Experimental predictions for strongly correlating liquids

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1 Experimental predictions for strongly correlating liquids A new look at the Prigogine-Defay ratio 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, Thomas B. Schrøder and Jeppe C. Dyre April 3, 2009 Ulf R. Pedersen (Roskilde Unv.) April 3, / 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 pv = Nk B T + W Fluctuations of conf. parts: U(t) = U(t) U W (t) = W (t) W Constant V and T simulations. [Pedersen et al. (2008) PRL 100:015701] W γ U at constant VT Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

3 Relating to experiments Strongly correlating liquids (SCL) have W γ U at constant VT : ar n + br + c U pair LJ Two important numbers Degree of correlation: R W U VT ( W ) 2 VT ( U) 2 VT General problem Recall isomorphs: F (ρ γ /T ) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

4 Relating to experiments Strongly correlating liquids (SCL) have W γ U at constant VT : ar n + br + c U pair LJ Two important numbers Degree of correlation: R W U VT ( W ) 2 VT ( U) 2 VT Apparent exponent: γ W VT U VT = n/3 General problem Recall isomorphs: F (ρ γ /T ) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

5 Relating to experiments Strongly correlating liquids (SCL) have W γ U at constant VT : ar n + br + c U pair LJ Two important numbers Degree of correlation: R W U VT ( W ) 2 VT ( U) 2 VT Apparent exponent: γ W VT U VT = n/3 General problem Subtract off ideal gas -terms: W = pv Nk B T and U = E K. Recall isomorphs: F (ρ γ /T ) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

6 Relating to experiments Strongly correlating liquids (SCL) have W γ U at constant VT : ar n + br + c U pair LJ Two important numbers Degree of correlation: R W U VT ( W ) 2 VT ( U) 2 VT Apparent exponent: γ W VT U VT = n/3 General problem Subtract off ideal gas -terms: W = pv Nk B T and U = E K. Constant VT -ensemble (where we understand SCL) vs. constant pt -ensemble (experiments) Recall isomorphs: F (ρ γ /T ) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

7 γ related to crystallization γ = W a W b U a U b β V ( ) p T V = ( ) S V T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

8 γ related to crystallization γ = W a W b = Vr (pa p b ) U a U b E a E b β V ( ) p T V = ( ) S V T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

9 γ related to crystallization γ = W a W b U a U b = Vr (pa p b ) E a E b = Vr (pa p b ) T r (S a S b ) β V ( ) p T V = ( ) S V T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

10 γ related to crystallization β V ( ) p T V = ( ) S V T γ = W a W b U a U b = Vr (pa p b ) E a E b = Vr (pa p b ) T r (S a S b ) (1) Cons. pt melting (2) compress liq.: p a p b = 0 Vr + fus V (p b ) V r S a S b = fus S(p b )+ VK a T dv Vr + fus V (p b ) V r β b V dv Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

11 Pressure and energy fluctuations of viscous liquids. C EE (t) = C EE (t)/c EE (0), C EE (t) = E(0) E(t) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

12 Pressure and energy fluctuations of viscous liquids. C EE (t) = C EE (t)/c EE (0), C EE (t) = E(0) E(t) C EE (t 0) = C EE = ( E) 2 liquid ( E) 2 solid. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

13 Pressure and energy fluctuations of viscous liquids. C EE (t) = C EE (t)/c EE (0), C EE (t) = E(0) E(t) C EE (t 0) = C EE = ( E) 2 liquid ( E) 2 solid. Subtracting off ideal gas terms: C EE = C UU C pp = V 2 C WW C pe = VC WU. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

14 Pressure and energy fluctuations of viscous liquids. C EE (t) = C EE (t)/c EE (0), C EE (t) = E(0) E(t) C EE (t 0) = C EE = ( E) 2 liquid ( E) 2 solid. Subtracting off ideal gas terms: C EE = C UU C pp = V 2 C WW C pe = VC WU. R C pe C pp C EE Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

