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, France2 Dipartimento di Fisica “Enrico Fermi”, Universita’ di Pisa, Pisa, Italia

3 IPCF/CNR, UoS Pisa, Italia

Debenedetti and Stillinger, 2001

Structural arrest

< u2 >1/2

Random walk: cage effect

Structural arrest and particle trapping in deeply supercooled states

Log h

(Poi

se)

Debenedetti and Stillinger, 2001

Structural arrest

OUTLINE

• Cage scaling: ta , h vs. Debye-Waller factor <u2>

< u2 >1/2

Structural arrest and particle trapping in deeply supercooled states

Log h

(Poi

se)

Random walk: cage effect

Debenedetti and Stillinger, 2001

Structural arrest

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

< u2 >1/2

Structural arrest and particle trapping in deeply supercooled states

Log h

(Poi

se)

Random walk: cage effect

Debenedetti and Stillinger, 2001

Structural arrest

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

< u2 >1/2

Structural arrest and particle trapping in deeply supercooled states

Log h

(Poi

se)

Random walk: cage effect

Debenedetti and Stillinger, 2001

Structural arrest

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

< u2 >1/2

Structural arrest and particle trapping in deeply supercooled states

Log h

(Poi

se)

Random walk: cage effect

<u2> = f(G/T ) <u2> = y(rg/T )

ta = F[ f(G/T )] ta = F[y(rg/T ) ]

ta = F[ <u2> ]

Elastic scaling

“universal” master curve

Thermodynamic scaling

material-dependent master curve

< u2 >1/2

Cage scaling

ta = F[ <u2> ]

< u2 >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 <u2>

Log t*F. Puosi, DL, JPCB (2011)

Log ta

Cage scaling: evidence from the Van Hove function

< u2 >1/2

MSD(t*) = <u2>

Log t

Log

MSD

Log <u2>

Log t*F. Puosi, DL, JPCB (2011)

Log ta

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

MSD(t*) = <u2>

Log t

Log

MSD

Log <u2>

Log t*F. Puosi, DL, JPCB (2011)

Log ta

Cage scaling: evidence from the Van Hove function

Polymer melt

Gs(X) (r, t*) = Gs

(Y) (r, t*) Gs(X) (r, ta ) = Gs

(Y) (r, , ta )

X, Y : generic states

< u2 >1/2

MSD(t*) = <u2>

Log t

Log

MSD

Log <u2>

Log t*F. Puosi, DL, JPCB (2011)

Log ta

Cage scaling: evidence from the Van Hove function

Polymer melt

Gs(X) (r, t*) = Gs

(Y) (r, t*) Gs(X) (r, ta ) = Gs

(Y) (r, , ta )

Jumps !

X, Y : generic states

< u2 >1/2

MSD(t*) = <u2>

Log

MSD

Log <u2>

F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013)

Binary mixture

Log tLog t* Log ta

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

MSD(t*) = <u2>

Log

MSD

Log <u2>

Log tLog t* Log ta

Cage scaling: implications

Polymer melt

< u2 >1/2

t*

MSD(t*) = <u2>

A. Ottochian, C. De Michele, DL, JCP (2009)

Binary mixture, polymer melt

Cage scaling: implications

“rule of thumb 1”

Log

MSD

Log <u2>

Log tLog t* Log ta

< u2 >1/2

MSD(t*) = <u2>

C. De Michele, E. Del Gado, DL, Soft Matter (2011)

Cage scaling: implications

“rule of thumb 1”

Log

MSD

Log <u2>

Log tLog t* Log ta

< u2 >1/2

Colloidal gel

MSD(t*) = <u2>

C. De Michele, DL, unpublishedF. Puosi, DL, JPCB (2011)

Binary mixturePolymer 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: t0 or h0

<u2> = f(G/T )

ta = F[ f(G/T )]

ta = F[ <u2> ]

Elastic scaling

< u2 >1/2

Cage scaling

Elastic models: see RMP review by Dyre (2006)

Log t

G(t)

Gp = 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*) = <u2>

Transient shear modulus

Log t

G(t)

