Does cooperation lead to relaxation? Slow dynamics in disordered systems. Nathan Israeloff Northeastern University Boston

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1 Does cooperation lead to relaxation? Slow dynamics in disordered systems Nathan Israeloff Northeastern University Boston

2 Collaborators Ezequiel Vidal Russell Tomas Grigera Konesh Sinnathamby Shomeek Mukhopadhyay Phil Crider Michael Rose Luke MacDonald

3 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

4 :KDWLVDJODVV" A solid that lacks structural order? 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 T g which depends on the rate of cooling. Edmund Di Marzio, NIST

5 Other Disordered or Glassy Systems Spin Glass Frozen paramagnet with built-in (quenched) disorder Phase transition confirmed Mean-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 T G decreases with cure

6 '\QDPLFVDWWKH*ODVV7UDQVLWLRQ Viscosity and relaxation times grow by near the glass transition Fragile liquids, relaxation times τ, diverge at T 0 < T G Vogel-Tamman-Fulcher τ = τ 0 exp[a(t-t 0 ) -1 ] Is this a phase transition? 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)

7 Near T G : Nonexponential relaxation. Dielectric susceptibility ε (ω) Polarization Broadened response Debye Log (ω) stretched exponential (glassy) Dielectric Response exponential time glassy Kohlrausch-Williams- Watt (KWW) P = P 0 exp[-(t/ τ) β ] Rough energy landscape? Aging: glasses out of equilibrium have time-dependent properties

8 Cooperativity Strong glass formers: e.g. SiO 2 activation is Arrhenius, almost certainly due to breaking of single Si-O bonds. Fragile liquids: apparent activation energies for relaxation exceed bond energies near T G ---cannot derive from single molecule motions. Cooperativity postulated (Adam-Gibbs Model)

9 Adam-Gibbs Model Ensemble of small independent, equivalent cooperatively relaxing regions (CRR). Connects relaxation time to thermodynamic quantities. G. Adam and J. H. Gibbs, J. Phys Chem, 1965 Smallest CRR have z molecules (~ 3 near T g ) with two-states ξ CRR ~ z 1/3 Energy barriers U ~z, τ=τ o exp(u/k B T) entropy of CRR s c = k B ln2 S = s c N a /z decreases through glass transition Find ξ CRR grows weakly with decreasing temperature explains growth in relaxation times (Vogel-Fulcher!)

10 Recent Theories Mode-Coupling (Gotze, Leutheusser, 1984) gives insight into molecular caging for T>>T G, breaks down near T G Frustration Limited Domains (S. Kivelson, 1995) can account for heterogeneity, but not uniquely Random First-Order Transitions (Wolynes, 2000) predicts heterogeneity length scales in very rough agreement with experiment Defect Diffusion (Shlesinger, 2001) ionic conductivity connected with other properties Many, many more (Parisi ~ 5/year)

11 Recent Developments:Spatially Heterogeneous Dynamics? Mark Ediger, Annu. Rev. Phys. Chem :99 128

12 Heterogeneous broadening? Dielectric susceptibility ε (ω) glassy Debye α-peak Log (ω) ε ω

13 We ll need some Fluctuation-Dissipation Relations (FDR) Stokes-Einstein Relation D= k B T /6πη 0 R. Stokes-Einstein-Debye D rot = k B T /8πη 0 R 3 Brownian motion: Diffusion constant scales inversely with viscosity Rotational diffusion Nyquist Relation S V = 4k B TRe(Z) Voltage noise scales with resistance

14 Derivation of an FDR Derive Nyquist s relation for a resistor S V = 4k B TR Every resistor has some stray capacitance, C, in parallel Equipartition theorem: Average thermal energy stored on capacitor ½ k B T = ½C<V 2 > Decay time for voltage τ = RC frequency bandwidth f ~ [2πRC] -1 Thus spectral density: S V = <V 2 >/ f = 2πk B TR ~ 4k B TR Real Derivation, see: The mathematics of Brownian motion and Johnson noise, Am. J. Phys. 64, 225 (1996)

15 Enhanced Translational Diffusion in supercooled liquids and polymers Stokes-Einstein Relation D= k B T /6πη 0 R violated near the glass transition. tris-naphthylbenzene Phys. Rev. Lett. 90, (2003) Stephen F. Swallen, Paul A. Bonvallet, Robert J. McMahon, and M. D. Ediger

