Lecture "Molecular Physics/Solid State physics" Winterterm 2013/2014 Prof. Dr. F. Kremer Outline of the lecture on

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1 Lecture "Molecular Physics/Solid State physics" Winterterm 013/014 Prof. Dr. F. Kremer Outline of the lecture on The spectral range of Broadband Dielectric Spectroscopy (BDS) (THz - <=mhz). What is a relaxation process? Debye relaxation What is the information content of dielectric spectra? What states the Langevin equation Examples of dielectric loss processes

2 The spectrum of electro-magnetic waves UV/VIS IR Broadband Dielectric Spectroscopy (BDS)

3 What molecular processes take place in the spectral range from THz to mhz and below?

4 Basic relations between the complex dielectric function ε* and the complex conductivity σ* The linear interaction of electromagnetic fields with matter is described by one of Maxwell s equations D curl H = j + t D = ε ε E j = σ E 0 (Ohm s law) (Current-density and the time derivative of D are equivalent) = ( ) = + i ε ω ε ε ( ) i σ ω σ σ σ = iωε ε 0

5 Effect of an electric field on a unpolar atom or molecule: In an atom or molecule the electron cloud is deformed with respect to the nucleus, which causes an induced polarisation; this response is fast (psec), because the electrons are light-weight Electric Field

Effects of an electric field on an electric dipole μ : An electric field tries to orient a dipole μ; but the thermal fluctuations of the surrounding heat bath counteract this effect; as

6 Effects of an electric field on an electric dipole μ : An electric field tries to orient a dipole μ; but the thermal fluctuations of the surrounding heat bath counteract this effect; as result orientational polarisation takes place, its time constant is characteristic for the molecular moiety under study and may vary between 10-1s 1000s and longer. Electric field +e -e

7 Effects of an electric field on (ionic) charges: Charges (electronic and ionic) are displaced in the direction of the applied field. The latter gives rise to a resultant polarisation of the sample as a whole Electric field

8 What molecular processes take place in the spectral range from THz to mhz and below? 1. Induced polarisation. Orientational polarisation 3. Charge tansport 4. Polarisation at interfaces

9 What is a relaxation process? Relaxation is the return of a perturbed system into equilibrium. Each relaxation process can be characterized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp( t / τ ). In many real systems a Kohlrausch β law with exp( t / τ ) and a streched exponential β is observed.

10 What is the principle of Broadband Dielectric Spectroscopy?

11 A closer look at orientational polarization: Capacitor with N permanent dipoles, dipole Moment μ Debye relaxation * ε ε s ε = ε + (1 + iωτ ) ε(t)=(p(t) - P ) / ΔE E (t) 1 0 ΔE ε S Δε = ε S - ε 00 orientational polarization ε 00 induced polarization ε s ε ε ( ω) = ω τ 1 + ( ωτ) ε' ε'' 3,6 3,,8,4,0 0,8 0,6 0,4 0, 0,0 ε'=ε s Δε=ε s -ε? ε'=ε? ε'' max ω max ω [rad s -1 ] Time complex dielectric function P( ω) = ε ( ε*( ω, T) 1) E o ε *( ω, T ) ε ( ω) = ε + 1+ ( ω τ ) = + ε s ε ε ( ω) ω τ 1 ( ωτ) ε ε s

12 P. Debye, Director ( ) of the Physical Institute at the university of Leipzig (Nobelprize in Chemistry 1936)

13 What are the assumptions of a Debye relaxation process? 1. The (static and dynamic) interaction of the test-dipole with the neighbouring dipoles is neglected.. The moment of inertia of the molecular system in response to the external electric field is neglected.

14 The counterbalance between thermal and electric ennergy Capacitor with N permanent Dipoles, Dipole Moment μ CH 3 C CH Polarization : 1 N P = μ i + P = μ + P V V C O CH O ε(t)=(p(t) - P ) / ΔE E (t) ε S Δε = ε S - ε 00 orientational polarization ε 00 induced polarization Time ΔE Mean Dipole Moment: Boltzmann Statistics: Mean Dipole Moment W th = kt Thermal Energy μ = Counterbalance 4π 4π CH 3 W el μ E μ exp( )dω kt μ E exp( )dω kt Dipole moment = μ E Electrical Energy The factor exp(μe/kt) dω gives the probability that the dipole moment vector has an orientation between Ω and Ω + dω.

