1

DIELECTRIC RELAXATION IN POROUS MATERIALS

Yuri Feldman

Tutorial lecture 5 in Kazan Federal University

2

Initial sodium borosilicate glass of the following composition (% by weight): 62.6% SiO2, 30.4% B2O3, 7%Na2O

heat treatment at 6500C for 100hheat treatment at 4900C for 165himmersion in deionised water

0.5N HCL

drying at 2000Crinsing in deionized water

additional treatment in 0.5N KOH

drying at 2000Crinsing in deionized water

Porous borosilicate glass samples

3

additional treatment in 0.5M KOH

dryingrinsing in deionized water

drying

bithermal heat treatment treatment at 650 0C and at 530 0Cthermal treatment at 5300C

immersion in deionised water3M HCL

rinsing in deionized water

Commercial alkali borosilicate glass DV1 of the following composition (mol.%):

7% Na2O, 23% B2O3, 70% SiO2

4

SamplesPorosity ,

%Pore diameter,

nmPresence of

silica-gelHumidity h,

%

A 38 40-70 With 1.2

B 48 40-70 Very Small 1.4

C 38 280-400 Small 3.2

D 50 300 Very Small 1.6

I 26.5 5.4 With 3.6

II 42.5 88 No 0.63

III 25.5 11 With 3.39

Structure parameters and water content

10-4

10-2

100

102

Perm

ittivity'' []

10-4

10-2

100

102

Perm

ittivity'' []

1

Sample C3 Sample C after heating

Dielectric response of the porous glass materials

6

10-4

10-3

10-2

10-1

100

101

102

Perm

ittiv

ity'' [

]

10-4

10-2

100

102

Perm

ittiv

ity'' [

]

1

3

3-D PLOTS OF THE DIELECTRIC LOSSES FOR THE POROUS GLASS MATERIALS

Sample C

Sample II

7

-100 0 100 200 300

0

20

40

60

80

100

'

Temperature ( 0C )

-100 0 100 200 300

0

10

20

30

40

50

''

Temperature ( 0C )

Low frequency behaviour ~20 Hz

-100 0 100 200 300

6

5

4

3

2

'

Temperature ( 0C )-100 0 100 200 300

10-3

10-2

10-1

100

101

''

Temperature ( 0C )

High frequency behaviour ~ 100 kHz

A

B

CA

B

C

8

10-1 100 101 102 103 104 105 106100

101

102

' ''

', '

'Frequency (Hz)

12

*( ) = B* n-1, >> 1

*( ) = -i0/0

1)

Jonscher

Conductivity

*( ) = / [1 + ( i ) ] +

2) Havriliak-Negami

The fitting model

9

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8

10-7

10-6

10-5

10-4

10-33.3 3.5 3.7 3.9 4.1 4.3 4.5 4.7

Sample I II III Ice

, [s

]

1000/T, [K-1]

A - 50 kJ/mol B - 42 kJ/mol

C - 67 kJ/mol D - 19 kJ/mol

Ice - 60 kJ/mol

I - 64 kJ/mol II - 36 kJ/mol

III - 61 kJ/mol

Ice - 60 kJ/mol

1st Process

4.6 4.8 5.0 5.2 5.4 5.6 5.8

10 -7

10 -6

10 -5

10 -4

10 -33.5 3.7 3.9 4.1 4.3 4.5 4.7

Sample

A

B

C

D

Ice, [s

]

1000/T, [K -1]

10

-17 -16 -15 -14 -13 -12 -11 -10 -9 -8

0.36

0.39

0.42

0.45

0.48

0.51

0.54

0.57

0.60

0.63

A B C D I II III

ln()

SamplesHumidity h,

%

II 0.63

A 1.2

B 1.4

D 1.6

C 3.2

III 3.39

I 3.6

Dependence of the Cole-Cole parameter from ln()

11

170 180 190 200 210 2200.01

0.1

1

A

B

C

D

Temperature, [K]

170 180 190 200 210 220

0.1

1 I

II

III

Temperature, [K]

235 240 245 250 255 260 265 270

60

63

66

69

72

75

Ice

Temperature, [K]

Temperature dependence of the dielectric strength

12

170 180 190 200 210 220 230

101

102

(A)

(B)

(D)

(II)

B(T

)

Temperature, [K]

)()32(

11

111 TBT

Parallel and anti-parallel orientation

170 180 190 200 210 220

6

8

10

12

14

16 B(T)C

B(T)I

B(T)III

B(T

)*10

2

Temperature, [K]

B( T

)

anti-parallel

Temperature

Orientation of the relaxing dipole units

parallel non-correlated

system

13

The symmetric broadening of dielectric spectraThe Empirical Cole-Cole law (1941 )

)(1 i

(1-) / 2

Character of interaction

Temperature

Structure

etc

is a phenomenological parameter

is the relaxation time

?

is the dielectric strength

?

