1. Elasticity, caged dynamics and thermodynamics: three (related) scalings of the relaxation in glassforming systems Francesco Puosi 1, Dino Leporini 2,3 1 LIPHY, Université Joseph Fourier, Saint Martin d’Hères, France 2 Dipartimento di Fisica “Enrico Fermi”, Universita’ di Pisa, Pisa, Italia 3 IPCF/CNR, UoS Pisa, Italia

2. Debenedetti and Stillinger, 2001 Structural arrest 1/2 Random walk: cage effect Structural arrest and particle trapping in deeply supercooled states Log  (Poise)

3. Debenedetti and Stillinger, 2001 Structural arrest OUTLINE Cage scaling:  ,  vs. Debye-Waller factor 1/2 Structural arrest and particle trapping in deeply supercooled states Log  (Poise) Random walk: cage effect

4. Debenedetti and Stillinger, 2001 Structural arrest OUTLINE Cage scaling:  ,  vs. Debye-Waller factor Elastic scaling:  ,  vs. elastic modulus G -Elastic scaling and cage scaling: vs. G/T 1/2 Structural arrest and particle trapping in deeply supercooled states Log  (Poise) Random walk: cage effect

5. Debenedetti and Stillinger, 2001 Structural arrest OUTLINE Cage scaling:  ,  vs. Debye-Waller factor Elastic scaling:  ,  vs. elastic modulus G -Elastic scaling and cage scaling: vs. G/T Thermodynamic scaling:  ,  vs.   /T, (density  and temperature T ) -Thermodynamic scaling and cage scaling: vs.   /T 1/2 Structural arrest and particle trapping in deeply supercooled states Log  (Poise) Random walk: cage effect

6. Debenedetti and Stillinger, 2001 Structural arrest OUTLINE Cage scaling:  ,  vs. Debye-Waller factor Elastic scaling:  ,  vs. elastic modulus G -Elastic scaling and cage scaling: vs. G/T Thermodynamic scaling:  ,  vs.   /T, (density  and temperature T ) -Thermodynamic scaling and cage scaling: vs.   /T Conclusions 1/2 Structural arrest and particle trapping in deeply supercooled states Log  (Poise) Random walk: cage effect

7. =  (G/T ) =  (   /T )   = F[  (G/T )]   = F[  (   /T ) ]   = F[ ] Elastic scaling “universal” master curve Thermodynamic scaling material-dependent master curve 1/2 Cage scaling

8.   = F[ ] 1/2 Cage scaling …echoes the Lindemann melting criterion Hall & Wolynes 87, Buchenau & Zorn 92, Ngai 2000, Starr et al 2002, Harrowell et al 2006, Larini et al 2008…

9. Log t Log MSD Log Log t* F. Puosi, DL, JPCB (2011) Log   Cage scaling: evidence from the Van Hove function 1/2 MSD(t*) =

10. Log t Log MSD Log Log t* F. Puosi, DL, JPCB (2011) Log   Cage scaling: evidence from the Van Hove function G s (X) (r, t*) = G s (Y) (r, t*) G s (X) (r,   ) = G s (Y) (r,,   ) X, Y : generic states 1/2 MSD(t*) =

11. Log t Log MSD Log Log t* F. Puosi, DL, JPCB (2011) Log   Cage scaling: evidence from the Van Hove function Polymer melt G s (X) (r, t*) = G s (Y) (r, t*) G s (X) (r,   ) = G s (Y) (r,,   ) X, Y : generic states 1/2 MSD(t*) =

12. Log t Log MSD Log Log t* F. Puosi, DL, JPCB (2011) Log   Cage scaling: evidence from the Van Hove function Polymer melt G s (X) (r, t*) = G s (Y) (r, t*) G s (X) (r,   ) = G s (Y) (r,,   ) Jumps ! X, Y : generic states 1/2 MSD(t*) =

