:.
44
i .' ) v )
1“ 3‘; I‘m-1n.
'!

PE ‘ ’
21,, 1;

I
l
"Ii“ v

I §
4‘"
a ‘5

i:

“i: J:
_W
m“.

>
1
. ,
x
,
x
1
1

nu... no.
u

....... u. . u.
..-.., as.
..--_o«- .-

r
u

4»

v «- -
rm.’..,..~u -v
a.“ mu..- .. ,
um”. ”.13."... «y

' no) a

. .vu. .
a“ M-

£5

.4 ,w

n
1..»
.

m luau-07
.m
u

:Nflra,»:e5'&_

4....-
‘-,«c‘~

u4-ww v

.-.-.—-n

train"

 

P5}?

\.
1‘;

i. _:
@c

J
9;? §
.‘4-

uad

a.“

@‘iezse
1h!-

a
.-

L
w:-

“31‘:
Emma

‘t

\u’ ' k‘firh ‘

UNIVERSITY LIBRAR

IIIIIIIIIIIII IIII IIIIIII IIIIIIIIIIIII

3 129I3 010468 597

 

 

 

 

 

 

 

 

 

 

 

 

             

This is to certify that the

dissertation entitled

Transport and Dielectric Breakdown in Composites
with Complex Inclusion Microstructures.

presented by

Heungjin Bak

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Physics flfisffc’70“‘3

mu»;

Major professor

 

 

Date Sep. 7 1994

MSU is an Affirmatiw Action/Equal Opportunity Institution 0- 12771

 

 

 

LIBRARY

Michigan State
University

 

 

 

PLACE N RETUM BOXto roman“ Mom

imam your record.
To AVOID FINES Mum on or baton duo duo.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Transport and Dielectric Breakdown
in Composites with
Complex Inclusion Microstructures

By

Hem-in Bak
A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1994

ABSTRACT

Transport and Dielectric Breakdown in Composites with Complex
Inclusion Microstructures

by
Heungj in Bak

A series of experiments have been performed on the dielectric and breakdown
properties of random continuum composites. The experiments are divided into two
categories: those performed on samples with static interfaces(e. g. steel balls in paraffm
wax) and those with dynamic interface(e. g. polystyrene microspheres in suspension).
The critical parameter for the stainless steel spheres in wax departed considerably from
the predicted parameter. Since the theory is based on the intrinsic conductivity of the
metal being infinite, the departures for the stainless steel spheres were ascribed to this
cause. The higher conductivity Cu "spheres" gave better values, but the critical
concentration of the spheres was much lower than anticipated, possibly due to large
departures from sphericity for the Cu "spheres".

The dielectric breakdown measurements were performed for similar samples stainless
steel particles in wax and compared with theory. The effect of a conducting defect is
more drastic in dielectric breakdown than in capacitance. The statistics describing the
dielectric and breakdown data are different. For the breakdown data the modified
Gumbel distribution and the Weibull distribution have been suggested as possible
distributions. 3‘00 failure tests indicated that at least two breakdown mechanisms were
present, and that the part ascribed to intrinsic breakdown is better described by the
modified Gumbel distribution.

Attempts were made to measure the dielectric properties of colloidal suspensions of
polystyrene spheres. Several methods designed to reduce or correct for electrode
polarization were performed. In a study of the effect of electrode material, we found that
bare Pt electrodes were superior to conventionally recommended Pt black and sand
blasted Pt electrodes.

The experimental results for the real part of the dielectric constant still exhibited
electrode polarization effects below 1 KHz. Other serious problems included time
variation and the separation of AC and DC conductivities necessary for the extraction of
the imaginary part of the dielectric constant.

The experimental results were compared with the theory developed by Sen and Chew.
Large discrepancies in the real part of the dielectric constant seemingly stemmed from the
electrode polarization effect. The results for the imaginary part yielded characteristic
frequencies proportional to the inverse square of the particle radius as expected from

theory.

To my parents

iv

ACKNOWLEDGEMENTS

I wish to thank professor Peter Schroeder for his support and patient guidance
throughout my thesis work. Without his help it would not have been possible to finish this
thesis.

I would like to thank professors J. Cowen, P. Duxbury, T. Pinnavaia and J. Pumplin for
their service as my doctoral guidance committee. I am thankful to Dr. B. Cukier for his
commitment as a substitute for P. Duxbury during his sabbatical leave.

I am also thankful to Dr. M. Dubson for letting me use HP Impedance Analyzer. I
am also indebted to R. Loloee, R. Joy and P. Kaviratna for their help and advice.

I would also like to thank Dr. A. Rudin, the Professor of Chemistry at University of
Waterloo, Canada and his graduate student, G. Shouldice, for providing the polystyrene
samples.

My special thank goes to my parents for their prayer and support. Most of all I
would like to thank my wife, Junghee, and daughter, Jihae, for their love, prayer and

encouragement.

Contents

Motivation ............................................................................................ 1
1. Capacitance ....................................................................................... 4
1.1 Introduction ................................................................................................ 4
1.2 Theory ........................................................................................................ 4
1.2.1 Low loading limit ............................................................................... 6
1.2.2 High loading limit ............................................................................... 6
1.2.3 Frequency dependence ....................................................................... 11
1.3 Experiments and discussion ......................................................................... 12
2. Dielectric breakdown ........................................................................ 29
2.1 Introduction ................................................................................................ 29
2.2 Theory ........................................................................................................ 29
2.3 Experiments and discussion ......................................................................... 32
3. Comparison of capacitance and dielectric breakdown ........................ 38
3.1 Introduction ................................................................................................ 3 8
3.2 Local field distribution ................................................................................ 38
3 .3 Statistics ..................................................................................................... 41

4. Dielectric properties of colloidal suspensions
- Theoretical reviews ........................................................................................ 47

4.1 Introduction ............................................................................................... 47

4.2 Complex permittivity .................................................................................. 47

4.2a Debye equations .................................................................................. 49
4.3 Maxwell-Wagner theory ............................................................................. 50
4.4 Thin double layer approximation ................................................................. 57

5. Dielectric properties of colloidal suspensions

- Experimental attempts .................................................................................... 64
5.1 Introduction ................................................................................................ 64
5.2 Electrode polarization ................................................................................. 64
5.3 Review of other experiments ....................................................................... 72
5.4 Experimental setup ..................................................................................... 76
5.4.1 Capacitance cell ................................................................................. 76
5.4.2 Electrodes .......................................................................................... 77
5.4.3 Microsphere samples .......................................................................... 79
5.4.4 Measurement setup ............................................................................ 82
5.5 Experiments and discussion ......................................................................... 82
5.5.1 Long cell ............................................................................................ 82
5.5. la Efl‘ect of electrode material ......................................................... 84
5.5.2 Small cell ........................................................................................... 9O
5.5.2a Results using Pt black electrode .................................................. 95
5.5.2b Variation of measurement with time ........................................... 99

5.5.2c Sand blasted electrode ................................................................ 105

5.5.2d Results with Pt electrode ............................................................ 111

6. Summary ........................................................................................... 121

6.1 Capacitance and dielectric breakdown ......................................................... 121

6.2 Dielectric properties of polystyrene latex suspensions .................................. 122

Appendix 1: Derivation of Eqs. (2.2.1) and (2.2.2) ................................ 127
Appendix 2: Derivation of equations in Sec.4.4 ..................................... 132

viii

List of Tables

1.1 Conversion table of each equivalent circuit mode .............................................. 18
5.1 Comparison of the previous experimental results ............................................... 76
5.2 The quality factor of 1 mM KCl ........................................................................ 89

5 .3 Comparison of conductivity values obtained from direct measurement(Gm)

and from using analysis(G) ............................................................................... 96

ix

List of Figures

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Two-dimensional examples of model microstructures: (a) a regular array
of metal particles in a continuum background; (b) a site percolation model
composed of metal spheres and insulating spheres and (c) a random
configuration of metal particles in a continuum background ........................

