The dielectric properties of glassy ion-conducting materials

Size: px

Start display at page:

Download "The dielectric properties of glassy ion-conducting materials"

Bernard Williams
6 years ago
Views:

1 The dielectric properties of glassy ion-conducting materials F.Kremer Co-authors: J.R. Sangoro, A.Serghei, C. Iacob, S. Naumov, J. Kärger

2 Example for ion-conducting glassy materials: Ionic Liquids (ILs) Ionic liquids are salts between an organic cation and a simple anion: BMIM BF BMIM SCN -butyl-3-methylimidazolium tetrafluoroborate -n-butyl-3-methylimidazolium thiocyanate Properties of ILs: Low glass transition temperature ( K) High conductivity at room temperature Low viscosity even at low temperature ( K) Novel solvent Applications of ILs: Medium in fuel cells, super-capacitors, batteries and solar cells Catalysis, extraction and separation processes Lubricants and fuels

3 Structures of some of the ILs studied: HMIM Cl EMIM BF HMIM PF MOIM BF HMIM BF BMIM BF HEMIM BF MMIM MePO MMIM EtPO 3-MEP EtSO3

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 Maxwell s equations D D curl H = j t = σ (Ohm s law) = ε ε E j E (Current-density and the time derivative of D are equivalent) = ( ) = i ε ω ε ε ( ) i σ ω σ σ σ = iωε ε

5 Broadband dielectric spectra of an ionic liquid (MMIM) MePO log ε ' log σ' [S/cm] K 58 K 8 K 38 K 8 K 8 K 8 K 8 8 log f [Hz] Strong temperature dependence of the charge transport processes and electrode polarisation log ε'' log σ'' [S/cm]

6 Two temperature dependent regimes: Electrode polarisation and bulk charge transport The spectral range where electrode polarisation dominates depends strongly on temperature

7 Basic relations between rotational and translational diffusion Stokes-Einstein relation: D kt ζ = D: diffusion coefficient, ζ (=π η a) : frictional coefficient,t: temperature, k: Boltzmann constant, a: radius of molecule Maxwell s relation: η =τ η G Einstein-Smoluchowski relation: D λ ω c = G : instantaneous shear modulus (~ 8 - Pa) η: viscosity,τ η : structural relaxation time λ : characteristic (diffusion) length ω c : characteristic (diffusion) rate Basic electrodynamics and Einstein relation: qd = = c σ qn µ n σ ω kt σ : : dc conductivity, µ: mobility ; q: elementary charge, n: effective number density of charge carriers

8 Predictions to be checked experimentally:.:.: 3.:.: ωc = P ωη = kt P π G a λ s D λ ω c = σ ~ ωc ωc: characteristic (diffusion or hopping ) rate ωη: structural relaxation rate G : instantaneous shear modulus (~ 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;

9 Dielectric spectra of MMIM MePO ionic liquid log ε ' log σ' [S/cm] demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo 8 K 58 K 8 K 38 K 8 K 8 K 8 K demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo 8 8 log f [Hz] log ε'' log σ'' [S/cm] Strong temperature dependence of the charge transport processes and electrode polarisation

10 Scaling of dielectric spectra with characteristic rate ωc log ε' log σ'/σ - demo demo demo demo demo demo slope -.5 demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo -8 - log ω/ω c demo demo demo demo demo demo 8 K 58 K 8 K 38 K 8 K slope - demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo demo -8 - log ω/ω c - log ε'' log σ''/σ All charts collapse into one if scaled with respect to ωc. Identical thermal activation of charge transport and electrode polarisation

11 Description of the data: 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= σ ln( iωτ ) e τ e is the characteristic time related to attempt frequency to overcome the largest barrier determining the Dc conductivity

12 Random Barrier Model (RBM) used to fit the conductivity spectra of ionic liquid (MMIM MePO) σ ' (S/cm) f (Hz) 8 K 58 K 8 K 38 K 8 K 8 K 8 K 98 K 88 K Lines represent fits by RBM The random barrier model quantitatively describes the measured dielectric spectra describing charge transport

