Web Course Physical Properties of Glass. Range Behavior

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1 Web Course Physical Properties of Glass Glass Transformation- Range Behavior Richard K. Brow Missouri University of Science & Technology Department of Materials Science & Engineering Glass Transformation-1

2 Glass Transformation-Range Behavior Structural relaxation and the glass transition Rheology and configurational entropy Thermal history effects on glass properties Glass Transformation-2

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 (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, *AK Varshneya, Fundamentals of Inorganic Glasses, 2 nd Ed (2006) Glass Transformation-3

4 Why should we care? Glass properties depend on thermal history The nature of the glass transition is the deepest and most interesting unsolved problem in solid state theory -Philip W. Anderson, 1995 Glass Transformation-4

5 Class Exercise- Consider the question What is the glass transition? Glass Transformation-5

6 Structural relaxation and the glass transition Glass Transformation-6

Glass transition: Glass transition: Changes in the structure of a supercooled liquid fall out of equilibrium as the liquid is cooled Relaxation time is long

7 Glass transition: Glass transition: Changes in the structure of a supercooled liquid fall out of equilibrium as the liquid is cooled Relaxation time is long compared with the observational time Properties (including T g ) depend on thermal history; viz., quench rate through the glass transition region Glass Transformation-7

8 Structural relaxation Average structure specifies thermo-dynamic state of the liquid (T,P,V specified) At equilibrium, average structure is timeindependent 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 Glass Transformation-8

9 Stress Relaxation- Maxwell Model Strain (ε) ε 0 ε 0 Stress (σ) Mathematical form: σ t = σ 0 exp(-gt/η) G = shear modulus (Pa) (η/g) = τ: time for stress to decay to (1/e) σ 0 = (0.367 σ 0 ) = Relaxation Time Time σ 0 =G 0 ε 0 σ(t)=g(t)ε 0 σ =0 Time σ t = σ 0 exp(-t/τ) Exponential relaxation curve Glass Transformation-9

10 Voigt-Kelvin Element- Anelastic behavior Glass Transformation-10

11 Burger elementpermanent deformation Note: viscoelastic substances may be modeled by a distribution of mechanical elements representing a distribution of structural features Glass Transformation-11

12 Consider isothermal relaxation after ΔT elastic viscous Note: bonds break & reform From Moynihan, 1995 Fictive Temp. Glass Transformation-12

13 Fictive Temperature Introduced by Tool, NBS (1946) Describes the contribution of structural relaxation to a property, expressed in units of temperature In equilibrium: T fic =T and dt fic /dt=1 For glass with frozen structure : T fic =const. and dt fic /dt=0 Not a fundamental property, but a conceptual view Temperature, T, T fic ΔT T T fic time Fictive: feigned, sham (from fiction) Glass Transformation-13

14 Volume Temp 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 order rate constant (k=1/t) and depends on deviation of volume from the new equilibrium value (at T 2 ): ( ) Fast (glassy) response d V Ve = k( V Ve ) dt V V T T e fic 2 Slower (viscous) response Φ( t) = V0 Ve T1 T2 t Φ( t) = exp( kt ) = exp τ Φ(t) is a relaxation function time Φ(t) = 1 at t=0, Φ(t) = 0 at t= Relaxation Time T 1 T 2 V 1 V 0 V e Glass Transformation-14

15 Assumption 3: for small departures from equilibrium and small ΔT, the temperature dependence of τ is described by the Arrhenius relationship: τ =τ exp Δ 0 where ΔH* is the activation enthalpy ( H * RT ) ΔH*>0, so as T decreases, τ increases and the rate of structural relaxation decreases Glass Transformation-15

16 Relaxation during heating and cooling τ>>t, H e is reached τ<<t, no relaxation (glassy behavior) Equilibrium property Deviation between experiment and equilibrium From Moynihan, 1995 Glass Transformation-16

17 Some observations: On cooling, H>H e after Δt On heating, H<H e after Δt Note on heating: H initially decreases when approaching H e, 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 Glass Transformation-17

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 Glass Transformation-18

19 Fast cooling rate: fall out of equilibrium at greater temperature (shorter relaxation time), greater limiting fictive temperature (T f )- the intersection of the H-values for the glass and liquid Recall Heat Capacity C p =dh/dt T g is observed when t Δt=ΔT/q From Moynihan, 1995 Note: sigmoidal shape of C p (T) is a consequence of the hysteresis in the ΔH/ΔT due to the relaxation kinetics Glass Transformation-19

