Broadband Dielectric Spectroscopy and its Relationship to Other Spectroscopic Techniques

Transcription

1 Broadband Dielectric Spectroscopy and its Relationship to Other Spectroscopic Techniques A. Schönhals BAM Federal Institute for Materials Research and Testing, Unter den Eichen 87, 105 Berlin 6 th International Conference on Broadband Dielectric Spectroscopy and its Applications Satellite Courses, Madrid, Spain

2 Content Introduction, Dielectric Spectroscopy Basics of Relaxation Spectroscopy Examples of Relaxation Spectroscopy Thermal spectroscopy Neutron spectroscopy (Scattering) Application of Relaxation Spectroscopy

3 Dielectric Relaxation Spectroscopy Maxwell s Equations: r rot E = t r B r div B= 0 Electric field Magnetic induction r r r rot H = j + D div D r = ρ t Density of charges Magnetic field Dielectric displacement Current density Material Equation: Small electrical field strengths: ε 0 permittivity of vacuum r D =ε* ε0 r E r r r P D D0 = ( ε* 1) ε0 r E = Polarization Complex Dielectric Function ε* is time dependent if time dependent processes taking place within the sample

4 Dielectric Relaxation Spectroscopy A Special Case of Linear Response Theory Input E(t) System under investigation ε(t) Output P(t) Simplest Case : Step-like Input The polarization measured as response to a step-like disturbance ε(t) = P(t) P ΔE P S E(t); Input ΔE P(t); Output Relaxational Contribution P Instantaneous response Time Time

5 Dielectric Relaxation Spectroscopy Arbitrary input Input E(t) A Special Case of Linear Response Theory System under investigation ε(t) Approximation of the Input by step-like changes Output P(t) ΔE * Δ t Δt E(t); Input dx dt E(t) E(0) + *dt P(t) = P + ε t 0 ε(t ΔE * Δt Δt d E(t') t') dt' dt' Time Basics of Linear Response Theory Linearity Causality

6 Dielectric Relaxation Spectroscopy A Special Case of Linear Response Theory Special input Periodical input (t) = E exp( iωt) = E sin( t) E 0 0 ω Phase shift δ P(t) = P0 sin( ωt + δ) ω - angular frequency Complex Numbers E(t) ; P(t) P( ω) =ε0[( ε iε ) 1]E( ω) Time In-Phase with E(t) 90 -shifted to E(t) Complex Dielectric Function ε* ( ω) = ε'( ω) i ε''( ω) Complex Compliance

7 Dielectric Relaxation Spectroscopy A Special Case of Linear Response Theory Complex Compliance ε* ( ω) = ε'( ω) i ε''( ω) Real Part Imaginary or Loss Part Related to the energy stored reversibly during one cycle Related to the energy dissipated during one cycle ε J ε* J* Complex Plane ε'' tan δ = ε' δ Phase shift ε J

8 Dielectric Relaxation Spectroscopy A Special Case of Linear Response Theory Real Part Loss Part Related to the energy stored reversibly during one cycle Related to the energy dissipated during one cycle Law of energy conservation Relationship between both quantities Kramers Kronig Relationships ε'( ω) ε = 1 π ε''( ξ) ξ ω dξ ε''( ω) = 1 π ε'( ξ) ε ξ ω d ξ

9 Dielectric Relaxation Spectroscopy A Special Case of Linear Response Theory P(t) t d E(t') = P + ε0 ε(t t') dt' Linear Equation Can be inverted dt' ε(t) Output Input System under investigation E(t) P(t) M(t) Electrical Modulus δ(t) Dirac function ε ( t t') M(t')dt' = δ(t) Relationship between ε(t) and M(t) Fourier Transformation M *( ω) ε*(ω) = 1

10 Dielectric Relaxation Spectroscopy A Special Case of Linear Response Theory Thermodynamic quantities are average values. Because of the thermal movement of the molecules these quantities fluctuate around their mean values Correlation function of Polarization Fluctuations < ΔP( τ) ΔP(0) > ( τ) = < ΔP > Φ Spectral Density < ΔP > ( ΔP ) ω = π r P 1 V r = μi Φ( τ)exp(iωτ) dτ Φ( τ) = i < μ (0) μ ( τ) > + i i < ( i μ i i ) i< j > < μ (0) μ Measure for the frequency distribution of the fluctuations i j ( τ) > Fluctuation Dissipation Theorem Φ(τ ) = ( ΔP ) ω = 1 k T 1 kt ε (τ ) 1 Δε ε''( ω) 1 πω Links microscopic fluctuations and macroscopic reactions For a small (linear) disturbance a system reacts only in the way as it fluctuates