13 The rates ωc, ωm and /τe coincide and have - over decades - a similar temperature dependence log ω c [s - ], log ω e [s - ], log ω M" [s - ] 8 - ω M" ω e ω c HMIM PF HMIM BF HMIM I,,5 5, /T [K - ] The different rates ωc, ωm and /τe describe the same mechanism of charge transport

14 The rates ωc, ωm and /τe nearly coincide and have - over 5 decades - a similar temperature dependence 7 log(ω c, ω M, /τ e ) 5 3 ω c ω M /τ e T [K] F. Kremer and A. Schönhals (3), Broadband Dielectric Spectroscopy

15 Scaling of invers viscosity /η and conductivity σ with temperature (/η) (Pa - s - ) /η (Pa - s - ) - log [/η ] (Pa - s - ) HMIM BF HMIM Cl HMIM PF -8 HMIM Br HMIM HMIM I I HMIM HMIM Br PF HMIM Cl - Calorimetric T g Full symbols: /η , 3, T,,,8 5, g /T,8, T The invers viscosity /η /T has (K g /T an - ) identical temperature dependence as σ and scales with T g. σ ( S/cm)

16 Effect of structural variation on viscosity - /η (Pa - s - ) HMIM Cl HMIM Br HMIM I HMIM PF Calorimetric T g 3, 3,,,,8 5, /T (K - ) log [/η ] (Pa - s - ) -8 -,8, T g /T Fluidity (/η) strongly depends on the anion (PF < I < Br < Cl)

17 Correlation between translational and rotational diffusion ω c (s - ), ω η (s - ) 8 - ω c ω η BMIM BF HMIM PF HMIM BF 3,,,,8 5, 5, /T (K - ) Measured G inf BMIM BF (.7 GPa) HMIM PF (.7 GPa) HMIM BF (. GPa) Stokes-Einstein, Einstein- Smoluchowski and Maxwell relations: ω P c = = P ω η kt π G λ a s Typically: G. Gpa; λ. nm; a s. nm; J. R. Sangoro et al. Phys. Chem. Chem. Phys, DOI:.39/b8b,(9).

18 Correlation between translational and rotational diffusion log ω η (s - ) Stokes-Einstein, Einstein- Smoluchowski and Maxwell relations: ω c = P ω η log ω c (s - ) P = kt π G λ a s Typically: G. Gpa; λ. nm; a s. nm; J. R. Sangoro et al. Phys. Chem. Chem. Phys, DOI:.39/b8b,(9).

19 Comparison with Pulsed-Field-Gradient NMR and deter - mination of diffusion coefficients from dielectric spectra From Einstein and Einstein-Smoluchowski relations and assuming the root mean square diffusion distance similar to the order of Pauling diameters, i.e λ ~. nm follows (idea: Joshua Rume Sangoro): log D [m s - ] qd( T) n( T) q λ σ ( T) = nt ( ) qµ ( T) = nt ( ) q = ~ ωe kt kt τ ( T ) D NMR D E 3,5,,5 5, /T (K - ). Quantitative agreement between PFG-NMR measurements and the dielectric determination of the diffusion coefficients.. In ILs mass diffusion (as measured by PFG- NMR) equals charge transport as measured by BDS but in a much wider spectral range. 3. The empirical BNN-relation is an immediate consequence. h

20 Further diffusion coefficients from dielectric spectra HMIM BF -9 D BDS - -3 D PFG NMR D (m s - ) ,8 3, 3,,,,8 5, /T (K - ) Charge transport is mass transport in ionic liquids. Probing Flouride and Hydrogen yield practically the same diffusion coefficient! PFG NMR measurements performed by Sergej Naumov

21 Separation of n(t) and µ(t) from σ (T) log D [m s - ] MMIM MePO ,5,,5 5, /T [/K] log N [/m 3 ] D NMR D E 3,5,,5 5, /T (K - ) µ log µ [m V - s - ]. µ(t) shows a VFT-type temperature dependence. n(t) shows an Arrhenius-type temperature dependence 3. σ (T) derives its dependence from µ(t) J. Sangoro, et al., Phys. Rev. E 77, 5 (8)

