Slow Dynamics from Gels to Glasses and Granular mat erials

Antonio Coniglio

Università di Napoli “Federico II”

International School of Physics "Enrico Fermi" - Cou rse CLXXVI

"Complex materials in physics and biology" Varenna (Lake Como) 29 June - 9 July 2010

Università di Napoli “Federico II”

T. Abete Univ di Napoli Federico II, ItalyA. De Candia Univ di Napoli Federico II, ItalyA. Fierro Univ di Napoli Federico II, ItalyE. Del Gado ETH SwitzerlandM. Nicodemi Warwick University UKD. D. Grebenkov Ecole Polytechnique Orsay, FranceR. Pastore Univ di Napoli Federico II,Italy

Collaborators:

1

JAMMING TRANSITIONS

• GLASSES• CHEMICAL GELS

There are many systems, which, by changing the con trol parameters, exhibit a slow dynamics followed by structural arrest (Jamm ing).

• CHEMICAL GELS• COLLOIDAL GELS • GRANULAR MATERIALS

Understanding Jamming transition is still one of th e major problem in condensed matter. For example there is still no consensus whether the glass transition is associated with a r eal thermodynamic phase transition and what are the differences and u niversalities in the jamming transitions.

Developments of the old theory of Adam and Gibbs ha s suggested the existence of dynamical heterogeneities, which p lay the role of critical fluctuations in critical phenomena .

The concept of Dynamical heterogeneities seems very promising to distinguish between competing theories and understa nding differences and universalities in the jamming trans ition.

Outline

1) Dynamical heterogeneities in glasses

2) Geometrical description of Dynamical hetrogeneit ies in gels to describe quantitatively the sol gel transition, which occur at very low volume fraction

4) Jamming transition in granular materials, which occur in the opposite limit at high volume fraction.

5) Physical picture of the Dynamical heterogeneitie and mechanism for the origin of dynamical correlations in glasses.

Glass Transition

Debenedetti and Stillinger Nature 2001

ANGELLPLOT

Glass Transition

Debenedetti and Stillinger Nature 2001

Arrhenius

(Strong)

VFT

= nT

CAexpη

=T

EAexpη

−=

0

expTT

BAη

Super Arrhenius

(Fragile)

MODE COUPLING THEORY

Gotze and Sjogren Rep Prog Phys 1992Gotze J. Phys Cond Matt 1999

From first-principles an equation for the time depe ndent densityautocorrelation function is derived, which makes a number of precise dynamical prediction. dynamical prediction.

See i.e.Van Megen and Underwood 1993 Colloidal Hard Spheres Kobe and Andersen 1994 MD Voightman, Puertas and Fuchs 2004 MD

MODE COUPLING THEORYGotze and Sjogren Rep Prog Phys 1992Gotze J. Phys Cond Matt 1999

Intermediate scattering function

High temperature TNormal liquidExponential decay

Low temperature T, deep in the supercooled region

γτ −−≈ )( CTTA

)21(

)1(

)21(

)1( 22

b

b

a

a

+Γ+Γ=

−Γ−Γ

Kob and Andersen PRL 1994

Lennard -Jones mixture MD simulationsGlass transition

Self intermediate scattering function Mean square displacement

CAGE EFFECT

A3

Diapositiva 9

A3 Antonio; 25/06/2008

Weeks, Crocker, Lewitt,Schofield,Weitz, Science 2000

Evidence of cage effect in colloidalsuspension

A typical trajectory for volume fraction 0.56

DYNAMICAL HETEROGENEITIESCicerone,Blackburn,Ediger Macromolecules (1995)

Donati, Douglas, Kob, Poole,Plimpton, Glotzer PRL(1998)

COOPERATIVELY REARRANGING REGIONS Adam and Gibbs (1965)Kirkpatrick Thirumalai Wolynes (1989 )

the decay towards equilibrium of a density fluctuat ion is due to a cooperative rearrangement of correlated re gions

Dynamical Heterogeneities in Glasses

group of particles which are dynamically correlated, whose size grows as the glass transition is approached .

