Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern University Boston

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 Dielectric susceptibility Signatures of glassy systems: Slow- nonexponential relaxation. Rough energy landscape? exp[-(t/  )  ] Broadened response Diverging relaxation times below T g (fragile glasses) Aging after T-quench Cooperative dynamics –jamming

Fluctuation-Dissipation Relations (FDR) Stokes-Einstein Relation D= k B T /6  0 r Nyquist Relation S V = 4k B TR 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? T eff =S V /4k B R Energy

Time-dependent FDR violations and effective temperature t wait t obs For t obs << t w looks like equilibrium FDR holds T eff = T k B T t= t w +t obs C(t,t w )= noise  (t,t w ) =O(t)/h(t w ) susceptibility  (t,t w ) = [1/k B T][C(t w,t w )-C(t,t w )] Slope=-1/k B T h(t)  (t,t w )

Time-dependent FDR violations and effective temperature t wait t obs For t obs ≥ t w looks non-equilibrium FDR fails T eff > T k B T t= t w +t obs C(t,t w )=  (t,t w ) =O(t)/h(t w )  (t,t w )  (t,t w ) = [1/k B T eff ][C(t w,t w )-C(t,t w )] Slope=-1/k B T eff Slope=-1/k B T h(t) mean-field models  (t,t w )

Frequency-dependent FDR violations and effective temperature t wait t obs For  t w < 1 looks non-equilibrium FDR fails T eff > T k B T  /t obs h(t) Difficult to access low ft w – need rapid quench ft w T 1 Mean-field T 2

Evidence from simulations p-spin Ising model Cugliandolo, Kurchan, 1997 Lennard-Jones Barrat, Kob 1998 Domain growth- infinite T eff Barrat, 1998

Experiment on FDR in aging supercooled liquid Oscillator as thermometer: E osc = ½k B T eff Cugliandolo et. al Resonant circuit driven by thermal fluctuations in dielectric sample C = k B T eff 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’=C 0  ’ C”=C 0  ”  t w ~ 10 5

FDR violations in spin glasses Herisson and Ocio PRL 2002

FDR violations in Laponite and polymer glass Electrical: large FDR violations and non-Gaussian Teff ~10 6 K Buisson, Bellon, Ciliberto, J. of Phys.: Cond Mat But these samples are macroscopic: Spikes require the coherent fluctuation of entire 10 cm 3 sample! In any case, these measurements are tricky and extrinsic noise is challenging. Large violations due to non-Gaussian spikes. Attributed to intermittency Intermittency found in simulations of mesoscopic glass models: Sibani, PRE 2006

Summary of experimental results Material Property FDR violations? t w Ref. Glycerol electrical small short-moderate Grigera, 1999 Spin glass magnetic large short Herisson, 2002 Laponite 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 noise polymer glass: PVAc, T g =308 K  ’  i  ”  FDR: S i =4k b TC 0  ”

Aging of dielectric susceptibility Rapid quench 330K to 300K ft w scaling

Current noise measurements Ultra-low-noise current amplifier 0.5 fA/√Hz FDR prediction:

Equilibrium noise and T eff

Two temperature quench profiles T(K) time (s) Initial dT/dt=0.15 K/s “fast” “slow” aging Initial dT/dt=8 K/s cooling

Current noise during and after rapid quench coolingaging TgTg Average of 840 quenches

Dielectric response measurements Conventional measurement Apply V=V 0 sin(  t) Measure I with Lock-in → Admittance Y=I/V But fails for highly non-stationary early t w 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 t w =t Q -5 from 1.5s to 400 s Slower decay than ft w scaling expected Shape also disagrees with mean-field models

Equilibrium 318 K t Q (s) Spectrum of response,  ”(f), is distorted during quench  ”C 0

Time evolution of spectrum: noise and response Frequency (Hz) t Q (s) Equilibrium 318 K during quench t Q = during aging response noise response noise One interpretation:  for response is lower than  for noise

FDR violations in aging Lennard- Jones Barrat and Kob 1998 Correlation Response

t(MCS) Correlation 1-k B T ·Response Noise·  /k B T Susceptibility  ”(arb. units) ft w Frequency domain susceptibility and noise for aging Lennard-Jones Barrat and Kob 1998 t w =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 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 T eff ~ T  ”(peak)/  ”(earliest t w ) ~ 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 ft w scaling. Noise is Gaussian Interesting results: Apparent  response <  corr noise much less stretched T eff < T regime observed, disagrees with mean-field models but consistent with Lennard-Jones Summary of FDR violation experiments

C r is correlation function (noise)  r is response function Local aging is heterogeneous in a model spin glass Castillo, Chamon, Cugliandolo, Kennett PRL 2002 Castillo, Parsaeian, Nature Physics 2007 FDR violations heterogeneous Non-Gaussian distributions and possibly intermittent noise Chamon et. al. PRE 2003 Crisanti and Ritort cond-mat/

Local dielectric spectroscopy F=dU/dz UHV SPM Electric Force Microscopy Probed depth 20 nm +-+- (susceptibility ) (polarization, charge) Select 1  or 2  with lockin

Time (s) V P / V P (0) Relaxation after a dc bias reduction

Polarization images in PVAc near T g 600 x 600 nm t=0 t=17 mint= 48 min K, we find rms spatial = 23±4 mV K =28±4 mV

Time (s) position (nm) 700 Imaging spatio-temporal dipolar fluctuations near T g =308 K Longer time correlations at lower temperatures seen. Hints of dynamical heterogeneity and web-like structures Can study various correlation Functions e.g. global C(t) K K

Time (s) C(t) C(x) X (nm)

Local Response vs. Correlation Q=C eff V P C eff = 7.2x F R(t) =A-Q(t)/V C(t)= T (K)-1/k B slope ± ± ± K K K

Four-point space-time correlation functions Various four-point space-time correlation functions have been studied in simulations. A recurring one is g 4 (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 1999 Bouchaud et al 2006 Cipelletti et al 2006

Variance of C(t)  2 (C)

Local non-contact dielectric spectroscopy – PVAc shows a small reduction in T g 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. Crider H. Oukris M. E. Majewski J. Zhang T. S. Grigera E. Vidal Russell NSF-DMR- ACS-PRF