20 Measuring the limiting fictive temperature - one measure of T g T f ' T >> T T << T ( C ) = ( ) p( e) C p( gl) dt C p C p( gl) g Area I T >> T g g Area II Glass Transformation-20

21 The glass transition temperature depends on the thermal history Borosilicate crown glass T g (and T fic ) increases with increasing quench rates (q c ) CT Moynihan, et al., JACerS, (1976) Glass Transformation-21

22 d(ln q c )/d(1/t f ) = -ΔH*/R Note: Over small temperature intervals, centered on T r, the activation energies for enthalpy relaxation are equivalent to those from shear-viscosity measurements (solid line) and from volume relaxation measurements (Ritland) CT Moynihan, et al., JACerS, (1976) Glass Transformation-22

23 Shear viscosity relaxation (closed symbols) and enthalpy relaxation (open symbols) processes have the same activation energies Volcanic obsidian glasses Dingwell, 1995 Glass Transformation-23

24 Enthalpy Relaxation vs. Viscous Flow From Moynihan, 1995 Glass Transformation-24

25 Example: Geospeedometry Volcanic obsidian natural cooling rate Dingwell, 1995 Glass Transformation-25

26 Example: Geospeedometry Natural extrapolated cooling rates 1º/week 1º/day 1º/hour Calibration range 1º/minute Glass Transformation-26

27 Kinetics of structural relaxation Qualitative first order kinetic model t Φ( t) = exp τ τ = τ 0 exp Δ ( H * RT ) Quantitative model must account for Non-linear character of the relaxation function Non-exponential character of the relaxation function Glass Transformation-27

28 Non-linear isothermal relaxation Note that the relaxation time depends on the instantaneous structure (fictive temp) τ = τ 0 x ΔH * (1 x) ΔH * exp + RT RT f Tool-Narayanaswamy (TN) Equation x is the nonlinear parameter (0<x<1) From Moynihan, 1995 Glass Transformation-28

29 Relaxation is non-exponential Normalized property Normalized time beta=1.0 beta=0.8 beta=0.6 beta=0.4 Φ( t) t τ = exp Stretched exponential function (KWW)- β is the nonexponentiality parameter (0<β<1) β Glass Transformation-29

30 Distribution of relaxation times t exp τ p β N k = 1 w k t exp τ k Broader distributions are associated with smaller values of β Scherer, 1991 Glass Transformation-30

31 Is there a physical source for non-exponentiality exponentiality? Consider a distribution of relaxation times (τ i ) incorporated into the relaxation function with a weighting coefficient (g i, where Σg i =1), and each relaxation time described by the T-N form: Φ ( t) = i g i exp xδh * τ i = τ 0 exp + RT t 0 dt' / τ i (1 x) ΔH RT f * 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 of structural differences (differences in configurational entropy from μ-region to μ- region) Glass Transformation-31

32 Structural relaxation models Four adjustable parameters: τ 0, ΔH*, x, β Φ( t) = P( t) P P P 0 e e T fic T ( t) T 1 T 2 2 From Moynihan, 1995 Glass Transformation-32

33 From Moynihan, 1995 B 2 O 3 Glassreheated at 10K/min Cooling rates 0.62K/min 2.5K/min 10K/min 40K/min Glass Transformation-33

34 Structural Relaxation Summary The glass transition is a kinetic phenomena Thermal history dependence Thermal history effects on glass properties described using the fictive temperature concept. T fic represents the contribution of structural relaxation to the property of interest, expressed in temperature units MA Debolt et al., 1976 Glass Transformation-34

35 Structural Relaxation Summary The relaxation function Φ(t) 1. Is non-linear Up-quench down-quench relaxation rates Φ(t) depends on instantaneous structure (T fic ) Tool-Narayanaswamy non-linearity parameter x x ΔH * (1 x) ΔH * τ = τ 0 exp + RT RT f 2. Is non-exponential Stretched exponential function (KWW)- β Modeled by a distribution of relaxation times Is their a microscopic explanation? Φ( t) t τ = exp β Glass Transformation-35

36 How is enthalpy relaxation connected to viscosity? d(logη) = d( T / T ) ΔH 2.3 η * g RT g Log (viscosity in poise) Strong Fragile From C. A. Angell, Science, 267, (1995), T g /T Glass Transformation-36

37 How is enthalpy relaxation connected to viscosity? xna 2 O (1-x)B 2 O 3 glasses Fragile melt behavior (greater E η /T g ) correlated with larger ΔC p at T g d(logη ) E = d( T / T ) 2.3 g RT g Related to greater structural changes as glass is heated through transition range? η Moynihan, 1995 Glass Transformation-37

