Effective Temperatures in DrivenSystems near Jamming

Andrea J. LiuDepartment of Physics & Astronomy

University of Pennsylvania

Tom Haxton Physics, UPennYair Shokef Physics, UPennTal Danino Chemistry, UCLAIan Ono Chemistry, UCLACorey S. O’Hern Mechanical Engineering, Yale Univ.Douglas Durian Physics, UPenn.Stephen Langer NISTSidney R. Nagel James Franck Inst., U Chicago

•We understand a lot about the collective properties ofmany-particle systems in thermal equilibrium

•But flowing glassy systems are far from equilibrium

Modern Challenge

{systems near thermal equilibriumstatistical mechanics

{ }}

systems far from equilibrium??

macroscopicmicroscopic

¿ Can fluctuations be described by effective temperature?¿ If so, so what?

Driven Systems Near Jamming

95% gas5% liquid

Unsheared foam(D. J. Durian)

energy barrier for bubblerearrangements

!

"106kBT

Sheared foam(M. Dennin)

Shearing drives systemover energy barriers

Testing Effective Temperature

•We and others conducted numericalsimulations of simple models understeady-state shear

•Calculate Teff in several independent

ways

•In an equilibrium system, all calculations

must yield same result

•Are they the same in a driven system?

D. J. Durian, PRL 75, 4780 (1995).

Model: Equations of Motion

!

md2r r

i

dt2

=r F

i

e+"

r F

i

v #$%r v

i

Fluid

!&

+ •

•

•

!

vx

= ˙ " Limage

image

!

vx

= 0

!

vx

= " ˙ # L

1. Sheared; massless; athermal 2. Sheared; massive; athermal3. Sheared; massive; thermal4. Equilibrium

!

˙ " = 0;m # 0;$ = 0;T > Tg!

˙ " > 0;m = 0;# = 0

!

"r v i =

r v i # ˙ $ yi

ˆ x

Elastic forces

Viscousforces thermostat

!

˙ " > 0;m # 0;$ = 0

!

˙ " > 0;m # 0;$ = 0;T > 0

Constant Shear-rate b.c.’s

Types of simulations:

L

•Thermal diffusion

Temperatures from Linear Response

!

"r(t)( )2

= 6Dt D. A. Weitz

0.5µm

!

D =T

"

temperature

drag

!

F /"

!

" = 6#$a

!

Teff " D

v /F ,

#V( )2

V$T , . . .

%

& '

( '

)

* '

+ '

size of fluctuations (correlation)

ease of creating fluctuation (response)

•Response to Force F

Particles move atspeed where is drag

Other Linear Response Relations

Adiabatic compressibility Pressure Fluctuations

Viscosity Shear Stress Fluctuations

Heat Capacity Energy Fluctuations

! =A

Tdt " xy(t)" xy (0)

0

#

$c

! U

!T=1

T2U " U( )

2d U

!U( )2" =

dT

T2"

!S"1 =

A

Tp " p( )

2

Langer and Liu, Europhys. Lett.49, 68 (2000).

•4 indep. def’ns of Teff yield the same result•Effective temperature appears to be a useful concept!

•AND there’s more….

Results

TpTxyTU

I. K. Ono, C. S. O’Hern, D. J.Durian, S. A. Langer, A. J. Liu andS. R. Nagel, PRL 89, 095703 (2002).Teff

0

0.002

0.004

0.006

0.008

Bidisperse

T

0

0.005

0.01

0.0001 0.001 0.01 0.1 1

Polydisperse

T

!.

0

0.002

0.004

0.006

0.008

Bidisperse

T

0

0.005

0.01

0.0001 0.001 0.01 0.1 1

Polydisperse

T

!.

Berthier & Barrat, PRE, 012503(2001); J Chem Phys, 116, 6228(2002); PRL 89, 095702 (2002).

•Monte-Carlo results for 1/T=dS/dU:

•Shear makes system nearly ergodic! BUT…….

Textbook Definition of Temperature

-100

-80

-60

-40

-20

0

N=163664100144

log10(!)

0.001

0.01

0.1

1

0.001 0.01 0.1 1

U/N

TS T

UTU from shearing runs

I. K. Ono, C. S. O’Hern, D. J.Durian, S. A. Langer, A. J. Liuand S. R. Nagel, Phys. Rev.Lett. 89, 095703 (2002).

H. Makse, J. Kurchan,Nature 415, 614 (2002).

L. F. Cugliandolo, J. Kurchan, L. Peliti, PRE 55, 3898 (1997).L. Berthier, J.-L. Barrat, J. Kurchan, PRE 61, 5464 (2000).