15 Pressure and energy fluctuations of viscous liquids. C EE (t) = C EE (t)/c EE (0), C EE (t) = E(0) E(t) C EE (t 0) = C EE = ( E) 2 liquid ( E) 2 solid. Subtracting off ideal gas terms: C EE = C UU C pp = V 2 C WW C pe = VC WU. R γ C pe C pp CEE V 2 Cpp CEE Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

16 R and γ from (static) response functions R = C pe C pp C EE = p E liquid q VT p E solid VT [ ( E) 2 liquid VT ( E)2 solid VT ][ ( p)2 liquid VT ( p)2 solid VT ] Fluctuation dissipation theorem [Allen & Tildesley (1987)] Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

17 R and γ from (static) response functions R = C pe C pp C EE = p E liquid q VT p E solid VT [ ( E) 2 liquid VT ( E)2 solid VT ][ ( p)2 liquid VT ( p)2 solid VT ] Fluctuation dissipation theorem [Allen & Tildesley (1987)] Definitions: c V 1 ( E ) ( ) ( ) V T V, K T V p V and β V p T T ( E) 2 VT = k B T 2 Vc V V. R W U ( W ) 2 ( U) 2 Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

18 R and γ from (static) response functions R = C pe C pp C EE = p E liquid q VT p E solid VT [ ( E) 2 liquid VT ( E)2 solid VT ][ ( p)2 liquid VT ( p)2 solid VT ] Fluctuation dissipation theorem [Allen & Tildesley (1987)] Definitions: c V 1 ( E ) ( ) ( ) V T V, K T V p V and β V p T T ( E) 2 VT = k B T 2 Vc V ( p) 2 VT = k BT V [ 2Nk BT 3V + p VT K T + X VT V ], V. R W U ( W ) 2 ( U) 2 Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

19 R and γ from (static) response functions R = C pe C pp C EE = p E liquid q VT p E solid VT [ ( E) 2 liquid VT ( E)2 solid VT ][ ( p)2 liquid VT ( p)2 solid VT ] Fluctuation dissipation theorem [Allen & Tildesley (1987)] Definitions: c V 1 ( E ) ( ) ( ) V T V, K T V p V and β V p T T ( E) 2 VT = k B T 2 Vc V R ( p) 2 VT = k BT V [ 2Nk BT 3V + p VT K T + X VT V ], U p = k B T 2 (β V N V k B), W U ( W ) 2 ( U) 2 V. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

20 R and γ from (static) response functions R = C pe C pp C EE = p E liquid q VT p E solid VT [ ( E) 2 liquid VT ( E)2 solid VT ][ ( p)2 liquid VT ( p)2 solid VT ] Fluctuation dissipation theorem [Allen & Tildesley (1987)] Definitions: c V 1 ( E ) ( ) ( ) V T V, K T V p V and β V p T T ( E) 2 VT = k B T 2 Vc V R ( p) 2 VT = k BT V [ 2Nk BT 3V + p VT K T + X VT V ], U p = k B T 2 (β V N V k B), W U ( W ) 2 ( U) 2 V. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

21 R and γ from (static) response functions R = C pe C pp C EE = p E liquid q VT p E solid VT [ ( E) 2 liquid VT ( E)2 solid VT ][ ( p)2 liquid VT ( p)2 solid VT ] Fluctuation dissipation theorem [Allen & Tildesley (1987)] Definitions: c V 1 ( E ) ( ) ( ) V T V, K T V p V and β V p T T ( E) 2 VT = k B T 2 Vc V R γ ( p) 2 VT = k BT V [ 2Nk BT 3V + p VT K T + X VT V ], U p = k B T 2 (β V N V k B), W U ( W ) 2 ( U) 2 = ( W ) 2 ( U) 2 (β liquid V βv solid ) q T KT solid )(c liquid V cv solid )/T = V. β V KT c V /T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

