Analysis and modelling of dielectric relaxation data using DCALC

Transcription

1 Analysis and modelling of dielectric relaxation data using DCALC Michael Wübbenhorst Short course, Lodz, 8 October 2008 Part I dielectric theory, representation of data

2 Schedule 1 st session (8 October 2008): 1) A brief introduction into DRS including dielectric relaxation theory, presentation of dielectric relaxation data in f- and T-domain, empirical relaxation functions and their temperature dependence (~ 45 min). 2) Presentation of the internal structure of DCALC, import/export of data, 2D and 3D graphical presentation capabilities and quick data analysis, example: PVAc. 2 nd session (9 October 2008): 3) General fit-functions in the frequency domain, Fit-functions and options in DCALC, demonstration case PVAc or P2VP 4) "Hands-on" session (Please bring your own laptop with you and have DCALC installed and tested in advance) 3 rd session (date t.b.a.): 5) Advanced numerical algorithm of DCALC: 1D and 2D derivatives, KK-transform, conductivity analysis 6) Fits using ε der and fitting in temperature domain; 2D-fits and temperature depending HNparameters Laboratorium voor Akoestiek en Thermische Fysica Michael Wübbenhorst

3 Outline Electrostatic relations Polarization microscopic approach From macroscopic to macroscopic quantities The complex permittivity, the f-domain d i experiment Relaxation time (spectra) Empiric dielectric functions Temperature dependence of the dielectric properties Experimental implementation if DRS Representation of dielectric relaxation data Laboratorium voor Akoestiek en Thermische Fysica Michael Wübbenhorst

4 Dielectric response Electrostatic relations on a parallel-plate capacitor: Electrical field E between plates in vacuum: E = V d Charges +Q and -Q on surface: Q = ε EA Vacuum capacitance C 0 : 0 C = Q V 0 Dielectric constant of the material ε: C Q+ PA ε = = C Q 0 or ε = ε 0E + ε E 0 P ε 0= As/Vm As [ C] = F ( Farad) = V P : polarisation

5 Dielectric response (2) Total electrical polarisation in a dielectric: ( ε 1ε ) E Ps P + = 0 induced polarisation all dielectric materials with ε > 1 spontaneous polarisation pyroelectricity, piezoelectricity ferroelectricity passive dielectric properties active further: χ el = electrical susceptibility = ε 1 χ el

6 Polarisation microscopic approach P = N α N 0 : concentration of dipoles 0 E L molecular polarisability α E L : local electric field P = N ) E 0( α e + αa + αo orientational atomic electronic α o α e Electronic and atomic polarisability: α a L + _ + No field fast displacements of electrons and atoms at optical frequencies + _ + - E (almost) T-independent property of all nonpolar materials refractive index n > 1

7 Polarisation microscopic approach (2) DRS characteristic time scales α o αe α e α a Relaxation phenomena IR VIS/UV Orientational polarisability

8 Orientational polarisation No field, E=0: Direction of mobile molecular dipoles arbitrary and driven by thermal energy kt no average orientation m=0 μ E>0: Change of angular distribution due to directing electrical field, energy is me L m μ E Effective (induced) dipole moment m: m 2 μ E 3kT L m : molecular dipole moment

9 Orientational polarisation Langevin function x x m e + e 1 μe ( ) L Lx = = with x= x x μ e e x kt L( (x) Lx ( ) saturation Practical (low-field) conditions: E = 1V/50μm T = 300K μ = 2D x μe = L μe ! 3 3kT kt 0.0 linear approximation holds x=æe/kt

10 From microscopic to macroscopic quantities P N ) E = local field external field ( 0 α e + α a + α o) L Clausius-Mosotti relation: ε 1 MW N Aα = ε + 2 ρ 3ε M W : molecular weight ρ: density N A : Avogadro s number 0 local field considerations see e.g. Blythe, Electrical Properties of Polymers 10

11 From microscopic to macroscopic quantities (2) Dielectric experiment capacity C d Aεε = Alternating voltage: d C 0 ε = C C 0 A V ( t) = V e 0 Current response of capacitor C= εc 0 ε V I 0 iωt dv ( t) t) = ε C0 = iωεc V ( t) dt ( 0 90 phase shift

