Sub -T g Relaxation in Thin Glass
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1 Sub -T g Relaxation in Thin Glass Prabhat Gupta The Ohio State University ( Go Bucks! ) Kyoto (January 7, 2008) 2008/01/07 PK Gupta(Kyoto) 1
2 Outline 1. Phenomenology (Review). A. Liquid to Glass Transition (LGT or GT). B. Structural Relaxation (SR). 2. New developments (Enthalpy landscape view). 3. Size, surface, and interface effects. 2008/01/07 PK Gupta(Kyoto) 2
3 Glass transition phenomena Liquid to glass transition (LGT) Glass (Non-equilibrium) Liquid (Equilibrium) Structural relaxation (SR) 2008/01/07 PK Gupta(Kyoto) 3
4 LGT - Phenomenology
5 LGT: Enthalpy (H) vs. Temp (T) H H G (T,T f ) = H L (T f ) + T Tf ( dh dt ) G dt LIQ GLS (B h ) Fixed P B = Cooling rate GLS (B l ) B h > B l SCL C = Heat Capacity T f = Fictive Temp. XTL T f (B l ) T f (B h ) T m T 2008/01/07 PK Gupta(Kyoto) 5
6 LGT: Heat Capacity (C) vs. Temp (T) C L SCL LIQ C C(T f ) [C L (T f ) - C G (T f ) ] > 0 C G ~ C X GLS T f T 2008/01/07 PK Gupta(Kyoto) 6
7 LGT: Ergodic (eqbm, liq) to Non-ergodic (frozen, glass) T f increases with increase in cooling rate (B). LXT Ln B LGT Ln B 2008/01/07 PK Gupta(Kyoto) 7 Tf
8 Rayleigh Scattering in silica glass 2008/01/07 PK Gupta(Kyoto) 8
9 LGT:Generic Features Not a thermodynamic transition. Not a spontaneous process. T f (B) as B. V = 0, H = 0 (No latent heat!). C > 0, κ > 0, β >,=,< 0 PDR (ΔC)(Δκ ) TV(Δβ) 2 > 1 κ = (1 / V ) V P T,β = (1 / V ) V T P,V = volume Glasses have intrinsic nano-scale heterogeneities. 2008/01/07 PK Gupta(Kyoto) 9
10 Relaxation - Phenomenology 2008/01/07 PK Gupta(Kyoto) 10
11 H Isothermal SR Endo (T > T f ) LIQ GLS SR SR Exo (T < T f ) T f T 2008/01/07 PK Gupta(Kyoto) 11
12 Non-linear Relaxation Kinetics Soda-Lime Glass T f Hara and Suetoshi (1955)
13 Sub-Exponential Relaxation M(t,T) = Relaxation Function M (t,t ) T (t,t ) T f T f (0) T Ln M 1 b 0.5 Stretched Exponential t M (t,t ) = exp[ { τ o (T,T f ) }b(t ) ] 0 < b(t) < 1 τ av = (η /modulus) = (τ o / b)γ(1 / b) t/ τ o 2008/01/07 PK Gupta(Kyoto) 13
14 Adam-Gibbs (AG) Model of Viscosity A Ln[η(T )] = Ln[η ] + [ TS c (T ) ] A, η = Parameters independent of T. S c (T) Configurational entropy of the liquid S S L (T) - S G (T).
