Does cooperation lead to relaxation?Slow dynamics in disordered systems

Nathan Israeloff

Northeastern University

Boston

Collaborators

Ezequiel Vidal RussellTomas GrigeraKonesh SinnathambyShomeek MukhopadhyayPhil CriderMichael RoseLuke MacDonald

Outline

• Introduction to glassy dynamics– Old idea: cooperative molecular dynamics explains slow,

glassy phenomena.

– Fluctuation-Dissipation-Relation (FDR)

• Recent Developments– Nanoscale dynamical heterogeneity

– Cooperativity observed in simulations and model systems

– FDR in non-equilibrium glassy systems

• Low frequency noise experiments– Probe local dielectric fluctuations

– Test FDR violations

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Thermodynamic Definition

…….an ordinary liquid at high temperatures and whose thermodynamic extensive quantities, volume V, and entropy S, fall out of equilibrium as we lower the temperature past some temperature Tg which depends on the rate of cooling.

Edmund Di Marzio, NIST

A solid that lacks structural order?

Other Disordered or Glassy Systems

•Spin Glass Frozen paramagnet with built-in (quenched) disorder

Phase transition confirmedMean-Field Ising model solved (Mezard, Parisi, 1982)

hierarchical arrangement of states

•Relaxor Ferroelectric

•Proteins (Fraunfelder, 1986) Solvent driven??

•Colloids (e.g Weitz) density driven glass transition

•Gels

•Epoxies TG decreases with cure

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Recent Reviews: Mark Ediger, Annu. Rev. Phys. Chem. 51, 99 (2000).

Pablo Debenedetti and Frank Stillinger, Nature 410, 259 (2001)

Austen Angell, Science 267:1924 (1995)

Viscosity and relaxation times grow by 1012 near the glass transition

Fragile liquids, relaxation times

τ, diverge at T0 < TG

Vogel-Tamman-Fulcherτ = τ0exp[A(T-T0)-1]

Is this a phase transition?

0

0.2

0.4

0.6

0.8

1

1 10 100 1000 10000

Dielectric Response

time

exponentialstretchedexponential(glassy)

Log (ω)

ε”(ω)

Debye

glassy

Die

lect

ric

susc

e ptib

ility

Near TG: Nonexponential relaxation.

Rough energy landscape?

Kohlrausch-Williams-Watt (KWW)P = P0exp[-(t/ τ)β]

Pola

riza

tion

Broadened response

Aging: glasses out of equilibrium have time-dependent properties

Cooperativity postulated (Adam-Gibbs Model)

Cooperativity

Fragile liquids: apparent activation energies for relaxationexceed bond energies near TG---cannot derive from singlemolecule motions.

Strong glass formers: e.g. SiO2 activation is Arrhenius, almost certainly due to breaking of single Si-O bonds.

G. Adam and J. H. Gibbs, J. Phys Chem, 1965

Adam-Gibbs Model

Ensemble of small independent, equivalentcooperatively relaxing regions (CRR). Connectsrelaxation time to thermodynamic quantities.

Smallest CRR have z molecules (~ 3 near Tg)

with two-states ξCRR ~ z1/3

Energy barriers U ~z, τ=τ o

exp(U/kB

T)

entropy of CRR sc = kBln2

S = scNa/z decreases through glass transition

Find ξCRR grows weakly with decreasing temperatureexplains growth in relaxation times (Vogel-Fulcher!)

Recent Theories

• Mode-Coupling (Gotze, Leutheusser, 1984) gives insightinto molecular caging for T>>TG, breaks down near TG

• Frustration Limited Domains (S. Kivelson, 1995) canaccount for heterogeneity, but not uniquely

•Random First-Order Transitions (Wolynes, 2000) predictsheterogeneity length scales in very rough agreement withexperiment

•Defect Diffusion (Shlesinger, 2001) ionic conductivityconnected with other properties

•Many, many more (Parisi ~ 5/year)

Recent Developments:Spatially Heterogeneous Dynamics?

Mark Ediger, Annu. Rev. Phys. Chem. 2000. 51:99–128

ω

α-peak

ε’’

Heterogeneous broadening?

Log (ω)

ε”(ω)

Debye

glassy

Dielectric susceptibility

We’ll need some Fluctuation-DissipationRelations (FDR)

Stokes-Einstein RelationD= kBT /6πη0R.