Dynamical Heterogeneities in glass transition have been recently described also within Mode Coupling Theory,

Franz and Parisi (2000). Biroli and Bouchaud (2005) ,Biroli, Bouchaud, Myazaki, Reichman (2006 ),Toninelli et al 2005,Berthier et al (2007)

Lecheneau et al 2008

Dynamical heterogeneities in colloidal suspension measured using confocal microscopy

Weeks et al Nature 2000

Evidence for dynamical heterogeneities

Cicerone et al 1995

Russel and Israeloff 2000

Richert 2002

Berthier et al.2005 Science

Duri and Cipelletti 2006

Location of fastest particles

Volume fraction 0.56

Pitard et al 2007

Dalle- Ferrier et al 2007

Ballesta, Duri,Cipelletti 2008

How to quantify the dynamical heterogeneities?

DYNAMICAL SUSCEPTIBILITYFluctuation of the self Intermediate Scattering Function

[ ]224

)0()((

),(|),(|),(

1

),(

tktkNtk

eN

tk

ss

i

rtrkis

ii

Φ−Φ=

=Φ ∑ −⋅

χ

rrr

>Φ=< ),(),( tktkF ss

Self intermediate scattering functionCharacterizes the size over which particle motion is correlated

Franz,Parisi 2000 (p-spin glasses)

Lenard-Jones mixture

As the temperature T decreases the peak increases a nd shifts to longer time. The decay to zero is a consequence of the transient nature of dynamical heterogeneitie s

Lacevic, Schroder, Starr and Glotzer 2003MD simulation Lennard –Jones mixture

Kob and Andersen PRL 1994

Can we give a geometrical representation of the dynamical heterogeneities?

•Geometrical representation of the dynamical heterogeneities in chemical gels.

•Show how the arrested transition in chemical

CHEMICAL GELS

•Show how the arrested transition in chemical gels can be explained quantitatively in terms of dynamical heterogeneities.

CHEMICAL GELATION

Trimethoyl benzene

A simple illustration from Gordon and Ross-Murphy 1975

Two monomers canreact to form a dymer

The transition to the gel phase occurs when a macroscopic polymer infinite when a macroscopic polymer infinite in spatial extent is formed .At the gel transition the viscosity diverges and the shear modulus goes to zero.

Percolation model for chemical gels

The divergence of the cluster size and the formation of a percolating network isresponsible for the sol-gel transition.

(Flory, Stockmayer ,de Gennes…)

CHEMICAL GELATIONMonomers interact and form permanent bonds.

SOL GEL

CHEMICAL GELATIONAbete, De Candia, Del Gado, Fierro, AC, PRL (2007)

For alternative models Del Gado,Fierro, de Arcangelis, AC,Eur. Lett. 2003

Saika-Voivod, Zaccarelli,Sciortino, Buldyrev, Tartaglia PRE 2004

See also review by E. Zaccareli J.Phys 2008

3d OFF LATTICE PERCOLATION MODEL

Kremer and Grest 1990( application to polymer chain)

Weeks-Chandler-Andersen soft sphere potential 1971

At time t=0 any pair of particles within a distance R is bonded. At later time t>0 no more bonds are introduced

UWCA UFENE

R

PERCOLATION TRANSITION

Off lattice model 3d

mean cluster size

connectednesslength

cluster size distribution

random percolation exponents

SELF INTERMEDIATE SCATTERING FUNCTION

Off lattice model 3d

cϕϕ =

cs ttkF ϕϕτ β <−≈ )/exp(),( min

Dynamical Transition

Dynamical transition at the percolation threshold: from stretched exponential to power law behaviour.

cϕϕ <

zc

−−≈ )( ϕϕτ

cc

s ttkF ϕϕ =≈ −),( min

Lk

π2min =

DYNAMICAL SUSCEPTIBILITY IN CHEMICAL GELS

DYNAMICALSUSCEPTIBILITY

Abete, De Candia, Del Gado, Fierro, AC PRL (2007)

Off lattice model

It can be shown that only sites i and j in the same clusters contributes

∑=∞→

)(),(lim 24

0snsk

kχ (mean cluster size )

SKETCH OF THE PROOF

Abete et al. PRL (2007)

cluster) same in the and ( ji

clusters)different in and ( ji

(mean cluster size)

Abete, De Candia, Del Gado, Fierro, AC PRL (2007)MD SIMULATION CONFIRM THE RESULT

