DIELECTRIC PROPERTIES OF ION - CONDUCTING MATERIALS F. Kremer Coauthors: J. Rume, A. Serghei,

DIELECTRIC PROPERTIES OF ION -CONDUCTING MATERIALS

F. Kremer

Coauthors: J. Rume, A. Serghei,

The relationship between the complex dielectric function and the complex conductivity

Phenomenology of the conductivity of charge – conductingmaterials

The dielectric properties of zwitterionic polymethacrylate

The dielectric properties of „Ionic Liquids“

Theoretical descriptions of the observed frequencyand temperature dependemce of the complex conductivity

The spectral range of Broadband Dielectric Spectroscopy (BDS) and its information content

for studying dielectric relaxations and charge transport.

Dcurl H j

t

0D E j E (Ohm‘s law)

The linear interaction of electromagnetic fields with matter is described by Maxwell‘s equations

Current-density and the time derivative of D are equivalent

0i

i i

Dielectric spectroscopy

2( )1s

2( )1

s

*

(1 )s

i

Debye relaxation

2.0

2.4

2.8

3.2

3.6

'=s

'==s-

'10

010

110

210

310

410

510

6

0.0

0.2

0.4

0.6

0.8

''max

max

'' [rad s-1]

complex dielectric function

electric field E

E

t0

polarization P

PD( )

P

PS

P

t0

10-2

10-1

100

101

102

103

104

105

106

100

101

102

103

104

105

235 K220 K

205 K

190 K

propylene glycol

´

frequency [Hz]

10-2

10-1

100

101

102

103

104

105

106

100

101

102

103

104

´

frequency [Hz]

Analysis of the dielectric spectra

(sample amount required < 5 mg)

The spectral range (10-3 Hz to 1011 Hz) of Broadband Dielectric Spectroscopy (BDS)

Brief summary concerning Broadband Dielectric Spectroscopy (BDS)

1. The spectral range of BDS ranges from 10-3 Hz to 1011 Hz.

2. Orientational polarisation of polar moieties and charge transport are equivalent and observed both.

3. The main information content of dielectric spectra comprises for fluctuations of polar moieties the relaxation- rate, the type of its thermal activation, the relaxational strength and the relaxation-time distribution function. For charge transport the mean attempt rate to overcome the largest barrier determining the d.c.conductivity and its type of thermal activation can be deduced

Phenomenology of the conductivity of charge – conducting materials

407K417K427K438K448K

458K468K478K488K491K

1 2 5 6-10

-9

-8

-7

-6

4

log ( [H z ])3

log

([S

cm])

’-1

Frequency and temperature dependence of the conductivity of a mixed alkali-glass 50LiF-30KF-20Al(PO3)3

3 4 5 6 7-11

-10

-9

-8

-7

-6

log( [rad/s])

100 mol%

log

(' [

Scm

-1])

396 K 391 K 382 K 373 K 364 K 355 K 346 K

Frequency and temperature dependence of the conductivity of a zwitterionic polymer

T [K]

295285

262250

236

223

210

1 3 5 7

-4

-2

log ( [H z ])

-6

log

([S

cm])

’-1

Frequency and temperature dependence of the electronic conductivity of poly(methyl-thiophene)

p

0 .1 00

0 .0 75

0 .0 50

0 .0 30

0 .0 20

0.015

0.012 5

0 .0 10

2 3 4 5 6 7 8 9

log( [Hz])

1

0

-1

-2

-3

-5

-4

-6

-7

-8

-9

-10

Frequency and concentration dependence of the electronic conductivity of composites of carbonblack

and poly(ethylene terephthalate)

-2 0 2 4 6

0

1

2

3

4

- 396 K - 391 K - 382 K - 372 K - 364 K - 355 K

log

('/

o)

log()

Mixed alkali-glass: Scaling with temperature is possible

-4-6-8-10-12-1

0

1

2

3

4

5

log e 2kT 0

log

’ 0

poly(methyl-thiophene): Scaling with temperature is possible

2 4 8 10 12

log(a [Hz])p

3

2

1

0

composites of carbonblack and poly(ethylene terephthalate): Scaling with concentration is possible

-1 0 1 2 3 4 5 6 7 8

-12

-10

-8

-6

-4

-2

log

(0

[Scm

-1])

log(1/e [Hz])

- 0 mol% - 100 mol% - 200 mol% - conductor-polymer composite - mixed alkali glasses - polymer

The Barton-Nakajima-Namikawa (BNN) – relationship holds for all materials examined:

Experimental findings

In all examined materials the conductivity shows a similarfrequency and temperature (resp. concentration) dependence

There is no principle difference between electron – and ion –conducting materials

The conductivity „scales“ with the number of effective charge-carriers as determined by temperature or concentration

A characteristic frequency exists where the frequencydependence of the conductivity sets in

With increasing number of effective charge-carriers the conductivity increases.