Stokes-Einstein-DebyeDrot= kBT /8πη0R3

Nyquist RelationSV = 4kBTRe(Z)

Brownian motion: Diffusion constant scales inversely with viscosity

Voltage noise scales with resistance

Rotational diffusion

‘Derivation’ of an FDR

Derive Nyquist’s relation for a resistor SV = 4kBTR

Every resistor has some stray capacitance, C, in parallel

Equipartition theorem: Average thermal energystored on capacitor

Decay time for voltage τ = RC

frequency bandwidth ∆f ~ [2πRC]-1

Thus spectral density:

½ kBT = ½C<V2>

SV = <V2>/ ∆f = 2πkBTR ~ 4kBTR

Real Derivation, see: The mathematics of Brownian motion and Johnson noise,Am. J. Phys. 64, 225 (1996)

Enhanced Translational Diffusionin supercooled liquids and polymers

Stokes-Einstein Relation D= kBT /6πη0R violated near the glass transition.

Phys. Rev. Lett. 90, 015901 (2003) Stephen F. Swallen, Paul A. Bonvallet, Robert J. McMahon, and M. D. Ediger

tris-Naphthylbenzene

Enhanced Translational Diffusion relative toRotational Diffusion

Stokes-Einstein-Debye(rotational diffusion)

Drot= kBT /8πη0R3

Not violated

D/Drot ~ not constant aspredicted

Evidence for growing dynamical heterogeneity near Tg

Cicerone and Ediger, J. Chem. Phys, 1996;

Chang, Fujara, Silescu et. al. J. Non-Cryst Sol 1994

Enhanced Translational Diffusion:Evidence for dynamical heterogeneity near Tg ?

The fastest diffusion coefficients dominatebecause percolating paths allowmolecules to go around slow regions.Ediger (2000)

Correlation between non-exponentialparameter β and enhanced translation

If relaxation rates are broadly spread--get enhanced translation

ε”(ω)

Log (ω)

Other evidence for dynamical heterogeneity:Dynamically selective experiments

Bohmer et al, J. Non Cryst Sol, 1998;Ediger, Ann Rev Phy Chem,2000

Hole-Burning: If you remove, or burn, some molecules,say faster relaxing ones. Do remaining molecules have

the full distribution?

No: the slow molecules remain slow, for aperiod τR

Techniques: Photo-bleaching Multi-dimensional NMR Dielectric hole-burning

Heterogeneity lifetime studies

How does recovery time τR compare with alpharelaxation time τα?

ε”(ω)

Log (ω)

ε”(0)- ε”(tw)

tw

Fluorescence of probe molecules: slow recovery τR ~100 ταMacro (Ediger, 1995-2000)Single Molecule (Vanden Bout, 2000)

NMR (Spiess, Heuer 1995-2000): rapid recovery τR ~ ταDielectric hole burning (Bohmer, Chamberlin, 1996): τR ~ τα

at high frequencies (Richert, 2003): τR < τα

Perhaps lifetime increases with decreasing temperaturemight explain discrepancies

But dielectric hole burning near Tg finds τR ∼ τ α

Heterogeneity Length Scales

NMR of PVAc at Tg +10

ξhet = 3 nm

i.e. ~ 200 monomers

Tracht et. al., PRL 1998

Similar analysis on glycerol ξhet = 1.4 nm (Reinsberg et al 2001).

Is ξhet = ξCRR?

Other experiments: ultra-thin free-standing PS films show TG reduction

10-6

10-5

10-4

10-3

0 50 100 150

δε"

tw

(s)

Aexp(-(t/τ)0.4)

tburn τ

16 s 6.9 s8 s 4.8 s2 s 1.6 s

fburn

=90Hz

fmeas

=350Hz

Dielectric Hole Burning in High Frequency Wing

PVAc 312K

HV

Rat

io T

rans

form

er

Lockinδε’δε”

0

0.5

1

1.5

2

0.01 0.1 1 10 100 1000

ε"

Freq. [Hz]

317.5K

312.5K

Dielectric Susceptibility PVAc Thin Film

Parallel Plate Capacitor d=0.5 �m Apply 15-50 V sinusoidal burn

τR increases with burn time--approaches τα

Cooperative dynamics observed in model glassysystems

Colloidal glasses• Spatially heterogeneous

dynamics• Transient mobile clusters

Weeks, Weitz et. al. Science (2000);

MD simulations of binaryliquids

• Growing dynamicalcorrelation lengths

Donati et al. PRL 1998, Glotzer,Nature, 2000

Probing ultra-short time dynamics at T >>TG

Questions about cooperativity and heterogeneity:

•Heterogeneity explained by small densityfluctuations? Or more sophisticated model?