22 R and γ from (static) response functions R = C pe C pp C EE = p E liquid q VT p E solid VT [ ( E) 2 liquid VT ( E)2 solid VT ][ ( p)2 liquid VT ( p)2 solid VT ] Fluctuation dissipation theorem [Allen & Tildesley (1987)] Definitions: c V 1 ( E ) ( ) ( ) V T V, K T V p V and β V p T T ( E) 2 VT = k B T 2 Vc V R γ ( p) 2 VT = k BT V [ 2Nk BT 3V + p VT K T + X VT V ], U p = k B T 2 (β V N V k B), W U ( W ) 2 ( U) 2 = ( W ) 2 ( U) 2 = (β liquid V βv solid ) q T KT solid )(c liquid V cv solid )/T T KT solid ) T (cv liquid c solid = KT v ) T c v = V. β V KT c V /T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

23 A standard Roskilde liquid - DC704 silicon oil (a SCL) K S (ω): [Tina Hecksher, unpub.] T ref = 214K. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

24 A standard Roskilde liquid - DC704 silicon oil (a SCL) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] T ref = 214K. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

25 A standard Roskilde liquid - DC704 silicon oil (a SCL) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

26 A standard Roskilde liquid - DC704 silicon oil (a SCL) γ ( W ) 2 ( U) 2 = T KT solid ) T (cv liquid cv solid ) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

27 A standard Roskilde liquid - DC704 silicon oil (a SCL) γ ( W ) 2 ( U) 2 = c V = c p T α 2 pk T T KT solid ) T (cv liquid cv solid ) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

28 A standard Roskilde liquid - DC704 silicon oil (a SCL) γ ( W ) 2 ( U) 2 = c V = c p T α 2 pk T K T = K S c V cp T KT solid ) T (cv liquid cv solid ) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

29 A standard Roskilde liquid - DC704 silicon oil (a SCL) γ ( W ) 2 ( U) 2 = c V = c p T α 2 pk T c K T = K V S cp ( c V = c p 1 + T α2 p K S c p T KT solid ) T (cv liquid cv solid ) ) 1 K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

30 A standard Roskilde liquid - DC704 silicon oil (a SCL) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) γ ( W ) 2 ( U) 2 = c V = c p T α 2 pk T c K T = K V S cp ( c V = c p 1 + T α2 p K S c p ( K T = K S 1 + T α2 p K S c p T KT solid ) T (cv liquid cv solid ) ) 1 ) 1 Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

31 A standard Roskilde liquid - DC704 silicon oil (a SCL) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) γ ( W ) 2 ( U) 2 = c V = c p T α 2 pk T c K T = K V S cp ( c V = c p 1 + T α2 p K 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 c p = 0.13 = 0.01 T KT solid ) T (cv liquid cv solid ) ) 1 ) 1 Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

32 A standard Roskilde liquid - DC704 silicon oil (a SCL) K S (ω): [Tina Hecksher, unpub.] C p (ω): [Bo Jacobsen, unpub.] α liquid P αp solid T ref = 214K. = K 1 = K 1 (guess) γ ( W ) 2 ( U) 2 = c V = c p T α 2 pk T c K T = K V S cp ( c V = c p 1 + T α2 p K 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 T KT solid ) T (cv liquid cv solid ) ) 1 ) 1 Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

33 A method of extrapolation [Takahara et al. (1999)] OTP-OPP data. R = q γ = T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

34 A method of extrapolation [Takahara et al. (1999)] OTP-OPP data. R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

35 A method of extrapolation [Takahara et al. (1999)] OTP-OPP data. R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) c V = c p T α 2 pk T )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

36 A method of extrapolation [Takahara et al. (1999)] OTP-OPP data. R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) c V = c p T α 2 pk T β V = α p K T )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

37 A method of extrapolation [Takahara et al. (1999)] OTP-OPP data. R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) c V = c p T α 2 pk T β V = α p K T R = 0.79 )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

38 A method of extrapolation [Takahara et al. (1999)] OTP-OPP data. R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) c V = c p T α 2 pk T β V = α p K T R = 0.79 γ = 6.0 )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

39 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 [Takanhara et al., 1999]: K T T c v = 6.0 γ scaling Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

40 A method of extrapolation. What are the uncertainties? R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) c V = c p T α 2 pk T β V = α p K T R = 0.79 γ = 6.0 (c liquid p c liquid V )/c liquid V = 26% )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