12 From microscopic to macroscopic quantities (3) Non-ideal capacitor: polarisation build-up requires finite time complex dielectric constant ε * = ε' iε" this leads to 2 components of the current: I C = iωc0ε ' V I R = ωc0ε " V imaginary (capacitive) component, real (resistive) component, I and V are out of phase energy dissipation term additional phase lag δ beteen V and I tanδ = ε ε 12

13 Single-Time Relaxation Process thermal agitation Relaxation of P 0 into thermal equilibrium i 1.2 Characteristic time to attain thermal equilibrium: P(t), E(t) E(t) t Pt () = P0 exp τ relaxation time τ 0.2 τ= time 13

14 Single-Time Relaxation Process Statistical nature of single-exponential relaxation E=0 m = t 0 E>>0 time t m > t 0 thermal energy: rotational diffusion of individual dipoles time τ involves many small-step rotations

15 Dielectric relaxations Alternating electrical field at angular frequency ω: complex dielectric constant ε ω ε ω ε ω ' ( ) = ( ) i "( ) All molecules have the same τ Debye relaxation ε' ωτ = 1 ε" (loss) ( ) ' ε ω εs ε " = ε + ε ( ω) ω τ equivalent circuit: τ = RC εs ε = 1 + ωτ 2 2 ωτ ( ) C 0 εs ε R ε CC log(ω)

16 Relaxation time spectra deviation of peak shape from Debye type no single relaxation time behaviour formal approach: Debye equation: Δε ε * ( ω) = ε iωτ distribution in relaxation time L(τ) ( ) ε * ω = ε +Δε lnτ 1+ iωτ L ( τ ) d with L ( τ ) d ln τ = 1

17 Relaxation time spectra molecular reasons: real distribution in molecular potentials causes distribution in relaxation time e.g.: secondary peaks cooperativity it many-body interactions between relaxing units e.g.: α-relaxation (segmental process) t=t 0 t=tt 0 +Δ t

18 Relaxation time spectra model functions Debye equation: Δε ε * ( ω) = ε iωτ ε ω = ε + * Cole/Davidson: ( ) CD asymmetric broadening + Δε ( 1+ iωτ ) CD β ε ω ε * Cole-Cole: ( ) CC symmetric broadening Δε = ( iωτ ) 1 CC β

19 Relaxation time spectra ε ω = ε + * Havliliak-Negami: ( ) HN 2 shape parameters: β: symmetric broadening γ: asymmetry log(ε) Δε ( β 1+ ( iωτ ) ) HN γ empirical function nevertheless, widely used to model dielectric spectra m= β n= -βγ log(ω)

20 Relaxation time spectra (HN-function)

21 Relaxation time spectra (model-free) distribution in relaxation time L(τ) ( τ ) L ε *( ω) = ε +Δε d lnτ 1+ iωτ Computation of L(τ) by numerical techniques possible, e.g. Tikhonov regularization

22 Temperature dependence of the relaxation time Molecular processes usually thermally activated processes Arrhenius law ε' α E β α J β T T dielectric relaxations: gain in orientational polarisability with temperature mechanical relaxations: gain in softness with temperature 22

23 Thermally activated processes Temperature dependence of relaxation time: Arrheniuslaw thermal activation of τ log f α (Hz) max α β τ β-process p E = kt ( ) a T τ exp /T (1/K) T-dependence τ(t) ofα α and β- process in a poly(ether ester) [Mertens et al. 1999, pol. 1a]

24 Thermally activated processes Physical meaning of the Arrheniuslaw thermal activation of a motion over energy barrier Energy U Barrier model: T-independent height of energy barrier E a (activation energy) ΔU typical Arrhenius processes in polymers: rotation angle local (secondary) relaxations in glassy state dynamics in liquid crystalline polymers

25 Dipoles in polymers Some dipole moments of simple molecules Compound Dipole moment Cm Di l t i l H 2 O 6.1 HF 6.4 HCl 3.6 PVC - + HBr 2.6 H C - Cl H CH 4 0 C μ CCl 4 0 C H CO 2 0 Cl NH CH 3 Cl PVDF C 2 H 5 OH 5.7 H + (C 2 H 5 ) 2 O 3.8 F C - C C 6 H 5 Cl 5.7 F C 6 H 5 NO H F Dipole moments in polymers C - F Remember: dielectric response 2 μ

26 Dipoles in polymers Location of dipole moments in the polymer chain: Type A: dipole moment along chain axis Type B: dipole moment perpendicular to chain axis Cl Cl CH 2 C CH 2 C H H poly(vinyl chloride) Type C: dipole moment in side group CH 3 CH 2 CH 2 C C CH 2 H cis-1,4-polyisoprene CH 3 CH 3 CH 2 C CH 2 C O C O C O O CH 3 CH 3 PMMA

