Experimental predictions for strongly correlating liquids

Transcription

1 Experimental predictions for strongly correlating liquids A new look at the Prigogine-Defay ratio, and how to estimate γ Ulf Rørbæk Pedersen Glass and Time - DNRF centre for viscous liquids dynamics, IMFUFA, Department of Science, Systems and Models, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark Collaborators: Nicholas Bailey, Nicoletta Gnan, Tina Hecksher, Bo Jakobsen, Thomas B. Schrøder and Jeppe C. Dyre urp@ruc.dk May 11, Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

2 Strong W-U correlations in the Lennard-Jones liquid The well-studied Lennard-Jones liquid: [ ( U pair (r) = 4ε σ ) 12 ( r σ ) ] 6 r E = K + U p = Nk B T + W Fluctuations of conf. parts: U(t) = U(t) U W (t) = W (t) W Constant and T simulations. [Pedersen et al. PRL 100:015701, 2008] W γ U at constant NT Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

3 Relating to experiments Strongly correlating liquids (SCL) have W γ U at constant NT : ar 3γ + br + c U pair LJ Two important numbers Degree of correlation: R W U NT ( W ) 2 NT ( U) 2 NT Apparent exponent (r 3γ ): γ ( W ) 2 NT ( U) 2 NT Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

4 Relating to experiments Strongly correlating liquids (SCL) have W γ U at constant NT : ar 3γ + br + c U pair LJ Two important numbers Degree of correlation: R W U NT ( W ) 2 NT ( U) 2 NT Apparent exponent (r 3γ ): γ ( W ) 2 NT ( U) 2 NT General problem Subtract off ideal gas -terms: W = p Nk B T and U = E K. NT -ensemble (where we understand SCL) vs. NpT -ensemble (experiments) Recall density scaling: F (ρ γ /T ). Lots of γ scale in literature. Here, independent way(s) of getting γ. Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

5 k B T 2 c (ω) = ( E) 2 iω 0 E(0) E(t) e iωt dt c (ω) = ω C k B T 2 ω { E(0) E(t) } C ω {f (t)}: cosine transform Fluctuations on slow 1/ω timescale. Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

6 k B T 2 c (ω) = ( E) 2 iω 0 E(0) E(t) e iωt dt c (ω) = ω C k B T 2 ω { E(0) E(t) } C ω {f (t)}: cosine transform Fluctuations on slow 1/ω timescale. K ω T (ω) = k B T C ω{ p(0) p(t) NT } β (ω) = ω C k B T 2 ω { E(0) p(t) NT } Kinetic terms average out: U W E p SCL: c (ω) K T (ω) β (ω) Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

7 k B T 2 c (ω) = ( E) 2 iω 0 E(0) E(t) e iωt dt c (ω) = ω C k B T 2 ω { E(0) E(t) } C ω {f (t)}: cosine transform Fluctuations on slow 1/ω timescale. K ω T (ω) = k B T C ω{ p(0) p(t) NT } β (ω) = ω C k B T 2 ω { E(0) p(t) NT } Kinetic terms average out: U W E p SCL: c (ω) K T (ω) β (ω) Dynamic Prigogine-Defay ratio Λ NT (ω) c (ω)k T (ω) 1, R Λ 1/2 T [β (ω)]2 T γ(ω) = K T (ω) Tc (ω), γ(ω) γ Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

8 A new look at the Prigogine-Defay ratio Old: Λ = 1 single parameter -liquids (and Λ > 1?). New: 1 ΛT R Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

9 A new look at the Prigogine-Defay ratio Old: Λ = 1 single parameter -liquids (and Λ > 1?). New: 1 ΛT R Independent of ensemble: 1 = Λ T = Λ pt = Λ S = Λ ps Λ pt (ω) c p (ω)κ T (ω) T [α p (ω)] 2 [Ellegaard et al., JCP 126, (2007)] :-( unfortunately experimental data is not available (yet) :-) another approach can be made [PRE 77, (2008)] Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

10 Pressure and energy fluctuations of viscous liquids. ( E) 2 iω k B T 2 c (ω) = 0 E(0) E(t) e iωt dt. Liquid, c liquid = c (ω 0): k B T 2 c liquid = ( E) 2 Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

11 Pressure and energy fluctuations of viscous liquids. ( E) 2 iω k B T 2 c (ω) = 0 E(0) E(t) e iωt dt. Liquid, c liquid = c (ω 0): k B T 2 c liquid = ( E) 2 Solid-like, c solid = c (ω ): k B T 2 c solid = ( E) 2 ( E) 2 slow Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

12 Pressure and energy fluctuations of viscous liquids. ( E) 2 iω k B T 2 c (ω) = 0 E(0) E(t) e iωt dt. Liquid, c liquid = c (ω 0): k B T 2 c liquid = ( E) 2 Solid-like, c solid = c (ω ): k B T 2 c solid = ( E) 2 ( E) 2 slow ( ) ( E) 2 slow = k B T 2 c liquid c solid ( ) ( U) 2 slow = k B T 2 c liquid c solid Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

