Experimental tests of the Fluctuation-Dissipation-Relation in aging glassy

systems

collaborators:

Hassan OukrisPhil Crider

Matt Majewski

Northeastern UniversityBoston

Outline

• Nonequilibrium Fluctuation-Dissipation-Relation (FDR) Concept, Theory,

Simulations

• Experiments thus far: a mixed bag

• New results on a polymer glass.

– Try to “catch it in the act” of falling out of equilibrium

• Can we measure local correlation and response functions?

– Test local FDR violations

– Space-time correlation functions and dynamical heterogeneity

Log ()

”()

Debye

glassy

Die

lect

r ic

susc

epti

bil i

tySignatures of glassy systems: Slow- nonexponential relaxation.

Rough energy landscape?

exp[-(t/ )]

Broadened response

Diverging relaxationtimes below Tg

(fragile glasses)

Aging after T-quench

Cooperative dynamics –jamming

Fluctuation-Dissipation Relations (FDR)

Stokes-Einstein Relation D= kBT /60r

Nyquist RelationSV = 4kBTR

Violations expected in systems far from equilibrium

Brownian motion: Diffusion constant scales inversely with viscosity (1906)

Voltage noise scales with resistance (1928)

Aging glass: ideal system to study non-equilibrium FDR Cugliandolo and Kurchan, PRL 1993, PRE 1997, …

Configuration coordinate

•Universality in the violations?•Model dependent?•Effective temperature useful?

Teff =SV /4kBR

Ener

gy

Time-dependent FDR violations and effective temperature

twait tobs

tobs

For tobs<< tw looks like equilibrium

FDR holds Teff = T

kB T

t= tw +tobs

C(t,tw)=<O(tw )O(t)> noise

(t,tw) =O(t)/h(tw) susceptibility

(t,tw) = [1/kB T][C(tw,tw )-C(t,tw)]

R(t,t

w)

C(t,tw)

Slope=-1/kBT

h(t)

(t,t

w)

Time-dependent FDR violations and effective temperature

twait tobs

tobs

For tobs ≥ tw looks non-equilibrium

FDR fails Teff > T

kB T

t= tw +tobs

C(t,tw)=<O(tw )O(t)>

(t,tw) =O(t)/h(tw)(t,tw)

(t,tw) = [1/kB Teff][C(tw,tw )-C(t,tw)]

R(t,t

w)

C(t,tw)

Slope=-1/kBTeff

Slope=-1/kBT

h(t)

mean-field models(

t,tw)

Frequency-dependent FDR violations and effective temperature

twait tobs

tobs

For tw < 1 looks non-equilibrium

FDR fails Teff > T

kB T

/tobs

)(2

COSo

)(")(')( i

)("

Bk

ST O

eff

h(t)

Difficult to access low ftw –

need rapid quench

0.1 1 10

ftw

T1

Mean-field T2

Evidence from simulations

p-spin Ising modelCugliandolo, Kurchan, 1997

Lennard-JonesBarrat, Kob 1998

Domain growth- infinite Teff

Barrat, 1998

Experiment on FDR in aging supercooled liquid

Oscillator as thermometer:Eosc = ½kBTeff Cugliandolo et. al. 1997

Resonant circuit driven by thermal fluctuations in dielectric sample

C<V2> = kBTeff <V2> is integrated noise power under resonance

Grigera and Israeloff, PRL 1999

Small Long-Lived FDR Violations Observed

Violations persisted up to the average relaxation time of the material, suggested series or stringy kinetics

C’=C0’ C”=C0” tw ~ 105

FDR violations in spin glasses

Herisson and Ocio PRL 2002

FDR violations in Laponite and polymer glassElectrical: large FDR violations and non-Gaussian Teff ~106 K

Buisson, Bellon, Ciliberto, J. of Phys.: Cond Mat. 2003

But these samples are macroscopic:

Spikes require the coherent fluctuation of entire 10 cm3 sample!

In any case, these measurementsare tricky and extrinsic noise is challenging.

