Analysis and modelling of dielectric relaxation data using DCALC Michael Wübbenhorst Short course, Lodz, 8 October 2008 Part I dielectric theory, representation of data
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
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
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= 8.85 10-12 As/Vm As [ C] = F ( Farad) = V P : polarisation
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
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
Polarisation microscopic approach (2) DRS characteristic time scales α o αe α e α a Relaxation phenomena IR VIS/UV Orientational polarisability
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
Orientational polarisation Langevin function x x m e + e 1 μe ( ) L Lx = = with x= x x μ e e x kt L( (x) 1.0 0.8 0.6 0.4 0.2 Lx ( ) saturation Practical (low-field) conditions: E = 1V/50μm T = 300K μ = 2D x μe = L μe 5 3.2 10! 3 3kT kt 0.0 linear approximation holds 0 1 2 3 4 5 6 7 8 x=æe/kt
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
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
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
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) 10 1.0 0.8 0.6 0.4 E(t) t Pt () = P0 exp τ relaxation time τ 0.2 τ=1 00 0.0 0 2 4 6 8 time 13
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
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 12 10 8 ε' ωτ = 1 ε" (loss) 10 2 10 1 10 0 ( ) ' ε ω εs ε " = ε + ε ( ω) 2 2 1 + ω τ equivalent circuit: τ = RC εs ε = 1 + ωτ 2 2 ωτ 6 4 10-1 10-2 ( ) C 0 εs ε R ε CC 0 2 10-3 0-4 -2 0 2 4 log(ω) 10-4 15
Relaxation time spectra deviation of peak shape from Debye type no single relaxation time behaviour formal approach: Debye equation: Δε ε * ( ω) = ε + 1 + iωτ distribution in relaxation time L(τ) ( ) ε * ω = ε +Δε lnτ 1+ iωτ L ( τ ) d with L ( τ ) d ln τ = 1
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
Relaxation time spectra model functions Debye equation: Δε ε * ( ω) = ε + 1 + iωτ ε ω = ε + * Cole/Davidson: ( ) CD asymmetric broadening + Δε ( 1+ iωτ ) CD β ε ω ε * Cole-Cole: ( ) CC symmetric broadening Δε = + 1 + ( iωτ ) 1 CC β
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)
Relaxation time spectra (HN-function)
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
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
Thermally activated processes Temperature dependence of relaxation time: Arrheniuslaw thermal activation of τ log f α (Hz) max 6 4 2 α β τ β-process p E = kt ( ) a T τ exp 0-2 3 4 5 1000/T (1/K) T-dependence τ(t) ofα α and β- process in a poly(ether ester) [Mertens et al. 1999, pol. 1a]
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
Dipoles in polymers Some dipole moments of simple molecules Compound Dipole moment 10-30 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 3 4.9 CH 3 Cl 53 5.3 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 2 14.0 H F Dipole moments in polymers C - F Remember: dielectric response 2 μ
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
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
Experimental techniques for DRS RLC meter HP4284A (Agilent) 20 Hz 1MHz Medium resolution in loss Dielectric spectrometer (Novocontrol) Alpha-Analyser (high resolution) 0.001001 Hz 10 MHz
Experimental techniques for DRS Time-domain setup Prof. Yuri Feldman, Jerusalem Up to 10 GHz Time domain data need transformation ti to frequency space
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
Dielectric instrumentation: Subject of separate lectures Last part of this part: Various representation of complex dielectric spectra DRS on ultra-thin films
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:
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
Representation of relaxation spectra 2) KK-transform effect of frequency spacing! less is more! DCALC algorithm optimized for geometric f- series (f, 2f, 4f )
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
Representation of relaxation spectra 3) ε deriv effect of peak shape
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
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)
Representation of relaxation spectra 5) The Cole-Cole plot Nyquist Diagram General Nyquist diagram Nyquist diagram for impedance
Representation of relaxation spectra 5) Dielectric modulus definition Note that tanδ is the same as for ε* M
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
Representation of relaxation spectra 5) Dielectric modulus in CC-representation Conduction peak α-peakp α-peak
Representation of relaxation spectra 6) Conductivity representation σ ε ε ω = 0 Increasing T
Representation of relaxation spectra 6) Conductivity representation filtered data Special function of DCALC: Exclusive display of (nearly) frequency independent σ-values
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. 40-49, 2002. 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. 50-58, 2002. 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