15 The Langevin-function Spherical Coordinates: Only the dipole moment component which is parallel to the direction of the electric field contributes to the polarization μ = π 0 μ μ E cosθ cosθexp( ) kt π μ E cosθ 1 exp( ) kt 0 sin 1 sin θdθ θdθ x = (μ E cos θ) / (kt) a = (μ E) / (kt) Λ(a) Λ(a)=a/3 Langevin function μe a = < 0.1 kt cos θ = 1 a a a a x exp(x)dx exp(a) + exp( a) 1 = = Λ(a) exp(a) exp( a) a exp(x) dx a Langevin function kT E < μ Λ(a) a/3 μ μ = E 3k T a Debye-Formula ε S ε = 1 3ε 0 μ k T N V ε 0 - dielectric permittivity of vacuum = As V -1 m -1

16 Analysis of the dielectric data propylene glycol K 0 K 10 4 ε ε K frequency [Hz] K frequency [Hz]

18 Brief summary concerning the principle of Broadband Dielectric Spectroscopy (BDS): 1. BDS covers a huge spectral range from THz to mhz and below.. The dielectric funcion and the conductivity are comlex because the exitation due to the external field and the response of the system under study are not in phase with each other. 3. The real part of the complex dielectric function has the character of a memory function because different dielectric relaxation proccesses add up with decreasing frequency 4. The sample amount required for a measurement can be reduced to that of isolated molecules. (With these features BDS has unique advantages compared to other spectroscopies (NMR, PCS, dynamic mechanic spectroscopy).

19 1. Dielectric relaxation (rotational diffusion of bound charge carriers (dipoles) as determined from orientational polarisation )

20 Analysis of the dielectric data propylene glycol K 0 K 10 4 ε ε K frequency [Hz] K frequency [Hz]

21 Relaxation time distribution functions according to Havriliak-Negami

22 Analysis of the dielectric data log 10 (1/τ max [Hz]) 4 0 experimental data: propylene glycol bulk VFT-fit: 1/τ=A exp(dt 0 /(T-T 0 )) /T [K -1 ]

23 Summary concerning Broadband Dielectric Spectroscopy (BDS) as applied to dielectric relaxations 1.: BDS covers a huge spectral range of about 15 decades from THz to below mhz in a wide range of temperatures..: The sample amount required for a measurement can be reduced to that of isolated molecules. 3.: From dielectric spectra the relaxation rate of fluctuations of a permanent molecular dipole and it s relaxation time distribution function can be deduced. The dielectric strength allows to determine the effective number-density of dipoles. 4.: From the temperature dependence of the relaxation rate the type of thermal activation (Arrhenius or Vogel-Fulcher- Tammann (VFT)) can be deduced.

24 . (ionic) charge transport (translational diffusion of charge carriers (ions))

25 Basic relations between the complex dielectric function ε* and the complex conductivity σ* The linear interaction of electromagnetic fields with matter is described by Maxwell s equations D 0 D curl H = j + t = ε ε E j = σ E (Ohm s law) (Current-density and the time derivative of D are equivalent) = ( ) = + i ε ω ε ε ( ) i σ ω σ σ σ = iωε ε 0

26 Dielectric spectra of MMIM MePO4 ionic liquid log ε ' log σ' [S/cm] K 58 K 48 K 38 K 8 K 18 K 08 K log f [Hz] log ε'' log σ'' [S/cm] Strong temperature dependence of the charge transport processes and electrode polarisation

27 σ ( T, ω ) for the ionic liquid (OMIM NTf ) OMIM NTf log (σ' /S cm -1 ) K 60 K 40 K 0 K 10 K 00 K 190 K ω c σ 0 log (σ'/σ 0 ) log (ω/s -1 ) log (ω/ω c ) There are two characteristic quantities σ0 and ωc which enable one to scale all spectra!