11 Dielectric Spectroscopy Normal Mode Relaxation 04.5 K 33.8 K PPG, M w = 4000 g/mol Dynamic Glass Transition, α- relaxation 33.7 K 53.3 K 73. K K log ε'' Coaxial log ( f [Hz]) Reflectometer Time Domain Spectrometer Frequency Response Analysis

12 Dielectric Spectroscopy.. PMPS, M w = 1000 g/mol log ε'' log(f p [Hz]) VFT- equation: log f p A = log fp T T 0? - * ε HN log / T (f[k[hz]) -1 ] HP4191A Alpha-Analyzer ( ω) =ε Δε + (1 + (iωτ HN Mean Relaxation Rate f p Dielectric Strength Δε ) β ) γ

13 Linear Response Theory Input Disturbance x(t) Molecular Motions System under Investigation Material function Output Reaction y(t) Electric field E Polarization P Shear tension σ Shear angle γ Magnetic field H Magnetization M Temperature T Entropy S

14 Basics of Relaxation Spectroscopy Molecular Motions Input Output System under Investigation Disturbance x(t) Reaction y(t) Disturbance Reaction Material function Intensive - independent of V Extensive ~ V Generalized Modulus G(t) Tensile Strain Strain Elastic Modulus Relaxation Extensive ~ V Intensive - independent of V Generalized Compliance J(t) Strain Tensile Strain Elastic Compliance Retardation

15 Input x(t) System under Investigation Output y(t) Type of Experiment Disturbance x(t) Response y(t) Compliance J(t) or J*(ω) Modulus G(t) or G*(ω) Dielectric Electric field E Polarization P Dielectric Susceptibility χ*(ω)=(ε*(ω)-1) Dielectric Modulus Mechanical Shear Shear tension σ Shear angel γ Shear compliance J(t) Shear Modulus G(t) Isotropic Compression Pressure p Volume V Volume compliance B(t) V Compression Modulus K(t) Magnetic Magnetic field H Magnetization M Magnetic susceptibility α*(ω)=(μ*(ω)-1) - Heat Temperature T Entropy S Heat capacity c p (t) Temperature Modulus G T (t)

16 Specific Heat Spectroscopy Input T(t) A Special Case of Linear Response Theory System under investigation c p (t) Output S(t) Temperature Modulated DSC The modulated temperature results in a modulated heat flow Temperature Furnace Thermocouples T(t) = T*t & + Asin( ω* t) The heat flow is phase shifted if time dependent processes take place in the sample C* p (ω)= C p(ω) i C p(ω) Time Ch. Schick, Temperature Modulated Differential Scanning Calorimetry - Basics and Applications to Polymers in Handbook of Thermal Analysis and Calorimetry edited by S. Cheng (Elsevier, p 713. Vol. 3, 00).

17 Specific Heat Spectroscopy Example Temperature Modulated DSC A Special Case of Linear Response Theory Polystyrene Due to heating rate Specific Heat Capacity in J g -1 K T g (q) total c p c p ' T α (ω) c p '' Gaussfit Due to modulation Temperature in K Hensel, A., and Schick, C.: J. Non-Cryst. Solids, (1998)

18 Specific Heat Spectroscopy Example Temperature Modulated DSC A Special Case of Linear Response Theory Glass forming liquid crystal E7 Mixture of different LC s Real Part 4 Frequency Loss factor Phase Transition: Frequency Glass Transition: T N/I = 333 K T g = 11 K c p [Jg -1 K -1 ] 1.6 Heat Flow [mw] mhz 3.33 mhz 6.66 mhz 13.3 mhz 6.6 mhz 53. mhz Phase Transition Glass Transition corrected phase angle / rad mhz 3.33 mhz 6.66 mhz 13.3 mhz 6.6 mhz 53. mhz T [K] T [K] Frequency range : 10 T [ C] -3 Hz. 5 *10 - Hz Heating and cooling rates must be adjusted to meet stationary conditions!

