The Temperature Dependence of the Relaxation Time in ultraviscous liquids
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1 The Temperature Dependence of the Relaxation Time in ultraviscous liquids Is there evidence for a dynamic divergence in data? Tina Hecksher, Albena I. Nielsen, Niels B. Olsen, and Jeppe C. Dyre DNRF Centre Glass & Time, Roskilde University Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 1
2 Fragility log (τ [s]) 4 EH BN dbaf DEP ER Gly Cum PHIQ SB Temperature [K] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p.
3 Fragility log (τ [s]) Arrhenius behavior: m=16 m T g /T Arrhenius equation: ( ) E τ = τ exp k B T Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p.
4 Fragility log (τ [s]) Arrhenius behavior: m=16 m T g /T Arrhenius equation: ( ) E τ = τ exp k B T Most liquids: ( ) E(T) τ = τ exp k B T Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p.
5 Fragility log (τ [s]) Arrhenius behavior: m=16 m T g /T Fragility index: m d log τ d(t g /T) Tg Arrhenius equation: ( ) E τ = τ exp k B T Most liquids: ( ) E(T) τ = τ exp k B T Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p.
6 Empirical equations VFT equation [H. Vogel (191), G. S. Fulcher (195), G. Tammann, (195)]: ( ) A τ = τ exp E(T) T, T < T g T T T T Avramov equation [G. Harrison (1976), I. Avramov (5)]: ( ) B τ = τ exp T n E(T) T n+1, n 1 Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 3
7 Theoretical models Derivations of the VFT-equation: J.H. Gibbs & E.A. DiMarzio, J. Chem. Phys., 8, p (1958) G. Adam & J.H. Gibbs, J. Chem. Phys., 43, p (1965) S.F. Edwards, Polymer, 17, p (1976) P. W. Anderson, Ill-Condensed Matter, p , North-Holland (1979) S.F. Edwards & T. Vilgis, Phys. Scr. (1986) J.P. Sethna, Europhys. Lett., 6, p (1988) T.R. Kirkpatrick et al, Phys. Rev. A, 4, p (1989) U. Mohanty, Physica A, 177, p (1991) S.F. Edwards, Int. J. Mod. Phys. B, 6, p (199) H.-O. Carmesin, Physica A, 1, p. 5-9 (1993) H. Tanaka, J. Chem. Phys., 15, p (1996) T. Kitamura, Physica A, 6, p (1999) J. Rault, J. Non-Cryst. Sol., 71, p () J.-P. Bouchaud & G. Biroli, J. Chem. Phys., 11, p (4) R.R. Nigmatullin, Physica B, 358, p (5) J.S. Langer, Phys.Rev. E, 73, 4154 (6) V. Lubchenko & P.G. Wolynes, Annu. Rev. Phys. Chem., 58, p (7) Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 4
8 Theoretical models Entropy-model: E 1 S c (T) [Gibbs & DiMarzio (1958), Adam & Gibbs (1965) ] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 4
9 Theoretical models Entropy-model: E 1 S c (T) [Gibbs & DiMarzio (1958), Adam & Gibbs (1965) ] Free volume-model: [Cohen & Turnbull (1959)] E 1 v free (T) Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 4
10 Theoretical models Entropy-model: E 1 S c (T) [Gibbs & DiMarzio (1958), Adam & Gibbs (1965) ] Free volume-model: [Cohen & Turnbull (1959)] E 1 v free (T) Shoving-model: E G [Dyre, Christensen, & Olsen (1996) ] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 4
11 Theoretical models Entropy-model: E 1 S c (T) [Gibbs & DiMarzio (1958), Adam & Gibbs (1965) ] Free volume-model: [Cohen & Turnbull (1959)] E 1 v free (T) Shoving-model: E G [Dyre, Christensen, & Olsen (1996) ] Other elastic models E [Eyring (1936), Dyre (6)] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 4
12 How do we test the existence of a finite T? Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 5
13 How do we test the existence of a finite T? A dynamic divergence is probable the VFT-equation fits data better than other fitting functions with the same number of parameters Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 5
14 Data used in the analysis Dielectric data: the most precise and most abundant data K 13.4K 15.4K 17.5K 19.5K 1.5K 3.5K 5.5K 7.6K 9.6K 31.6K 33.6K 35.6K 37.6K 39.7K τ := 1 fp log (frequency) [Hz] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 6 DC74 log (ε ")