Large violations dueto non-Gaussian spikes.Attributed to intermittency Intermittency found in simulationsof mesoscopic glass models: Sibani, PRE 2006

Summary of experimental resultsMaterial Property FDR violations? tw Ref.Glycerol electrical small short-moderate Grigera, 1999Spin glass magnetic large short Herisson, 2002Laponite electrical large short-moderate Buisson, 2003 Laponite rheological none Buisson, 2004

“ “ large long Abou, 2004 “ “ large long Strachan, 2006

“ “ large long Bartlett, 2006“ “ none Jabbari-Farouji, 2007

Poly-carbonate electrical large short-moderate Buisson, 2005

Measure dielectric susceptibility and current noisepolymer glass: PVAc, Tg =308 K

’i” FDR: Si =4kbTC0”

Aging of dielectric susceptibility

Rapid quench 330K to 300K

ftw scaling

Current noise measurements

Ultra-low-noise current amplifier 0.5 fA/√Hz

"4 CTkS BI

FDR prediction:

Equilibrium noise and Teff

1.E-32

1.E-31

1.E-30

1.E-29

1.E-28

0.1 1.0 10.0 100.0

S (

A/H

z)

Frequency (Hz)

0

100

200

300

400

500

0.1 1.0 10.0 100.0

Tem

per

atur

e (K

)

Frequency (Hz)

"4 CkST

B

Ieff

Two temperature quench profiles

T(K)

time (s)time (s)

Initial dT/dt=0.15 K/s

“fast”“slow”

aging

300

305

310

315

320

325

330

0 2 4 6 8 10 12 14

T fictive 13.3 Hz

TInitial dT/dt=8 K/s

cooling

Current noise during and after rapid quench

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50

I (pA)

t(s)

cooling aging

Tg

Average of 840 quenches

Dielectric response measurements

Conventional measurement Apply V=V0sin(t)

Measure I with Lock-in → Admittance Y=I/V

But fails for highly non-stationary early tw

V is white noise, measure I noise

FT- I, V and Admittance Y=I/V

Slow quench: effective temperature

No clear FDR violations found for slow quench

Effective temperature during fast quench

Scaling of effective temperature in aging regime

100

200

300

400

500

600

700

800

0.01 0.1 1 10 100

Tef

f(K

)

ftw0.45

tw =tQ -5 from 1.5s to 400 s

Slower decay than ftw scaling expected Shape also disagrees with mean-field models

1E-12

1E-11

1E-10

0.01 0.1 1 10

C''

Frequency (Hz)

0.470.60.91.522.75.1102060100200300450

Equilibrium 318 K tQ (s)

Spectrum of response, ”(f), is distorted during quench

”C0

Time evolution of spectrum: noise and response

0

0.5

1

1.5

2

2.5

0.1 1 10

"

Frequency (Hz)

tQ (s)

0.9

1.5

2

2.75

Equilibrium 318 K

during quench

0.07

0.7

0.01 0.1 1 10

"

tQ = 5

10 20 80

200

during aging

responsenoise

responsenoise

One interpretation: for response is lower than for noise

FDR violations in aging Lennard- Jones

Barrat and Kob 1998

Correlation

Response

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10 100 1000 10000 100000

0

0.5

1

1.5

2

2.5

0.1 1 10 100 1000

t(MCS)Co

rrel

ation

Correlation

1-kBT ·Response

Noise·/kBT

Susceptibility

”(a

rb. u

nits

)

ftw

Frequency domain susceptibility and noise for aging Lennard-Jones

Barrat and Kob 1998

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1000000

tw =40000

Noise is Gaussian even when FDR violated

Large extrinsic spikes (> 5 do occur, but very rarely, and are removed

FDR violations during cooling and aging

fdt

dT

T

m

g

gTdT

dm

/

log

Hypotheses:

•Noise decorrelates faster during cooling and aging due to energy lowering transitions

•significant violations when quench rate, dT/dt, is high

•E.g. when fragility index

• Nonequilibrium noise saturates at ~ equilibrium -peak noise –this is reasonable since there are a finite number of dipoles.

• Practical upper limit on Teff ~ T ”(peak)/”(earliest tw) ~ 3T

Caught polymer melt in the act of falling out of equilibrium

Moderate FDR violations observed: but only for high quench rates.