∑=∞→

)(),(lim 24

0snsk

kχ

Inset:Dynamical susceptibility (square) and mean cluster size (red)

For small k, scaling arguments give

Percolation exponents

By measuring the asymptotic Dynamical Susceptibilit y it is possible to measure percolation exponents

EXPRESS THE COMPLEX DYNAMICAL BEHAVIOUR OF CHEMICAL GELS AS SUPERPOSITION OF DYNAMICAL BEHAVIOUR OF SINGLE CLUSTERS

DYNAMICAL BEHAVIOUR OF SINGLE CLUSTERS

Self ISF for clusters of size s

65.0)( ≅≈ xss xτ

Each cluster of size s is expected to relax for large t with a simple exponential function

dimension fractal 2.5 D

/)1(

)(with

))(/exp(),( min

≅−=

≈

−≈

DDx

ss

sttkfx

s

ττ

Self ISF for clusters of size s=1,2,4,10,21,52 for Φ =0.09.Φ =0.09.Φ =0.09.Φ =0.09.

Fit with exponential function.

Inset:

SELF INTERMEDIATE SCATTERING FUNCTIONIn terms of self ISF of single clusters

]),()([),( minmin ∑=s

ss tkfssntkF

Fierro, Abete, A.C. Jour Chem Phys 2009

xs sssttkf ≈−≈ )())(/exp(),( min ττ

)/exp()( ∗− −≈ ssssn τ

SELF INTERMEDIATE SCATTERING FUNCTIONIn terms of self ISF of single clusters

[ ] [ ])()(exp1

)()(exp1

),( ortrkiN

ortrkiN

tk iis C Cii

iiself

s s

rrrrrr−=−=Φ ∑∑∑∑

∈

[ ]),( tkF selfself Φ=

Fierro, Abete, A.C. Jour Chem Phys 2009

]),()([),( minmin ∑=s

ss tkfssntkF

xs sssttkf ≈−≈ )())(/exp(),( min ττ

)/exp()( ∗− −≈ ssssn τ

Self Intermediate Scattering Function predictions compared with numerical simulations

cz

self

cc

self

zttkF

tttkF

ϕϕ

ϕϕτ β

=Γ≈

<−≈−

−

)(),(

)/exp(),(

min

min

For large t

Abete, Fierro, AC, J. Chem, Phys (2009)

4.1)-( c ≅=≈= −∗ νϕϕτ xDfs fx

60.0)1/(1

31.0/)2(

36.0)2/3(

≅+=

≅−=

≅−=

x

xz

c

per

perper

βτ

τβ

Φ = .06, .07,.08, .085, .09Φ = .06, .07,.08, .085, .09Φ = .06, .07,.08, .085, .09Φ = .06, .07,.08, .085, .09

cc

self tAtPtkF ϕϕτ β >−+≈ −∞ )/exp(),( min

60.0)1/(1

36.0)2/3(

≅+=

≅−=

x

c perper

βτβ

ISF above the percolation threshold

Φ>ΦΦ>ΦΦ>ΦΦ>Φc

60.0)1/(1 ≅+= xβ

cc

self tAtPtkF ϕϕτ β >−+≈ −∞ )/exp(),( min

60.0)1/(1

36.0)2/3(

≅+=

≅−=

x

c perper

βτβ

ISF above the percolation threshold

60.0)1/(1 ≅+= xβ

Dynamical susceptibility: Fluctuation of the self ISF expressed In terms of simple cluster dynamics

Abete, Fierro, AC, J. Chem, Phys (2009)