19 Specific Heat Spectroscopy Input T(t) AC Chip calorimetry A Special Case of Linear Response Theory System under investigation c p (t) Output S(t) Pressure gauge, Xsensor Integration 10 mm mm 0.1 mm Differential setup Increase of sensitivity Fast heating and cooling H. Huth, A. Minakov, Ch. Schick, Polym. Sci. B Polym. Phys. 44, 996 (006).

20 Specific Heat Spectroscopy Example AC calorimetry A Special Case of Linear Response Theory PMPS, M w = 1000 g/mol C' [au] 80 f=40 Hz 0.5 δ Corr [deg] T [K] Frequency range : 10 Hz Hz A. Schönhals et al. J. Non-Cryst. Solids 353 (007) 3853

21 Combination of Dielectric Spectroscopy.. PMPS, M w = 1000 g/mol and Specific Heat Spectroscopy 8 Specific Heat 6 4 AC Chip Calorimetry log (f p [Hz]) 0 Dielectric TMDSC / T [K -1 ] Both methods measure the same process. Glassy dynamics of PMPS A. Schönhals et al. J. Non-Cryst. Solids 353 (007) 3853

22 Combination of Dielectric Spectroscopy.. Nematic Isotropic Glass forming liquid crystal E7 1 Questions: log ε'' 0 Temperature dependence of both modes? -1 - δ-relaxation Tumbling-mode log (f [Hz]) Which mode corresponds to glassy dynamics? A.R. Bras et al. Phys Rev. E 75 (007) 61708

23 Combination of Dielectric Spectroscopy.. Glass forming liquid crystal E7 10 T I/N Tumbling-mode log (f p [Hz]) T g The temperature dependence of the relaxation rates of both the δ- and the tumbling mode is curved vs. 1/T. Close to T g the temperature dependencies of both mode seems to collapse. 0 δ-relaxation / T [K -1 ] Derivative analysis A.R. Bras et al. Phys Rev. E 75 (007) 61708

24 Combination of Dielectric Spectroscopy.. Glass forming liquid crystal E7 VFT- Similar equation: to the decoupling of rotational and translation diffusion in glass-forming liquids A log f = log f p p T T Similar to the decoupling of 0 segmental and chain dynamics in polymers Derivative 1/ d log f p 1 = d T 1/ Can be explained by different A averaging of modes with different length scales ( T T ) 0 {d log (f p [Hz]) / dt [K] } -1/ The temperature dependence of the relaxation rates of both the δ- and the tumbling mode follow the VFT-equation. Difference in ΔT 0 = 30 K!!!! T 0,δ T 0,Tumb δ-relaxation Tumblingmode Which mode is responsible for Straight glassy dynamics? Line T [K] A.R. Bras et al. Phys Rev. E 75 (007) 61708

25 Combination of Dielectric Spectroscopy.. Relaxation Map Derivative Plot Glass forming liquid crystal E7 and Specific Heat Spectroscopy log (f p [Hz]) T I/N δ-relaxation Tumbling-mode Stars Thermal 10 data {d log (f p [Hz]) / dt [K] } -1/ / T [K -1 ] T [K] The thermal data agree perfectly with that of the tumbling with regard to the absolute values and its temperature dependence A.R. Bras et al. Phys Rev. E 75 (007) T g Tumbling-mode δ-relaxation Thermal data Tumbling mode is responsible for glassy dynamics!

26 Neutron Scattering Neutrons are parts of the atomic nuclei Isotopes: Elements with the same number of protone but with a different number of neutrons. Examples: Hydrogen, Deuterium, Tritium Neutrons Have the same mass than protons Electrically neutral Have a spin Neutrons can be obtained by nuclear fission processes Atomic reactor, Neutron spallation source

27 Neutron Scattering Theoretical Consideration Neutron Cross Sections, Correlation Functions k i Nuclei ki wave vector of the incoming neutron k f q k f wave vector of the scattered neutron Scattering vector: q = k f -k i θ - scattering angle Momentum transfer: ħq = ħ( k f k i ) Energy transfer: ħω = ħ ( k f k i ) r Elastic scattering: k f k i q = q = sin θ Inelastic scattering: k f k i Time dependence 4π λ 0 Time and spatial information

28 Comparison of Scattering Vector and Energy Ranges of Different Inelastic Scattering Methods r / Å E /mev Raman Brilloui n scatt. Laser PCS X-R PCS Q / Å -1 inelastic X-R NMR t / ps