13. Log MSD Log F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013) Binary mixture Log t Log t* Log   Cage scaling: evidence from the Van Hove function G s (X) (r, t*) = G s (Y) (r, t*) G s (X) (r,   ) = G s (Y) (r,,   ) X, Y : generic states 1/2 MSD(t*) =

14. Log MSD Log Log t Log t* Log   Cage scaling: implications Polymer melt 1/2 t* MSD(t*) =

15. A. Ottochian, C. De Michele, DL, JCP (2009) Binary mixture, polymer melt Cage scaling: implications “rule of thumb 1” Log MSD Log Log t Log t* Log   1/2 MSD(t*) =

16. C. De Michele, E. Del Gado, DL, Soft Matter (2011) Cage scaling: implications “rule of thumb 1” Log MSD Log Log t Log t* Log   1/2 Colloidal gel MSD(t*) =

17. C. De Michele, DL, unpublishedF. Puosi, DL, JPCB (2011) Binary mixture Polymer melt Cage scaling: implications “rule of thumb 2” t

18. Cage scaling: experimental evidence L. Larini et al, Nature Phys. (2008) Master curve taken from MD simulation 1 adjustable parameter:  0 or  0

19. =  (G/T )   = F[  (G/T )]   = F[ ] Elastic scaling 1/2 Cage scaling Elastic models: see RMP review by Dyre (2006)

20. Log t G(t) G p = G(t*) Initial affine response, total force per particle unbalanced F.Puosi, DL, JCP 041104 (2012) Elastic scaling in polymer melts N.B.: MSD(t*) = Transient shear modulus

21. Log t G(t) G p = G(t*) “Inherent” dynamics: particle moved to the local potential energy minimum Initial affine response, total force per particle unbalanced Fast mechanical equilibration F.Puosi, DL, JCP 041104 (2012) Elastic scaling in polymer melts N.B.: MSD(t*) = Transient shear modulus

22. G(t) G∞G∞ GpGp t* ~ 1-10 ps Log t  Affine elasticity F.Puosi, DL, JCP 041104 (2012) Elastic scaling in polymer melts

23. G(t) G∞G∞ GpGp Log t  F.Puosi, DL, JCP 041104 (2012) Elastic scaling in polymer melts t* ~ 1-10 ps

24. Master curve: Log   =  +  G/T +  [ G/T ] 2 ,  constants Modulus term matters: evidence from one isothermal set Not another variant of the Vogel-Fulcher law   = f(T)… Elastic scaling in polymer melts No adjustments

25. 1/ Elastic scaling: building the master curve MD simulations: polymer G/ T The elastic scaling works for the Debye-Waller factor, F.Puosi, DL, arXiv:1108.4629v1, to be submitted

26. 1/ MD simulations: polymer G/ T The elastic scaling works for the Debye-Waller factor, Elastic scaling: building the master curve F.Puosi, DL, arXiv:1108.4629v1, to be submitted

27. 1/   = F[ ] =  (G/T ) MD simulations: polymer G/ T   = F[  (G/T )] The elastic scaling works for the Debye-Waller factor, Elastic scaling: building the master curve F.Puosi, DL, arXiv:1108.4629v1, to be submitted

28.   = F[  (G/T )] 1/ G/ T   = F[ ] =  (G/T ) Experiments G/T ( T g /G g ) The elastic scaling works for the Debye-Waller factor, the experimental master curve follows from the MD simulations Elastic scaling: building the master curve F.Puosi, DL, arXiv:1108.4629v1, to be submitted

29. =  (   /T )   = F[  (   /T ) ]   = F[ ] Thermodynamic scaling 1/2 Cage scaling Thermodynamic scaling: see review by Roland et al, Rep. Prog. Phys. (2005)

30. Thermodynamic scaling in Kob-Andersen binary mixture F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013) The thermodynamic scaling works for the Debye-Waller factor,   /T