The BNN-relationship is fulfilled

The dielectric properties of zwitterionic poly-methacrylate: poly{3-[N-[-oxyalkyl)-N,N-

dimethylammonio]propanesulfonate}

H C3 C C O O

C H 2

(C H )2 m

x

N +

C H 3

C H 3

(C H )2 3 S O 3

Dielectric data as displayed for the complex dielectric function T)

3 4 5 6 7 8101

102

103

log( [rad/s])

396 K 391 K 382 K 373 K 364 K 355 K 346 K

'

3 4 5 6 710-1

100

101

102

103

104 396 K 391 K 382 K 373 K 364 K 355 K 346 K

"

log( [rad/s])

Dielectric data as displayed for the complex conductivity T)

3 4 5 6 7

-9,5-9,0-8,5-8,0-7,5-7,0-6,5-6,0-5,5-5,0

log( [rad/s])

log

(' [

Scm

-1])

3 4 5 6 7

-9

-8

-7

-6

-5

396 K 391 K 382 K 373 K 364 K 355 K 346 K

log(

" [S

cm-1

])log( [rad/s])

3 4 5 6 7 8

0.00

0.02

0.04

0.06

0.08

0.10

M'

Log (Rad/s)3 4 5 6 7 8

0.00

0.01

0.02

0.03

0.04

M"

log( [rad/s])

Dielectric data as displayed for the complex

electrical modulus M*T) =1/ T)

Dyre‘s random free energy barrier model

Hopping Conduction in a spatially randomly varying energy barrier :

22 2 2

0 arctan1

ln 1 arctan4

e e

e e

2 2

22 2 2

0 ln 1

1ln 1 2 arctan

2

e e

e e

Fits using the Dyre theory „work well“

3 4 5 6 7 8

-9,5-9,0-8,5-8,0-7,5-7,0-6,5-6,0-5,5-5,0

log( [rad/s])

lo

g('

[S

cm-1])

340 350 360 370 380 390 4002

3

4

5

6

7

log

(c,

M, 1

/ e)

T [K]

c

M

1/e

The rates c, M and 1/e nearly coincide and have - over 5 decades - a similar temperature dependence

-1 0 1 2 3 4 5 6

-13

-12

-11

-10

-9

-8

-7

-6

log

( 0 [S

cm-1])

log(1/e [Hz])

- 0 mol% - 100 mol% - 200 mol%

The BNN-relationship holds for varying the charge carrier concentration

Summary

The dielectric properties of the zwitterionic poly-methacrylate:poly{3-[N-[-oxyalkyl)-N,N-dimethylammonio]propanesulfonate} are characterized by a pronounced frequency -and temperature dependence.

It should be analysed in terms of the complex dielectric function T), the complex conductivity T) and the complex electrical modulus M*T) =1/ T)

The data can be well described by Dyre‘s random freeenergy barrier model

The BNN-relation is fulfilled

At low frequencies electrode polarisation effects show up

The dielectric properties of „Ionic Liquids“

BMIM BF4 BMIM SCN

1-n-butyl-3-methylimidazolium thiocyanate

1-butyl-3-methylimidazolium tetrafluoroborate

Temperature dependence

Imaginary and real part of the complex dielectric function are strongly temperature dependent

10-1 101 103 105 107101

103

105

107

109

280 K 270 K 260 K 250 K 240 K 230 K'

Frequency (Hz)

BMIM BF4

10-1 101 103 105 10710-1

101

103

105

107

280 K 270 K 260 K 250 K 240 K 230 K

" Frequency (Hz)

BMIM BF4

Temperature dependence

The complex conductivity of the ionic liquid BMIM BF4 is also strongly temperature dependent

10-1 101 103 105 107

10-6

10-5

10-4

10-3

280 K 270 K 260 K 250 K 240 K 230 K

' (S/c

m)

Frequency (Hz)

BMIM BF4

10-1 101 103 105 10710-9

10-8

10-7

10-6

10-5

10-4

10-3

280 K 270 K 260 K 250 K 240 K 230 K

" (S/c

m)

Frequency (Hz)

BMIM BF4

Broadband dielectric measurements displayedfor the complex dielectric function T)

10-2 100 102 104 106 108 101010-210-1100101102103104105106107

268 K 258 K 248 K 238 K

"

Frequency (Hz)

MMIM Me2PO

4

Thickness= 50µm

10-2 100 102 104 106 108 1010100

102

104

106

268 K 258 K 248 K 238 K'

Frequency (Hz)

MMIM Me2PO

4

Broadband dielectric measurements displayedfor the complex conductivity T)

10-2 100 102 104 106 108 101010-8

10-7

10-6

10-5

10-4

10-3

268 K 258 K 248 K 238 K

' (S/c

m)

Frequency (Hz)

MMIM Me2PO

4

10-2 100 102 104 106 108 1010

10-8

10-7

10-6

10-5

10-4

10-3

268 K 258 K 248 K 238 K

" (S/c

m)

Frequency (Hz)

MMIM Me2PO

4

Scaling with temperature possible

10-10 10-8 10-6 10-4 10-2 100 102 104100

101

102

103

104

105

106

107

e

268 K 258 K 248 K 238 K

'