Violations are short-lived: but modified ftw scaling.

Noise is Gaussian

Interesting results:

Apparent response < corr noise much less stretched

Teff < T regime observed, disagrees with mean-field models but consistent with Lennard-Jones

Summary of FDR violation experiments

Cr is correlation function (noise)

r is response function

Local aging is heterogeneous in a model spin glassCastillo, Chamon, Cugliandolo, Kennett PRL 2002

Castillo, Parsaeian, Nature Physics 2007

FDR violations heterogeneous

Non-Gaussian distributions and possibly intermittent noiseChamon et. al. PRE 2003 Crisanti and Ritort cond-mat/0307554.

PVAc

Au film

V

Glass substrateDens ity P lot: |E |, V /m

2.145e-001 : > 2.200e-0012.090e-001 : 2.145e-0012.035e-001 : 2.090e-0011.980e-001 : 2.035e-0011.925e-001 : 1.980e-0011.870e-001 : 1.925e-0011.815e-001 : 1.870e-0011.760e-001 : 1.815e-0011.705e-001 : 1.760e-0011.650e-001 : 1.705e-0011.595e-001 : 1.650e-0011.540e-001 : 1.595e-0011.485e-001 : 1.540e-0011.430e-001 : 1.485e-0011.375e-001 : 1.430e-0011.320e-001 : 1.375e-0011.265e-001 : 1.320e-0011.210e-001 : 1.265e-0011.155e-001 : 1.210e-001< 1.100e-001 : 1.155e-001

Local dielectric spectroscopy

resresres fVz

C

kf

z

F

kf

k

kf 2

02

2

8

1

4

1

4

1

2

2

1VCU tip

F=dU/dz

UHV SPM

Electric Force Microscopy

Probed depth 20 nm

+

-

tVV sin0

20

202

20 tsin2

2

tcos21

4 PP VVVVdz

Cd

k

fdf

(susceptibility ) (polarization, charge)

Select 1 or 2 with lockin

Time (s)

VP / VP(0)

Relaxation after a dc bias reduction

Polarization images in PVAc near Tg

600 x 600 nm

t=0 t=17 min t= 48 min

303.5 K, we find rms spatial <VP > = 23±4 mV . 305.5 K <VP > =28±4 mV

Tim

e (s

)

0

2500

0

2500

0 position (nm) 700

Imaging spatio-temporal dipolar fluctuations near Tg =308 K

Longer time correlations at lower temperatures seen.

Hints of dynamicalheterogeneity and web-like structures

Can study various correlationFunctions

e.g. global C(t)

301.5 K

305.5 K

Time (s)

C(t)

C(x)

X (nm)

Local Response vs. Correlation

0

0

R(t

)

C (t)

Q=Ceff VP

Ceff = 7.2x10-18 F

R(t)=A-Q(t)/V

C(t)=<Q(t’)Q(t’+t)>

T (K) -1/kB slope

305.5 262 ± 15303.5 258 ± 30302.5 253 ± 40

305.5 K

303.5 K302.5 K

Four-point space-time correlation functions

Various four-point space-time correlation functions have been studied in simulations. A recurring one is

g4(x,t) = <V(0,0)V(0,t)V(x,0)V(x,t)> - <V(0,0)V(0,t)><V(x,0)V(x,t)>

When integrated over all x, a generalized susceptibility, 4(t), is obtained.

4(t) is variance of C(t) Glotzer et al PRL 1999Bouchaud et al 2006 Cipelletti et al 2006

Variance of C(t)

2 (C)

Local non-contact dielectric spectroscopy –

PVAc shows a small reduction in Tg and narrowing of the distribution of relaxation times in 20 nm free

surface layer. No suppression of glassy dielectric response

Spatio-temporal fluctuation images

Quantitative agreement with equilibrium thermal noise will allow study of local FDR violations.

Various x-t correlation functions can be studied

Summary

Acknowledgements:

P. S. CriderH. Oukris M. E. MajewskiJ. ZhangT. S. GrigeraE. Vidal RussellNSF-DMR-ACS-PRF