•At short times, system sees Tbath

•At long times, system sees Teff

•What happens if harmonic oscillator is inserted into system?

Two Time-Scale/Two Temperatures

!

kBTspr = kspr < y2

>

T. Danino0

0.001

0.002

0.003

0.004

0.005

0.001 0.01 0.1 1 10 100kspr/k

Tspr

Teff

Tbath

!

C(t) = "r k ,t( )" #

r k ,0( )

!

"r k ,t( ) = e

ir k #

r r j (t )

j=1

N

$

!

R(t) ="

r k ,t( ) # "

r k ,0( )

h"

!

R(t) =[1"C(t)]

T

e.g. FD relation for density

!

R(t) =C(0)

T1"

C(t)

C(0)

#

$ %

&

' (

!

R(t) =R(t)

C(0)

!

C(t) =C(t)

C(0)

hρ= perturbation

Time-Dependent Linear Response

Fluctuation-DissipationTheorem

correlation

response

Intercept vs. Long-Time Slope

•Previous definitionsbased on static linearresponse

R-intercept•Definitions based ontime-dependentlinear response

slope

!

R(t) =[1"C(t)]

T

!

R(t =") =1/T

slope

intercept

System in thermal equilibrium

Sheared Glassy Systems

The response-correlation curve is not a straight line!

•short-time slope: TS is bath temperature•long-time slope: TL depends on shear rate

Because curve is not straight, TL TI

J. –L. Barrat and L. Berthier,Phys. Rev. E 63, 012503(2000).

!

"

At short times,system fluctuateswithin local minimum

At long times,system canexplore otherminima

Cugliandolo and Kurchan,PRL 71, 173 (1993)

Density fluctuations: Def. 4c. f. Berthier & Barrat, PRL 89, 095702 (2002)

•TL from density fluctuations matches Defs. 1-3 based onstatic linear response!

m>0;athermal

!

˙ " = 0.01

!

"r k ,t( ) = e

ir k #

r r j (t )

j=1

N

$

0 0.2 0.4 0.6 0.8 1

CP(t)

0

100

200

300

400

0 0.2 0.4 0.6 0.8 1

C!(k)(t)

0

100

200

300

400

500

600

(a)

�1/TS

�1/TL

(b)

�1/TS

1/T"

-

-k=4.5

k=9

k=11

•Intercept, not slope, corresponds to consistent Teff

m>0;thermal

!

˙ " = 0.01

!

P = Pj

j=1

N

"

k=0 quantity

Pressure fluctuations

0 0.2 0.4 0.6 0.8 1

CP(t)

0

100

200

300

400

0 0.2 0.4 0.6 0.8 1

C!(k)(t)

0

100

200

300

400

500

600

(a)

�1/TS

�1/TL

(b)

�1/TS

1/T"

--1/Ts

This result is robust. We have varied• System (athermal with dissipation, thermal with no dissipation)

• Potential• Density• Shear rate• Bath temperature

• Static and dynamic linear response give consistent Teff

• Cugliandolo-Kurchan picture is conceptually incomplete

TI(p)=TL(ρk)

C. S. O’Hern, A. J. Liu, S. R. Nagel, PRL93, 165702 (2004).

•For most observables, can choose consistent set

But how to know which definition to use???k=0 quantities: TI

k>0 or local quantities: TL

Others don’t agreepzz, (2pxx-pyy-pzz)/3, config. temps, force temp. (Shokef)

The Complications

Quantity TI TL

p(k=0)σ(k=0)p(ky)ρ(ky)

!

X

!

X

!

X

!

X

•Altogether, 8 indep. def’ns of Teff yield same value– Tp, Tσ, Tpk, TU, Tρ, TE, Tspr,TK TS is consistent

•Effective temp. appears to be reasonable concept

Summary

0

0.002

0.004

0.006

0.008

Bidisperse

T

0

0.005

0.01

0.0001 0.001 0.01 0.1 1

Polydisperse

T

!.

0

0.002

0.004

0.006

0.008

Bidisperse

T

0

0.005

0.01

0.0001 0.001 0.01 0.1 1

Polydisperse

T

!.

L. Berthier & J.-L. Barrat, PRE012503 (2001); J Chem Phys 116,6228 (2002); PRL 89, 095702(2002).

I. K. Ono, C. S. O’Hern, D. J.Durian, S. A. Langer, A. J. Liu andS. R. Nagel, PRL 89, 095703 (2002).

C. S. O’Hern, A. J. Liu, S. R. Nagel,PRL 93, 165702 (2004).

Why should we care?

•If I tell you the temperature of a material, youdon’t get very excited.