13

N. Shinyashiki, S. Yagihara, I. Arita, S. Mashimo, JPCB,102 (1998) p. 3249

10 20 30 40 50 60

0.75

0.80

0.85

0.90

0.95

1.00 PAA PVA PEI PAIA PEG PVP PVME

, [ps]

What is behind the relationship ()?

How can we use experimental knowledge about and ?For instance does their temperature or concentration dependencies explain the nature of dipole matrix interactions in complex systems?

14

The Traditional Theoretical Models

)]([10 tfD

dtdf

Fractional Cole-Cole equationfor relaxation function f(t)

Anomalous Diffusion

)],([),( 10 rtfMD

dtrtdf

r

Dipole-Matrix interactions

Fractal set

dftMtfdtd t

)()()(0

Due to space averaging both space and time fractal properties are incorporated in parameters .

Continuous time random walk (CTRW) model.

The random Energy Landscape

r

Levy flights

R.Metzler, J. Klafter, Physics Reports, 339 (2000) 1-77W.T. Coffey, J. Mol. Liq. 114 (2004) 5-25R.Hilfer , Phisica A, 329 (2003) 35-40

15

16

Dipole-matrix interaction The symmetric broadening of dielectric spectra

)ln()ln(

N

0

0.04 0.11 0.31 0.83 2.26 6.14 16.70 45.40 123.41 335.46

0.4

0.6

0.8

s

Ryabov et al J. Chem. Phys. 116 (2002) 8611.

)/ln() ln(

2 0 sGd

dftMtfdtd t

)()()(0

Gdss G

R

D20

Fractal set

,0xx

BA

All dependences for different CS can be described by Universal function

0

N

N is the average number of relaxation acts in the time interval t=

is the macroscopic relaxation time 0 is the cutoff relaxation time

- fractal dimension of the relaxation acts in time

,0

0

A

NN

00 ln,ln

xx

A

x0=ln0

x=ln

<0 >0

A is the asymptotic value of fractal dimension not dependent on temperature

, and N depends on temperature, concentration, etc

is a minimum number of relaxation acts

BeN 0

If is a monotonic function

Scaling relations

700 N

18ps10

,0

0

A

NN

Sample C

A0.19 is the fractal dimension of the time set of interactions

Sample Porous Size, nm

Specific porous area, m2/g

Porosity% H,%

C 280-400 9.880 38 3.2

Rich water content

A

x0=ln0

x=ln

The total number of the relaxation acts during the time

-18 -15 -12 -90.40

0.48

0.56

ln

>0

t0 0

During the time of 1 ps, 70 relaxation acts occurs.

The density of the relaxation acts on the time interval

170 180 190 200 210 220

107

108

109

1010

n

Temperature , K

(a)

Nn

A

x0=ln0

x=ln

Sample D

Poor water content

s40 1056.6

531.00 N

-14 -13 -12 -110.52

0.54

0.56

0.58

ln

<0

A=0.495

t0

0

<

Sample Porous Size, nm

Specific porous area, m2/g

Porosity% H,%

D 300 8.74 50 1.2

t0 019

170 180 190 200 210 220

1

1.5

2

2.5

3

3.54

n x1

04

Temperature, K

(b)

20

How can we link the numbers of the relaxation acts in time and the molecular structure, in which they occurred ?

ands,Additional parameters should be considered :which can be incorporated by using the Kirkwood-Froehlich approach ,M

VkTB

s

s 2

0

13

13

2

Kirkwood-Froehlich approach

N

iimM

1

cosNmNM n 122

Temperature

B

Orientation of the relaxing dipole units

anti-parallel parallel

non-correlated

system

170 180 190 200 210 220

194

196

198

200

202

204

206

208

210

212

214

B

To K

anti-parallel parallel

is the average dipole moment of the i -th cellim

<…> indicate a statistical averaging over all possible configurations.

Θ is the angle between the dipole moment of a given cell and neighboring ones, Nn is the number of the nearest cell dipoles.

200.0

210.0

0.40

0.45

0.50

0.55

-16-14

-12-10

-8

ln(B

21

cos122nNmNM

2

0

13

1)( NmVk

TBB mm

,cos1 nm NBTB

For water molecules in porous glasses

mncl B

BNN 1cos

The effective number of the correlated water molecules is

Sample C

Tm195 K

θ is the angle between the dipole moment of a given cell and neighboring ones, Nn is the number of the nearest cell dipoles.

reflect the system state with balanced parallel and anti parallel dipole orientations . The corresponding values of parameters are :

0cos,coscos The maximum conditions:

5.0 mm T sec107.1 6 mm T

Sample C:l The kinetic and structural properties

The CC relaxation process is associated with the anomalous sub-diffusion.

2R

R. Metzler and J. Klafter, Phys. Rep., 339,1(2000).R. Hilfer, Applications of Fractional Calculus in Physics,Ed. By R. Hilfer ,(World Scientific, Singapore,2000).