16 Enhanced Translational Diffusion relative to Rotational Diffusion Stokes-Einstein-Debye (rotational diffusion) D rot = k B T /8πη 0 R 3 Not violated D/D rot ~ not constant as predicted Cicerone and Ediger, J. Chem. Phys, 1996; Chang, Fujara, Silescu et. al. J. Non-Cryst Sol 1994 Evidence for growing dynamical heterogeneity near T g

17 Enhanced Translational Diffusion: Evidence for dynamical heterogeneity near T g? The fastest diffusion coefficients dominate because percolating paths allow molecules to go around slow regions. Ediger (2000) Correlation between non-exponential parameter β and enhanced translation If relaxation rates are broadly spread-- get enhanced translation

18 Other evidence for dynamical heterogeneity: Dynamically selective experiments ε (ω) Log (ω) 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 a period τ R Techniques: Photo-bleaching Multi-dimensional NMR Dielectric hole-burning Bohmer et al, J. Non Cryst Sol, 1998; Ediger, Ann Rev Phy Chem, 2000

19 Heterogeneity lifetime studies How does recovery time τ R compare with alpha relaxation time τ α? ε (ω) ε (0)- ε (t w ) Log (ω) t w Fluorescence of probe molecules: Macro (Ediger, ) Single Molecule (Vanden Bout, 2000) NMR (Spiess, Heuer ): slow recovery τ R ~100 τ α rapid recovery τ R ~ τ α Dielectric hole burning (Bohmer, Chamberlin, 1996): τ R ~ τ α at high frequencies (Richert, 2003): τ R < τ α

20 Perhaps lifetime increases with decreasing temperature might explain discrepancies But dielectric hole burning near T g finds τ R τ α

21 Heterogeneity Length Scales NMR of PVAc at T g +10 ξ het = 3 nm i.e. ~ 200 monomers Tracht et. al., PRL 1998 Similar analysis on glycerol ξ het = 1.4 nm (Reinsberg et al 2001). Other experiments: ultra-thin free-standing PS films show T G reduction Is ξ het = ξ CRR?

22 2 Dielectric Hole Burning in High Frequency Wing Dielectric Susceptibility PVAc Thin Film ε" Ratio Transformer 312.5K Lockin 317.5K Freq. [Hz] δε δε HV δε" Parallel Plate Capacitor d=0.5 mm Apply V sinusoidal burn f burn =90Hz f meas =350Hz PVAc 312K Aexp(-(t/τ) 0.4 ) t w (s) t burn τ 16 s 6.9 s 8 s 4.8 s 2 s 1.6 s τ R increases with burn time--approaches τ α

23 Cooperative dynamics observed in model glassy systems Colloidal glasses Spatially heterogeneous dynamics Transient mobile clusters Weeks, Weitz et. al. Science (2000); MD simulations of binary liquids Growing dynamical correlation lengths Donati et al. PRL 1998, Glotzer, Nature, 2000 Probing ultra-short time dynamics at T >>T G

24 Questions about cooperativity and heterogeneity: Heterogeneity explained by small density fluctuations? Or more sophisticated model? Cooperative dynamics observed in simulations and colloids relevant to molecular glasses near T G? Lifetime of CRR? Local relaxation exponential? Detailed dynamical processes? Spatial structure and length scale of CRR?

25 Local Probes of Glassy Dynamics Probe dielectric susceptibility of a nano-volume of glass Macroscopic volume ε (ω) Mesoscopic volume Log(ω) ε (ω) Russell and Israeloff, Nature, 408, 695 (2000) Russell et. al. Phys. Rev. Lett 81, 1461(1998) Walther et. al. Phys Rev. B57, R15112 (1998) Log(ω) see also Vanden Bout, Science 2001, single-molecule fluorescence

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27 Sample [Dielectric material] Conducting substrate Sample ----[---C H----- C H---] n O C ==O C H 3 Electrostatic Force Microscopy (EFM) Poly-vinyl-acetate (PVAc) Average mol. Wt Glass transition temperature Tg ~ 305 K V F e = -du/dz = -(1/2)V 2 dc/dz Measure variations in tip-sample capacitance C = C 0 ε Cantilever resonance frequency is more sensitive: ω 2 = k eff /m k eff =k+d F e /dz = k-(1/2)v 2 d 2 C/dz 2