(a) Cross section of a cubic array of conducting spheres of radius a with
distance c between centers. The dotted line is the cross section of a cube
of side 6. (b) Enlarged view of the gap between a plane midway between
two spheres, and one of the spheres ............................................................
Critical behaviors of Eqs.(1 .2. 12)(El) and (1.2.13)(A). Using f.=0.6 .........
Cell used to make paraffin disks .................................................................

(a) and (b) Optical photographs of cross sections of metal loaded paraffin
discs. (d) Copper particles. (d) Stainless steel particles ...............................

(a) Principle of guard ring to assure uniform sample field. (b) Block
diagram of HP4192A impedance analyzer ...................................................
Capacitances of paraflin wax containing (a) copper particles and (b)
stainless steel particles against their volume concentrations. Error bars
indicate minimum and maxrnum values .......................................................

The capacitance ratio, C(O)/C(t) as a function of the volume fraction of
metal inclusions. Experimental values using stainless stee1(Cl) and using
copper(O). Numerical sirnulations(+) calculated using a simple cubic
lattice .........................................................................................................

Test oqu.(l.3.3) for largef. Ifthe Eq.(l.3.3) is correct the data plotted

10
14

15

l6

19

21

1.10

1.12
2.1

2.2
2.3

2.4

3.1

3.2

here should be linear. f.’s used are 0.65 for stainless steel(El) and 0.41
for copper(A). The solid lines are a guide to the eye ..................................

If Eq.(l.3.3) is correct the data plotted here should be linear. The best
fitting parameters are f.=0.65 and or=-0. 15 for stainless steel(El), and
f.=0.41 and a=-0.86 for copper(A). The solid lines are guide to the eye...

(a) Real part and (b) imaginary part of the dielectric constant of paraffin
wax containing stainless particles. Numbers in the legend are the
concentrations of stainless particles ............................................................

The frequency dependence of stainless steel particles ................................ ‘..
A field-enhancing cluster of conducting bonds(———) in a background of
insulating bonds(---) ................................................................................

A schematic digram of breakdown experiment ............................................
(a) The dielectric breakdown field ratio of experimental results on samples
of parafin wax containing stainless steel balls. (b) The dielectric
breakdown field ratio calculated using the square lattice dielectric
breakdown model[2]. Each point is an average over at least 50
configurations of 100 x 100 lattices. Solid lines are a guide to the eye ........
A test of Eq.(2.2.5) with or=3/2(+) and ct=1(l:l). (a) Experimental data
same as that in Fig.2.3(a). (b) Numerical simulation data same as that in

Fig.2.3(b). The full curves are a guide to the eyes. ....................................
The capacitance ratio C(O)/ C( f ) , (E1 and A) and the dielectric

breakdown field ratio E, (f )/ E b ( f ) , (+). (a) The results calculated using

the square lattice dielectric breakdown model. Each point is an average
over at least 50 configurations of 100 x 100 lattices[3]. (b) Experimental
results on samples of paraflin wax containing stainless steel balls(E1) and
copper particles(A). The full curves are a guide to the eye .........................
A cumulative probability of failure of paraflin discs against breakdown

22

23

25

26

31
33

35

36

4O

3.3

4.1

4.2

4.3

4.4
4.5

4.6
5.1

5.2

5.3

voltage .......................................................................................................
Tests of breakdown statistics. (a) If the modified Gumbel distribution
(Eq.(3.3.3)) is correct the data as plotted here should be linear; (b) If the
Weibull distribution (Eq.(3.3.2)) is correct this way of plotting the data
should yield a linear relationship. The full curves are a guide to the eye.
The data are from 300 samples of pure paraflin(0.3mm thick) .....................
Real (8') and imaginary (8") parts of the complex dielectric constant of a
suspension of polystyrene particles as a firnction of frequency: particle
diameter 0.188um, particle concentration 30% ..........................................
(a) The model of a binary heterogeneous system. (b) The equivalent circuit
of the system ..............................................................................................
Real part(EI) and imaginary part(A) of dielectric behavior of Eq.(4.3.4). In
this plot a. = 8030, 8,, =230, k, = 1x10’5, k, =1.5x10‘°, andjE3/500 are
used ...........................................................................................................
Wagner's model for a dilute suspension of spherical particles ......................
(a) A negatively charged particle in ionic solution acquires positively
charged counterions in the absence of an external electric field. (b) In the
presence of an external electric field E() the charge distribution in the
double layer changes and polarization charges develop within the particle...
Dielectric behavior of Eq.(4.4.9) ................................................................
Idealized view of the polarization due to dipoles and ions: (a) unpoiarized;
(b) orientation and conduction begins; (c) firlly polarized. In actual
materials, both processes can occur simultaneously .....................................
The apparent dielectric constants of several liquids which are difi‘erent in
conductivity ...............................................................................................
(a) Equivalent circuit of dielectric cell. (b) Equivalent circuit of the total
measured addmittance ................................................................................

xii

43

44

48

51

53
55

58

62

66

67

68

5.4
5.5

5.6

5.7

5.8

5.9

5.10

5.11

5.12

5.13

5.14

Cylindrical teflon sample cell S with metallic coaxial shield M .....................
Components of the apparent dielectric constant as function of electrode
distance ......................................................................................................
Dielectric properties(real part(Cl) and imaginary part(A)) from (a) Springer
and Lyklema and (b) Lirn and Franses (c) Myers and Saville .......................
Long cylindrical dielectric cell. H and L are the high and low terminal. G
is ground. EU and EL are the upper and low electrodes. M is the
micrometer .................................................................................................
Titration data of (a) 213.8nm (b) 364.1nm (c) 367.7nm (d) 624.3nm
suspensions ................................................................................................
(a) Four terminal pair measurement principle. (b) Actual four terminal pair
connections ................................................................................................
Apparent dielectric constants of (a) distilled water and (b) 1mM KCl
solution using brass electrode .....................................................................
Apparent dielectric constants(l:l). (a) Real dielectric constant(O) after
subtraction of stray efi‘ects of distilled water at IOOKHz. (b) Real
dielectric constant(V) afier subtraction of electrode efi‘ects of 1mM KCl at
lKHz. (c) Real dielectric constant(+) after subtraction of stray effects of
1mM KCl at IOKHz ...................................................................................
Apparent dielectric constants of distilled water at (a) lKHz and (b)
IOOKHz using brass(El), p1atinum(0), and platinum b1ack(A) electrodes,

respectively ................................................................................................
Conductance vs. V! of 1mM KCl solution using Pt(A) and Pt black(O)

electrodes at lOOKHz. The solid line is a guide to the eye ..........................
Dielectric cell with guard ring arrangement. H and L are the high and low
terminals respectively. G is ground, and EU and EL are the upper and

lower electrodes .........................................................................................