13 Pressure and energy fluctuations of viscous liquids. ( E) 2 iω k B T 2 c (ω) = 0 E(0) E(t) e iωt dt. Liquid, c liquid = c (ω 0): k B T 2 c liquid = ( E) 2 Solid-like, c solid = c (ω ): k B T 2 c solid = ( E) 2 ( E) 2 slow ( ) ( E) 2 slow = k B T 2 c liquid c solid ( ) ( U) 2 slow = k B T 2 c liquid c solid ( W ) 2 slow = k B T (K liquid T W U slow = k B T 2 (β liquid K solid T ) β solid ) Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

14 R and γ from (static) response functions ( U) 2 slow = k B T 2 ( c liquid ( W ) 2 slow = k B T (K liquid T W U slow = k B T 2 (β liquid R R R γ W U ( W ) 2 ( U) 2 W U slow ( W ) 2 slow ( U) 2 slow q (K liquid T (β liquid β solid ) K solid )(c liquid T ( W ) 2 ( U) 2 Recent note: [arxiv: ] c solid )/T (K liquid T KT solid ) T (cv liquid cv solid ) ) c solid K solid T ) β solid ) Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

15 DC704 silicone oil (a SCL) K S (ω): [Tina Hecksher, unpub.] Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

16 DC704 silicone oil (a SCL) γ ( W ) 2 ( U) 2 = (K liquid T KT solid ) T (cv liquid cv solid ) K S (ω): [Tina Hecksher, unpub.] c L (ω): [Bo Jacobsen, unpub.] α liquid P αp solid ref = 214K. = K 1 = K 1 (guess) Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

17 K S (ω): [Tina Hecksher, unpub.] c L (ω): [Bo Jacobsen, unpub.] α liquid P αp solid ref = 214K. DC704 silicone oil (a SCL) = K 1 = K 1 (guess) γ ( W ) 2 ( U) 2 = c = c p T α 2 pk T c K T = K S cp ( c = c p 1 + T α2 pk S c p ( K T = K S 1 + T α2 p K S T (α liquid p ) 2 K liquid S T (α solid p c liquid p ) 2 KS solid c solid p γ = 6.6 c p = 0.13 = 0.01 (K liquid T KT solid ) T (cv liquid cv solid ) ) 1 ) 1 Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

18 A method of extrapolation [Takahara et al. (1999)] OTP-OPP mixture. R = q γ = (K liquid T K T = 1/κ T (β liquid β solid ) KT solid )(cv liquid cv solid (K liquid T KT solid ) T (cv liquid cv solid ) c = c p T α 2 pk T β = α p K T )/T Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

19 A method of extrapolation Note on arxiv [arxiv: ] [Takahara et al. (1999)] OTP-OPP mixture. R = q γ = (K liquid T K T = 1/κ T (β liquid β solid ) KT solid )(cv liquid cv solid (K liquid T KT solid ) T (cv liquid cv solid ) c = c p T α 2 pk T β = α p K T R = 0.79 γ = 6.0 Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14 )/T

20 Density scaling OTP-OPP τ = f (ρ γ /T ) From [Roland et al., Rep. Prog. Phys. 68, 1405 (2005)] Reanalyzing OTP-OPP glass transition data from [Takahara et al., 1999]: K T T c v = 6.0 γ scaling Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

21 The classical Prigogine-Defay ratio The method of extrapolation is similar to what is done when evaluating the classical Prigogine-Defay ratio: Π NpT c p κ T T g ( α p ) 2, where refers to the change at the glass transition temperature. Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

22 The classical Prigogine-Defay ratio The method of extrapolation is similar to what is done when evaluating the classical Prigogine-Defay ratio: Π NpT c p κ T T g ( α p ) 2, where refers to the change at the glass transition temperature. Since Π NT c K T T g ( β ) 2 R 2 and that single parameter -ness is ensemble independent, R = 1 = Π NT = Π NpT, we expect strongly correlating liquids to have Π NpT close to unity. Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

23 Literature values of the (NpT) Prigogine-Defay ratio Material Π NpT Π 1/2 NpT Pure SiO 2 glass Glycerol, C 3 H 5 (OH) GeO 2 6.9± ± Na 2 O10B 2 O 3 74SiO B 2 O Ca NO 3 0.6KNO Glucose, OC 6 H 7 (OH) Se ZrTiCuNiBe Na 2 O74SiO Polyvinylacetate 2.2(1.7) 0.67(0.77) n-propanol, C 3 H 7 OH Polyvinylchlorid, PC Polystyrene, PS 1.3± ±0.03 OTP(75%)-TPCM(25%) OTP(67%)-OPP(33%) 1.20± ±0.02 Polystyrene, PS Phenoxy Polycarbonate Polysulfone Polyarylate Polyisobutene Π NpT 1 for van der Waals bonded glass formers.... not liquids with directional bonds. Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

24 Conclusions R γ W U ( W ) 2 ( U) 2 ( W ) 2 ( U) 2 (β liquid β solid ) q (K liquid T KT solid )(c liquid c solid )/T (K liquid T KT solid ) T (cv liquid cv solid ) New look at Prigogine-Defay ratio: Λ 1/2 T = R Π NpT cp κ T T g ( α p) 1 for SCL 2 Strongly correlating liquid (SCL) W γ U at constant NT R W U NT ( W ) 2 NT ( U) 2 NT γ W NT U NT = n/3. Ulf R. Pedersen (urp@ruc.dk) Glass and Time. Roskilde Uni. May 11, / 14

25 Thank you for your attention. Ulf R. Pedersen Glass and Time. Roskilde Uni. May 11, / 14