31. Thermodynamic scaling in Kob-Andersen binary mixture   /T F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013) The thermodynamic scaling works for the Debye-Waller factor, Cage scaling fails for   < 1

32. Thermodynamic scaling in Kob-Andersen binary mixture   /T F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013) =  (   /T )   = F[  (   /T )]   = F[ ] Cage scaling fails for   < 1 The thermodynamic scaling works for the Debye-Waller factor,

33. propylen carbonate F. Puosi, O. Chulkin, S. Capaccioli, DL to be submitted The master curve of the thermodynamic scaling follows from the MD simulations with one adjustable parameter: the isochoric fragility Thermodynamic scaling from Debye-Waller factor: comparison with the experiment preliminary results

34. 1/2 Conclusions Cage scaling (   vs ): -Results suggest that is a “universal” picosecond predictor of the  relaxation. -Tested on different MD models: polymers, binary atomic mixtures, colloidal gels… - The MD master curve fits (with one adjustable parameter) the scaling of the experimental data covering over ~ 18 decades in   drawn by glassformers in the fragility range 20 ≤ m ≤ 190.

35. 1/2 Conclusions Cage scaling (   vs ): -Results suggest that is a “universal” picosecond predictor of the  relaxation. -Tested on different MD models: polymers, binary atomic mixtures, colloidal gels… - The MD master curve fits (with one adjustable parameter) the scaling of the experimental data covering over ~ 18 decades in   drawn by glassformers in the fragility range 20 ≤ m ≤ 190. Elastic scaling (   vs G/T): - Intermediate-time shear elasticity and are highly correlated. - MD master curve   vs G/T drawn by using the cage scaling. - The MD master curve fits (with one adjustable parameter) the scaling of the experimental data covering over ~ 18 decades in   drawn by glassformers in the fragility range 20 ≤ m ≤ 115.

36. 1/2 Conclusions Cage scaling (   vs ): -Results suggest that is a “universal” picosecond predictor of the  relaxation. -Tested on different MD models: polymers, binary atomic mixtures, colloidal gels… - The MD master curve fits (with one adjustable parameter) the scaling of the experimental data covering over ~ 18 decades in   drawn by glassformers in the fragility range 20 ≤ m ≤ 190. Elastic scaling (   vs G/T): - Intermediate-time shear elasticity and are highly correlated. - MD master curve   vs G/T drawn by using the cage scaling. - The MD master curve fits (with one adjustable parameter) the scaling of the experimental data covering over ~ 18 decades in   drawn by glassformers in the fragility range 20 ≤ m ≤ 115. Thermodynamic scaling (   vs   /T ) - scales with   /T. Extensive MD simulations in progress - MD master curve   vs   /T drawn by using the cage scaling. - Good comparison with the experimental data on a single glassformer (13 decades in   ) by adjusting the isochoric fragility only. Work in progress…

37. Collaborators: C. De Michele, Ric TD Roma L. Larini, Ass. Prof.Rutgers University A. Ottochian, Postdoc ’Ecole Centrale Paris F. Puosi, Postdoc Univ. Grenoble 1 S. BerniniPhDPisa O. ChulkinPostdocOdessa M. BaruccoGraduatePisa Credits

38. 1/ G/ T    / T

39. t* ~ 1-10 ps Log t Log   Log Log t* Log t Log F s (q max, t) 1/2

40. C. De Michele, F. Puosi, DL, unpublished F. Puosi, DL, JPCB (2011)

41. MD simulations Density  Temperature T Chain length M (polymer) Potential: p, q

42. 10 17 s (eta’ dell’universo)    ~ 10 26 s 1/2 First “universal” scaling: structural relaxation time   or viscosity  vs.Debye-Waller factor (rattling amplitude in the cage)

43. Log MSD Log Log t Log t* Log   Cage scaling: implications G s (X) (r, t*) = G s (Y) (r, t*) G s (X) (r,   ) = G s (Y) (r,,   ) Polymer melt