•Cooperative dynamics observed in simulationsand colloids relevant to molecular glasses near TG?

•Lifetime of CRR?

•Local relaxation exponential?

•Detailed dynamical processes?

•Spatial structure and length scale of CRR?

Log(ω)

Macroscopic volume

ε” (ω)

Local Probes of Glassy Dynamics

Probe dielectric susceptibility of a nano-volume of glass

Mesoscopic volume

Log(ω)

ε” (ω)

see also Vanden Bout, Science 2001, single-molecule fluorescence

Russell and Israeloff, Nature, 408, 695 (2000) Russell et. al. Phys. Rev. Lett 81, 1461(1998)

Walther et. al. Phys Rev. B57, R15112 (1998)

Sample

Poly-vinyl-acetate (PVAc)Average mol. Wt. 167 000

Glass transition temperature Tg ~ 305 K

----[---C H ----- C H ---]---- 2 | n O | C = = O | C H 3

Sample [Dielectric material]

Conducting substrate

V

Electrostatic Force Microscopy (EFM)

Fe = -dU/dz = -(1/2)V2dC/dz

Measure variations intip-sample capacitance C = C0 ε

Cantilever resonance frequency is more sensitive:

ω2 = keff/mkeff =k+d Fe /dz = k-(1/2)V2d2C/dz2

V

Vbias

Vpiezo

oscillator Phasedetector

Non- contact frequency-modulation EFM measurement

Constant A, f0

Pre-amp

Z

δf

Local dielectric relaxation

• Dielectric relaxationmeasured in 50 nmregion of polymer filmvia cantileverresonance.

• Noise and possiblediscrete steps inrelaxation observed.

Walther, Israeloff, Vidal Russell,Gomariz Phys. Rev. B57, R15112(1998).

71840

71860

71880

0 20 40 60 80 100Time (s)

PVAc

T = 306 K

72180

72200

72220

72240

72260

72280

400 500 600 700 800 900Time (s)

T= 303 K PVAc

S f ~ f −α

Power spectrum vs. Temperature

Time series of PVAc polarization fluctuations

Power law

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 500 1000 1500 2000 2500 3000 3500

10-8

10-7

10-6

0.2 0.4 0.6 0.8 1 3 5 7

Sv-302K

Sv-305K

Sv-310K

Sv-315

Spe

ctra

l den

sity

f [ Hz ]

In a nano-volume, expectfluctuations to beimportant

Fluctuation-Dissipation Relation.Noise spectral density:

Sv = 4kBT(ε”/C0ωε2)

�

0

0.5

1

1.5

2

0.01 0.1 1 10 100 1000

ε"

Freq. [Hz]

317.5K

312.5K

Dielectric Susceptibility PVAc Thin Film

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

300 302 304 306 308 310 312 314 316

Susceptibility

Noise

Sp

ectr

al E

xpo

nen

t

Temperature (K)

Sf~4kBT((f2-f02 )/f)2ε”/ωC0ε2V2

For cantilever resonancefrequency fluctuations:

Spectral exponent fromnoise and susceptibility

Noisepower

Log(ω)

Log(ω)

ε” (ω)

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Nano noise contains same information as susceptibility (FDT)

Heterogeneous picture:Expect spectral features

Evolution of noise spectra

Transient Lorentzian-likefeatures show dynamicalheterogeneity

Local fit to f −α with spectral

exponent α

Evolution of noise spectral exponent

α vs. time shows transientappearance of dynamicalheterogeneities

Autocorrelation function measureslifetime of dynamical heterogeneities.

Heterogeneitylifetime comparableto usual α relaxationtime.

Why?

Similar to NMRresults at T>TG

Heuer, Spiess, et.al.