28 Basic relations between rotational and translational diffusion Stokes-Einstein relation: D = kt ζ D: diffusion coefficient, ζ (=6π η a) : Maxwell s relation: η =τ η G Einstein-Smoluchowski relation: D λ ω c = frictional coefficient,t: temperature, k: Boltzmann constant, a: radius of molecule G : instantaneous shear modulus (~ Pa) η: viscosity,τ η : structural relaxation time λ : characteristic (diffusion) length ω c : characteristic (diffusion) rate Basic electrodynamics and Einstein relation: qd 0 0 c σ = qn μ = n σ ω kt σ 0 : : dc conductivity, μ: mobility ; q: elementary charge, n: effective number density of charge carriers

29 Predictions to be checked experimentally: 1.:.: 3.: ωc = P ωη P = kt π G λ as D λ ω c = σ0 ~ ωc ωc: characteristic (diffusion) rate ωη: structural relaxation rate G : instantaneous shear modulus (~ 10 8 Pa for ILs), a s : Stokes hydrodynamic radius ~λ, k: Boltzmann constant, λ: characteristic diffusion length ~. nm D : molecular diffusion coefficient (Barton-Nakajima-Namikawa (BNN) relation) Measurement techniques required: Broadband Dielectric Spectroscopy (BDS); Pulsed Field Gradient (PFG)-NMR; viscosity measurements;

30 1. Prediction: ωc = P ωη with P ~ 1 Broadband Dielectric Spectroscopy (BDS) Mechanical Spectroscopy ωc: characteristic (diffusion) rate ωη: structural relaxation rate G : instantaneous shear modulus (~ 108 Pa for ILs),

31 Random Barrier Model (Jeppe Dyre et al.) Hopping conduction in a spatially randomly varying energy landscape Analytic solution obtained within Continuous-Time- Random Walk (CTRW) approximation The largest energy barrier determines Dc conduction The complex conductivity is described by: iωτ ( ) e σω= σ0 ln( 1 iωτ ) + e τ e is the characteristic time related to the attempt frequency to overcome the largest barrier determining the Dc conductivity. Hopping time.

32 Random Barrier Model (RBM) used to fit the conductivity spectra of ionic liquid (MMIM MePO4) OMIM NTf log (σ' /S cm -1 ) K 60 K 40 K 0 K 10 K 00 K 190 K ω c σ log (ω/s -1 ) The RBM fits quantitatively the data; ω = 1/ τ c e

33 Scaling of invers viscosity 1/η and conductivity s 0 with (1/η) (Pa -1 s -1 ) /η (Pa -1 s -1 ) log [1/η ] (Pa -1 s -1 ) HMIM BF HMIM Cl 4 HMIM HMIM PF 6 Br HMIM HMIM I I HMIM HMIM BrPF 6 HMIM Cl -8-1 temperature Calorimetric T g Full symbols: 1/η , 0.6 3,6 4, ,4 4, , T 0,8 1,0 g /T T The invers viscosity 1/η 1000/T has (K g /T an -1 ) identical temperature dependence as σ 0 and scales with T g. σ 0 ( S/cm)

34 Correlation between translational and rotational diffusion ω c (s -1 ), ω η (s -1 ) ω c ω η BMIM BF 4 HMIM PF 6 HMIM BF 4 3,6 4,0 4,4 4,8 5, 5,6 1000/T (K -1 ) Measured G inf BMIM BF4 (0.07 GPa) HMIM PF6 (0.07 GPa) HMIM BF4 (0.04 GPa) Stokes-Einstein, Einstein- Smoluchowski and Maxwell relations: Typically: G 0.1 GPa; λ. nm; a s.1 nm; ω P c = P ω kt = 1 G π G λ a η s J. R. Sangoro et al.(009) Phys. Chem. Chem. Phys.

35 Correlation between translational and rotational diffusion log ω c (s -1 ) 6 4 log ω η (s -1 ) Stokes-Einstein, Einstein- Smoluchowski and Maxwell relations: ω P c = P ω kt = G π G λ a η s 1 0 Typically: G 0.1 GPa; λ. nm; a s.1 nm; J. R. Sangoro et al.phys. Chem. Chem. Phys, DOI: /b816106b,(009).