xiii

7O

73

75

78

81

83

85

86

87

91

92

5.15

5.16

5.17

5.18

5.19

5.20

5.21

5.22

5.23

5.24

5.25

(a) A detailed electric circuit of the cell. (b) A measured equivalent circuit.
The subscripts r, e, m stand for residual, electrode and measure,
respectively. R, is lead wire resistance ........................................................
Real part(Cl) and imaginary part(A) of dielectric constants of (a) distilled
water and (b) 624.3nm suspensions with platinum black electrode ..............
Real part(EI) and imaginary part(A) of dielectric constants of (a)367.7nm,
(b)364. lnm and (c) 213 .8nm suspensions with platinum black electrode .....
Real part(EI) and imaginary part(A) of dielectric constant of (a) distilled
water and (b) 1mM KCl solution with brass electrode covered with tape. . ..
Real part(CI) and imaginary part(A) of dielectric constants of (a) 367.7 nm
and (b) 364.1nm suspensions with brass electrode covered tape ..................
Time variation of dielectric behavior of 1mM KCl solution using small cell
with Pt electrodes .......................................................................................
Time variation of capacitance of 624.3nm suspensions at (a) 5mm and (b)
7.6mm using Pt electrodes. For clarity only the data for 3 of the 5
frequencies studied are shown ....................................................................
Time variation of the conductance of 624nm suspensions at (a) 5mm and
(b) 7.6mm using Pt electrodes. For clarity only the data for 3 of 5
frequencies studied are shown ....................................................................
Time variation of 213.8nm suspensions with sand blasted electrodes at
1.9mm separation. For clarity only the data for 3 of the 5 frequencies
studied are shown .......................................................................................
Time variation of capacitance of 624.3nm suspensions at (a) 0.6mm and
(b) 1.9mm using sand-blasted Pt electrodes. For clarity only the data for 3
of the 5 frequencies studied are shown .......................................................
Time variation of conductance of 624.3nm suspensions at (a) 0.6mm and

(b) 1.9mm using sand-blasted Pt electrodes. For clarity only the data for 3

xiv

93

97

98

100

101

102

103

104

106

107

5.26

5.27

5.28

5.29

5.30

5.31

5.32

5.33

5.34

of the 5 frequencies studied are shown .......................................................
Time variation of the capacitances of suspensions of (a) 213.8nm, (b)
364.1nm, (c) 367.7nm and (d) 624.3nm using sand-blasted Pt electrodes at
0.05" separations ........................................................................................
Time variation of the conductances of suspensions of (a) 213.8nm, (b)
364.1nm, (c) 367 .7nm and (d) 624.3nm using sand-blasted Pt electrodes at
0.05" separations ........................................................................................
(a) Real parts , (b) conductivities and (c) imaginary parts of suspensions
using sand-blasted Pt electrodes .................................................................
(a) The conductivity of 367. 7nm suspensions vs. frequency. (b) The
imaginary part of 367 .7nm suspensions with choice of different DC
conductivity. The numbers in the legend correspond to the frequencies
where the DC conductivity was taken .........................................................
Dielectric behavior of 624.3nm suspension using small cell with Pt
electrodes ...................................................................................................
Dielectric behavior of 367.7nm suspension using small cell with Pt
electrodes ...................................................................................................
Dielectric behavior of 364.1nm suspension using small cell with Pt
electrodes ...................................................................................................
Dielectric behavior of 213.8nm suspension using small cell with Pt
electrodes ...................................................................................................
The characteristic frequencies vs. reciprocals of the square of the particle

diameter .....................................................................................................

108

109

110

112

113

115

116

117

118

119

Motivation

In the accompanying flow diagram we give the original sources of interest in the
work of this thesis. On one side there was the interest in the clay-water system originating
from prior work in the chemistry and physics departments, and on the other the theoretical
interest of Dr. Duxbury. The explanation of the dielectric behavior of the clay water
system is not easy because the platelet shape of the clay particle is difficult to deal with
mathematically. Thus it is better to study simple ideal systems which are mathematically
tractable. The lessons learned from these systems may be then be applied to the
complicated systems. We consider two interface systems.

First for a static interface system, where there is no mass transport involved, we
made metal loaded dielectric samples. Microspheres in a continuum background
constitute the simplest system to deal with mathematically. In contrast with percolation
theory, in which mathematical inclusions can overlap, real systems have a hard core
repulsion between inclusions. The object was to study the effect of such hard core
repulsion on the capacitance of metal particles in a pure dielectric. In dielectric
breakdown the concern is with low concentrations where the difference between
percolation theory and hard core repulsion is minimal. The dependence on local field
distribution provides a great contrast with the capacitance treatment. The object was to
study the contrasting relations using the guidance of Duxbury theory.

Second for a dynamic interface system we studied the properties of polystyrene
spheres. in ionic solution. This is a prototypical model for the effect of dynamic interfaces
on dielectric response. Surprisingly despite recent theoretical interest, the only

comprehensive measurement on this system dates from 1962. Our experimental program

is to first control the polarization effect at the electrodes, and then to measure the
dielectric response of the polystyrene system as a firnction of frequency, ionic strength of

the solution and type of polystyrene surface functional group.

 

 

Structure-Property Relationships
in
Random Continuum Composites.

 

 

 

 

 

Clay-Water
Systems

 

 

Theory of Composite];

 

 

 

 

Simple "Ideal" Systems.
Proposed Subject matter for this Thesis

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Static Interfaces Dynamic Interfaces
Steel Balls in Paraffin Polystyrene Microspheres
in Suspension
Capacitance Dielectric
Breakdown 1.Repeat Earlier Work
. . . 2.Modify Theory for
Cntrcal Behavrour . High Concentrations
Hard Core Interactions
between Inclusions 3.Chemical Effects of
Surface Groups

 

 

 

 

 

 

 

Extreme Statistical
Behaviour Distributions

 

 

 

 

 

 

 

Chapter 1

Capacitance

1.1 Introduction

There is an enormous variety of physical systems in which random inhomogeneities
occur. It is not therefore surprising that the electrical properties of randomly
inhomogeneous materials have been of interest for a long time. One way of studying the
electrical properties of random systems is to investigate the capacitance of such a system.
Dielectrics which contain metal inclusions are usefitl for this purpose.

In this chapter we review the theoretical predictions regarding the capacitance of such
dielectrics. To test the theoretical predictions we describe experiments on discs of paraffin

wax which contained stainless steel particles and copper spherical particles.

1.2 Theory

When a metal inclusion is added to a simple insulator the capacitance of the system is
increased. Fig.1.] shows three models of rnicrostructures we consider. Fig. 1. 1(a)
illustrates a regular array of metal particles. Fig.1.1(b) shows a mixture of two different
kinds of particles, which can randomly occupy the available sites. A crystalline random
alloy fits in this category. Fig. 1.1(c) depicts a random distribution of inclusions in a
continuum host. We consider the capacitance of a system consisting of metal spheres

embedded in parafiin wax, so that a model of the form shown in Fig. 1.1(c) is appropriate.

 

 

 

 

 

 

 

 

 

 

(C)

Fig. 1 . 1: Two—dimensional examples of model microstructures: (a) a regular array of metal
particles in a continuum background; (b) a site percolation model composed of metal

spheres and insulating spheres and (c) a random configuration of metal particles in a

continuum background.

 

The effective capacitance of a metal loaded dielectric is proportional to the dielectric
constant of the background and depends upon f, the fi'actional volume occupied by the

spheres.

