Nonlinear Dielectric Effects in Simple Fluids

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1 Nonlinear Dielectric ffects in Simple Fluids High-Field Dielectric Response: Coulombic stress Langevin effect homogeneous heating heterogeneous heating Ranko Richert LINAR RSPONS - THORY AND PRACTIC Søminestationen, Holbæk, Denmark - 8 July 7

2 definitions of dielectric and electric quantities permittivity of vacuum F m A s - m S s cm - voltage I current Q charge D displacement electric field P polarization dielectric function M electric modulus χ susceptibility j current density σ conductivity ρ resistivity steady state relations ( ) ( ) t d t I Q d A Q D D P D M D t χ χ ρ ρ ρ σ σ σ ω ρ σ ρ ω σ + + i im M M i i i M M i ˆ ˆ ˆ ˆ / ˆ ˆ ˆ / ˆ ˆ / ˆ ˆ ˆ D j j D j j j j A I j & & + + free charges ρ σ σ σ

3 ''(ω) dielectric relaxation techniques log(ωτ) '(ω) D * + iωτ s ( ω) + { Debye} dielectric displacement D Q A d electric field * ( ω) s + Havriliak α γ [ + ( iωτ ) ] - Negami

4 typical case near the glass transition propylene glycol 74 K - 98 K (3K) 5 ' 4 3 '' - 3 ν [Hz] - 3 ν [Hz]

5 non-linear susceptibility () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) K K symmetry 3 3 symmetry causality symmetry: causality: electret χ χ χ χ χ χ χ χ χ P P P P P ( ) ( ) [ ] ( ) s s s s s λ λ λ Δ Δ : Piekara factor ln ( ) ( ) ( ) ( ) t t d t dp t P θ θ ϕ θ χ θ superposition principle : ( ) ( ) dt t J J kt L F L J J L e e e e F F e F F the equilibrium fluctuations in the conjugate flux auto - correlation function of are related to Kubo (95's) : linear transport coefficients R. Green, M.S.

6 variety of high field techniques voltage voltage voltage time time time high DC field low AC field mainly ω high AC field no bias ω + 3ω low AC field on high pulse ND(t)

7 from nonlinearity to cooperativity length scale? Physical Review B 7, 644 (5) Nonlinear susceptibility in glassy systems: A probe for cooperative dynamical length scales Jean-Philippe Bouchaud and Giulio Biroli We argue that for generic systems close to a critical point, an extended fluctuation-dissipation relation connects the low frequency nonlinear (cubic) susceptibility to the four-point correlation function. In glassy systems, the latter contains interesting information on the heterogeneity and cooperativity of the dynamics. Our result suggests that if the abrupt slowing down of glassy materials is indeed accompanied by the growth of a cooperative length l, then the nonlinear, 3ω response to an oscillating field (at frequency ω) should substantially increase and give direct information on the temperature (or density) dependence of l. The analysis of the nonlinear compressibility or the dielectric susceptibility in supercooled liquids, or the nonlinear magnetic susceptibility in spin-glasses, should give access to a cooperative length scale, that grows as the temperature is decreased or as the age of the system increases. Our theoretical analysis holds exactly within the modecoupling theory of glasses.

8 Coulombic stress

9 Coulombic stress exercise µm qq qq r F r 3 cgs 4 π r r FSI 3 Q C nergy Q Q C A C d top electrode 6 mm Ø bottom electrode 3 mm Ø ring outer: mm Ø ring inner: 4 mm Ø calculate force resulting from voltage () t sin( ωt) calculate effect on measured assume Teflon spacer with Young's modulus GPa mechanical modulus [ stress] L F [ strain] ΔL A ( t)

10 Coulombic stress solutions µm ( ) Q d A C Q C r qq SI 3 4 π r F ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] [ ] [ ] A F L L strain stress t t d A t d A F d A d A d d A F d A C C C Q Q Q d F Q energy Fd work dc dc dc ac dc Δ modulus Young's cos sin sin ω ω ω () () ( ) [ ] ln.67 MPa 3 N (Teflon) GPa 4-6 mm) d (Teflon ring, m ln cos 4 Δ Δ + Δ Δ d d a F F Y a ay A ay F d d t A t A t F s s s ω

11 impedance approach to mechanical loss in a cantilever ac x I ac cm h d f d p L t dc R. Richert, Rev. Sci. Instrum. 78 (7) 539