22 Separation technique applied to BMIM BF log D (m s - ) BMIM BF log n (m -3 ) 8 7 3,,,,8 5, 3,,,,8 5, 5, /T (K - ) D BDS D NMR µ BDS 3,,,,8 5, /T (K - ) log µ (m V - s - ) Agreement between dielectric and PFG NMR diffusion coefficients J. R. Sangoro,...S. Naumov, J. Kärger,..., J. Chem. Phys. 8, 59 (8)

23 Temperature dependence of Dc conductivity of ILs log σ (S/cm) -8 - log σ (S/cm),5 3, 3,5,,5 5, 5,5 /T (K - ) -8 - HMIM BF (87 K) HMIM PF (9 K) HMIM I (8 K) HMIM Br ( K) HMIM Cl ( K),5,75, T g /T Excellent scaling with calorimetric glass transition temperature

24 Temperature dependence of characteristic diffusion rate log ω e (s - ) - T g by DSC log ω e (s - ) - HMIM BF HMIM PF HMIM I HMIM Br HMIM Cl T g by DSC,,,8 5, 5, /T (K - ),85,9,95, T g /T Characteristic diffusion rate shows a strong dependence on the anion (BF < PF < I < Br < Cl)

25 log σ (S/cm) Universality of charge transport in ionic liquids HMIM Cl HMIM I HMIM Br HMIM BF MMIM Me PO EMIM Et PO BMIM BF HMIM PF DMIM Et PO HEMIM BF MOIM BF 3-MEP EtSO 3-OMA BTA TBP Br log ω c (s - ). Universality despite significant structural variation (imidazolium-, ammonium-, pyridinium and phosphoniumbased ionic liquids). BNN relation holds J. R. Sangoro et al. PCCP DOI:.39/b8b,(9). Despite the structural variation, universal response is observed over decades.

26 Predictions checked experimentally:.:.: 3.:.: ωc = P ωη kt P = π G λ a D λ ω c = σ ~ ωc s ωc: characteristic (diffusion) rate ωη: structural relaxation rate G : instantaneous shear modulus (~ 8 Pa for ILs), a s : Stokes hydrodynamic radius ~λ, k: Boltzmann constant, λ: characteristic diffusion length ~. nm D : molecular diffusion coefficient (Barton-Nishijima-Namikawa (BNN) relation)

27 Electrode polarization (EP) in Ionic Liquids (ILs) experimental features and quantitative description Questions to be adressed:. What is the signature of electrode polarisation (EP) and how can it be analysed quantitatively?. What quantitative information can be deduced from fits of the EP? 3. What is the quantitative model to describe EP?. What information can be deduced from EP?

28 Experimental features of EP in ILs - frequency dependence HMIM PF ( K, µm cell thickness) 7 5 ε'τe ε'' τ e : hoping time σ'' [S/cm] σ' [S/cm] log(f)

29 Experimental features of EP in ILs - frequency dependence 7 5 HMIM PF ( K, µm cell thickness) slope - ε'τe ε'' τ e : hoping time σ : DC conductivity σ'' [S/cm] -5-7 σ -5-7 σ' [S/cm] log(f)

30 Experimental features of EP in ILs - frequency dependence HMIM PF ( K, µm cell thickness) 7 5 slope - f on ε'τe ε'' τ e : hoping time σ : DC conductivity f on : onset of electrode polarization σ'' [S/cm] -5-7 σ -5-7 σ' [S/cm] log(f)

31 Experimental features of EP in ILs - frequency dependence 7 5 slope - f on ε'τe 3-3 HMIM PF ( K, µm cell thickness) slope ε'' τ e : hoping time σ : DC conductivity f on : onset of electrode polarization slope of.8 in ε for f < f on σ'' [S/cm] -5-7 σ -5-7 σ' [S/cm] log(f)