1.2.1. Low loading limit
At a low concentration of inclusions, interactions between inclusions are negligible so
the detailed form of the rnicrostructure is not important. When an inclusion is embedded
in a dielectric medium the effective dielectric constant can be obtained by using the
Clausius-Mossotti approximation[ 1], i.e.,
e - a, a, - e,

= _._, 1.2.1
8+2r»:l fs,+23l ( )

 

where e is the efi‘ective dielectric constant, a, and 82 are the dielectric constants of the

background and the inclusion respectively and f is the volume fraction of the inclusions.
(This equation is identical to Maxwell-Wagner equation developed in Section 4.3.) In the
case of conducting inclusions like metal, 82 >> 81 , and we can approximate Eq.(1.2. 1) for
the special cases2 —> oo , i.e.,[2]

8'“ =f. (tan
a+2g

 

For small f the ratio of dielectric constants becomes
§L~1-3f. (r23)
8

This equation indicates that the ratio of dielectric constant varies linearly with the

inclusion concentration.

1.2.2 High loading limit
As the volume fraction of metal particles is increased one must consider the
interactions. This is especially important near the percolation point at which the

inclusions first form a connecting path across the sample. Near that point the spheres are

very close to each other, and are separated by a very narrow neck. Duxbury argues that, if
the neck size distribution is narrow, critical behavior in the same universality class as that
of a regular array of spheres might be observed[3]. If this is true the critical behavior can
be obtained in the following way.

We consider a cubic array of perfect conducting, identical spheres embedded in a
continuum background as depicted in Fig. 1.2(a). Fig. 1.2(b) shows the enlargement of the
critical part of a unit cell. The distance between the centers of two spheres is denoted by c
and a is the radius of the sphere. When an external field is applied no current flows across
the symmetry planes parallel to the field (dotted in Fig. 1.2(a)). Since we consider the
volume fraction of spheres near the random packing point, spheres are nearly touching.
Therefore most of the current flows across the narrow gaps separating adjacent spheres
along the z-axis. The capacitance between two spheres is like that between two plates of
non-uniform separation. [4] If we neglect the charge in the region r > a the capacitance

can be obtained approximately fiom

1 Zrmir
2!: c/2-x/az—r2

 

Near the random packing point the gap between spheres is very narrow so that the
capacitance between two spheres is dominated by this narrow gap. Thus we may

approximate
C=1tzre rdr 2
° c/2-a+r /Za

 

z —1rsaln[c/20- 1] (1.2.5)
In the above equation we have neglected terms like In a which remain finite when c/2a
approaches unity. The volume fi'action is given by f = (47r/3)(a/c)3. Eq.(l.2.5) can be

rewritten as
C a: —rcea1n[-l—(4rt/3 f)* —- 1] (1.2.6)

This equation diverges at f = 7t/6 so that we may write this

 

.................

.................

 

 

 

(b)

Fig. 1.2: (a) Cross section of a cubic array of conducting spheres of radius a with distance
c between centers. The dotted line is the cross section of a cube of side c. (b) Enlarged

view of the gap between a plane midway between two spheres, and one of the spheres.

C = -1taaln[(1t/6-f)] (1.2.7)
This last form is valid only when 7r/ 6 — f << 1. Similarly we can get the capacitance of

the array of identical conducting discs. The diameter of each disc is a and the distance
between two discs is c. We then calculate the capacitance between two adjacent discs like

Eq.(l.2.4), i.e.,

C dx

=-;- 5 a 1 . (1.2.8)
“ Ja‘-x‘+c/2

This integral can be done by introducing new variable x = a cos¢ and we get

Car—c- 2 ., c/2+a
20 c2/4—a2 c/2—a

 

(1.2.9)

When c/2a approaches unity
Ira

‘Icz/4—a2

This equation can be rewritten in terms of volume fraction f = 7r(a2/cz) as

(1.2.10)

Ca:

1
sz’L 1 (1.2.11)

c 7r/4— f
The Eqs.(1.2.7) and (1.2.11) are valid only if c/Za approaches unity. Thus we may

express the critical behaviors:

two dimension

C( f) z C(o)(f. — f)"/2 ~ (1.2.12)
three dimension
C(f) as -C(O)ln(|f. -f|) (1.2.13)

Here f. is the volume fi'action at which the capacitance diverges. The critical behaviors of
these equations are illustrated in Fig.1.3. The two dimensional model gives more rapid
divergence than does the three dimensional one near the critical point. But we will see

that it is very dificult to verify these critical behaviors experimentally due to sample to

sample variability in f.. This is because the metal spheres do not achieve an equilibrium or

liquid-like structure, but are in a random packing state. There are two extreme random

 

10

packing states, random close packing and random loose packing. The random close

packing state

 

 

 

 

 

 

 

10 g
F ..
9 I: Ci I .
r . I
I- 9 I
8 f_ ° I
L ° W I
r: I . f=0.6 I
'E 7 L S’ I
3 : 5 'I I
cc 6 t. I
b I .3 .
g 5 :_ 5
V ,_ A
0 :
4 L
3 '_
2 I. L l l l l 4 J l l l l J i
0.4 0.5 0.6 0.7

' 11
Fig. l .3: Critical behaviors oqus.(1.2.12)(U) and (1.2.13)(A). Using f. =06.

can be observed by pouring ball bearings into a container with shaking, while the random
loose packing state occurs when the container is not shaken. We consider the random
close packing state. Random close packing can be defined by the following two
conditions[5]: (1) random packing, i.e., a packing containing no statistically significant
short- or long-range order, is so dense that any increase in density can be achieved only by
a statistically significant increase in short-range order; and (2) any decrease in density from
the random close-packed density leads to ensembles of particles which need not be close
packed, i.e., a given particle is not necessarily in contact with another particle in the
absence of interparticle forces and gravitational potentials. Since the random close
packing state is not an equilibrium state f. does not approach a single limit value. The

observed f. is in the ranges 0.82 < f.“ < 0.89 and 0.58 < f." < 0.66 for identical spherical

particles[6,7].

1.2.3 Frequency dependence

Eq.(l .2. 1) does not include the interaction of inclusions so that the validity of the
equation is for low concentration of inclusions. We can obtain the efi’ective dielectric
constant at high concentration using effective medium theory which is the most commonly
invoked approximation in treating an inhomogeneous medium. Let us consider a two

component medium with dielectric constants c, and a, respectively. Take one of the

components type 1 and consider it as spherical, and assume it is embedded in a uniform

medium with an effective dielectric constant em. Ifthe external electric field is EO then the

 

induced dipole moment is[8]
- 3 e -e
=—V ‘ "' E , 1.2.14
p 4n e,+28,_ ° ( )

where V is the volume of the sphere. Thus the field inside a sphere is given by
E = 150 - 531‘- P

 

 

12

 

ago—(81‘8" )EO. (1.2.15)

31 + 28...
The second term is the depolarization field due to uniform polarization of a sphere. P is

the polarization which is the electric dipole moment per unit volume. After the completion

of the exchanges the average8 deviation of E must be zero. Therefore

(1- f)§—— +f—— =,0 (1.2.16)

8, + 28: 32 + 2:
where f is the volume fraction of the component 2. If the two components have complex

dielectric constants then Eq. (1 2.16) can be rewritten by

e a
1-8'" —2—L_=0. 1.2.17
( f);- +28; +fe§+28,,, ( )

Here the dielectric constant is defined by
e:=e;—j-q-‘—,i=l,20rmandj=J-_l. (1.2.18)
a)

The solutions of Eq.(l.2.l7) was given by Springett[9] assuming 8,, 82, o, and 62 are

independent of frequency and are
e~=eiez[(A2+Bz) )"2 -WA] tan62, (12.1%)
1/2
omzoioz[(A2+-1--1-B2)V2+A] , (12.1%)
where
l o 2 lo a 2 18
=— —— +——-‘-— — +——‘- tan5 '2 , 1.2.20
2{[°2] 20: [(82) 232]( 2) } ( a)
B=[l.91.+_ 3° +lfl](m52)", (1.2.20b)
402 8262 4:32
a=-}(3f—1)a,+-}(2-—3f)a,,a=core, (1.2.20c)
tanli2 =02/(oez. (1.2.20d)

According to effective medium theory the effect of fi'equency occurs through the factor
tan52 , a; and 6;.