particles s ofcluster a of ISF ))(/exp(),(

)),(1()(),(

min

min22

min4

sttkf

tkfsnstk

s

ss

τ

χ

−≈

−=∑

cc tAtStk ϕϕτχ β <−−≈ ))/2exp(1(),( min4

≈= −∗ ϕϕτ

Dynamical susceptibilityPrediction compared with MD simulation

S mean cluster size

Large t

60.0)1/(1

24.0)2/5(

≅+=

≅−=

x

c perper

βτβ

4.1

)-( c

≅=≈= −∗

νϕϕτ

xDf

s fx

Φ=.06, .07, .08, .085Φ=.06, .07, .08, .085Φ=.06, .07, .08, .085Φ=.06, .07, .08, .085

Diffusivity properties of Chemical Gel – Off lattice model

Abete et al PRE 2008

Mean square displacement

φ=.05

φ=.12

Apparent simplicity

Chemical Gel – Off lattice model

Self Part of the Van Hove DistributionAbete et al 2008

−−= ∑

=

)()(|(1

),(1

ortrrN

trG i

N

iiself

rrδ

For a system of particles diffusing with the same diffusion coefficient D

functiongaussian are lines the

09. and t timeincreasingfor

r offunction as ),(

=ϕtrGself

)4/exp(4

1),( 2

2/3

DtrDt

trGself −

=π

Chemical Gel – Off lattice model

Non gaussianity parameter

1)(5

322

4

2 −><><=

r

rα

φ = .05 . . . 0.12

Diffusion coefficient of single clusters

66.01 )()( −− ≈∝ sssD τ

Origin of non Gaussianity

Distribution of diffusion coefficient

Each cluster diffuses with a diffusion coefficient D(s)

66.01 )()( −− ≈∝ sssD τ

Self part of the Van Hove distribution for chemical gels in terms of distribution of single clusters

),( trG

∑ −−=i

iiself rtrrN

trG |)0()(|(1

),(vvδ

),( trGself

Analytical expression of

1/2 >tr

Scaling collapse

A= 1-t +3x/2 =-0.3

),( trGself

Inset shows an effective behaviour Indistinguishible from exponential

Numerical data compared with the prediction

Origin of Breakdown of Stokes-Einstein relation

x

s

s

ssDs

sssn

sDssnD

≈∝

=

=

−

∑

∑

)()(

)()(

)()(

1τ

ττ

ττττ

xssDs ≈∝ − )()( 1τ

Differently from D only the largest cluster contributes to ττττ

D

22.1)( −Φ−Φc

minkk =

Relaxation time as funcion of volume fraction for increasing values of k

7=k

ANALOGY WITH SPIN GLASSESGoldbart et al J.Phys Cond Matt (2000)Fierro et al J.Phys Cond Matt (2009)

(Mean cluster size)

To be compared with

Fierro et al J.Phys (2009)

Contribution to the static structure factor S(k) (black triangles) of connected (red triangles) and disconnected (squares) particles

COMPARISON BETWEEN

GELS AND GLASSES

SELF INTERMEDIATE SCATTERING FUNCTIONKob and Andersen PRL 1994

Lenard-Jones mixtureCHEMICAL GEL OFF LATTICE MODEL

Abete et al 2007

DYNAMICAL SCEPTIBILITY Lacevic, Schroder, Starr and Glotzer 2003

Non gaussianity parameter

Lennard Jones mixtureChemical gelation off lattice model

Abete, De Candia Fierro, A.C. 2008 Kob et al PRL 1997

The gel transition is similar to the spin glass tra nsition and is characterized by

1) dynamical transition

2) static thermodynamic transition

3) The complex dynamics can be quantitatively explaine d as superposition of contributes coming from the heterogeneities (clusters)

In hard sphere glasses bonds play no role .

Very intriguing is the case of gels formed in attra ctive colloids which are in between these two extremes.

Colloid consists of solid particles in a liquid.

COLLOIDS

From Cheng, Chaikin 2001Experimental study of Colloidal Hard SpheresPusey Van Megen Nature 1986Theory: Parisi, Zamponi 2006

COLLOIDS

Adding polymers to colloidal suspensions induce an effective attraction U (DEPLETION EFECT)

Colloid consists of solid particles of the order of 10 nanometers in a liquid.

Hard sphere glass

Attractive glass

High

Low

U

Tk B

U

Tk B

However hard sphere + attraction leads to gel forma tion due to an interrupted phase separation.

