 2 0.37 1.4 NaO-CaO-SiO 2 0.27 low m 8 Log  0 (Pa s) 6 1.2 high m C p / C pg  C p 4 1.0 2 T g 0 0.8 -2 0.4 0.6 0.8 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 T / T g (K/K) T g / T   0  A m C ( 1 ) p T m g Smedskjaer, et al. J. Phys. Chem. B , 2011

Viscosity models

Vogel-Fulcher-Tamman (VFT) Model    A ' exp( )   T T 0 where  ∞ is the high temperature limit of viscosity, and A and T 0 are constants. Or      A log log  T T 0 T 0 = T k ? This is a debating problem. Vogel, Phys. Zeit. 22 (1921) 645; Fulcher, J. Am. Ceram. Soc. 8 (1925) 339 Tammann, Hesse, Z. Anorg. Allg. Chem. 156 (1926) 245

Adam-Gibbs (AG) Model (Entropy model )    B ' exp( )  TS ( T ) c where  ∞ is the high temperature limit of viscosity, B is constant, and S c ( T ) is the configurational entropy as a function of temperature:   T     This is a problem too. ( ) ln S T C c p   T K Adam and Gibbs, J. Chem. Phys. 43 (1965)139

Avramov-Milchev (AM) Model     T g F log log B ( )  AM T where  ∞ is the high temperature limit of viscosity, B AM constant, and T g the glass transition temperature, and F is a measure of liquid fragility. F = m / B AM , where m is the Angell fragility index Avramov and Milchev, J. Non-Cryst. Solids 104 (1988) 253

Angell-Rao (AR) model Angell and Rao, JCP (1972) 𝑚𝑝𝑕𝜃 = 𝑚𝑝𝑕 𝜃 ∞ + 𝐵𝑓𝑦𝑞(𝐶 𝑈 − 𝐷) 16 data 14 Angell-Rao model 12 log  = -3.45+0.434exp(3931/T-(-0.09)) 10 log  (Pa s) 8 6 4 2 0 Anorthite -2 1000 1500 2000 2500 T (K) This 4-parameters model with fits the data excellently and bears physical meaning.

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)   K exp     C log log  T T B 3     Adam-Gibbs expression log log  TS c   S c fNk ln The configurational entropy    exp  H Topological degrees of freedom f 3 A simple two-state system kT 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:   T relation for oxide, ionic and molecular liquids    log   log    B T exp C 12  SiO2 Window glass   T 10 Corning aluminosilicate Basalt 8 Anorthite Glycerol 6 log  (Pa s) Propylene carbonate Triphenylethe 4 O-terphenyl ฀ 4Ca(NO 3 ) 2 -6KNO 3 2 0       fragility T T   m           g  g    -2 log log 12 log exp 1 1             T  12 log T   -4 0.0 0.2 0.4 0.6 0.8 1.0 T g / T (K/K)

The new model is physically reasonable. (Fitting results based on 1000 glasses) log   =-3 160 high T Current Model 140 120 100 Count VFT AM 80 60 40 low T 20 0 -6 -5 -4 -3 -2 -1 0 Log[  (Pa-s)]  New model: • log   : the narrowest distribution • S c converges at T =  • log   =-3: A universal value? • S c = 0 at T =0

The new model is practically useful. 12 16 12.1 K RMS Error (K) Predicted 12 8.7 K 10 Isokom 8 5.6 K 4 Log[Viscosity (Pa-s)] 8 Used for 0 VFT AM Current Model Fitting (b) 6 16 4 12 9.4 K Average Error (K) Measured 8 2 VFT 4 AM Current Model Avramov 0 New Model VFT -0.5 K 0 -4 0.5 0.6 0.7 0.8 0.9 1.0 -8 -5.6 K 1000/[T (K)] -12 (c) 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

T g,vis (from viscosity) and T g,DSC (from DSC) 14 data measurements AM fit 12 log  (Pa s) 1400 linear fit 10 Silica? Ca(PO 3 ) 2 melt 8 1200 6 4 T g,DSC (K) 1000 2 0 800 silicate 1.8 upscan 600 C p (Jg -1 K -1 ) 1.6 downscan NaPoLi 1.4 400 1.2 200 water? Tg 1.0 200 400 600 800 1000 1200 1400 q h = q c =10 K/min 0.8 T g,vis (K) 600 800 1000 1200 1400 T(K) T g,10K/min = T log  =12 Y. Z. Yue, J. Non-Cryst. Solids 2008, 2009

Practical use of the MYEGA log    d B exp C        m  log log  T Tg   d ( Tg / T ) T T         T T m            g  g    log log log log exp 1 1       Tg        T log log T      Tg For inorganic systems For inorganic systems η ∞ ≈ 10 -3 Pa s η Tg ≈ 10 12 Pa s       T T m g  g            log 3 15 exp 1 1       15   T  T  Now, only two parameters, m and T g , remain. Meaning: the entire log  ~T relation can be estimated just by DSC!

Be careful with the difference between m vis and m DSC A model:       •      m vis > m DSC m m m m 1 f m m vis 0 DSC 0 DSC 0 • m vis - m DSC due to Arrhenian approximation of non-Arrhenius behavior m vis = 1.289( m DSC - m 0 )+ m 0 • m vis – m DSC increases as fragility increases 28 Zheng, Mauro, Yue, J. Non-Cryst. Solids . 2017