32 Experimental features of EP in ILs - frequency dependence σ'' [S/cm] 7 5 fon ε'τe HMIM PF ( K, µm cell thickness) f max slope -.8 slope - σ ε'' σ' [S/cm] τ e : hoping time σ : DC conductivity f on : onset of electrode polarization slope of.8 in ε for f < f on f max : full development of electrode polarization log(f)

33 Experimental features of EP in ILs - frequency dependence σ'' [S/cm] 7 5 fon ε'τe HMIM PF ( K, µm cell thickness) slope -. f max slope -.8 slope - σ ε'' σ' [S/cm] τ e : hoping time σ : DC conductivity f on : onset of electrode polarization slope of.8 in ε for f < f on f max : full development of electrode polarization slope of. in ε for f < f max log(f)

34 Experimental features of EP in ILs - frequency dependence σ'' [S/cm] 7 5 fon ε'τe HMIM PF ( K, µm cell thickness) slope -. f max slope change in slope log(f) slope - σ ε'' σ' [S/cm] τ e : hoping time σ : DC conductivity f on : onset of electrode polarization slope of.8 in ε for f < f on f max : full development of electrode polarization slope of. in ε for f < f max change in slope of σ for f < f max

35 Dielectric spectra of ionic liquids - temperature dependence (experiment) - log(ε') HMIM-PF log(ε'') 9 K 8 K K K 5 K 7 K 88 K 3 K 3 K log(σ'') [S/cm] log(σ') [S/cm] log(f) [Hz] log(f) [Hz]

36 Dielectric spectra of ionic liquids - temperature dependence (experiment) - log(σ') norm HMIM-PF K K K 5K 7K 88 K 3 K 3 K 33 K log(σ'') norm log(f) norm. perfect scaling with temperature over many orders of magnitude in time

37 Dielectric spectra of ionic liquids - temperature dependence (experiment) - log(f max ), log(f on ) [Hz] HMIM-PF - 5 f max -3 f on σ DC 3-5 /T [/K] 3, 3,,, log(f max ) norm log(f on ) norm. - log(σ DC ) norm , 3, 3, 3, 3,8,,,,,8 /T [/K] log(σ DC ) [S/cm] f max, f on and σ DC scale similarly with temperature

38 Dielectric spectra of ionic liquids- dependence on the length of the sample cell (experiment) - HMIM-PF (5 K) log(σ') [S/cm] log(σ'') norm. - - log(f) norm sample thickness 9 µm 9 µm 7 µm -5 log(σ'') [S/cm] log(f) [Hz]

39 Dielectric spectra of ionic liquids- dependence on the length of the sample cell (experiment) - f on :. ±. f on [Hz] f max [Hz] f max :. ± /L [/µm] No scaling of f on and f max with respect to /L!

40 Dielectric spectra of ionic liquids effect of the electrode material (experiment) - -MIMM-MEPO log(σ') [S/cm] σ'' norm f norm log(σ'') [S/cm] -5-7 brass stainless steel aluminium 3 5 log(f) [H z] Pronounced effect of electrode material!

41 Experimental features of EP in ILs:. EP shows a complex dependence on frequency, temperature, concentration, sample length, material of the electrodes.. EP does not depend below a certain (low) treshold, i.e. V/cm on the applied electric field. 3. EP does not depend on the roughness of the electrodes, as long as: roughness << sample thickness. Over many decades in frequency, scaling with respect to variation of temperature and concentration 5. No scaling with respect to variation of the length of the sample cell and material of the electrodes.