-0 .15

-0 .1

-0 .05

0

0.05

0 .1

0.15

0 500 1000 1500 2000 2500 3000 3500 4000

FF tn -302M -6.tim es

Res

onan

ce fr

eque

ncy

(Hz)

t im e series

-0 .15

-0 .1

-0 .05

0

0 .05

0 .1

0 .15

0 500 1000 1500 2000 2500 3000 3500 4000

Res

onan

ce fr

eque

ncy

(Hz)

-0 .1

-0 .05

0

0 .05

0 .1

0 .15

0 .2

0 500 1000 1500 2000 2500 3000 3500 4000

Res

onan

ce fr

eque

ncy

(H

z)

-0 .2

-0 .15

-0 .1

-0 .05

0

0 .05

0 .1

0 500 1000 1500 2000 2500 3000 3500 4000

Res

onan

ce f

requ

ency

(H

z)

t im e se ries

Random-telegraph-signals (RTS)in polarization time-series.

CRR dipole-momentFluctuations~ 10µmonomer

Direct evidence forcooperativemolecular dynamics

Recent Simulations(Berthier, Garrahan,Chandler, 2003) showqualitatively similar RTS

Multi-state CRR

µ Simple model: CRR dipole momenthas 4 favorite orientations.

Do individual CRR relax exponentially or non-exponentially?

Long-lived 2-state RTS:Distribution of times spent in each state--nonexponential

Exponential behavior found in some short stretches

Overall tendency towards exponential relaxation(β =1) with decreasing observation times

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 500 1000 1500 2000 2500 3000 3500 4000Time(s)

T =302 K

-3

-2.5

-2

-1.5

-1

-0.5

0

-0.1 -0.05 0 0.05 0.1 0.15 0.2

"U"

Polarization

1

10

100

1000

-0.1 -0.05 0 0.05 0.1 0.15 0.2

PolarizationHistogram

Polarization

Probing energy landscape properties

Polarization histogram

“Landscape”

3

3.5

4

4.5

5

-0.05 0 0.05 0.1

4.8 V4 V

Polarization

Effect of Field on Landscape

Effect of electric field on landscape

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Summary

• Nanoscale dipolar fluctuations probed in a polymer glass.

• The dynamics of individual cooperatively relaxing regions(CRR) observed.

• CRR repeatedly revisit a handful (2-4) configurations(telegraph noise).

• Lifetime of CRR comparable to average dielectricrelaxation time near TG (short-lived dynamicalheterogeneity)

• Evolution from exponential to nonexponential CRRkinetics seen (coupled TLS?).

FDR violations in disordered systems

Proposed failure below spin-glass transition: Sompolinsky PRL, 1981

Experiments on spin-glasses, SQUID-based magnetization noise and susceptibiliy measurements. Ocio, Bouchiat, and Monod, 1985; Reim et. al 1986; Bouchiat, Ocio, 1988

All found agreement with FDR ω

ωδ ")( 2 Χ>≈< Tk

M B

Cugliandolo and Kurchan (PRL 1993, Phys. Rev. E. 1997)Parisi + many more (minor industry)

FDR should be violated in slowly evolving systems such asaging spin glasses and glasses, and sheared

Defined an Effective Temperature in terms of usual FDR

kBTeff(t, tw) = C(t,tw)/R(t,tw) Fluctuations/Response

Main Point: violations should occur when observation time (t) and age (tw) ofsystem are comparable.

FDR Violations Theory

tw

Energy t

FDR test in an aging supercooled liquid

Resonant circuit driven bythermal fluctuations in dielectric sample

<V2> = kBT /C FDR prediction

integrated power under resonance

Aging of dielectric susceptibility following temperature quench

time (s)

glycerol

Small Long-Lived FDR Violations Observed

Violations persisted up to 105 times the correlation time of degrees of freedom under study,but comparable to the average relaxation time of the material.

Suggests possible series kinetics: energy flows from slower to faster relaxing modes.

Recently: Spin Glass FDR Violations Ocio et. al. PRL (2002)

Conclusions

A number of old and new experimental/computational findingsin glasses which need to be explained:

•Spatially heterogeneous dynamics

•Lifetimes of heterogeneity (both short and long)

•Details of cooperative processes, small number of states

•Cooperative length scales (growing?)

•Long-lived FDR violations

•Phase transition??

•MD and colloidal cooperativity = cooperativity near TG?