•What does Teff tell us about the system (otherthan the relation between correlation andresponse)?

•Does Teff tell us anything about the dynamics ofthe system near jamming?

A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998).

•But this is a weird phase diagram– Shear stress is nonequilibrium axis– Can we replace shear stress with Teff?

Jamming Phase Diagram

unjammed

Temperature

Shear stress

1/Density

jammedTeff(shear stress)

Glass Transition

1.0

log τ(s)

Tg/T

C. A. Angell,J. Non-Cryst.Solids 131-133, 13(1991).

Super-Arrhenius dependence of relaxation time

Viscosity vs. TT. Haxton

Super-Arrhenius dependence of viscosity on T

equil

1/φ

T

σ

!

" ="0eA /(T#T0 )

!

T0" 0.0013

•System has apparent yield stress for T<T0

•Apparent yield stress increases with decreasing T•Power-law rheology for (SGR)

Steady-state shear rheology

Dashed: T>T0

Solid: T<T0

!

" xy

!

˙ "

Solid: T<T0

Dashed: T>T0

!

" xy

!

"

!

T " T0

equil

1/φ

T

σ

Behavior of Teff

Teff

!

˙ " !

"

1/Teff

Solid: T<T0

Dashed: T>T0

•For T>T0,Teff -> T as -> 0•For T< T0,Teff -> Teff,g as ->0•η vs. Teff is super-Arrhenius•Teff,g< T0 !

˙ "

!

˙ "

•Let Vr = typical size of rearrangement events•Suppose that between rearrangements, local strain canbuild up to due to to activated crossing ofenergy barriers ΔE with Teff.•But•This yields .

What do shear-induced fluctuations do?Data collapse for T<<T0or large σxy

1/Teff

!

" xy

Limiting behavior:

!

" xy #" 0 exp($%E /Teff )

!

1/Teff " #1

$Elog % xy /% 0[ ]

!

"max # exp $E /Teff( )

!

" xy#maxVr $ %E

!

"E #$ xy%maxVr

Summary

•Steady shear appears to have two effects– It lowers typical energy barriers

– But it also gives rise to fluctuations, which allowactivated crossing of the barriers

•The typical energy barrier should depend on shear rate(Ashwin, Brumer, Reichman, Sastry, J Phys Chem B 108, 19703 (2004).)

•Still ahead: analysis of inherent structures

Second law of thermodynamics

•Do 2 systems in “thermal” contact equilibrate to same Teff?•Recall earlier results:

– Small and large particles in binary sheared glass havesame Teff (Berthier & Barrat, J Chem Phys, 116, 6228 (2002))

– Massive particle has same Teff as small particles(Berthier & Barrat, PRL 89, 095702 (2002).)

– Harmonic oscillator has same Teff as system•Requires matching of time scales

– Massive particle– Low-frequency oscillator

•What about two large systems? How can we place them incontact?

Preliminary studies of 2nd law

•Separate by flexible chainwith kchain

•This allows– No particle exchange– Propagation of

rearrangement events– Equilibration of shear

stress if averagehorizontal position is notfixed

– Equilibration of pressureif average verticalposition is not fixed

!&

!&"

T. Danino

Preliminary results

•Final effective temps of both systems are equal iffinal pressures are equal.•Does Teff depend only on pressure? (N. Xu & C. S. O’Hern)

X

XXXXX

Teff

equilibrates?

Shear stressallowed to

equilibrate?

Pressureallowed to

equilibrate?

•If Teff were just a function of p, then all data fordifferent T, would collapse on single curve•No, Teff does not just depend on pressure.•So why does equilibration of p allow equilibration of Teff???

Does Teff does just depend on p?

Teff

p

!

˙ "

T. Haxton

Conclusions

•Teff seems to be reasonable for sheared glassy systems•Experiments need to measure as many Teff as possible(Durian)

• Einstein relation• Harmonic oscillator (optical or magnetic trap)

•Teff seems to control barrier hopping w/ Boltzmann prob.

•When is Teff useful? (2 time-scale/2 temp?)

•Second law?• How is “effective heat” exchanged?• Is configurational entropy conjugate to Teff even when

particles have inertia?

Song, et al. (PNAS 05)

AcknowledgementsTom Haxton Physics, UPennYair Shokef Physics, UPennTal Danino Chemistry, UCLAIan Ono Chemistry, UCLACorey S. O’Hern Mechanical Engineering, Yale Univ.Douglas Durian Physics, UPenn.Stephen Langer NISTSidney R. Nagel James Franck Inst., U Chicago

Supported by NSF-DMR-0087349, 0605044