36 Prediction checked experimentally: 1.: ωc = P ωη ωc: characteristic (diffusion) rate ωη: structural relaxation rate G : instantaneous shear modulus kt (~ 10 8 Pa for ILs), P = 1 G π G λ a a s s : Stokes hydrodynamic radius ~λ, k: Boltzmann constant, λ: characteristic diffusion length ~. nm

37 . Prediction: λ DT ( ) = ~ ωc τ ( T) e Pulsed-Field-Gradient NMR Broadband Dielectric Spectroscopy (BDS)

38 Comparison with Pulsed-Field-Gradient NMR and determination of diffusion coefficients from dielectric spectra Based on Einstein-Smoluchowski relation and using PFG NMR measurements of the diffusion coefficients enables one to determine the diffusion length λ : λ DT ( ) = ~ ωc τ ( T) e log D [m s -1 ] D NMR D E 3,5 4,0 4,5 5,0 1000/T (K -1 ) Quantitative agreement between PFG-NMR measurements and the dielectric determination of diffusion coefficients. Hence mass diffusion (PFG-NMR) equals charge transport (BDS)..

39 Comparison with Pulsed-Field-Gradient NMR and determination of diffusion coefficients from dielectric spectra From Einstein-Smoluchowski and basic electrodynamic definitions it follows: ( ) ( ) 0( ) ( ) q ( ) nt ( ) q qdt nt q λ σ T = nt μ T = = ~ ω kt k c T τ ( T) (n(t):number-density of charge carriers; µ(t):mobility of charge carriers) e log D [m s -1 ] D NMR D E 3,5 4,0 4,5 5,0 1000/T (K -1 ) 1. The separation of n(t) and µ(t) from σ0 (T) is readily possible.. The empirical BNN-relation is an immediate consequence.

40 log D [m s -1 ] Separation of n(t) and µ(t) from σ 0 (T) MMIM MePO4 3,5 4,0 4,5 5,0 1000/T [1/K] log N [1/m 3 ] D NMR D E 3,5 4,0 4,5 5,0 1000/T (K -1 ) μ log µ [m V -1 s -1 ] 1.: µ(t) shows a VFT temperature dependence.. n(t) shows Arrheniustype temperature dependence 3.: the σ 0 (T) derives its dependence from µ(t) J. Sangoro, et al., Phys. Rev. E 77, 0510 (008)

41 Predictions checked experimentally:.: D λ ω c = ωc: characteristic (diffusion) rate λ: characteristic diffusion length ~. nm D : molecular diffusion coefficient σ0 : Dc conductivity Applying the Einstein-Smoluchowski relation enables one to deduce the root mean square diffusion distance λ from the comparison between PFG-NMR and BDS measurements. A value of λ ~. nm is obtained. Assuming λ to be temperature independent delivers from BDS measurements the molecular diffusion coefficient D. Furthermore the numberdensity n(t) and the mobility μ(t) can be separated and the BNN-relation is obtained. 3.: σ0 ~ ωc (Barton-Nishijima-Namikawa (BNN) relation)

42 Summary concerning Broadband Dielectric Spectroscopy (BDS) as applied to ionic charge transport 1.: The predictions based on the equations of Stokes-Einstein and Einstein-Smoluchowski are well fullfilled in the examined ionconducting systems..: Based on dielectric measurements the self-diffusion coefficient of the ionic charge carriers and their temperature dependence can be deduced. 3.: It is possible to separate the mobility μ(t) and the effective numberdensity n(t). The former has a VFT temperaturedependence, while the latter obeys an Arrehnius law. 4.: The Barton-Nishijima-Namikawa (BNN) relation turns out to be a trivial consequence of this approach.

43 Kontrollfragen Was ist ein Relaxationsprozess? Wie unterscheidet er sich von einer Schwingung? 103. Welche Annahmen liegen der Debye Formel zugrunde? 104. Was besagt die Langevin Funktion? 105. Was ist der Informationsgehalt dielektrischer Spektren? 106. Nennen Sie Beispiele für dielektrisch aktive Verlustprozesse.

Dielectric spectra dominated by charge transport processes, as examplified for Ionic Liquids. F.Kremer

Dielectric spectra dominated by charge transport processes, as examplified for Ionic Liquids F.Kremer Content 1. Reminder concerning Broadband Dielectric Spectroscopy (BDS).. What are Ionic Liquids (IL)?