Lu,Zaccarelli, Ciulla, Schofield,Sciortino,Weitz Na ture 2008

Adding electric charges induces a long range screened repulsion which supresses phase separation

Groenewold and Kegel 2003Sciortino Mossa Zaccarelli,Tartaglia2004Wu,Liu,Chen W,Chen S. 2005Coniglio, de Arcangelis, Del GadoFierro, Sator 2004Imperio, Reatto 2004

See review by E. Zaccareli J.Phys 2008 for diffe rent routes to colloidal gelation

A model for interacting colloids

DLVO model Sciortino Tartaglia and Zaccarelli J. Chem Phys 2005

Campbell, Anderson, van Duijneveldt, Bartlett PRL 2005(experiments)

De Candia, Fierro, De Candia, Fierro, Del Gado, AC PRE 2006

COLLOIDAL GELATION

MD results for DLVO model

Self intermediate scattering function

Fierro, De Candia, Del Gado,Coniglio J. Stat Mech 2008

0U

Bonds have finite lifetime

T=0.15

= 0.01, …..0.13 minkk =

Phase Diagram DLVO model

DLVO model

Relaxation time comparedwith bond lifetimeFor T= 0.15 (bleu ) and 0.25 (red)

COLLOIDAL GELATION

RELAXATION TIME

Bond lifetime

Fierro, De Candia, Del Gado,A.C. J. Stat Mech 2008

T=0.15

For T=0.15 and low volume fraction,the relaxation time follows a power lawlike in irreversible gels,as if the bonds were permanent .

Relaxation time

T=0.25

Dynamical heterogeneities in colloidal gelation

DLVO model

13.0=ϕ

ϕDynamical susceptibility for T = 0.15 and different volume fraction =0.01, … 0.13

The plateau is reached in a time Interval of the order of the relaxation time and starts to decay after a time Interval of the order of the bond life time.

01.0=ϕτ

bτ

Dynamical susceptibility (red) compared with the time dependent cluster size (bleu) of clusters made of bonds, which have survived up to time t

12.0=ϕ

Geometrical representation of dynamical heterogeneities in colloidal gels :

Clusters made of bonds survived in the time interval (0, t)

Note the crossover from the cluster dominated regim e to a crowding dominated regime

∑= ),()( 2 tsnstSm

05.0=ϕ

For experimental evidence of such crossover in coll oids seeMallamace, Chen et al, (2002,2006)

•The complex dynamics in chemical gels has been enti rely quantified in terms of dynamical heterogeneities. These are clus ters made of monomers connected by permanent bonds.

•In colloidal gels the bonds have a finite lifetime, the clusters are made of persistent bonds. They are time dependent and deca y.

•As the bond lifetime decreases there is a crossover towards glass like behavior.

Conclusions

behavior.

Jamming Transition in Granular Materials

•Granular materials are systems made of a large numb er of particles whose size typically is larger than a micron.

•Example of granular materials are powder, sand, cor nfleaks, cement…

•They are very important in farmaceutic and food ind ustry. In fact they are among the products most manipulate d In fact they are among the products most manipulate d in industry. Therefore it is extremely important to study their properties

•Due to their large mass they can be considered part icles at zero temperature T=0

•At low density they behave like a fluid that you ca n poure in a box. In this case if you apply a shear force the system flows.

•Above a crtical density the system behave like a d isordered solid. If you apply a shear stress the system does not flo w.

•The critical density corresponds to the jamming tr ansition

JAMMING PHASE DIAGRAM

Liu, Nagel Nature 2000

O’Hern, Silbert,Liu,Nagel PRE 2003O’Hern, Silbert,Liu,Nagel PRE 2003O’Hern, Silbert,Liu,Nagel PRE 2003O’Hern, Silbert,Liu,Nagel PRE 2003

Temperature

1/Density

Shear stress

A

Fx=σ

Modelling Granular Material

x16 d

MD simulations

Control parameters:

Shear Stress : σσσσVolume Fraction: Φ

Pica Ciamarra A.C. 2009,Pica Ciamarra A.C. 2009,Pica Ciamarra A.C. 2009,Pica Ciamarra A.C. 2009,Pica Ciamarra, Pastore, Nicodemi A.C 2009Pica Ciamarra, Pastore, Nicodemi A.C 2009Pica Ciamarra, Pastore, Nicodemi A.C 2009Pica Ciamarra, Pastore, Nicodemi A.C 2009

See also Heussinger and Barrat PRLSee also Heussinger and Barrat PRLSee also Heussinger and Barrat PRLSee also Heussinger and Barrat PRL2009200920092009((((2222D at fixed shear strain)D at fixed shear strain)D at fixed shear strain)D at fixed shear strain)

Grebenkov, Pica Ciamarra, Nicodemi, A.C,, PRL 100, 128001, 2008.