27 Experimental techniques for DRS measurement of (complex) capacitance or resistance by e.g. bridge techniques frequency response analyser A dielectric sample cell Z x A V phase sensitive measurement of voltage and current sample configurations: comb electrode: parallel plate config.: E E

28 Experimental techniques for DRS RLC meter HP4284A (Agilent) 20 Hz 1MHz Medium resolution in loss Dielectric spectrometer (Novocontrol) Alpha-Analyser (high resolution) Hz 10 MHz

29 Experimental techniques for DRS Time-domain setup Prof. Yuri Feldman, Jerusalem Up to 10 GHz Time domain data need transformation ti to frequency space

30 Specific application: DRS on ultrathin film "capacitors" Top electrode Polymer film Lower electrode Substrate Clean room for thin film preparation 1. thermal evaporation of lower metal electrode on glass substrates. 2. spin-coating of very dilute (filtred) polymer solutions 3. drying + annealing in vacuum 4. 2 nd evaporation patterned top electrode DRS on ultra-thin films

31 Dielectric instrumentation: Subject of separate lectures Last part of this part: Various representation of complex dielectric spectra DRS on ultra-thin films

32 Representation of relaxation spectra 1) Spectra of the real (ε ) and imaginary part (ε ) of the permittivity Generally: ε ω ε ω ε ω ' ( ) = ( ) i "( ) 2 spectra that carry (almost) equivalent information Kramers-Kronig integral transforms:

33 Representation of relaxation spectra 2) KK-transform numerically computed by DCALC = ε ε KK Note: ε KK lacks conduction term Good reproduction of ε in the peak region Additional features at low frequencies revealed

34 Representation of relaxation spectra 2) KK-transform effect of frequency spacing! less is more! DCALC algorithm optimized for geometric f- series (f, 2f, 4f )

35 Representation of relaxation spectra 3) An approximation of the KK-transform ε deriv loss-free part of ε Good reproduction of ε in the peak region for broad processes For narrow (Debye) processes extra-sharp peak in ε der Additional features at low frequencies revealed

36 Representation of relaxation spectra 3) ε deriv effect of peak shape

37 Representation of relaxation spectra 4) Loss tangent a mixed quantity ε tan δ = ε normalized dielectric ect c loss eliminates dimensional problems peak in tanδ shifted compared to ε

38 Representation of relaxation spectra 5) The Cole-Cole plot ε vs ε Allows easy consistency check between e and e data (KK relation fulfilled) Frequency not explicit displayed Easy graphical inspection of peak symmetry and broadening Commonly used in impedance spectroscopy (Nyquist-plot)

39 Representation of relaxation spectra 5) The Cole-Cole plot Nyquist Diagram General Nyquist diagram Nyquist diagram for impedance

40 Representation of relaxation spectra 5) Dielectric modulus definition Note that tanδ is the same as for ε* M

41 Representation of relaxation spectra 5) Dielectric modulus Imaginary part shows conduction peak in the conduction region, Maximum yields Ohmic relaxation time: Relaxation peaks appear shifted in M -representation This shift is twice of the shift in tanδ because of M

42 Representation of relaxation spectra 5) Dielectric modulus in CC-representation Conduction peak α-peakp α-peak

43 Representation of relaxation spectra 6) Conductivity representation σ ε ε ω = 0 Increasing T

44 Representation of relaxation spectra 6) Conductivity representation filtered data Special function of DCALC: Exclusive display of (nearly) frequency independent σ-values

45 Representation of relaxation spectra More options, e.g. Admittance Impedance Capacitance etc. Further reading: M. Wübbenhorst and J. van Turnhout, "Analysis of complex dielectric spectra. I. Onedimensional derivative techniques and three-dimensional modelling.," J. Non-Cryst. Solids, vol. 305, pp , J. van Turnhout and M. Wübbenhorst, "Analysis of complex dielectric spectra. II. Evaluation of the activation energy landscape by differential sampling.," J. Non-Cryst. Solids, vol. 305, pp , Mario F. García-Sánchez et al., An Elementary Picture of Dielectric Spectroscopy in Solids: Physical Basis, Journal of Chemical Education, Vol. 80, 2003, pp. 1062