42 What is the microscopic mechanism of EP?

43 Charge transport in the bulk τ e log(σ') -7 σ τ e log(f) [Hz]. hoping time in bulk: τ e E = τ exp kt. BNN-relation: σ ~ log(σ ) τ e 8 log(/τ e )

44 electrode Charge transport at interfaces E >> kt ~ nm electrolyte Due to the coulombic interactions increase by many orders of magnitude in the hoping time at the interface. hoping time at the interface: E E E i Since: b τ e(interface) τ (bulk) e σ this implies: ~ τ σ (interface) e E exp >> kt (BNN relation) << σ (bulk)

45 Microscopic model of sample cell thickness L bulk ( ε, σ, τ e ) L d L d = ε ε ε i * * * meas i b (,, ) ε = f ε σ τ * bulk bulk b i d i (,, ) ε = f ε σ τ * i i i interfaces with: τ e(interface) τ (bulk) e σ (bulk) σ (interface) >>

46 L d L d = ε ε ε i * * * meas i b Electrode polarization ( ) b i ( xε yε ) ( xε yε ) ( x ) ( ) ( ) ε = ε ε xε yε meas b b ( x ) ( ) ( ) ε = ε ε b i b i xε yε ( ) b i ( xε yε ) ( xε yε ) meas b b where: x = with * ε L d d * σ = and iε ω i i i y = b i b i ( ε b) ( ε b) ( ε ) ( ε ) * σ ( ω) σ i i iωτ e = ln( iωτ e) Dyre functions

47 EP - consequences of the model d dω ( ) σ = meas f on σ ε ε ε bi d i L Consequences: σ d f i max ε πε L i. f on ~f max ~σ. f on ~, while f max ~ L L 3. f on ~ d i, while f max ~di ( σ ) d( ε ) d meas meas. f=f max solution for both = and = dω dω ( f ) on 5. σ =πεεs f max

48 Comparison of experiment and calculations - dielectric spectra of ionic liquids experiment calculations log(σ'') [S/cm] K K K 5 K 7 K 88 K 3 K 3 K log(σ'') [S/cm] log(f) [Hz] 3 5 log(f) [Hz]

49 Comparison of experiment and calculations - dielectric spectra of ionic liquids experiment calculations log(σ'') norm log(f) norm. K K K 5K 7K 88K 3K 3K 33K log(σ'') norm log(f) norm.

50 What information can be deduced from EP a novel formula. log(σ'') [S/cm] log(ε') -5-7 f max f on ε S 3 5 log(f) [Hz] σ =? σ =πε ε S ( f ) on f max discovered by Anatoli Serghei et al.

51 Test of the validity of the novel formula σ =πε ε S ( f ) f on max -MIMM-MEPO - log(σ'') [S/cm] -5-7 brass stainless steel aluminium log(σ DC ) [S/cm] -8 - σ DC measured σ DC calculated (brass) σ DC calculated (steel) σ DC calculated (alum) 3 5 log(f) [Hz],,8 3, 3,,,,8 5, /T [/K]

52 Dielectric spectra of ionic liquids - length dependence (MIMM-MePO, experiment and calculations) - Calculations experiment -5-5 log(σ') [S/cm] sample length 9 µm 9 µm 7 µm log(σ') [S/cm] sample length 9 µm 9 µm 7 µm -5-5 log(σ') [S/cm] log(σ'') [S/cm] log(f) [Hz] log(f) [Hz]

53 Dielectric spectra of ionic liquids - length dependence, scaling - experiment calculations log(σ'') norm µm 7 µm log(f) norm. log(σ') norm µm 7 µm log(f) norm. d σ d d ( ) σ i σ = f and f i dω meas on ε L max L ε ε ε πε bi i f ~, while f ~ on max L L

54 Dielectric spectra of ionic liquids - length dependence, scaling - experiment f on :. ±. f on [Hz] f max [Hz] f max :. ± f ~ slope of.5 on L /L [/µm] f ~ slope of. max L

55 Summary for EP. What is the signature of electrode polarisation (EP) and how can it be analysed quantitatively? EP has a peculiar signature in the complex dielectric function and conductivity but as well the length of the sample cell.. What quantitative information can be deduced from fits of the EP? With a novel formula discovered by A. Serghei σ can be deduced from f min and f max of σ (f). 3. What is the quantitative model to describe EP? A model adding the complex impedances at the interface and in the bulk.. What information can be deduced from EP? A detailed understanding of the geometry and the complex dielectric function of the interphase is in reach.

56 Thanks to...dfg for finacial support and to you for attention

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)?