elasticity, caged dynamics and thermodynamics: three (related) scalings of the relaxation in...
TRANSCRIPT
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, t a ) = Gs
(Y) (r, , t a )
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, t a ) = Gs
(Y) (r, , t a )
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, t a ) = Gs
(Y) (r, , t a )
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, t a ) = Gs
(Y) (r, , t a )
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 t a = 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 t a 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 t a 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 t a 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 t a 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 t a 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 t a ) 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
< D
r 2 (t
) >
Log <u2>
Log t*
Log t
Log
F s (
q m
ax , 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, t a ) = Gs
(Y) (r, , t a )
Polymer melt