1.3 Experiments and discussion

 

13

The metal loaded dielectric we have used is paraffin wax containing stainless steel
particles and copper particles. The size of the steel particles which are manufactured by
Duke Scientific Inc. ranges from 10 um to 65pm diameter. The Copper particles(Aldrich
Chemical Co.) range from SOum to 90pm diameter. The cell used to make a mixture is
depicted in Fig.1.4. First we weighed the particles and poured them into the cell. We
heated the whole cell above the melting point of the paraffin wax. The cell was filled with
molten parafin wax and then the whole cell was pumped with a fore pump to make the
sample without air bubbles. Finally it was sealed with a metal cylinder which contained
paraffin wax. A 45 um mesh sieve was used between the cell and the cylinder to prevent
migration of the particles into the cylinder. High pressure was maintained inside the
cylinder by a strong spring, so that high pressure liquid paraffin compensated the volume
contraction due to cooling. The whole cell was rotated by a slow motor at about 60 rpm
to ensure a random distribution of the steel particles while cooling. The capacitance
samples were 3/4” in diameter and 5mm in thickness. Typical optical photomicrographs of
a sample fabricated by this method are shown in Fig. 1.5, fi'om which it can be seen that the
steel spheres are fairly well distributed. Fig.5 (c) and (d) indicate that the steel balls are
spherical but the copper balls have gross departures from sphericity. Samples that contain
stainless steel and copper particles were made with volume fractions ranging from 0 to
0.62 and fi'om 0 to 0.41 respectively.

The cell used to measure the capacitance of the sample is a circular plane capacitor
with the guard ring arrangement shown schematically in Fig. 1.6(a). The guard ring
arrangement can be used to get a uniform sample field by eliminating the edge effect
provided that the total diameter of the electrode is large by comparison with the sample
thickness d. In addition to this, the center electrode and the guard ring must be at the
same potential. Thus a linear relationship between dielectric constant and capacitance,

i.e.,

 

 

(a)

 

(b)

Fig. l .4: Cell used to make paraffin disks.

 

(C) (d)

Fig. 1.5: (a) and (b) Optical photographs of cross section of metal loaded paraffin discs.

(d) Copper particles. ((1) Stainless steel particles.

l6

 

 

 

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
l I I) d
_L_

 

 

 

 

 

 

 

CH.A
{ ...... f :3 o—q , Vector Voltage Ratio Detector —————\
. : - F ‘t‘rf:-.'::r-o—CH'° — Process Amplifier ——..
.sIGNAL . - - - - PHASE
r- SOURCE ‘5 Hp 1
5H2~|3MH1 . Range
F1>~Q~J I OUT 1.: f“-
t‘Hc Lp .
""""""" "11"--“ P‘fU \fig?’ (3" DETECTION C .'
‘~. ‘~~ ~ ’ PHASE ‘ x
NULL _------- ‘~-- 1: ‘n’! x" 3' ‘5"
DETECTOR g ‘ ‘ fifty ‘ - J- ‘ . , .' SIGNAL 0‘»
Z “‘ -~‘-‘. : ‘l I" ,”
MIXER "3159- - - ‘. I i I 1 , '
, I l I | l : I
IF .’
,’ MICROPROCHSSOR BAsm mspuw
MODULATOR _, X DIGITAL CONTROL SECTION
.' , ____________________________ .m
........................ ' MEASUREMENT CONTROL PROGRAM
---------------------------- (
HP-IB INTERFACE H HP-IB EQUIPMENT

‘-— AUTO-BALANCE BRIDGE ——’

 

 

 

Fig. 1.6: (a) Principle of guard ring to assure uniform sample field. (b) Block diagram of
HP 4192A impedance analyzer.

 

17

S
C=aa—, 131
0 d ( )

can be achieved. Here 80 and a are the absolute pemrittivity of free space and the relative
permittivity of the sample respectively, S is the area of electrode and d is the sample
thickness. There are two equivalent circuits we may consider, i.e., parallel and series
equivalent circuits. Table 1.1 gives the relationship between the two circuit modes.

The capacitance was measured using a 4192A Hewlett Packard Impedance Analyzer.
The analyzer is set to measure parallel capacitance. A schematic diagram of the
measurement circuit is illustrated in Fig.1.6(b). The HP41'92A has three subsections (1)
the signal source, (2) the auto-balance bridge and (3) the vector ratio dector. It measures
the vector ratio between the applied test signal voltage and the current flowing through
the DUT(device under test). The bold lines in the block diagram show the main test signal
flow and the dashed lines show the balance control loop of the bridge circuit. In the auto-
balance bridge the null detector detects the current difference between DUT and the range
resistor. When it is not zero the modulator adjusts it to zero by applying a signal, Vr , of
the same frequency as the test signal to the range resistor. The vector voltage ratio
detector measures the vector voltage of a test signal Vs, and a signal applied to the range

resistor. The impedance of the DUT, 2,, can then be calculated by
X... .. .V_r
z " R ’

X f

(1.3.2)

where Rr is the resistance of the range resistor.

In actual measurement the guard ring is connected to ground and center electrode is
connected to the low terminal. When the bridge is balanced no current flows through the
detector so that the guard ring and the center electrode are kept at the same potential,

which is essential if a uniform field is required.

 

18

Table 1.1 Conversion table of each equivalent circuit mode.

 

 

 

 

Circuit Mode Dissipation Factor Conversion to Other Modes
CD G s 2
{31:1- 9: Cs=(1+D‘)C»R= D ~--'-
con " 1+ D' G
G

 

 

mCsW—‘R D=mC,R C 1 G D“ 1

 

 

 

 

 

 

The capacitance of the samples measured at lKHz are presented in Fig. 1.7. At each
volume concentration 10 samples were tested. For the stainless steel samples the critical
volume fraction f; is consistent with the random packing point as discussed before. But
for the copper samples the critical volume fraction 1; is not consistent with the random
packing point because they are not exact spheres so that they made a percolation path
before that point. Nevertheless capacitances of both samples diverge as f. is approached.
As the metal volume fraction is increased the capacitance values at each volume
concentration spread widely. It means that a small change of structure causes a large
change in the capacity of the sample. Capacitance change near the percolation point is
much more rapid than that at low loading range. This change is more drastic for copper
samples than for stainless steel samples because the conductivity of copper is much greater
than that of stainless steel. In deriving Eq.(1.2.l3) the assumption is made that the metals
are perfect conductors. It turned out that the stainless steel particles are not good
conductors and they are paramagnetic. The conductivity of the copper and stainless steel
particles was studied by filling the cell with metal by itself. The total resistance of the cell

filled with copper particles was the order of 10 but that with stainless steel particles was

l9

 

120 _

100 : ‘"

I—e—I

80

Capacitance(pF)
8

' I

40

20}

 

 

 

0.....A..,Q,I+B-.:er.'9'....12...1....1. 2-
0 0.1 0.2 0.3 0.4 0.5 0.6

 

ao_

50'—

40

YYTYITrTY

 

CapacitancflpF)

20

10

V'lfV—TTTjTYI

 

 

0:1LHI-m-1114'B-rrrn‘a'12.11..1.1.4111,,,,
0 0.1 0.2 0.3 0.4 0.5 0.6

 

Fig. 1.7: Capacitances of paraffin wax containing (a) copper particles and (b) stainless steel
particles against their volume concentrations. Error bar indicates minimum and maxrnum

values.