δγδ &−= kFnδ

Linear spring-dashpot model

8d

z

16 d

yF

Viscosity and Elastic modulus at small shear stress σσσσ

small σσσσ

MD simulations

Pica Ciamarra, A.C. 2009

G

LlG

l

z

z

δδσ

ση

=

= /vshear Viscosity

Elastic modulus45.0

5.1

)(

)(

φφφφη

−≈

−≈ −

J

J

G

645.0=Jφ

Structural signature at the jamming transitionResults from MD simulations for small values of applied stress σ,

number of contacts

P pressure

Different colours correspond to different values of σSharp discontinuity only at σ = 0

)(

)(

at

5.0

J2.0

J

Jc

P

ZZ

Z

φφφφ

φσφ

−≈−≈−

≈∂∂ −

Schematic behaviour at zero shear Σ=0Σ=0Σ=0Σ=0

Z – Ziso∼(φ - φJ )β

P=P0 (φ - φJ)ψ

G=G0 (φ - φJ)γ

η∼(φJ - φ)-x

viscosity

P. Olsson, S. Teitl, PRL 99, 178001 (2007) C. O’Her n et al., PRE 68, 011306 (2003);

Number of contacts

Pressure

Elastic modulus

Z

PG

G=G0 (φ - φJ) Elastic modulus

Nd = NZ/2 → Z= Ziso = 2d = 6

Mechanical equilibrium of N frictionless grains

Number of constraints: N dNumber of equations: NZ/2

P

Static transition at jφ

Dynamic Transition at the J point

Dynamic SusceptibilityIntermediate Scattering Function

Dynamic signature of the jamming transitionResults from MD simulations for small values of applied stress σ=0.002,

If particles i and j touch(do not touch)

>−−<= ∑ ))]0()((exp[1

),(j jj rtrik

NtkF ∑ ><=

jiijij ctc

NtZ

,

)0()(1

)(

)0(,1)( =tcij5.1)( −−≈≈ φφητ J

K = (0, 2π/d, 0)

An infinite cluster of localized particles character ized by a coordination number appears disco ntinuusly at the jamming transition.

The pressure and the elastic modulus are continuus

The jamming transition like in chemical gels

6≥Z

The jamming transition like in chemical gels is chracterized by a static and dynamic transition.

Note the difference beyween chemical gels where th e order parameter appears continuusly ,

O’Hern, Silbert,Liu,Nagel 2003 Pica Ciamarra, Nicodemi, A.C . 2010

Speculative Jamming Phase Diagram

SELF INTERMEDIATE SCATTERING FUNCTIONKob and Andersen PRL 1994

Lenard-Jones mixtureCHEMICAL GEL OFF LATTICE MODEL

Abete et al 2007

DYNAMICAL SCEPTIBILITY Lacevic, Schroder, Starr and Glotzer 2003

Dynamical susceptibility (red) compared with the time dependent cluster size (bleu) of clusters made of bonds, which have survived up to time t

12.0=ϕ

Geometrical representation of dynamical heterogeneities in colloidal gels :

Clusters made of bonds survived in the time interval (0, t)

Note the crossover from the cluster dominated regim e to a crowding dominated regime

∑= ),()( 2 tsnstSm

05.0=ϕ

For experimental evidence of such crossover in coll oids seeMallamace, Chen et al, (2002,2006)

.

In chemical gelation there is a static transition toghether with a dynamic transition. Above the gelation threshold a n infinite cluster of monomers with permanent bonds is present.

In granular materials there is also a static transi tion at the jamming transitionIn the jammed phase there is an iinfinite cluster o f particles with a number of

What is the physical origin of dynamical correlation in glasses?

permanent contacts Z>Zc (Zc=2d for frictionless pa rticles)

We ask now the question of what is the physical pic ture in the glass transition.In particular what is the origin of the of dynamic al correlations which give rise tothe dynamical heterogeneities and what is the dyna mical length.

I will present results on a simple dynamical model the Kob and Andersen modelalthough the results can be extended to other conti nuum non trivial model like soft spheres.