29 Neutron Scattering Theoretical Consideration Neutron Cross Sections, Correlation Functions The intensity of the scattered neutrons is described by a double differential cross section: r r { iq[ (t) r (0)]} σ 1 k N f r = dt exp( iωt) b jbk exp j Ω hω πh k j,k i k σ = Ω hω hk Solid angle Energy change Position of atom j Position of atom k at time t at time t =0 N Self Correlation Pair Correlation r r {[ b b ] Sinc(q, ω) + b S(q, ω) } k f i Scattering lengths Scattering lenght: r 1 N r r Hydrogen b H S= inc (q, ω) * = 10 dt exp(iωt) exp[ iqrj(0)]exp[iqrj(t)] -1 cm πn j r 1 N r r Deuterium b D S(q, = ω) = * 10-1 cm dt exp(iωt) exp[ iqrj (t)]exp[iqrk (0)] πn j,k Time Domain, Incoherent scattering S r (q,t) 1 N N exp [ r (t) r (0 ] inc = j j ) j r q 6

30 Neutron Scattering A Special Case of Linear Response Theory? Input x(t)=? System under Investigation J(t)=? Output y(t)=? S r q exp 6 [ r (t) r (0 ] r 1 N inc (q, t) = j j ) N j van Hove Correlation function S inc r r r (q, t) =< ρ(q, t) ρ(q) > Density correlation function Correlation function Fluctuation Dissipation Theorem Modulus, Compliance? Thermodynamics (t=0): κ( q) = S(q) k T N Isothermal compressibility S(q,t) is related to a q-dependent Compression Modulus W. Götze in Hansen et al. "Liquids, Freezing and the Glass Transition", North Holland 1990

31 Neutron Scattering IN 6 IN 16 Time of Flight - IN 6 / ILL Energy scale: mev Time scale: ps Backscattering - IN 16 / ILL and BSS Jülich Energy scale: μev Time scale: ns

32 S Inc (ω, 54 ) [a.u.] S inc (Q,t) Neutron Scattering.. IN PMPS 1 K IN 6 IN Å Å Å Å Å Å -1 PMPS - Bulk Examples Energy Transfer log(t [ps]) [mev] T=10 K Resolution of the spectrometer Broadband Neutron Spectroscopy S inc (Q,t) Quasielastic Scattering incoherent intermediate scattering function Spectral Shape of IN16 data similar but energy resolution μev Fourier Transformation of each data set Comparison in time domain

33 Overview of the molecular Dynamics of PMPS by elastics Scans Elastic scans on a backscattering instrument give an efficient overview over the temperature dependent dynamics. Mean square displacement u eff : I el / I 0 Q < u > = exp 3 eff I el elastically scattered intensity Increase of Mobility I 0 totally scattered intensity Analysis Methyl group rotation 0.3 <u > eff [Å ] <u > eff [Å ] Methyl Groups Glass Transition Methyl Groups Analysis Glass transition Harmonic Vibrations T [K] T [K]

34 Data Analysis Methyl Groups PMPS 8protons 3 protons in the CH 3 -group 5 protons in the phenyl ring 3/8 of the scattering is due to localized rotation of the CH 3 -group Analysis of the CH 3 group rotation! IN 6 IN 16 Debye-Waller-factor 1.0 PMPS - Bulk T=10 K Elastic incoherent structure factor (EISF) 0.9 S inc (Q,t) Å Å Å Å Å Å β t SInc(Q,t) = DWF* (1 A)exp τch 3 β=0.7! KWW-function CH3 + A log(t [ps])

35 Data Analysis Segmental Relaxation PMPS Above TFrom g the CH scattering 3 -Analysis results from the Fast Processes (DWF) β β CH3 CH 3 group rotation t t SInc(Q,t) = DWF* (1 A)exp + Aexp Segmental relaxation τ τch 3 seg seg From Dielectric S inc (Q,t) DWF CH 3 Contribution segmental PMPS Bulk T=50 K Q=1.16 Å -1 Segmental τ < τ >= KWW β 1 Γ β Comparison with other experiments Contribution CH 3 -group log (t [ps])

