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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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slide1

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

slide2

Structural arrest and particle trapping in deeply supercooled states

Structural arrest

Random walk: cage effect

Log h (Poise)

< u2 >1/2

DebenedettiandStillinger, 2001

slide3

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

slide4

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

slide5

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

slide6

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

slide7

< 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

slide8

< 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…

slide9

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)

slide10

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)

slide11

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)

slide12

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)

slide13

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)

slide14

Cage scaling: implications

Polymer melt

< u2 >1/2

t*

Log <u2>

Log MSD

MSD(t*) = <u2>

Log t*

Log ta

Log t

slide15

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)

slide16

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)

slide17

Cage scaling: implications

“rule of thumb 2”

Polymer melt

Binary mixture

t

F. Puosi, DL, JPCB (2011)

C. De Michele, DL, unpublished

slide18

Cage scaling: experimental evidence

  • Master curve taken from MD simulation
  • 1 adjustable parameter: t0 or h0

L. Larini et al, Nature Phys. (2008)

slide19

< 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)

slide20

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)

slide21

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)

slide22

Elastic scaling in polymer melts

G(t)

Affine elasticity

G∞

Gp

ta

t* ~ 1-10 ps

Log t

F.Puosi, DL, JCP 041104 (2012)

slide23

Elastic scaling in polymer melts

G(t)

G∞

Gp

ta

t* ~ 1-10 ps

Log t

F.Puosi, DL, JCP 041104 (2012)

slide24

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

slide25

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

slide26

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

slide27

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

slide28

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

slide29

< 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)

slide30

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)

slide31

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

slide32

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

slide33

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

slide34

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

slide35

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

slide36

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

credits
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
slide38

<u2>

1/ <u2>

rg/ T

G/ T

slide39

Log < Dr2 (t) >

< u2 >1/2

Log <u2>

Log t*

Log t

t* ~ 1-10 ps

Log Fs (qmax , t)

Log ta

Log t

slide40

F. Puosi, DL, JPCB (2011)

C. De Michele, F. Puosi, DL, unpublished

slide41

MD simulations

Density r

Temperature T

Chain length M (polymer)

Potential: p, q

slide42

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)

slide43

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