41 A method of extrapolation. What are the uncertainties? R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) c V = c p T α 2 pk T β V = α p K T R = 0.79 γ = 6.0 (c liquid p (c solid p c liquid V )/c liquid c solid V )/c solid V = 26% V = 3% )/T Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

42 A method of extrapolation. What are the uncertainties? R = q γ = T K T = 1/κ T (β liquid V βv solid ) KT solid )(cv liquid cv solid T KT solid ) T (cv liquid cv solid ) c V = c p T α 2 pk T β V = α p K T R = 0.79 γ = 6.0 (c liquid p (c solid p (c liquid V c liquid V )/c liquid c solid V )/c solid V = 26% V = 3% cv solid )/c liquid V = 19% Ulf R. Pedersen (Roskilde Unv.) April 3, / 14 )/T

43 Literature values of the (pt) Prigogine-Defay ratio Material Π pt Π 1/2 pt 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, PVC 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 VT = K T c V T ( β V ) 2 R = Π 1/2 VT Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

44 Literature values of the (pt) Prigogine-Defay ratio Material Π pt Π 1/2 pt 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, PVC 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 VT = K T c V T ( β V ) 2 R = Π 1/2 VT Π pt = κ T c p T ( α p) 2 1 = Π pt = Π VT [JCP 126, (2007)] Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

45 Literature values of the (pt) Prigogine-Defay ratio Material Π pt Π 1/2 pt 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, PVC 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 VT = K T c V T ( β V ) 2 R = Π 1/2 VT Π pt = κ T c p T ( α p) 2 1 = Π pt = Π VT [JCP 126, (2007)] Π pt 1 for van der Waals bonded glass formers. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

46 Literature values of the (pt) Prigogine-Defay ratio Material Π pt Π 1/2 pt 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, PVC 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 VT = K T c V T ( β V ) 2 R = Π 1/2 VT Π pt = κ T c p T ( α p) 2 1 = Π pt = Π VT [JCP 126, (2007)] Π pt 1 for van der Waals bonded glass formers.... not network forming. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

47 R and γ from imaginary parts of response c K β ω V (ω) = C Vk B T 2 ω { E(0) E(t) VT } T (ω) = ω Vk B T C ω{ p(0) p(t) VT } V (ω) = ω C Vk B T 2 ω { E(0) p(t) VT } Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

48 R and γ from imaginary parts of response c K β ω V (ω) = C Vk B T 2 ω { E(0) E(t) VT } T (ω) = ω Vk B T C ω{ p(0) p(t) VT } V (ω) = ω C Vk B T 2 ω { E(0) p(t) VT } R C ω{ E(0) p(t) VT } Cω{ E(0) E(t) VT }C ω{ p(0) p(t) VT } Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

49 R and γ from imaginary parts of response c K β ω V (ω) = C Vk B T 2 ω { E(0) E(t) VT } T (ω) = ω Vk B T C ω{ p(0) p(t) VT } V (ω) = ω C Vk B T 2 ω { E(0) p(t) VT } R C ω{ E(0) p(t) VT } Cω{ E(0) E(t) VT }C ω{ p(0) p(t) VT } R(ω) = β V (ω) K T (ω)c V (ω)/t Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

50 R and γ from imaginary parts of response c K β ω V (ω) = C Vk B T 2 ω { E(0) E(t) VT } T (ω) = ω Vk B T C ω{ p(0) p(t) VT } V (ω) = ω C Vk B T 2 ω { E(0) p(t) VT } R C ω{ E(0) p(t) VT } Cω{ E(0) E(t) VT }C ω{ p(0) p(t) VT } R(ω) = γ(ω) = β V (ω) K T (ω)c V (ω)/t K T (ω) Tc V (ω). Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

51 Conclusions Crystallization of SCL: γ = Vr (pa p b ) T r (S a S b ) at constant VT Strongly correlating liquid (SCL) W γ U at constant VT R W U VT ( W ) 2 VT ( U) 2 VT γ W VT U VT = n/3. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

52 Thank you for your attention. Ulf R. Pedersen (Roskilde Unv.) April 3, / 14

Experimental predictions for strongly correlating liquids

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,