36 Overall Molecular Dynamics PMPS Precise determination of glassy dynamics in a wide temperature range using different probes Neutrons 1 10 CH 3 -Rotation Dielectric log(f p [Hz]) Q Segmental - Thermal / T [K -1 ]

37 Analysis of the Temperature dependence Dielectric data cannot be described by the VFTequation in the whole temperature range VFT- equation: 10 8 log f p = log f p T A T 0 6 Derivative log (f p [Hz]) 4 0 d log f d T p 1/ = A 1 1/ ( T T ) / T [K -1 ] Straight Line, T 0 >0

38 Analysis of the Temperature dependence Derivative-Plot 8 low and high temperature regime T B { log (f p [Hz]) / dt [K] } -1/ T 0,high T 0,Low two VFT - dependencies Crossover temperature T B = 50 K Thermal and dielectric data have the same temperature dependence T [K] red : dielectric data blue: thermal data

39 Temperature dependence Dipole moment of the dielectric relaxation strength Debye Theory Δε = 1 3ε 0 g μ k B T N V Interaction parameter 80 T B Temperature dependence of ε is much stronger than predicted 60 Cooperative effects T*Δε 40 Change of the temperature dependence of ε at T B / T [K -1 ]

40 Temperature dependence of the dielectric relaxation strength The change in the temperature dependence of ε is not much pronounced Plot of ε vs. log f p 0.40 Sharp transition fro a low temperature to a high temperature branch T B The crossover frequency coresponds to T B Δε log (f p [Hz]) A. Schönhals Euro. Phys. Lett. 56 (001) 815

41 Overall Molecular Dynamics PMPS Neutrons 1 10 CH 3 -Rotation Dielectric log(f p [Hz]) Q Segmental - Thermal / T [K -1 ]

42 Dynamics PMPS: Q-Dependence Segmental Relaxation 5 Bulk 5 / β KWW homogeneous 4 4 log (τ KWW [ps]) 3 T=50 K 1 T=70 K T=300 K T=330 K Slope 3 heterogeneous log (Q [Å -1 ]) T [K] Theory : t Sinc(Q,t) = exp( Q Dt) = exp τ β β t Sinc(Q,t) = exp( Q At ) = exp τ τ~ Q τ~ Q / β

43 Overall Molecular Dynamics PMPS Neutrons 1 CH 3 -Rotation 10 Dielectric log(f p [Hz]) Q Segmental Temperature dependence Arrhenius-like E A =8.3 kj/mol In agreement with values found for other polymers Thermal / T [K -1 ]

44 Methyl Group Rotation in PMPS Threefold Potential A jump (Q) = sin( 3 Q r) Q r /8 of the scattering is due to localized rotation of the CH 3 -group A (Q) = (1 c )A(Q) + c App fix fix Structure: c fix =5/8=0.64 Experiment: c fix =0.55 EISF Methyl Methyl groups Q [Å ] Q [Å -1-1 ]]

45 Methyl Group Rotation in PMPS Q - dependence of the Relaxation Time.4 T=10 K Below T g the relaxation times are independent of Q.0 log (τ CH3 [ps]) 1.6 Slope: At T g the relaxation times become Q dependent 1. T=1 K = T g log(q [Å -1 ]) Onset of segmental dynamics

46 Methyl Group Rotation in PMPS Temperature dependence of the EISF K 170 K < T g Additional mobility above T g 1 K = T g EISF 0.8 EISF Methyl groups Dynamical Heterogeneities Islands of Mobility Q=1.81 Å -1 T g T B Q [Å -1 ] T [K] Higher quasielastic scattering Decrease of the EISF above T g Phenylring flip???

47 Conclusion. Precise Determination of the molecular dynamics of PMPS over 17 orders of magnitude in time α-relaxation (glassy dynamics) The temperature dependence of the relaxation times shows a crossover from a low to a high temperature range. Crossover temperature T B = 50K. In both ranges the temperature dependence of the relaxation rates is VFT-like with different Vogel temperatures Also the dielectric strength shows a crossover behavior at the same T B The dependence of the relaxation times on the momentum transfer indicates that the molecular dynamics is heterogeneous even above T B Methyl group Rotation The methyl group rotation can be described by jumps in a threefold potential considering the elastic scattering of the frozen chain. Above T g the EISF becomes temperature dependent and the relaxation times depends on momentum transfer. Thanks to the ILL and Research Center Jülich for enabling the neutron experiments.