15 AG model fits data extremely well! 2008/01/07 PK Gupta(Kyoto) 15
16 SR: Generic features SR is spontaneous (irreversible). SR is non-linear. SR is non-exponential (~ stretched exponential). Av. relaxation time approximately follows AG theory. Deviations from the Arrhenius behavior (fragility) of τ correlate with C. SR can be endothermic (at T > T f ). 2008/01/07 PK Gupta(Kyoto) 16
17 What is the entropy change during LGT? Entropy change : S(T f ) S L (T f ) - S G (T f ) S(T f ) cannot be measured experimentally because a) Experimental fact: no latent heat ( H(T f ) = 0). But H(T f ) T f S(T f ). b) S L (T f ) = S X (0) + S G (T f ) = S G (0) + Tf Tm o o C X T C G T dt dt + ΔH m T m + Tf Tm C L T dt [S G (0) is not known.] 2008/01/07 PK Gupta(Kyoto) 17
18 The Enthalpy Landscape View Gupta and Mauro,J. Chem. Phys. 126, , (2007) 2008/01/07 PK Gupta(Kyoto) 18
19 Rugged H-Landscape H=H(3N+1 dimensional Configuration, X) H (IS = Inherent states, minima, ) º Basin (-) Saddle (º) Basin Vib Freq. (ν) X 2008/01/07 PK Gupta(Kyoto) 19
20 Density of Inherent States, Ω(H) Ω(H) N! exp[ S c (H) / k ] º H 2008/01/07 PK Gupta(Kyoto) 20 X
21 Dynamics in H- Landscape Intra-basin relaxation: very fast ( ) (equilibration of a basin with the heat bath) Inter-basin transitions: slow ( ) H 2008/01/07 PK Gupta(Kyoto) 21 X
22 Inter-basin transitions An (i j ) transition is not allowed at T and t obs if: t obs < τ ij (T ) ( ε ν )exp[f ij kt ] F ij * = H ij * ktln(n ij * ) ε = a small constant number ν = attempt (vibrational) frequency (~same for all basins) k = Boltzmann s constant H* ij = Barrier enthalpy from basin i to j F* ij = Barrier free energy n* ij = Number of transition paths from basin i to j having barrier heights H* ij. * 2008/01/07 PK Gupta(Kyoto) 22
23 H Basin probability, p i L (T ) = Liquid State Equilibrium (ergodic) state: X i ba sin Ω i X i ba sin t obs > τ MAX (T ) exp[ H (X) kt ]dx exp[ H (X) kt ]dx i X Ergodic ensemble of basins 2008/01/07 PK Gupta(Kyoto) 23
24 Configurational Properties (Liquid) Enthalpy : Ω i H C L (T ) = H i.p i L (T ) Configurational Entropy (Gibbs): Not a property of a microstate. Ω i S C L (T ) = k p i L (T )Ln[ p i L (T )] 2008/01/07 PK Gupta(Kyoto) 24
25 Ergodicity Breaking ktln( νt obs ε ) < F * max Partitioning of X-space in to components such that no transitions among components each component is ergodic in itself. t obs(1) > t obs(2) > t obs(3) (Fixed T) H 1 T 1 > T 2 > T 3 (Fixed t obs ) 2a 2b 3 Components Meta-basins(MB) 2008/01/07 PK Gupta(Kyoto) 25 X
26 LGT and Broken Ergodicity At a fixed t obs (fixed cooling rate): T> T f (t obs ) No partitioning of X-space. (Ergodic system = Liquid) T=T f (t obs ) LGT partition starts. T<T f (t obs ) X-space is partitioned. (Broken ergodic system = glass) (System is trapped in a single MB) 2008/01/07 PK Gupta(Kyoto) 26
27 Glass is a constrained liquid! Constrained in a single meta-basin. The constraint is external: t obs < τ. Glass αth-mb Liquid (t obs > τ) H P L α (T f) 2008/01/07 PK Gupta(Kyoto) 27 X
28 The LGT-partition Total # of MBs = M(T f (t obs )) ( α = 1,,M) Probability of the αth MB: P α MB (T T f ) = (T f ) L p i i α Prob. of i th -basin: p i α (T,T f ) = P α MB (T f )g[ i α exp( H i / kt ) ] exp( H i / kt ) 2008/01/07 PK Gupta(Kyoto) 28
29 Glassy State, t obs < τ MAX (T ) Broken ergodic (or constrained eqbm) state: Q G (T,T f ) = MB α P α (T f ) Q i p i α i α (T,T f ) Q = a measurable property. 2008/01/07 PK Gupta(Kyoto) 29
