experimental tests of the fluctuation- dissipation-relation in aging glassy systems collaborators:...
TRANSCRIPT
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