 

20

the order of 1M!) at lKHz. In this test capacitance was out of the range of instrument
(more than 10uF) for the copper sample while it was still finite value for the stainless
sample. Therefore a good conductor is great for this purpose. Even though the copper is
good conductor it is hard to handle because it oxidizes easily. If the copper is oxidized it
acts as bad conductor. To keep the copper particles from oxidation they had to be stored
in a vacuum container after being cleaned by 20% diluted HCl.

Fig.1.8 shows capacitance ratio C(0)/ C( f ) as a function of metal volume fraction.
We also show numerical data calculated by McPhedran and McKenzie using a simple
cubic lattice of conducting spheres[10]. At very low loading (below 0.1) all data are very
close to each other but after that experimental capacitances are larger than those from
numerical simulations. This is because at very low loading the probability of making a
linear cluster in a random mixture is very low so that capacitance is not much increased.

When the metal fraction is increased the chance of making a linear cluster in a random
mixture also increases. This cluster enhances the local field and increases the capacitance
but in the simple cubic array metal inclusions do not touch each other up to f. . Both
experimental results below a volume fraction of 0.2 show linear behavior which is
consistent with Eq.(l .2.3).

Fig. 1.9 shows capacitance ratio against ln(f. -- f) for large f. If Eq.(1.2.l3) is
correct the data should be linear. The f. ’8 obtained from the least 1 2 fitting are 0.65 for
stainless steel and 0.41 for copper. Eq.(1.2.13) can be rewritten by

C(f)/C(°)~ln(f.-f|)a - (1.3.3)

Analytical derivation gives 0t=-1. For experimental data this exponent can be obtained by
plotting lnlq f )/ 610)] against ln[ln(l f. -fl)a] with or as a parameter. a‘s obtained

 

from the least zz-fitting are -0.15 for stainless steel and -0.86 for copper (See Fig. 1.10).

Comparing Figs. 1.7(a) and (b) we see that the high conductivity Cu gives a much sharper

rise near f. than does the low conductivity stainless steel. We therefore suspect that the

 

N

 

 

 

 

I
1 i
0.8 i ,
E E3 ‘:
r I
C + I
9 0.6 L ..
g~ I o I
0 E I
V +
0 E o I
0.4 C. + + I
r- +
: C1 + +
_ o
0'2 :— 0 U ++++,
’- 0 fit: I
._ 1:13,};- I
o
O a I 1.. l .%. II..DQQ@ I
O 01 02 03 04 05 05 07
f

Fig.1.8: The capacitance ratio, C(0)/C(f) as a function of the volume fraction of metal
inclusions. Experimental values using stainless steel(El) and using copper(O). Numerical

simulations(+) calculated using a simple cubic lattice.

22

140

 

120 I;

100 E.

TITIII

80

C(f)/C(0)

60L
I
40;

il- \
I- \.
.— ‘\‘
\
s
)—

“\-
. \
_ " ~. \
‘.\‘ \
I- "\,>
>- ‘i‘ A I I
)— .
J i J. l J l l 61 l L J

O IIIIIIIIII1.2411111141191111.

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5
1n(fi-f)

 

 

Fig. 1.9: Test of Eq.(l.3.3) for large f Ifthe Eq.(l.3.3) is correct the data plotted here

should be linear. f.’s used are 0.65 for stainless steel(U) and 0.41 for copper(A). The

solid lines are guide to the eye.

23

 

 

 

5 -
I- ‘\.
4.5 L
4 L
l'_| : A /
g 3.5 _ 1/ i
Q : CID
g“ 3 : / I
+— I
'E‘ . A .
2.5 L /
2 I— I
I I
— I
1.5 r11 [LI—LII II1 I IJILI L111 11414A111I1111I11 11I1- I

 

-2 -1.5 -1 -O.5 0 0.5 1 1.5 2 2.5

ln[ln(f. -f)a]

Fig.1. 10: If Eq.(l.3.3) is correct the data plotted here should be linear. The best fitting
parameters are f.=0.65 and a=-0. 15 for stainless steel(El), and f.=0.41 and ot=-0.86 for

copper(A). The solid lines are guide to the eye.

 

24

value of Ianl <1 is indicative of the low conductivity of the stainless steel balls. Stainless

steel particles do not obey the critical behavior represented by Eq.(1.2. 13) due to their
poor conductivity. Even though copper particles have good conductivity the behavior
near f. is not perfectly consistent with that predicted by Eq.(1.2. 13) due to considerable
sample to sample variability in f., and probably non-sphericity of the balls.

Fig.1.11 shows the frequency dependences of the sample contained stainless particles.
At low loading the dielectric constants are not dependent on frequency but at high loading

the dielectric constants are a function of frequency. For metal inclusions in the dielectric

material, 82 >> 81 and (32 >> 0, and tan52 cancels so that Eqs.(1.2.19) become

a,” ~ §(3f — l)az, (1.3.4a)

om~%(3f—l)oz. (1.3.4b)
Thus according to effective medium theory the complex dielectric constant of a metal
loaded dielectric is not a filnction of frequency (Remember f is the volume fraction of balls
not the fi'equency). To ascertain whether the fi'equency dependence was a property of the
balls by themselves, or whether it was property of the mixture we measured the frequency
dependence of the cell filled with stainless steel partilces by themselves. The surprising
results are shown in Fig.1. 12. For a homogeneous metal one would expect a very high
dielectric constant independent of fi'equency up to ~ I.R. region. On the other hand, the
Duxbury picture of high capacitance regions in narrow necks would apply if the stainless
steel balls were coated with an insulating oxide layer. Duxbury does not ascribe any
frequency dependence in his theory.

Since Eqs.(1.2. 19) are based on the mixture of elements whose dielectric properties do
not depend on fiequencies, the results in Fig. 1 . 12 indicate that the theory is not
appropriate for this mixture. There is no obvious relationship between the frequency
dependencies in Figs. 1 . 11 and 1.12.

