Post on 11-Jan-2016
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
Per
mitt
ivity
'' []
10-4
10-2
100
102
Per
mitt
ivity
'' []
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
ittivi
ty'' [
]
10-4
10-2
100
102
Per
mitt
ivity
'' []
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 30010
-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
106
100
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
)*1
02
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
dt
df
Fractional Cole-Cole equationfor relaxation function f(t)
Anomalous Diffusion
)],([),( 1
0 rtfMDdt
rtdfr
Dipole-Matrix interactions
Fractal set
dftMtfdt
d 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
dftMtfdt
d 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
x
x
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
18
ps10 ,
00
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)
N
n
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
3.5
4
nx1
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
,MVk
TBs
s 2
0
1
3
1
3
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(
21
cos122nNmNM
2
0
1
3
1)( Nm
VkTBB 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
kT
EA mol
kJE 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
L
l
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 L
R
LN
1B
The Kirqwood-Froehlich cell
BBconst mmm mm
mmm WBBconstW 2;
QF 1 QBBconstH mm
m / 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 volume
Vf 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
pV
Vff
~ exp*
V V Nf for VV
*
f1
N T NHkT
d( ) exp
0
, where ~1
p por f
kT
Hp a
or exp~
Si
O
Si
OO
Si
Orientation Defect
D-defect
0
*1N
V
VC
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 50 100 150 200
Temperature [°C]
020
4060
Perm
ittivit
y' []
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
V
Vp : 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 D
d 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 x
O
1 sDEsb
Static condition of renormalization 1 dDE
dbd
m
Dsl
L1