Gp = 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*) = <u2>

Transient shear modulus

G(t)

G∞

Gp

t* ~ 1-10 ps Log tta

Affine elasticity

F.Puosi, DL, JCP 041104 (2012)

Elastic scaling in polymer melts

G(t)

G∞

Gp

Log tta F.Puosi, DL, JCP 041104 (2012)

Elastic scaling in polymer melts

t* ~ 1-10 ps

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)…

Elastic scaling in polymer melts

No adjustments

1/ <

u2 >Elastic scaling: building the master curve

MD simulations: polymer

G/ T

• The elastic scaling works for the Debye-Waller factor <u2>,

F.Puosi, DL, arXiv:1108.4629v1, to be submitted

1/ <

u2 >

MD simulations: polymer

G/ T

• The elastic scaling works for the Debye-Waller factor <u2>,

Elastic scaling: building the master curve

F.Puosi, DL, arXiv:1108.4629v1, to be submitted

1/ <

u2 >

ta = F[ <u2> ]

<u 2> = f(G/T )

MD simulations: polymer

G/ T

ta = F[ f(G/T )]

• The elastic scaling works for the Debye-Waller factor <u2>,

Elastic scaling: building the master curve

F.Puosi, DL, arXiv:1108.4629v1, to be submitted

ta = F[ f(G/T )]

1/ <

u2 >

G/ T

ta = F[ <u2> ]

<u 2> = f(G/T )

Experiments

G/T • ( Tg /Gg )

• The elastic scaling works for the Debye-Waller factor <u2>,

• the experimental master curve follows from the MD simulations

Elastic scaling: building the master curve

F.Puosi, DL, arXiv:1108.4629v1, to be submitted

<u2> = y(rg/T )

ta = F[y(rg/T ) ]

ta = F[ <u2> ]

Thermodynamic scaling

< u2 >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 <u2>,

rg/T

Thermodynamic scaling in Kob-Andersen binary mixture

rg/T F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013)

• The thermodynamic scaling works for the Debye-Waller factor <u2>,

Cage scaling fails for ta < 1

Thermodynamic scaling in Kob-Andersen binary mixture

rg/T F. Puosi, C. De Michele, DL, JCP 138, 12A532 (2013)

<u 2> = y(r g/T )

ta = F[y(rg/T )]

ta = F[ <u2> ]

Cage scaling fails for ta < 1

• The thermodynamic scaling works for the Debye-Waller factor <u2>,

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

< u2 >1/2

Conclusions

• Cage scaling ( ta vs <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

Conclusions

• Cage scaling ( ta vs <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 ( ta vs G/T):- Intermediate-time shear elasticity and <u2> are highly correlated.

- MD master curve ta 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 ta drawn by glassformers in the fragility range 20 ≤ m ≤ 115.

< u2 >1/2

Conclusions

• Cage scaling ( ta vs <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 ( ta vs G/T):- Intermediate-time shear elasticity and <u2> are highly correlated.

- MD master curve ta 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 ta drawn by glassformers in the fragility range 20 ≤ m ≤ 115.

• Thermodynamic scaling ( ta vs rg/T )- <u2> scales with rg/T . Extensive MD simulations in progress- MD master curve ta vs rg/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…

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

Credits

1/ <

u2 >

G/ T

<u2 >

rg / T

t* ~ 1-10 psLog t

Log ta

Log

< Dr

2 (t

) >

Log <u2>

Log t*

Log t

Log

F s (

q max

, t)

< u2 >1/2

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

MD simulations

Density r

Temperature T

Chain length M (polymer) Potential: p, q

1017 s (eta’ dell’universo)t a ~ 10 26 s< u2 >1/2

First “universal” scaling: structural relaxation time ta or viscosity h vs.Debye-Waller factor < u2> (rattling amplitude in the cage)

Log

MSD

Log <u2>

Log tLog t* Log ta

Cage scaling: implications

Gs(X) (r, t*) = Gs

(Y) (r, t*) Gs(X) (r, ta ) = Gs

(Y) (r, , ta )

Polymer melt