12 ringdown of cantilever at resonance frequency..5 τ.6 s x(t) / x max time / s R. Richert, Rev. Sci. Instrum. 78 (7) 539

13 k' and k'' across 3.5 decades in frequency A(ν) / A A(ν) ν / Hz tan ϕ - - ϕ (deg) 'int' - ν / Hz 'ext' - ν / Hz -3 - ν / Hz mx && + ρx& + kx F e A ( ω ) tanϕ ( μ ) ( ω ) Dext A iωt + D μ μ, x ext μ Ae iωt mx && + kx ˆ mx & + k x + ik x A ( ω ) tanϕ A ( μ ) ( ω ) k k kμ + D int F e iωt, x Ae D D iωt μ ω ω ω m k ext int ρ mk k k R. Richert, Rev. Sci. Instrum. 78 (7) 539

14 dielectric saturation

15 expected non-linearity: dielectric saturation, Langevin effect cosθ cosθ e e μ μ cosθ cotanh kt kt μ Langevin cosθ L kt function μ with a kt cosθ L( a) cotanh( a) a if μ << kt or a << cosθ π π a μ 3 3kT μ cosθ kt μ cosθ kt dθ dθ L(a) cosθ a/3 L(a) a/3 cosθ a μ / kt

16 saturation in water at 93 K ( ) () ( ) ( ) ( ) ( ) vanleck Piekara Δ Δ μ λ s s s s s s s s kt N H. A. Kolodziej, G. Parry Jones, M. Davies, J. Chem. Soc. Faraday Trans. 7 (975) 69 J. H. van leck, J. Chem. Phys. 5 (937) 556 A. Piekara, Acta Phys. Polon. 8 (959) 36

17 Langevin and chemical ND's J. Malecki, J. Mol. Struct (997) 595

18 time resolved ND experimens M. Gorny, J. Ziolo, S. J. Rzoska, Rev. Sci. Instrum. 67 (996) 49

19 high field impedance SI-6 PZD-7 GN C geo 36 pf µm 6 propylene glycol - ' 4 R 5 Ω - R 5 Ω '' - 3 ν [Hz] S. Weinstein, R. Richert, Phys. Rev. B 75 (7) 643 S. Weinstein, R. Richert, J. Phys.: Condens. Matter 9 (7) 58

20 field effect results for propylene glycol [ 7-8 k/cm steady state harmonic measurement ] - propylene glycol Δ s 4 Nμ 45 ( van leck) ( kt ) 3 4 ( + ) s ( + ) ( + ) s s 4 Δ ln ' -4-6 Δ ln ' 5.6 Hz - Hz ' 6 4 propylene glycol -8 '' ν [Hz] S. Weinstein, R. Richert, Phys. Rev. B 75 (7) 643 S. Weinstein, R. Richert, J. Phys.: Condens. Matter 9 (7) 58

21 energy absorbtion

22 (less) expected non-linearity: sample heating via dielectric loss qeq π υ ( ω) 8. Mm qeq. J cm υ 3 g KWW (lnτ).3 β KWW log (τ / τ KWW ) heating c p 3 J K cm υ qeq ΔT.K c p

23 predicted heating/temperature effects temperature dependence : 3 A τ ( T ) τ exp kbt ΔT A Δ lnτ T k T B '' 5 5 x.5 A K kb ΔT K. Δ lnτ Δτ.5 τ 5 glycerol T 3 K 3 4 ν [Hz]

24 features of heterogeneous dynamics length scale (cooperativity) domain life time rate exchange time ξ R. Richert, J. Phys.: Condens. Matter 4 () R73

25 DILCTRIC HOL-BURNING IDA AND MTHOD

26 spectrally selective dielectric experiments g KWW (lnτ).3 β KWW.5.. heating log (τ / τ KWW ) g KWW (lnτ).3. β KWW.5 antihole spectrally selective 'heating' τ τ τ 3 ω b. hole log (τ / τ KWW ) ( t) b sin( ωbt) π ( ) ΔQ ω b b B. Schiener, R. Böhmer, A. Loidl, R.. Chamberlin, Science 74 (996) 75 B. Schiener, R.. Chamberlin, G. Diezemann, R. Böhmer, J. Chem. Phys. 7 (997) 7746 R.. Chamberlin, B. Schiener, R. Böhmer, Mat. Res. Soc. Symp. Proc. 455 (997) 7