15 Data used in the analysis Dielectric data: the most precise and most abundant data Requirements: - dynamical range [1 6 : 1 3 ]s - spanning at least 4 decades in relaxation time - at least 5 temperatures measured - no monoalcohols, polymers, or plastic crystals Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 6
16 Data used in the analysis Dielectric data: the most precise and most abundant data log 1 relaxation time [s] /T [K 1 ] Requirements: - dynamical range [1 6 : 1 3 ]s - spanning at least 4 decades in relaxation time - at least 5 temperatures measured - no monoalcohols, polymers, or plastic crystals 4 ultraviscous organic liquids Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 6
17 Examples of fits Both equations have parameters (prefactor fixed: τ = 1 14 s) VFT equation (solid line): «A τ = τ exp T T Avramov equation (dashed line): B τ = τ exp T n «log 1 relaxation time [s] 4 6 Solid line: VFT fit Dashed line: Avramov fit AFEH BePh BPIB DCHMMS mtcp DC74 TCP /T [K 1 ] Both fit data really nicely - but is one better than the other? Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 7
18 Goodness of the fits Variance: σ = 1 N n N i=1 ( log1 τ data i log 1 τi model ) Standard deviation, σ worst fit σ VFT σ Avramov σ VFT =.77 best fit σ Avramov =.1... the VFT equation on average fits better than the Avramov equation! Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 8
19 Temperature index 8 Definition [Dyre & Olsen, 4]: I E d ln E d ln T Connection to fragility: Temperature index m = 16(I E (T g ) + 1) Temperature [K] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 9
20 Temperature index 8 Definition [Dyre & Olsen, 4]: I E d ln E d ln T Connection to fragility: Temperature index m = 16(I E (T g ) + 1) Temperature [K] Calculated for models: I VFT = T T T, I Av. = n 1 Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 9
21 Functions without divergence FF1: I = 1 T 1 /T, FF: I = (T /T) Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 1
22 Functions without divergence FF1: I = 1 T 1 /T, FF: I = (T /T).4 Standard deviation, σ.3..1 Avramov: E T n+1 VFT: E T/(T T ) worst fit best fit Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 1
23 Functions without divergence FF1: I = 1 T 1 /T, FF: I = (T /T).4 Standard deviation, σ.3..1 worst fit best fit Avramov: E T n+1 VFT: E T/(T T ) FF1: E T exp ( T 1 ) T ) FF: E exp( T T Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 1
24 Index and model predictions Temperature index BPIB DC74 5 PPE TPE tnb mtol BPC VFT: full line FF1: Dashed line FF: Dash-dotted line (Avramov: constant) Temperature [K] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 11
25 Index and model predictions Temperature index PG Gly mtcp AFEH DMP TCP 3Sty MTHF VFT: full line FF1: Dashed line FF: Dash-dotted line (Avramov: constant) Temperature [K] Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 11
26 Summary the VFT equation on average fits better than the Avramov equation (with a fixed τ ) the temperature index of the activation energy is itself temperature dependent little evidence in the data for a dynamic divergence Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 1
27 Acknowledgements For kindly providing data to this study we thank: S. Benkhof T. Blochowicz L. F. del Castillo T. Christensen R. Diaz-Calleja L.-T. Duong K. Duvvuri G. Eska C. Gainaru A. Garcia-Bernabe S. Hensel-Bielowka W. Huang N. Ito B. Jakobsen E. Kaminska M. Köhler A. Kudlik A. Loidl P. Lunkenheimer D. V. Matyushov M. Mierzwa P. Medick K. L. Ngai K. Niss V. N. Novikov M. Paluch S. Pawlus L. C. Pardo S. Putselyk E. L. Quitevis J. R. Rajian R. Richert A. Rivera E. A. Rössler M. J. Sanchis N.V. Surovtsev C. Tschirwitz L.-M. Wang J. Wiedersich Thank you for your attention! Fragility of Viscous Liquids: Cause(s) and consequences, October 8th 8 p. 13
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