MMIM Me2PO

4

10-10 10-8 10-6 10-4 10-2 100 102 10410-2

100

102

104

106

268 K 258 K 248 K 238 K

"

e

MMIM Me2PO

4

Scaling with temperature as displayed in terms of the complex conductivity T)

10-10 10-8 10-6 10-4 10-2 100 102 10410-5

10-3

10-1

101

103

268 K 258 K 248 K 238 K

' /0

e

MMIM Me2PO

4

10-10 10-8 10-6 10-4 10-2 100 102 10410-4

10-2

100

102

104

268 K 258 K 248 K 238 K

" /0

e

MMIM Me2PO

4

All data collapse into a single characteristic curve

10-5 10-3 10-1 101 103 105102

104

106

108

s

"

317 mg/ml 44.63 mg/ml 4.09 mg/ml 0.52 mg/ml

NaCl

Scaling with concentration for NaCl solutions as displayed for the complex dielectric function

10-5 10-3 10-1 101 103 105101

103

105

107

109

s

' 317 mg/ml 44.63 mg/ml 4.09 mg/ml 0.52 mg/ml

NaCl

Scaling possible but deviations on the low frequency side

Scaling with concentration for NaCl solutions as displayed for the complex conductivity

10-5 10-3 10-1 101 103 10510-4

10-2

100

s

' /0

317 mg/ml 44.63 mg/ml 4.09 mg/ml 0.52 mg/ml

s is the angular frequency of the minimum in ´´

10-5 10-3 10-1 101 103 105

10-2

10-1

100

" (S

/cm

)

s

317 mg/ml 44.63 mg/ml 4.09 mg/ml 0.52 mg/ml

NaCl

D y r e ‘ s r a n d o m f r e e e n e r g y b a r r i e r m o d e l

H o p p i n g C o n d u c t i o n i n a s p a t i a l l y r a n d o m l y v a r y i n g e n e r g y b a r r i e r :

22 2 2

0 a r c t a n1

l n 1 a r c t a n4

e e

e e

2 2

22 2 2

0 l n 1

1l n 1 2 a r c t a n

2

e e

e e

Fits using the Dyre-model of conduction

10-2 100 102 104 106 108 101010-3

10-1

101

103

105

107

109 268 K 258 K 248 K 238 K 228 K 218 K 208 K 198 K 188 K

"

Frequency (Hz)

10-2 100 102 104 106 108 1010100

102

104

106 268 K 258 K 248 K 238 K 228 K 218 K 208 K 198 K 188 K

'

Frequency (Hz)

The Dyre –model describes the observed frequency-and temperature dependence; additionally electrodepolarization effects show up

Fits using the Dyre-model

10-2 100 102 104 106 108 101010-15

10-13

10-11

10-9

10-7

10-5

10-3

268 K 258 K 248 K 238 K 228 K 218 K 208 K 198 K 188 K

' (S/c

m)

Frequency (Hz)

10-2 100 102 104 106 108 101010-14

10-12

10-10

10-8

10-6

10-4

10-2

268 K 258 K 248 K 238 K 228 K 218 K 208 K 198 K 188 K

" (S/c

m)

Frequency (Hz)

Electrode polarization effects show up already at 100 kHz

The BNN Relation is fulfilled for 0 and e as obtained from Dyre-fits

10-2 100 102 104 106 10810-14

10-12

10-10

10-8

10-6

10-4

MMIM Me2PO

4

EMIM Et2PO

4

0(S/c

m)

1/e(Hz)

Alternative approach: Superposition of a thermally activated d.c. conductivity and „nearly constant loss“ contribution.

10-2 100 102 104 10610-15

10-13

10-11

10-9

10-7

0(T1)

0(T2)

0(T3)

0(T4)

' (S

/cm

)

Frequency (Hz)

e1

e2

e3

e4

e5

0(T5)

220 K210 K200 K190 k180 K170 K

' (S

/cm

) '

01

s

A

p

A:Near constant losscontribution

The BNN relation is a trivial consequence

Activation plots

Both 0 and 1/e show a VFT - dependence

3,6 3,8 4,0 4,2 4,4 4,6 4,8 5,010-14

10-12

10-10

10-8

10-6

10-4

MMIM Me2PO

4

EMIM Et2PO

4

0(S/c

m)

1000 K/Temperature3,6 3,8 4,0 4,2 4,4 4,6 4,8 5,0

10-2

100

102

104

106

108

MMIM Me2PO

4

EMIM Et2PO

4

1/ e(H

z)1000/Temperature (K-1)

Final Summary

The dielectric properties of „Ionic Liquids“ are similar to other ion - conducting systems

They should be analysed in terms of the complex dielectric function T), the complex conductivity T) and the complex electrical modulus M*T) =1/ T)

The data can be well described by Dyre‘s random freeenergy barrier model but as well a superposition a thermally activated d.c.conductivity,a power law and a „nearly constant loss“ contribution

The BNN-relation is fulfilled

At low frequencies electrode polarisation effects showup

A.

A.Serghei

Thanks to Joshua Rume

and

and financial support through the DFG