38 Relaxation time at T g sec Enthalpy Relaxation and Viscous Flow have similar activation energiessimilar structural mechanisms? Angell defines T g at Pa From Moynihan, 1995 Glass Transformation-38

39 Glass Transition Theories Free Volume-Viscosity Adam-Gibbs Cooperative Relaxations Configurational Entropy Glass Transformation-39

40 Free Volume Theory Turnbull, Cohen, J Chem Phys (1970); Cohen, Grest Phys Rev B, (1979) Consider ideal close-packed structure representing a thermodynamic minimum volume, V 0 Flow occurs by movement of molecules into voids or holes larger than a critical size, V h Thermal/density fluctuations open up voids Increase temperature, increase specific volume (V) of melt Free volume (V f ) within a structure becomes available to accommodate viscous flow hole = exp V η η 0 0 V f ( T, P,...) Glass Transformation-40

41 Free volume depends melt properties V f V V T, P 0 T0, P ( T, P) = V Δ 0 α PdT K T0, P0 T0, P0 α P =isobaric expansion coefficient K T =isothermal compressibility For constant pressure, f T 0 T0 T dp ( α α ) dt ( α α )( T T ) liq glass liq Substitute into the Arrhenian η equation to get the VFT eq.: logη = A + T B T 0 glass T 0 represents the temperature at which free volume disappears 0 Glass Transformation-41

42 Configurational Entropy Model G. Adam, JH Gibbs, J. Chem. Phys., (1965); GW Scherer, J. Am Ceram Soc, (1984) Fluidity of a system depends on the rate of disappearance of the configurational entropy A system at the ideal glass transition temperature (T 2 ) 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. Glass Transformation-42

Visualizing configurational entropy Lattice model version: glass regions imbedded in sea of liquidglass transition occurs (on cooling) when

43 Visualizing configurational entropy Lattice model version: glass regions imbedded in sea of liquidglass transition occurs (on cooling) when the percolation limit for imbedded regions is reached EA DiMarzio, J. Res. Natl. Inst. Stand. Technol. 102, 135 (1997) Glass Transformation-43

44 Adam-Gibbs (cont.) Probability for a single cooperative transition: p( T ) zδμ = Aexp kbt 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*): p( T ) = zδμ z *δμ Aexp A exp z= z* kbt kbt Entropy of the entire system (S c ) depends on the number (n) of rearranging units of size z and the entropy contribution of each unit: S c =ns c. For one mole of molecules, there are n=n A /z independent regions, so z*=s c *N A /S c and s c * k B ln2 is the configurational entropy of a minimally sized region. p( T ) * s = c N Aδμ C A exp = A exp kbtsc TSc Glass Transformation-44

45 Adam-Gibbs (cont.) Since fluidity (1/η) is proportional to the transition probability, then η = η 0 C exp TS c Note that η as S c decreases (Δs 0); viz., fewer configurations are accessible, more molecules must cooperate to permit flow. At some temp (T K ), Z and flow no longer occurs. S c T T K ΔC T p, ΔC p = D T Angell: Fragile liquids have small values for D, strong liquids have large values. S c 1 D T K 1 T η η 0 exp T B T K VFT-form Glass Transformation-45

46 A-G G Model fits viscosity data very well. Richet and Bottinga, 1995 Glass Transformation-46

47 A-G G Model fits viscosity data very well. Richet and Bottinga, 1995 Glass Transformation-47

48 Kauzmann Temperature (T K ) Does the extension of the supercooled liquid entropy below T g 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* T K provides a thermodynamic basis for the VFT relationship *Varshneya (p.328) Glass Transformation-48

49 Potential energy landscapes represent configuration distributions in a system From Adam-Gibbs, each transition probability depends on δμ, the energy barrier (per molecule) to rearrangement p( T ) zδμ = Aexp kbt Fragile liquids are characterized by many different configurations that are accessible at greater temps: greater S c, lower η. Strong liquids have few local minima- single barrier, Arrhenius dependence? Glass Transformation-49

50 Summary- Glass Transition Upon cooling a liquid through the supercooled region, viscosity rapidly rises Glass transition range: Pa-s Glass transition temperature sometimes defined at 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 T fic 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 Glass Transformation-50

SAMPLE ANSWERS TO HW SET 3B

SAMPLE ANSWERS TO HW SET 3B First- Please accept my most sincere apologies for taking so long to get these homework sets back to you. I have no excuses that are acceptable. Like last time, I have copied