30 Configurational entropy change during LGT S L c (T ) = P MB α (T ) S α C (T ) k P MB α (T )Ln(P MB α (T ) α The first term is the average configurational entropy of the system trapped in a MB. ΔS(T f ) = I(T f ) k P MB MB α (T f )LnP α (T f ) α α The second term - called complexity (I) - arises from inter-mb transitions. S C G (T,T f ) = P α MB (T f )S C α (T ) α 2008/01/07 PK Gupta(Kyoto) 30
31 ΔS during LGT ΔS = 0 (Current View) ΔS = I = k P α (T f )LnP α (T f ) (New view, ΔS >0) α Entropy loss at T f because the inter-mb transitions are frozen in a glass. 2008/01/07 PK Gupta(Kyoto) 31
32 2008/01/07 PK Gupta(Kyoto) 32
33 New View vs. Traditional View How can one distinguish between the two experimentally? 2008/01/07 PK Gupta(Kyoto) 33
34 Isothermal Relaxation in H-landscape SR occurs because time is no longer constrained so that some(or all) of the frozen inter-mb transitions begin to take place again. t > τ ij (T ) ( ε ν )exp[f ij* / kt ]
35 Relaxation: several time scales! Configurational changes (Inter-basin transitions): β process (intra-mb transitions) α process (inter-mb transitions, slowest) 2008/01/07 PK Gupta(Kyoto) 35
36 Hierarchy of Barriers Height Label Order of Order of T-dependence freezing thawing Highest α 1st (LGT) Last Non-Arrhenius (Primary) 2nd highest β 2nd 2nd to last More Arrhenius (Secondary) 3rd highest γ 3rd 3rd to last Small barriers unfreeze first. Thus relaxation is fast in the beginning and then becomes slower and slower as time increases.
37 Barrier Heights (MD simulation) Fragile system Av. barrier height Energy density of the adjacent IS. Parisi et al ( 2002) 2008/01/07 PK Gupta(Kyoto) 37
38 2008/01/07 PK Gupta(Kyoto) 38
39 SR: Far from equilibrium (Effect of T f ) [E glass, DSC] Bulk T f = 825K 10 um, Fiber T f = 1200K Huang and Gupta, 1992.
40 Sub-T g Relaxation δ T g T T g Stabilization (δ < 15 %) Sub-sub Tg Relaxation (δ > 15 %)
41 Effect of sub-tg isothermal annealing 2008/01/07 PK Gupta(Kyoto) 41
42 Stabilization C - C(ref) Moynihan et al (1985) Chen, Kurkjian (1983) 2008/01/07 PK Gupta(Kyoto) 42
43 Sub-sub Tg Relaxation Sub-Tg peak in C linear heating in DSC.
44 Size, surface, and interface effects Small samples (thin films, fibers, nano-particles) typically - but not always - show a decrease in T f (compared to that of the bulk glass). The magnitude of the change in T f depends on sample size (h) and the nature of confinement and interfaces. 2008/01/07 PK Gupta(Kyoto) 44
45 Nature of confinement (thin film) Film Substrate Substrate-T Film Substrate-B Free Supported Confined 2008/01/07 PK Gupta(Kyoto) 45
46 Size effect on T g (supported films) Keddie et al (1994) Kim et al (2000) 2008/01/07 PK Gupta(Kyoto) 46
47 Sputter deposited silica glass film (3um) bulk Hirose, Saito, Ikushima (2006) 2008/01/07 PK Gupta(Kyoto) 47
48 2-dim confined liquids in porous silica Trofymluk, Levchenko, Navrotsky (2005) 2008/01/07 PK Gupta(Kyoto) 48
49 Ideal glass transition T K in a fluid film model Mittal, Shah, and Truskett (2004) (SSMF = Soft sphere/ mean-field) 2008/01/07 PK Gupta(Kyoto) 49
50 Size effect and the AG model Adam Gibbs Model: T g S conf (T g ) = Cons. F(T, A,h) = F B (T ) + γ (T )A S = F T = S B (T ) A γ T s (S / V ) = s B (T ) (1 / h)γ T T f B T f (h) = [1 + { γ T hs B (T B g ) }] [T B f T f (h)] = K T f (h) h ha = V K = γ T s B
51 AG phenomenological model of size effect [T B f T f (h)] = K T f (h) h Eqn same as the theoretical result of the Truskett model (2004). Eqn same as the one obtained empirically by Kim et al (2000). Typically K > 0 since γ generally decreases with increase in T. K can be negative for some systems under certain conditions depending on the nature of the interface. 2008/01/07 PK Gupta(Kyoto) 51
52 Thanks and questions. 2008/01/07 PK Gupta(Kyoto) 52
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