scalings of the relaxation in

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27 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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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

2
Dipartimento

di
Fisica

“Enrico Fermi”,
Universita
’ di Pisa, Pisa, Italia

3

IPCF/CNR,
UoS

Pisa, Italia

Debenedetti

and

Stillinger
, 2001

Structural arrest

< u
2
>
1/2


Random walk: cage effect

Structural

arrest

and

particle

trapping

in

deeply

supercooled

states

Log
h
(Poise)

Debenedetti

and

Stillinger
, 2001

Structural arrest

OUTLINE



Cage scaling:
t
a

,
h

vs. Debye
-
Waller factor
<u
2
>

< u
2
>
1/2


Structural

arrest

and

particle

trapping

in

deeply

supercooled

states

Log
h
(Poise
)

Random walk: cage effect

Debenedetti

and

Stillinger
, 2001

Structural arrest

OUTLINE



Cage scaling:
t
a

,
h

vs. Debye
-
Waller factor
<u
2
>



Elastic scaling
:
t
a

,
h

vs.
elastic modulus G


-

Elastic scaling and cage scaling:
<u
2
> vs. G/T

< u
2
>
1/2


Structural

arrest

and

particle

trapping

in

deeply

supercooled

states

Log
h
(Poise
)

Random walk: cage effect

Debenedetti

and

Stillinger
, 2001

Structural arrest

OUTLINE



Cage scaling:
t
a

,
h

vs. Debye
-
Waller factor
<u
2
>



Elastic scaling
:
t
a

,
h

vs.
elastic modulus G


-

Elastic scaling and cage scaling:
<u
2
> vs. G/T



Thermodynamic scaling
:

t
a

,
h

vs.
r
g
/
T,
(density
r

and temperature T
)


-

Thermodynamic scaling
and cage scaling: <u
2
> vs.
r
g
/
T

< u
2
>
1/2


Structural

arrest

and

particle

trapping

in

deeply

supercooled

states

Log
h
(Poise
)

Random walk: cage effect

Debenedetti

and

Stillinger
, 2001

Structural arrest

OUTLINE



Cage scaling:
t
a

,
h

vs. Debye
-
Waller factor
<u
2
>



Elastic scaling
:
t
a

,
h

vs.
elastic modulus G


-

Elastic scaling and cage scaling:
<u
2
> vs. G/T



Thermodynamic scaling
:

t
a

,
h

vs.
r
g
/
T, (density
r

and temperature T )


-

Thermodynamic scaling
and cage scaling: <u
2
> vs.
r
g
/
T



Conclusions

< u
2
>
1/2


Structural

arrest

and

particle

trapping

in

deeply

supercooled

states

Log
h
(Poise
)

Random walk: cage effect

<u
2
> =
f
(G/T )

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

t
a

= F[
f
(G/T
)]

t
a

= F[
y
(
r
g
/T
)
]

t
a

= F[ <u
2
> ]

Elastic

scaling


“universal” master curve

Thermodynamic

scaling


material
-
dependent master curve

< u
2
>
1/2


Cage

scaling

t
a

= F[ <u
2
> ]

< u
2
>
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 <u
2
>

Log t*

F.
Puosi
, DL, JPCB (2011)

Log
t
a

Cage scaling: evidence from the Van Hove function

< u
2
>
1/2


MSD(t*) =
<u
2
>

Log t

Log MSD

Log <u
2
>

Log t*

F.
Puosi
, DL, JPCB (2011)

Log
t
a

Cage scaling: evidence from the Van Hove function

G
s
(X)

(r, t*) =
G
s
(Y)

(r, t*
)
G
s
(X)

(r
,
t
a
)
=
G
s
(Y)

(r, ,
t
a
)

X, Y : generic states

< u
2
>
1/2


MSD(t*) =
<u
2
>

Log t

Log MSD

Log <u
2
>

Log t*

F.
Puosi
, DL, JPCB (2011)

Log
t
a

Cage scaling: evidence from the Van Hove function

Polymer melt

G
s
(X)

(r, t*) =
G
s
(Y)

(r, t*
)
G
s
(X)

(r
,
t
a
)
=
G
s
(Y)

(r, ,
t
a
)

X, Y : generic states

< u
2
>
1/2


MSD(t*) =
<u
2
>

Log t

Log MSD

Log <u
2
>

Log t*

F.
Puosi
, DL, JPCB (2011)

Log
t
a

Cage scaling: evidence from the Van Hove function

Polymer melt

G
s
(X)

(r, t*) =
G
s
(Y)

(r, t*
)
G
s
(X)

(r
,
t
a
)
=
G
s
(Y)

(r, ,
t
a
)

X, Y : generic states

< u
2
>
1/2


MSD(t*) =
<u
2
>

Log MSD

Log <u
2
>

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

Binary mixture

Log t

Log t*

Log
t
a

Cage scaling: evidence from the Van Hove function

G
s
(X)

(r, t*) =
G
s
(Y)

(r, t*
)
G
s
(X)

(r
,
t
a
)
=
G
s
(Y)

(r, ,
t
a
)

X, Y : generic states

< u
2
>
1/2


MSD(t*) =
<u
2
>

Log MSD

Log <u
2
>

Log t

Log t*

Log
t
a

Cage scaling: implications

Polymer melt

< u
2
>
1/2


t
*

MSD(t*) =
<u
2
>

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

Binary mixture, polymer melt

Cage scaling: implications

“rule of thumb 1”

