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Francesco Puosi 1 , Dino Leporini 2,3

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

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Francesco Puosi 1 , Dino Leporini 2,3

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  1. Elasticity, caged dynamics and thermodynamics: three (related) scalings of the relaxation in glassformingsystems Francesco Puosi1, 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. Structural arrest and particle trapping in deeply supercooled states Structural arrest Random walk: cage effect Log h (Poise) < u2 >1/2 DebenedettiandStillinger, 2001

  3. Structural arrest and particle trapping in deeply supercooled states Structural arrest Random walk: cage effect Log h (Poise) < u2 >1/2 • OUTLINE • Cage scaling: ta , h vs. Debye-Waller factor <u2> DebenedettiandStillinger, 2001

  4. Structural arrest and particle trapping in deeply supercooled states Structural arrest Random walk: cage effect Log h (Poise) < u2 >1/2 • OUTLINE • Cage scaling: ta , h vs. Debye-Waller factor <u2> • Elastic scaling: ta, h vs. elastic modulus G • - Elastic scaling and cage scaling: <u2> vs. G/T DebenedettiandStillinger, 2001

  5. Structural arrest and particle trapping in deeply supercooled states Structural arrest Random walk: cage effect Log h (Poise) < u2 >1/2 • OUTLINE • Cage scaling: ta , h vs. Debye-Waller factor <u2> • Elastic scaling: ta, h vs. elastic modulus G • - Elastic scaling and cage scaling: <u2> vs. G/T • Thermodynamic scaling: ta, h vs. rg/T, (density r and temperature T ) • - Thermodynamic scaling and cage scaling: <u2> vs. rg/T DebenedettiandStillinger, 2001

  6. Structural arrest and particle trapping in deeply supercooled states Structural arrest Random walk: cage effect Log h (Poise) < u2 >1/2 • OUTLINE • Cage scaling: ta , h vs. Debye-Waller factor <u2> • Elastic scaling: ta, h vs. elastic modulus G • - Elastic scaling and cage scaling: <u2> vs. G/T • Thermodynamic scaling: ta, h vs. rg/T, (density r and temperature T ) • - Thermodynamic scaling and cage scaling: <u2> vs. rg/T • Conclusions DebenedettiandStillinger, 2001

  7. < u2 >1/2 <u2> = f(G/T ) <u2> = y(rg/T ) Cage scaling ta = F[ <u2> ] ta = F[y(rg/T ) ] ta = F[ f(G/T )] Thermodynamic scaling material-dependent master curve Elastic scaling “universal” master curve

  8. < u2 >1/2 Cage scaling ta = F[ <u2> ] …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. Cage scaling: evidence from the Van Hove function < u2 >1/2 Log <u2> Log MSD MSD(t*) = <u2> Log t* Log ta Log t F. Puosi, DL, JPCB (2011)

  10. Cage scaling: evidence from the Van Hove function Gs(X) (r, t*) = Gs(Y)(r, t*) Gs(X) (r, ta ) = Gs(Y) (r, , ta ) X, Y : generic states < u2 >1/2 Log <u2> Log MSD MSD(t*) = <u2> Log t* Log ta Log t F. Puosi, DL, JPCB (2011)

  11. Cage scaling: evidence from the Van Hove function Gs(X) (r, t*) = Gs(Y)(r, t*) Gs(X) (r, ta ) = Gs(Y) (r, , ta ) X, Y : generic states Polymer melt < u2 >1/2 Log <u2> Log MSD MSD(t*) = <u2> Log t* Log ta Log t F. Puosi, DL, JPCB (2011)

  12. Cage scaling: evidence from the Van Hove function Gs(X) (r, t*) = Gs(Y)(r, t*) Gs(X) (r, ta ) = Gs(Y) (r, , ta ) X, Y : generic states Polymer melt < u2 >1/2 Log <u2> Log MSD MSD(t*) = <u2> Jumps ! Log t* Log ta Log t F. Puosi, DL, JPCB (2011)

  13. Cage scaling: evidence from the Van Hove function Gs(X) (r, t*) = Gs(Y)(r, t*) Gs(X) (r, ta ) = Gs(Y) (r, , ta ) X, Y : generic states Binary mixture < u2 >1/2 Log <u2> Log MSD MSD(t*) = <u2> Log t* Log ta Log t F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013)

  14. Cage scaling: implications Polymer melt < u2 >1/2 t* Log <u2> Log MSD MSD(t*) = <u2> Log t* Log ta Log t

  15. Cage scaling: implications “rule of thumb 1” < u2 >1/2 Binary mixture, polymer melt Log <u2> Log MSD MSD(t*) = <u2> Log t* Log ta Log t A. Ottochian, C. De Michele, DL, JCP (2009)

  16. Cage scaling: implications “rule of thumb 1” < u2 >1/2 Colloidal gel Log <u2> Log MSD MSD(t*) = <u2> Log t* Log ta Log t C. De Michele, E. Del Gado, DL, Soft Matter (2011)

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

  18. Cage scaling: experimental evidence • Master curve taken from MD simulation • 1 adjustable parameter: t0 or h0 L. Larini et al, Nature Phys. (2008)

  19. < u2 >1/2 <u2> = f(G/T ) Cage scaling ta = F[ <u2> ] ta = F[ f(G/T )] Elastic scaling Elastic models: see RMP review by Dyre (2006)