Kob and Andersen model

Lattice Gas model of non interacting particles. Particles diffuse

under the following constraint: A particle can move if

1) it is surrounded by a number of particles less than a given number z2) if the landing site is surrounded by a number of particles

less than z+1

KA PREKA PRE19931993))

In 3d a reasonable choice is z=4

The detailed balance is satisfied. Consequently the static properties are the same as the non interacting lattice gas model.

A very simple model, with no static phase transition, which captures the physics of the cage in the glass phenomenology.

3d Kob and Andersen model

The diffusion coefficient D seems

to vanish at ρρρρc as a power-law.

γρρ )( −∝ cD

1.3=γ

88.0=ρ Kob and Andersen Kob and Andersen (PRE,(PRE,19931993))

In reality at ρ_c the diffusion coefficient exhibits a crossover towards an exponential of exponential, and goes to zero only at ρ =1.

)]]-(1exp[exp[c/ 1 ρ=−D

Toninelli Biroli Fisher PRL 2004

88.0=cρ

Dynamical Susceptibility χχχχdyn(t)

Overlap correlator:

])()([)(22 tqtqNtdyn −=χ

Power-law fits: Behavior of dyn(t) for different values of ⟩⟩⟩⟩

Glassy dynamics has an heterogeneous character involving regions of correlated motion whose extension and surviving time depends on control parametres. The dynamical susceptibility reveals and quantifies this character of heterogeneity.

4;b ;2

)()(

)()(max

max

==−=

−=−

−

a

t bc

acdyn

ρρρ

ρρρχIncreasing

ρ

Franz, Mullet and Parisi Franz, Mullet and Parisi (PRE, (PRE, 20022002))

Intermediate Scattering Function

Kob and Andersen model in 3d (PRE,(PRE,19931993))

>−−<= ∑ ))]0()((exp[1

),(j jjs rtrik

NtkF

It is often convenient to consider the sum over all values of k of the ISF:

)]0()([1

),()( ii

ik

ss rtrN

tktqrr −=Φ= ∑∑ δ

The sum is over all particles and )(tri

ris the position of particle i at time t,

SELF OVERLAP and DYNAMICAL SUSCEPTIBILITY

))]0()((exp[1

),( ∑ −−=Φj jjs rtrik

Ntk

0 fromdifferent or 0 x whether depending 0,1)( ==xδ

])()([)( 224 ><−><= tqtqNt ssχ

The dynamical susceptibility is given by

Kob and Andersen model

Self Overlap

100

102

104

106

time

0

0.2

0.4

0.6

0.8

1

<q s>

ρ = 0.75ρ = 0.80ρ = 0.84ρ = 0.85ρ = 0.86

Dynamical suseptibility: fluctuation of the overlap

Pastore, Pica Ciamarra, A.C 2010

time

100

101

102

103

104

105

106

107

time

0

5

10

15

20

25

30

χ4

ρ = 0.75ρ = 0.80ρ = 0.84ρ = 0.85ρ = 0.86

It is easy to show that the self overlap coincides with the fraction of particles which have been localized from time 0 to time t (a part from terms of the order of O(1/ Volume))

),0(1

)(

)(1

)(

==

><>=<

∑N

tnt

ttq fs

ρρ

ρρ

Physical picture of the glass transition in terms o f persistent localized particles

])()([)(

0. is otherwise , t][0, interval time theduring

i site from movednever has particle a if 1),0(

),0(1

)(

224 ><−><=

=

== ∑

ttV

t

tn

V

Ntn

Vt

ff

i

iif

ρρρ

χ

ρρ

Kob Andersen model

Self overlap and its fluctuation (red ) compared with the fraction of frozen particles and its fluctuation (black)

101

Pastore, Pica Ciamarra A.C. 2010

100

102

104

106 0

5

χ4

100

102

104

106

time

0

0.5q

fq

s

100

102

104

1060

0.2

0.4

0.6

0.8

1

ρ eff / ρ

ρ = 0.75ρ = 0.80ρ = 0.84ρ = 0.85ρ = 0.86

Kob and Andersen model

100

101

102

103

104

105

106

1070

5

10

15

20

25

30

χ4

ρ = 0.75ρ = 0.80ρ = 0.84ρ = 0.85ρ = 0.86

Glassy dynamics in terms of persistent ocalized particles

ρρ >< )(tf

100

102

104

106

time

0

Fraction of localised particles in the time interval [0,t]

Dynamical suseptibility: fluctuation of density of persistent localized particles

100

101

102

103

104

105

106

107

time

0

•On a time scale of the relaxation time the particles are localized (jammed) and the system behaves like a disordered solid able to bear stress.