27 dielectric (t) and electric M(t) hole-burning phase-cycle M M M 3 M 4 φ idle M M φ burn M M 4 () t () t M 3() t + M 8() t M 9 () t ( t ) M ( t ) + M ( t ) M ( t ) 3 () t 4 () t M 5() t + M 6() t M 7 () t ( t ) M ( t ) + M ( t ) M ( t ) , D M 5 M 6 3 M6 M 7 M 8 M 9 U(t) / U() - M M4 M7 M M3 - M time log (t/s)

28 dielectric hole-burning technique: what do we look for? P(t), P*(t)..5 selective heating P*(t) plain heating P*(t) original response P(t) log (t/s) Δ vertical difference Δ H P P () t * ( t) P( t) P ( t) horizontal difference P dp * ( t) P ( t) () t d ln( t s)

29 DILCTRIC HOL-BURNING XPRIMNTS

30 '(ω) dielectric relaxation and retardation in viscous glycerol log (ω / s - ) *(ω) versus M*(ω) '(ω), ''(ω) ''(ω) sample thickness: 6.4 μm '(ω) ''(ω) frequency domain data *(ω,t) T 9.8 K M'(ω) M''(ω) log (ω / s - ) T 83.5 K T 87.3 K T 9.8 K T 95.8 K T.5 K M'(ω), M''(ω) K. Duvvuri, R. Richert, J. Chem. Phys. 8 (3) 356

31 exploiting M(t) for high frequency hole-burning M n (t) n (t) T 83.5 K T 87.3 K T 9.8 K T 95.8 K T.5 K τ M τ t / s..8 n (t) T.5 K variable τ: τ M τ using parallel capacitance τ M τ s + + s par par M n (t), n (t).6.4. τ b M n (t), C par 5 nf M n (t), C par t / s K. Duvvuri, R. Richert, J. Chem. Phys. 8 (3) 356

32 DHB results for glycerol: vertical and horizontal differences 3 ΔM(t) ΔM(t) ΔH(t) T 87.3 K 3 vertical & horizontal signal at: n 6 t w s f b. Hz b 9 b 4 k/cm T 87.3 K ΔH(t) - - t / s K. Duvvuri, R. Richert, J. Chem. Phys. 8 (3) 356

33 DHB results for glycerol: burn-frequency dependence log (t max / s) - - T 83.5 K T 87.3 K T 9.8 K T 95.8 K T.5 K position of vertical hole versus burn frequency - - log (τ b / s) ''(ω) amplitude of horizontal hole versus relative burn frequency ''(ω) ΔH max log (ω τ HN ). ΔH max () ref / b K. Duvvuri, R. Richert, J. Chem. Phys. 8 (3) 356

34 DHB results for glycerol: waiting-time dependence ΔM max (t w ) / ΔM max () T 87.3 K t w / s f b. Hz f b.6 Hz f b. Hz f b.6 Hz f b. Hz f b 6. Hz log (τ mod / s) - pump - wait - probe t w ΔM ΔM max max τ mod τ b ( t ) ( ) w t exp w τ mod T 87.3 K T 9.8 K T 95.8 K T.5 K - - log (τ b / s) K. Duvvuri, R. Richert, J. Chem. Phys. 8 (3) 356

35 DILCTRIC HOL-BURNING MODL

36 calculation of energy loss (heating) for RC network external voltage X ( t) applied : R R R 3 R 4... R N C C C C3 C4 CN dc dt () t C I R () t X () t () t RC C... () Q () t C () t P t C C polarization g(lnτ) τ RC τ dtr dt () R t () t () t I () t T () t R B R c p R TR T () t R RC A kbt power R. Richert, Physica A 3 (3) 43

37 calculation of heating p(t) for RC network amplitude π π 3π 4π..5 p(t) x (t) P(t) t / s () t x steady state case: q eq π υ sin ( ω t) ( ω ), b t nπ ω t nπ ω b b b, otherwise t nπ ω b t > nπ ω b p () t υ + ω ( ω ) [ ( ) + ( ) ( )] b ω ωbτ sin ωbt cos ωbt exp t τ b bτ [ exp( nπ ω τ )] exp( t τ ) b,, t nπ ω t > nπ ω b b R. Richert, Physica A 3 (3) 43