  20. Elastic scaling in polymer melts Initial affine response, total force per particle unbalanced Transient shear modulus G(t) Gp= G(t*) Log t N.B.: MSD(t*) = <u2> F.Puosi, DL, JCP 041104 (2012)

  21. Elastic scaling in polymer melts Initial affine response, total force per particle unbalanced Transient shear modulus G(t) “Inherent” dynamics: particle moved to the local potential energy minimum Gp= G(t*) Log t N.B.: MSD(t*) = <u2> Fast mechanical equilibration F.Puosi, DL, JCP 041104 (2012)

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

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

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

  25. Elastic scaling: building the master curve 1/ <u2> G/ T MD simulations: polymer • The elastic scaling works for the • Debye-Waller factor <u2>, F.Puosi, DL, arXiv:1108.4629v1, to be submitted

  26. Elastic scaling: building the master curve 1/ <u2> G/ T MD simulations: polymer • The elastic scaling works for the • Debye-Waller factor <u2>, F.Puosi, DL, arXiv:1108.4629v1, to be submitted

  27. Elastic scaling: building the master curve <u2> = f(G/T ) 1/ <u2> G/ T ta = F[ <u2> ] MD simulations: polymer ta = F[ f(G/T )] • The elastic scaling works for the • Debye-Waller factor <u2>, F.Puosi, DL, arXiv:1108.4629v1, to be submitted

  28. Elastic scaling: building the master curve <u2> = f(G/T ) 1/ <u2> G/ T ta = F[ <u2> ] Experiments ta = F[ f(G/T )] • The elastic scaling works for the • Debye-Waller factor <u2>, • the experimental master curve • follows from the MD simulations G/T • ( Tg /Gg) F.Puosi, DL, arXiv:1108.4629v1, to be submitted

  29. < u2 >1/2 <u2> = y(rg/T ) Cage scaling ta = F[ <u2> ] ta = F[y(rg/T ) ] Thermodynamic scaling Thermodynamic scaling: see review by Roland et al, Rep. Prog. Phys. (2005)

  30. Thermodynamic scaling in Kob-Andersen binary mixture rg/T • The thermodynamic scaling works • for the Debye-Waller factor <u2>, F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013)

  31. Thermodynamic scaling in Kob-Andersen binary mixture • The thermodynamic scaling works • for the Debye-Waller factor <u2>, rg/T F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013) Cage scaling fails for ta < 1

  32. Thermodynamic scaling in Kob-Andersen binary mixture <u2> = y(rg/T ) ta = F[ <u2> ] ta = F[y(rg/T )] • The thermodynamic scaling works • for the Debye-Waller factor <u2>, rg/T F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013) Cage scaling fails for ta < 1

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

  34. Conclusions • Cage scaling ( tavs<u2> ): • Results suggest that <u2> is a “universal” picosecond predictor of the a 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 ta drawn by glassformers • in the fragility range 20 ≤ m ≤ 190. < u2 >1/2

  35. Conclusions • Cage scaling ( tavs<u2> ): • Results suggest that <u2> is a “universal” picosecond predictor of the a 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 ta drawn by glassformers • in the fragility range 20 ≤ m ≤ 190. • Elastic scaling ( tavsG/T): • - Intermediate-time shear elasticity and <u2> are highly correlated. • MD master curve tavsG/T drawn by using the cage scaling. • The MD master curve fits (with one adjustable parameter) the scaling • of the experimentaldata covering over ~ 18 decades in ta drawn by • glassformers in the fragility range 20 ≤ m ≤ 115. < u2 >1/2

  36. Conclusions • Cage scaling ( tavs<u2> ): • Results suggest that <u2> is a “universal” picosecond predictor of the a 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 ta drawn by glassformers • in the fragility range 20 ≤ m ≤ 190. • Elastic scaling ( tavsG/T): • - Intermediate-time shear elasticity and <u2> are highly correlated. • MD master curve tavsG/T drawn by using the cage scaling. • The MD master curve fits (with one adjustable parameter) the scaling • of the experimentaldata covering over ~ 18 decades in ta drawn by • glassformers in the fragility range 20 ≤ m ≤ 115. • Thermodynamicscaling ( tavsrg/T ) • <u2> scales with rg/T . Extensive MD simulations in progress • MD master curve tavsrg/T drawn by using the cage scaling. • Good comparison with the experimental data on a single glassformer (13 decades in ta ) • by adjusting the isochoric fragility only. Work in progress… < u2 >1/2

  37. Credits 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. Bernini PhD Pisa • O. Chulkin Postdoc Odessa • M. Barucco Graduate Pisa

  38. <u2> 1/ <u2> rg/ T G/ T

  39. Log < Dr2 (t) > < u2 >1/2 Log <u2> Log t* Log t t* ~ 1-10 ps Log Fs (qmax , t) Log ta Log t

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

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

  42. First “universal” scaling: structural relaxation time ta or viscosity h vs.Debye-Waller factor < u2> (rattling amplitude in the cage) ta ~ 10 26 s < u2 >1/2 1017 s (eta’ dell’universo)

  43. Cage scaling: implications Gs(X) (r, t*) = Gs(Y)(r, t*) Gs(X) (r, ta ) = Gs(Y) (r, , ta ) Polymer melt Log <u2> Log MSD Log t* Log ta Log t

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