One might also consider the stainless steel ball- oxide-stainless steel ball system as a

Maxwell-Wagner system(See section 4.3) in which case one might anticipate a frequency

 

 

25

 

 

 

 

 

 

 

   

 

 

r F-i
_ a 0.1 I
_ :1 0.2 I
+ 0.3; I
1,000 _ C, 0 4 I I
E ~th X “RAW/~20“ ‘-< 7 O.5 I I
: I X 053*. x 0.6
- 2* . _____) I
a ' x1, I
$1“:sz I
.w 9:13;}? A . I
‘ 'VICICA' 3‘ \. A \ A ‘ ‘ I
100 C I E" V “(”39 0 0.10111); I
i I
.. - I
Cum, .3. rm- ( - _ , ‘
: . ”Ll-ml (_1 I-I._‘1I9ml 1:1 13191131111 ..‘, 1...":':I ICCICE’ I
b T +++++++++~ + +++++H+ +++—+++H+ +++—H++++..
10 P £09 A AAAAImA A AAAmm A AAAAAIm A AW
E 5&3) 1301211111111] 0130311111111 Damn) 13133121113 I
I- 1 11111111 1 1 1111111 1 1 1111111_ 1 1 1111111 1 1 1111411 1
100 1,000 10,000 100,000 1E+006 1E+007
(a)
1,400
a 00 I III 0.4
1,200 ”_ o A 0.5
a 1 _
_ O a. $00 0.6
'- O Q ‘33 L_——_____J.
1,000 _’_ I}
__ Q d- ‘3
)— (D
" 0 (30 O
800 ”_ o
- .- (3
2° : 0.
600 ” ‘7';
_. , 11)
400 L A ‘
: 221.5 -1
1- 1:1‘
200 L A +€En
b A +_l
- m A
0 .. I A Am i. O ”A . {a
100 1,000 10,000 100,000 1E+OO6 1E+OO7

f(Hz)
(b)

Fig. l. 1 1: (a) Real part and (b) imaginary part of the dielectric constant of paraffin wax

containing stainless steel particles. Numbers in the legend are the concentrations of

stainless steel particles.

 

26

3,500

 

3,000

2,500

2,000

1,000

500

IlIrIIlrIleUTfTrjiiiitliiliifiT

 

1i . . c] i .
O 1 1 1111111 1 1 1111111 141111111 1 m OW
10 100 1,000 10,000 100,000 1E+006 1E+007

fIHzl

[ll

 

 

~

0.003

 

0.0025

0.002

I'VU'UUUr'YU'U
1 !

 

o(mholcm)
§

0.001

VIII

0.0005

 

VIII

 

 

 

O
p-

 

10 100 1,000 10,000 100,000 1E+006 1E+007

flHz)

~

Fig. l . 12: The fiequency dependence of stainless steel particles.

27

dependence similar to that in F ig.4.3. This is not observed, but the results suggest that the
next improvement on Duxbury theory would be to incoporate a Maxwell-Wagner

component.

 

Bibliography

[1] J. C. Garland and D. B. Tanner eds., Electrical Transport and Optical Properties of
Inhomogeneous Media, AIP Conf. Proc. 40 (1978)

[2] L. K. H. Van Beck, Progress in Dielectrics, 7, 69 (1967)

[3] P. M. Duxbury, P. D. Beale, H. Bak and P. A. Schroeder, J. Phys. D. Appl. Phys. 23,
1546(1990)

[4] J. B. Keller, J. Appl. Phys. 34, 991 (1962)

[S] J. G. Berryman, Phys. Rev. A. 27, 1053 (1983)

[6] W. M. Visscher and M. Bolsterli, Nature, 239, 504 (1972)

[7] G. Y. Onoda and E. G. Liniger, Phys. Rev. Lett. 64, 2727 (1990)

[8] J. D. Jackson, Classical Electrodynamics, 2nd Ed. pp151 (1975)

[9] B. E. Springett, Phys. Rev. Lett. 31, 1463 (1973)

[10] R. C. McPhedran and D. R McKenzie, Proc. R. Soc. Lond. A. 359, 45 (1978)

28

 

Chapter 2

Dielectric breakdown

2.1 Introduction

A metal particle in an electric field increases the local electric field. If the metal
particle has a sharp tip the enhancement of the local field is very large and a sharp metal
tip acts as a nucleation site for dielectric breakdown. Therefore such a flaw in the material
lowers its dielectric strength.

In this chapter the theoretical predictions of Duxbury are reviewed and experiments
performed using parafiin wax containing stainless steel spheres are described. Duxbury
proposed that in a metal loaded dielectric, metal particles can form a linear cluster which
acts as a sharp tip.[1-3] Increasing the volume fi'action of metal particles increase the
probability of finding a large cluster which reduces the dielectric breakdown field. The
theory predicts the field near the sharp metal tip as a function of metal volume fraction.

Experimental results supported this theoretical prediction.

2.2 Theory

The theory is based on these premises:
(1) Dielectric breakdown is directly related to flaws.

(2) In the metal loaded dielectric the flaws consists of linear clusters of particles which act

as a sharp tip.

29

30

(3) A long linear chain of metal particles can be regarded as a thin ellipse in two
dimensions or as a thin prolate spheroid in three dimensions. If we ignore space charge
effects solving the Laplace's equation gives the electric field near a metal inclusion of
length, or, and tip curvature, x.
(4) The breakdown is calculated using a square lattice dielectric breakdown model. The
derivation of the electric field near a thin prolate spheroid is given in Appendix 1.
The resulting electric field near the metal tip and the important asymptotic behaviors are
summarized below[4].

In these equations r is the distance from the metal tip in the direction of the applied

field, 11' = bz/Za is the tip curvature, and b arm a are the minor and major axes of the

ellipse(or spheroid). The applied electric field, Ea , is in the z direction.

Two dimensions

 

fl+a2,/r2 forr>a
Ez/Eo 5] 1+ {a/(a-b)}(a/2r)v2 forx <r <a (2.2.1)
L1+\/2(a/21c)v2 forr < 1:
Three dimensions
l+orM/r3 forr>a
E,/Eo 5+ {ln(r/2a) +a/2r}/B for x < r < a (2.2.2)
k1+(a/b)2 /ln(Za/b) for r < x

 

Here the coefiicients are
“24 =a(a-b)/2,
a3, = 2c3/3B,

B=l —-b—-)+£.
c+a a

Even when the metal particles are randomly placed in the material, there is finite
probability that somewhere in the material, several metal particles will be close to each

other. Within a lattice model, such a defect configuration is shown in Fig.2. 1. For f small,

31

Vext

 

fl _ - _ . _ — . _ _
_ - _ c - — - — -

_ - - - — — u . —
T---wuurnurruuuuurnunruluprurcruucuruurnus
_ . _ - _ - a _ -

. — - . . — c - _

- — . — — u . — -

I.III|I.IIII_IIII.IIII.II .III.I|II.IIII_IIII
'11 . . . . . II.I a . _
. - — _ . - u - -
. - _ — _ - . . -

— - _ — - - c - -
1111fllII.—Ill1.1ll1.1IIl.llllflllldllll.llll.llll
. . _ . _ _ - . .

. . - . . a . . -

. . - . . u . - -
rlllfilIIJIIIJIIII-llll—IlllflllldlllJllll.llll
_ . . _ _ . . . .

. . . — - _ . — _

_ - . . — — - . .
Illldulllnqlllq q 1 1 IIIJIIII_III.I
_ . . - - . . - —

. . . . - . . . —

. . . . . . u - _
YlllfilllJlllJllll-llllqllllfilllJlllJlIll-IIII
. . _ _ _ . . . -

. . . — - - g . _

. - _ — . . . _ .
IIIlflllnolllllIII-IIIITIII9III4IIIlllll—IIII
. . _ _ . - . . .

. - - _ _ - _ - -

_ . . - . . - . .
VIII?|IILIIULIIIITuIIITIII?III&IIILIIII_IIII
_ . . . — . . - _

_ - _ _ . - _ - —

. - - . - . . - .
IIUIPUIILIIILUIII_IUIlflllPlIlLlllLllll-I'II
_ . . — . . . - .

. _ _ — - . . - —

. . - . . . - _ -
rlllPIIILIIIL||||.IIII—lIIIPIIILIIIl—IIII.IIII

 

 

L___

) in a background of

A fieldoenhancing cluster of conducting bonds(

insulating bonds(---).