Log MSD

Log <u
2
>

Log t

Log t*

Log
t
a

< u
2
>
1/2


MSD(t*) =
<u
2
>

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

Cage scaling: implications

“rule of thumb 1”

Log MSD

Log <u
2
>

Log t

Log t*

Log
t
a

< u
2
>
1/2


C
olloidal gel

MSD(t*) =
<u
2
>

C. De Michele, DL, unpublished

F.
Puosi
, DL, JPCB (2011)

Binary mixture

P
olymer 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:
t
0

or
h
0

<u
2
> =
f
(G/T )

t
a

= F[
f
(G/T
)]

t
a

= F[ <u
2
> ]

Elastic

scaling

< u
2
>
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 041104
(2012
)

Elastic scaling in polymer melts

N.B.:

MSD(t*) =
<u
2
>


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 041104
(2012
)

Elastic scaling in polymer melts

N.B.:

MSD(t*) =
<u
2
>


Transient shear modulus

G(t)

G


G
p

t* ~ 1
-
10
ps

Log t

t
a

Affine elasticity

F.Puosi
, DL, JCP 041104
(2012
)

Elastic scaling in polymer melts

G(t)

G


G
p

Log t

t
a

F.Puosi
, DL, JCP 041104
(2012
)

Elastic scaling in polymer melts

t* ~ 1
-
10
ps

Master curve: Log
t
a

=
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/ <u
2
>

Elastic scaling: building the master curve

MD simulations: polymer

G/ T


The elastic scaling
works

for the

Debye
-
Waller factor <
u
2
>,

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

1/ <u
2
>

MD simulations: polymer

G/ T


The elastic scaling
works

for the

Debye
-
Waller factor <
u
2
>,

Elastic scaling: building the master curve

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

1/ <u
2
>

t
a

= F[ <u
2
> ]

MD simulations: polymer

G/ T


The elastic scaling
works

for the

Debye
-
Waller factor <
u
2
>,

Elastic scaling: building the master curve

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

1/ <u
2
>

G/ T

t
a

= F[ <u
2
> ]

Experiments

G/T


(
T
g

/
G
g

)


The elastic scaling
works

for the

Debye
-
Waller factor <
u
2
>,



t
he
experimental

master curve


follows from the MD
simulations

Elastic scaling: building the master curve

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

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

t
a

= F[
y
(
r
g
/T
)
]

t
a

= F[ <u
2
> ]

Thermodynamic

scaling

< u
2
>
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 <
u
2
>,


r
g
/T

Thermodynamic scaling in
Kob
-
Andersen binary mixture

r
g
/T

F.
Puosi
, C. De Michele, DL,

JCP
138
, 12A532 (2013)


The thermodynamic scaling
works


for the Debye
-
Waller factor <
u
2
>,


Cage scaling
fails for
t
a

< 1

Thermodynamic scaling in
Kob
-
Andersen binary mixture

r
g
/T

F.
Puosi
, C. De Michele, DL,

JCP
138
, 12A532 (2013)

t
a

= F[ <u
2
> ]

Cage scaling
fails for
t
a

< 1


The thermodynamic scaling
works


for the Debye
-
Waller factor <
u
2
>,


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

p
reliminary results

< u
2
>
1/2


Conclusions


Cage scaling (
t
a

vs

<u
2
>

):

-
Results suggest that <u
2
> 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.

< u
2
>
1/2


Conclusions


Cage scaling (
t
a

vs

<u
2
>

):

-
Results suggest that <u
2
> 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
(
t
a

vs


G/T):


-

Intermediate
-
time shear elasticity and <u
2
> are highly correlated.


-

MD master curve
t
a

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.

< u
2
>
1/2


Conclusions


Cage scaling (
t
a

vs

<u
2
>

):

-
Results suggest that <u
2
> 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
(
t
a

vs


G/T):


-

Intermediate
-
time shear elasticity and <u
2
> are highly correlated.


-

MD master curve
t
a

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
(
t
a

vs


r
g
/T
)

-

<
u
2
>
scales with
r
g
/T
. Extensive MD simulations in progress


-

MD
master
curve
t
a

vs

r
g
/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



E
cole
Centrale Paris


F.
Puosi
,



Postdoc


Univ
. Grenoble
1


S. Bernini



PhD



Pisa


O.
Chulkin



Postdoc


Odessa


M.
Barucco



Graduate


Pisa




Credits

1/ <u
2
>

G/ T

<u
2
>

r
g

/ T

t* ~ 1
-
10
ps

Log t

Log
t
a

Log <
D
r

2
(t) >

Log <u
2
>

Log t*

Log t

Log F

s

(q

max
, t)

< u
2
>
1/2


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

F.
Puosi
, DL, JPCB (2011)

MD
simulations

Density
r



Temperature
T


Chain
length M (polymer)



Potential: p, q

10
17

s (eta’
dell’universo
)

t

a

~ 10

26

s

< u
2
>
1/2


First

“universal” scaling
:
structural
relaxation time
t
a

or viscosity
h


vs.Debye
-
Waller factor < u
2
> (rattling amplitude in the cage)

Log MSD

Log <u
2
>

Log t

Log t*

Log
t
a

Cage scaling: implications

G
s
(X)

(r, t*) =
G
s
(Y)

(r, t*
)
G
s
(X)

(r
,
t
a
)
=
G
s
(Y)

(r, ,
t
a
)

Polymer melt