•The results from the jamming transition suggest that the localized particles should form a percolating network

0.4

0.6

0.8 ρf

P

ρ = 0.85

Density of localized particles ρρρρf(t) in (0,t)

Density of lòcalized particles in the infinite cluster P(t) in (0,t)

Kob and Andersen model

100

101

102

103

104

105

106

time

0

0.2

0.4 ρ = 0.85

All localized particles are part of the infinite cluster except when the infinite Cluster is about to disappear.

DYNAMICAL CORRELATION

It can be shown that

t][0, interval timein the rat

particles localized persistent ofdensity ),(

),(),0(),(),0(),(

),()(4

r

r

rr

rr

tr

trttrttrg

rdtrgt

f

ffff

ρρρρρ

χ

>><<−>=<

= ∫

t][0, interval timein the rat

Particles localized in the time interval [0,t] are correlated

A

A

A

A AA

AA

A AA

AA

AA

AA

A AA

AA

AA

AA

A

10-1

100

g/g(

r=0)

ρ = 0.75ρ = 0.80ρ = 0.84ρ = 0.85A A

ρ = 0.86

t = t*

AA

A

AA

A

A

A

A

A

0 5 10distance

10-3

10-2

g/g(

r=0)

A

A

AA A

AA

A A AA A A

10-2

10-1

100

g

t1= 10

2

t2= 5.3 10

3

t3= t

*= 2.6 10

5

t4= 2.8 10

6A A

t5= 4.6 10

6

ρ = 0.85

t3

A A AA

AA A

A AA A

AA A A

AA

AA

AA

AA

A

AA

A

0 5 10distance

10-4

10-3

10

t1

t2

t4

t5

1.5

2

2.5

ξ

10

χ4

ρ = 0.85

Correlation length and dynamical susceptibility

0

0.5

1

ξ

102

103

104

105

106

time

0

54

Surprisingly the correlation length does not decrease following 4χ

1.5

2

2.5

ξ5

10

χ4

ρ = 0.85

g(r=0)

0

0.5

1

102

103

104

105

106

time

0

5ξ

χ4

The decay of the dynamical Susceptibilty is not due to the decrease of ξξξξ but to the decay of the amplitude of g(r,t)

0.1

0.15

0.2

0.25g(

r=0)

ρ=0.85

102

103

104

105

106

time

0

0.05

0.1

g(r=

0)

ρ=0.85

)1()0( ><−>=<= ffrg ρρ

In the glass transition the defects, by diffusing, destroy the percolating cluster of frozen particles. Frozen particles left u ntouched are correlated as consequence of this process.

Mechanism for dynamical correlation

A particle frozen from time 0 to time t can become mobile if it is hit by a diffusing defect (which at low density is a hole ). Each time a partcilebecome mobile contributes to the decay of the autoc orrelation function

Each defect diffuses within a region of linear dim ension R, of the order of the distance betwen the defects.

Once each defect has explored such a distance R, it invades the area explored by one of the nearest defect and starts to decimate more particles.

Less particles contributeto the pair correlation fu nction g(r,t) while the correlation length does not decrease.

6

8

10le

nght

ξ∗

double-exp fit0.43 ξ

mob

0.7 0.75 0.8 0.85 0.9 0.95ρ

0

2

4leng

ht

This process, which corresponds to a reversed perco lation transition from a gel to a sol, was described by a model named “pacman percolation”. In this model a defect (the enzime) diffuses and cut s the bond until the percolating cluster vanishes.

Mechanism for dynamical correlation

PACMAN PERCOLATION

Interestigly it was shown that the particles left u ntouched where correlatedand the maximum of the correlation length is given by the distance R between the defects .

Pacman percolation Random percolation