38 dielectric and thermal relaxation times in glycerol RAL IMAG LOG ω N. O. Birge, S. R. Nagel, Phys. Rev. Lett. 54 (985) 674 R R R 3 R 4... R N c p... c p c p 3 c p 4 c p N C C C C3 C4 CN τ τ τ 3 τ 4... τ N... phonon bath c p g(lnτ) β KWW.64 β KWW.65 g(lnτ) τ τ K. Schröter and. Donth, J. Chem. Phys. 3 () 9

39 also assume: tau's are locally correlated τ c p... τ c p τ 3 c p 3 phonon bath τ 4 c p 4... c p τ N R R R 3 R 4... R N c p N Δc α p eff A HN.5 JK lnτ ( T ) 3.7, 75.7 s.95, γ cm HN 3 4 K.58, τ HN 85 s C C C C3 C4 CN... Model: locally correlated structural and thermal relaxation time heterogeneity g(lnτ) τ K.R. Jeffrey, R. Richert, K. Duvvuri, J. Chem. Phys. 9 (3) 65

40 DHB in glycerol: calculated vs. measured results 3 ΔM(t) ΔM(t) ΔH(t) 3 ΔH(t) vertical & horizontal signal at n 6 t w s f b. Hz b 9 b 4 k/cm - - t / s log (t max / s) log (τ b / s) K.R. Jeffrey, R. Richert, K. Duvvuri, J. Chem. Phys. 9 (3) 65

41 DHB in glycerol: calculated vs. measured results ΔM max (t w ) / ΔM max () log (τ mod / s). - - t w / s - t max vs. τ b - - log (τ b / s) K.R. Jeffrey, R. Richert, K. Duvvuri, J. Chem. Phys. 9 (3) 65

42 FURTHR MODL 'PRDICTIONS'

43 DHB: insight from the model [ hole amplitude and position ] 3 4 t w t max τ HN Δ P max 3 3 Δ P M, max Δ P, max log (τ b / s) log (t max / s) log g(ln(τ/s)) t w g HN (ln τ) log (τ b / s) τ HN log (τ / s) Δ P Δ P M Δ P M exp. S. Weinstein, R. Richert, J. Chem. Phys. 3 (5) 456

44 DHB: insight from the model [ hole recovery and accumulation ] 3 5 log (τ mod / s) β τ mod τ HN Δ P Δ P M log (τ b / s) τ mod τ b Δ P Δ P M Δ P M exp. Δ P max 3 log (t max / s) Δ P M, max Δ P, max Δ P Δ P t M w n f b. Hz f b. Hz f b. Hz S. Weinstein, R. Richert, J. Chem. Phys. 3 (5) 456

45 high field impedance

46 model prediction for 3 k/cm steady state harmonic field 8 ' / cm / cm / cm / cm '' log (ωτ HN ) α HN.95 γ HN.58 τ HN 85 s Δ D geo mm s geo 6.4 μm ΔC p.5 J/K/cm 3 T 87.3 K act K '' Δ ln '' log (ωτ HN ) R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (6) 9573

47 continuous harmonic (impedance) measurement (glycerol) µm k/cm SI-6 GN - - μ GLY.56 D 8. Mm T 3 K PZD-7 μ kt '' D.U.T. 5 Ω 5 Ω k/cm glycerol T 3 K 3 4 ν [Hz] R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (6) 9573

48 harmonic field model for high-field steady-state harmonic measurement ( ω t) steady state energy Q Q π ( ωb ) n power P averaged over period : ( ω ) Q ωb b ωb PR π period - averaged excesst dt dt ( dt dt ) b p ( ω ) Δ g( τ ) dτ c Δc g( τ ) for a single domain or branch with τ τ T R R b R P c R p TR τ T ω τ + sin ωbτ applied : n per period : τ T P c : T τ ( ωb ) ωbτ Δωbτ g( τ ) dτ ωbτ Δ ωbτ C Δc g( τ ) dτ + ω τ Δc + ω τ p b T R R p p R p C D : b p (ωτ) m / [ + (ωτ) ]. m m m log (ωτ) dτ configurational heat capacity! p b R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (6) 9573

49 high voltage impedance results for glycerol current model : '' glycerol T 3 K 4 k/cm Δ 7.8 τ HN.8 ms α HN.93 γ HN.65 8 k/cm * lnτ ˆ T lnτ T ( ω) + Δ g( τ ) * ( ω) A Δ ω τ Δc + ω τ + p dτ + iωτ * Havriliak - Negami fit : s [ + ( iωτ ) ] α γ 3 4 ν [Hz] R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (6) 9573

