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

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

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

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

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

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

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

  = 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…

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*) =

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*) =

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*) =

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*) =

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*) =

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

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*) =

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*) =

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

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

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

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

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 (2012) Elastic scaling in polymer melts N.B.: MSD(t*) = Transient shear modulus

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

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

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

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: v1, to be submitted

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: v1, to be submitted

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: v1, to be submitted

  = 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: v1, to be submitted

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

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

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

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,

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

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.

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.

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…

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

1/ G/ T    / T

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

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

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

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

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