The time-space scaling relationship

Anomalous sub-diffusion Arrhenius temperature dependence ;exp

kTE

A molkJE 67

kT

ER exp2 is a monotonically decreasing function of temperature throughout the temperature range

mTTAt )1 An anti parallel orientation of the cell dipoles, m, is stipulated by the influence of the porous matrix interface

Two main scales of cluster in the Ice-like layer on the matrix interface

Ll

L2 is the macroscopic scale of the matrix interface area

l 2 is the area of the mesoscopic scale of the Kirkwood-Froehlich elementary unit with an average dipole moment m

22 Ll

At T<<Tm BMLR 222

B

)( BBC mmm

mm Bm

170 180 190 200 210 220-10

-9

-8

-7

-6

-5

-4

-3

ln

Temperature, K170 180 190 200 210 220

-16

-14

-12

-10

-8

ln

Temperature , K

2RTm = 195K

LR

BMLR 222

BBconst mmm

222 LlR

RL

l

mTT mTT

mTmT

2

2

2 LRLN

1B

The Kirqwood-Froehlich cell

BBconst mmm mm

mmm WBBconstW 2;

QF 1 QBBconstH mmm / WQF /2

BBconstQ mmm max

- F1

- F2

- H

25

010

2030

Perm

ittivit

y'' []

Ewa C1 97-06-01 moisture=3.21%

2

220 240 260 280 300 320 340 360 380 40015

20

25

30

35

40

45

50

55

A

B

C

Temperature, [oK]

200 250 300 350 400 450 500 5500

10

20

30

40

50

60

70

80

I

III1

III2

Temperature, [oK]

Second Process

26

L -defect

V* is the defect effective volumeVf is the mean free volume for one defect

N is the number of defects in the volume of system V

0 exp

HkT C ea

HkT

d

pVVf

f

~ exp*

V V Nf for VV

*

f1

N T NHkT

d( ) exp

0

, where ~1

p por f

kTHp a

or exp~

Si

O

Si

OO

Si

Orientation DefectD-defect

0

*1 NVVC

27

Ha is the activation energy of the reorientationHd is the activation energy of the defect formationo is the reorientation (libration) time of the restricted water molecule in the hydrated cluster is the maximum possible defect concentration

2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6

10-5

10-4

10-3

10-2

A B C

, [s

]

1000/T, [K-1]

2.0 2.5 3.0 3.5 4.0 4.510-7

10-6

10-5

10-4

10-3

10-2

I III1 III2

, [s

]

1000/T, [K-1]

Sample Ha

[kJ/mol]

Hd

[kJ/mol]

0

[s]

A 55 39 310-14 910-7

B 54 31 310-15 810-6

C 42 30 910-13 210-5

I 41 29 810-13 510-5

III1 45 33 410-14 110-5

III2 39 22 410-12 210-4

The fitting results for the second process

28

( t / ) ~ e t / , Df = 3, where Df is a fractal dimension

Percolation: Transfer of electric excitation through the developed system of open pores

*

s

dd t tF

-100 -50 0 50 100 150 200Temperature [°C]

020

4060

Perm

ittivi

ty' [

]

Freq. [Hz]=5.10e+03 Freq. [Hz]=1.13e+04 Freq. [Hz]=3.24e+04

Dielectric relaxation in percolation

10-3 10-2 10-1 100

0.0

0.2

0.4

0.6

0.8

Sample A Sample B Sample C

Cor

rela

tion

func

tion

time ( s )

29

The Fractal Dimension of Percolation Pass

Sample AA BB CC DD II IIII IIIIII

Fractal dimension Df 00..9999 11..8899 11..3311 22..55 11..9966 22..44 22..22

30

d D

w A a exp

w : size distribution function

, , A: empirical parameters

VV

p : porosity of two phase solid-pore system

Vp : volume of the whole empty space

V : whole volume of the sample

, : upper and lower limits of self-similarity

D : regular fractal dimension of the system

,D w d1

= /

: scale parameter [,1]

Porous medium in terms of regular and random fractals

31

,

when a << 1, << 1

1

11

1

1

1

d Dd D

1

4 D

Sample

Fractal

dimension

Df

Porosity (%)

( obtained from relative

mass

decrement measurements )

Porosity (%)

( obtained from

dielectric

measurements )

A 0.99 38 33

B 1.89 48 47

C 1.31 38 37

D 2.5 50 68

I 1.96 26.5 49

II 2.4 42.5 63

III 2.2 25.5 56

Porosity Determination (A.Puzenko,et al., Phys. Rev. (B), 60, 14348, 1999)

1 << 3,=d

1 << ,0 1

32O

PercolationThe transition associated with the formation of a continuous path spanning an arbitrarily large ("infinite") range. The percolation cluster is a self-similar fractal.

yz

BC

E D

Q

A xO

1 sDEsb

Static condition of renormalization 1 dDE

dbd

m

DslL

1