50 high voltage impedance results for glycerol [details of frequency dependent field effects ] Δ ln '' 6 4 k/cm 77 k/cm 4 k/cm 3 4 ν [Hz] R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (6) 9573

51 summary of relevant non-linear effects (~ ) Δ ln ' Δ ln '' Δ ln ' 5.6 Hz - Hz Δ ln '' 4 Hz - 4. khz propylene glycol [ cm - ] Coulombic stress : Fd F s s s () t () t cos( ωt ) dielectric saturation : Δ s 4 Nμ 45 ( van leck) ˆ ( τ ) ( kt ) energy absorption : T e Q, Q C, C A A 3 Δ ωτ Δc + ω τ ( ω) + Δ g( τ ) p A s ( + ) ( + ) ( + ) 3 N s d T e dτ + iωτ * A 4 s 4 ωτ + ω τ S. Weinstein, R. Richert, Phys. Rev. B 75 (7) 643

52 situation with other liquids: propylene carbonate '' propylene carbonate T 66 K Δln'' k/cm Δc p.47 J K- cm -3 4 k/cm 77 k/cm 6 k/cm 4 k/cm 77 k/cm (a) (b) loss increases by up to % at electric field of 77 k/cm ΔT cfg.85 K 3 4 ν / Hz L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)

53 Adam-Gibbs: dynamics - thermodynamics link c τ ( T ) τ exp TS equivalent to FT if : S T K is Kauzmann temperature: c cfg ( T ) ( T ) S ( TK T ) lim S ( T ) T T K c On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming Liquids Adam G and Gibbs J H 965 J. Chem. Phys excess specific heat configurational modes + excess vibrational modes S S exc exc ( T ) is excess entropy : T ( ) fus melt cryst T ΔS C ( T ) C ( T ) fus p p d logt T

54 'forward' dynamic heat capacity experiment S(T) entropy glass T g T K liquid config crystal T m (a) N. O. Birge, S. R. Nagel, Phys. Rev. Lett. 54 (985) 674 C p (T) heat capacity glass T g liquid config (b) T crystal c p... c p c p 3 τ τ τ 3 τ 4 phonon bath c p 4... c p τ N c p N C p '(ω) liquid config ex.-vib. crystal (c) glass T T < T < T 3 log ω g(lnτ) C p ''(ω) (d) τ log ω

55 measuring the configurational heat capacity τ c p... τ c p c p3 τ 3 c p4 τ 4 phonon bath current model : * lnτ ˆ T lnτ T ( ω) + Δ g( τ ) A... c p τ N c pn Δ ω τ Δc + ω τ p dτ + iωτ * C p, cfg / C p, exc Δln'' k/cm PG GLY PD 4PD NMC (a) m PC (b) SORB L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)

56 Fourier analysis of signal with field change voltage..5. current time / s..5. time / s ( ω) Asin( ωt + ϕ) ω R π ω I π π ω π ω sin cos ( ωt) [ Asin( ωt + ϕ) ] Acos ( ϕ) ( ωt) [ Asin( ωt + ϕ) ] Asin ( ϕ) A R + I ϕ arctan π tanδ tan Δϕ I R dt dt L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)

57 time resolved nanoscopic heat-flow tan δ current 7 7 k/cm k/cm ν khz time / ms νδt ( ω) Asin( ωt + ϕ) ω R π ω I π π ω π ω sin cos ( ωt) [ Asin( ωt + ϕ) ] Acos ( ϕ) ( ωt) [ Asin( ωt + ϕ) ] Asin ( ϕ) A R + I ϕ arctan π tanδ tan Δϕ I R dt dt L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)

58 3ω signal during field change A 3ω / A ω current voltage time / ms (periods) ( ω ) Asin( ωt + ϕ ) ω R π ω I π A π ω R π ω sin + I ( 3ωt )[ Asin( ωt + ϕ )] Acos cos ( ϕ ) ( 3ωt )[ Asin( ωt + ϕ )] Asin ( ϕ ) ϕ arctan I R dt dt

Nonlinear Dielectric Effects in Simple Fluids

Nonlinear Dielectric Effects in Simple Fluids Nonlnear Delectrc ffect n Smple Flud NON-LINAR DILCTRIC XPRIMNTS Roma, Delectrc Tutoral 3 Augut 9 Hgh-Feld Delectrc Repone: Coulombc tre Langevn effect nergy aborpton Tme-reolved technque Ranko Rchert