28 Non- contact frequency-modulation EFM measurement oscillator Phase detector δf V V piezo V bias Pre-amp Z Constant A, f 0

29 Local dielectric relaxation Dielectric relaxation measured in 50 nm region of polymer film via cantilever resonance. Noise and possible discrete steps in relaxation observed PVAc T = 306 K Time (s) T= 303 K PVAc Walther, Israeloff, Vidal Russell, Gomariz Phys. Rev. B57, R15112 (1998) Time (s)

30 Time series of PVAc polarization fluctuations Power spectrum vs. Temperature 10-6 Sv-302K Sv-305K Sv-310K Spectral density 10-7 Sv-315 Power law S f ~ f α f [ Hz ]

31 In a nano-volume, expect fluctuations to be important ε" Dielectric Susceptibility PVAc Thin Film 317.5K Fluctuation-Dissipation Relation. Noise spectral density: S v = 4k B T(ε /C 0 ωε 2 ) K Freq. [Hz] Spectral exponent from noise and susceptibility 1.4 For cantilever resonance frequency fluctuations: Spectral Exponent Susceptibility S f ~4k B T((f 2 -f 0 2 )/f) 2 ε /ωc 0 ε 2 V Noise Temperature (K)

32 Nano noise contains same information as susceptibility (FDT) 'LHOHFWULF6XVFHSWELOLW\ ε (ω) Log(ω) 3RODUL]DWLRQQRLVH Heterogeneous picture: Expect spectral features Noise power Log(ω)

33 Evolution of noise spectra Transient Lorentzian-like features show dynamical heterogeneity Local fit to f α exponent α with spectral

34 Evolution of noise spectral exponent Autocorrelation function measures lifetime of dynamical heterogeneities. α vs. time shows transient appearance of dynamical heterogeneities

35 Heterogeneity lifetime comparable to usual α relaxation time. Why? Similar to NMR results at T>T G Heuer, Spiess, et.al.

36 0.15 FFtn-302M-6.times Random-telegraphsignals (RTS) in polarization timeseries. CRR dipole-moment Fluctuations ~ 10µ monomer Direct evidence for cooperative molecular dynamics Recent Simulations (Berthier, Garrahan, Chandler, 2003) show qualitatively similar RTS Resonance frequency (Hz) Resonance frequency (Hz) Resonance frequency (Hz) Resonance frequency (Hz) tim e series tim e series

37 Multi-state CRR µ Simple model: CRR dipole moment has 4 favorite orientations.

38 Do individual CRR relax exponentially or non-exponentially? Long-lived 2-state RTS: Distribution of times spent in each state--nonexponential

39 Exponential behavior found in some short stretches

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

41 Probing energy landscape properties 1000 Polarization histogram Polarization T =302 K Polarization Time(s) "U" Landscape Polarization

42 Effect of electric field on landscape Effect of Field on Landscape E µ V 4 V 6LPSOHVWPRGHO Polarization 8 δ( ORF δµ ] µ ] WR[ &P awrµ

43 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 dielectric relaxation time near T G (short-lived dynamical heterogeneity) Evolution from exponential to nonexponential CRR kinetics seen (coupled TLS?).

44 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 < δm ( ω) 2 kb Χ > T " ω

45 FDR Violations Theory Cugliandolo and Kurchan (PRL 1993, Phys. Rev. E. 1997) Parisi + many more (minor industry) FDR should be violated in slowly evolving systems such as aging spin glasses and glasses, and sheared Defined an Effective Temperature in terms of usual FDR k B T eff (t, t w ) = C(t,t w )/R(t,t w ) Fluctuations/Response Main Point: violations should occur when observation time (t) and age (t w ) of system are comparable. Energy t t w

46 FDR test in an aging supercooled liquid Resonant circuit driven by thermal fluctuations in dielectric sample <V 2 > = k B T /C FDR prediction integrated power under resonance

47 Aging of dielectric susceptibility following temperature quench glycerol time (s)

48 Small Long-Lived FDR Violations Observed Violations persisted up to 10 5 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)

49 Conclusions A number of old and new experimental/computational findings in 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 T G?