Fig.2. 1:

 

32

the number of times a defect cluster of size n will occur in a sample of size L" = N, is
approximated by, Nf "(here L is the linear dimension of the lattice). The electric field is
largest near the largest such cluster, and the typical size of this largest defect cluster is
estimated fi'om[5]

Nf""" ~l which implies nm ~ lnN/(—lnf). (2.2.3)
From Eqs.(2.2.2) and (2.2.3), the electric field near the end of this defect cluster is found
to scale as

15,, ~ Eo(l mg“) (2.2.4)
for n,m large, where %< (1 <1 in two dimensions and a =1 in three dimensions. If we
assume that the dielectric breakdown field, E, (f), scales as the inverse of the size of the
largest local electric field, E“, (f), then,

E, (f)/E,, (0) “’ Ear (0)/E,,., (f)
~ 1/[1+k,(1nNA1nf|)“] (22.5)

where kc is an undetermined constant.

2.3 Experiment and discussion

The samples we have used to measure the dielectric strength are paraffin waxes
containing stainless steel spheres which are similar to those used in the capacitance
experiments. The parafin discs are fabricated by the method discussed in section 1.3.
Thick samples were used in the capacitance measurement but we used thin samples
(thickness 1mm, diameter 19mm) in the measurement of dielectric strength because the
dielectric breakdown voltage is too high for the thicker samples. The power supply we
used is manufactured by Gamma High Voltage Research Inc. (max. IOOKV). The
experimental setup is illustrated in Fig. 2.2. At very low loading the dielectric breakdown

field of the sample is very high, so we must keep the ambient of the experimental setup

 

33

 

1. 3M!) Resistor

2. Teflon Rod

3. Upper Brass Electrode

4. Plastic Guide for Electrode
5. Sample

6. Lower Brass Electrode

Fig.2.2: A schematic diagram of breakdown experiment.

 

34

highly insulated in order to prevent unwanted discharge before the sample breaks. The
entire system shown in Fig. 2.2 is, therefore immersed in silicon oil (Dow 550 Fluid)
which has a very high dielectric strength (about 440KV/cm) so that the dielectric
breakdown process occurs only through the sample. Furthermore, three 1M0 resistors
were used to protect the power supply from a huge current just after a sample breakdown.
The applied voltage was increased by 2KV steps every 30 seconds, until failure[6]. 10
samples were tested for each concentration.

Fig.2.3(a) shows the measured dielectric breakdown field as a firnction of volume
fraction and Fig.2.3(b) shows the dielectric breakdown field calculated using the square
lattice dielectric breakdown model. It is seen that the dielectric strength can be affected by
a very small volume fi'action of conducting defects. Fig.2.4(a) is the test of Eq.(2.2.5)

using experimental results and Fig.2.4(b) is the result of numerical simulations done by
Duxbury. From both results a =% is the best fitting parameter rather than 0r=l which is

the analytical result. However the linearity with a = % for both experimental and
simulated systems suggest that Eq.(2.2.5) is the correct form. This discrepancy may result
fi'om finite size efi‘ect[5]. The analytical calculation was derived in the limit of large
sample size while the numerical simulation and the experiment were done on relatively

small sample sizes. But both results give the or as a free parameter in Eq.(2.2.5) is correct.

3S

 

 

 

 

 

I T 1'1fo 1YY'YI I ‘
1i?- 1
(a) <
b at
Z. _‘
F -I
I- -4
q
4
_
.
4
q
q
_.
.1
A 1
O a
v _:
g :
UJ 1
I
1
4
2
-l
<
3
1 All]. 1‘
Tl’ I rll
4
(b) d
4
fl
4
d
d
A “
o ,
.4
.4
V .4
w A
d

 

 

 

 

 

F ig.2.3: (a) The dielectric breakdown field ratio of experimental results on samples of
paramn wax containing stainless steel balls. (b) The dielectric breakdown field ratio
calculated using the square lattice dielectric breakdown model[2]. Each point is an average

over at least 50 configurations of 100 x 100 lattices. Solid lines are a guide to the eye.

36

 

 

 

 

 

 

 

 

 

f IYfT I IVY T I
5 _. +[IJ/ _
///
4 _ // 2
r +1 {:J/
c‘ t /
Y 3 _ A, C} a
g / " <
9; Z /"+ B D :
y 1
-/ .
1E ~
L I
. (a) .
0’.14.1....1....1..1.LL.
0 0.1 0.2 0.3 0.4 0.5
Iln/l"
I
// :
/ j
J
p’ .
e .. j
g ..
A
9
n _.
1.1.1 .
j
_ (b) «
05....1....44L..1....1..La
0 01 02 0.3 04 05

Fig.2.4: A test of Eq.(2.2.5) with or=3/2(+) and or=1(El). (a) Experimental data same as
that in Fig.2.3(a). (b) Numerical simulation data same as that in Fig.2.3(b). The full

curves are a guide to the eye.

Bibliography

[1] P.M.Duxbury, P.D.Beale and P.L.Leath, Phys. Rev. Lett. 57, 1052(1986)

[2] P.M.Duxbury, P.L.Leath and P.D.Beale, Phys. Rev. B 36, 367(1987)

[3] P.D.Beale and P.M.Duxbury, Phys. Rev. B 37, 2785(1988)

[4] Y.S.Li and P.M.Duxbury, Phys. Rev. B 40, 4889(1989)

[5] P.M.Duxbury, P.D.Beale, H.Bak and PA. Schroeder, J. Phys. D. Appl. Phys. 23,
1546(1990)

[6] R.W.Coppard, J.Bowman, R.T.Rakowski, R.T.Durharn and S.M.Rowland, 3rd

International Conference on Conduction and Breakdown in Solid Dielectrics. 1989

37

 

1“-

Chapter 3

Comparison of capacitance and dielectric
breakdown

3.1 Introduction

Capacitance and dielectric breakdown experiments were aimed at investigating the
effect of a conducting defect in an insulating background. The resulting behaviors are
different because breakdown fields are related to the extreme moments of the local field
distribution, while capacitance is related to the second moment of this distribution. [ 1] For
this reason the statistics of capacitance are described by the central limit form while the
statistics of dielectric breakdown fields obey the extreme statistics. The detailed
explanations of the difi'erences will be given. We will discuss the forms of statistical
distribution functions, i.e., the conventional Weibull distribution and the modified Gumbel
distribution. To distinguish between them we made 300 paraffin wax discs and tested
them for their dielectric strengths. Experimental results and the numerical data show that
the modified Gumbel distribution gives a better fit to the data than does the Weibull

distribution.

3.2 Local field distribution

38

 

39

Fig.3.l shows the difi‘erence between the low loading behavior of the dielectric
breakdown field, and that of the capacitance. It is seen that the experimental data is
qualitatively similar to the numerical data. From both the experimental and numerical data
we see that the dielectric breakdown field has much stronger defect sensitivity at low

loading than does the capacitance.

This difference may be understood from the study of the local field distribution, D(v).
D(v) is the probability that on application of an external voltage V; , the voltage across a
small region of fixed size and orientation inside the sample is v. If the sample is uniform,
the field everywhere inside sample would be the same and D(v) would have a simple delta
function form, but D(v) broadens in the presence of disorder. If we can determine the
local field distribution firnction, D(v) then we have obtained all possible information about

local electric field fluctuations. We usually cannot determine D(v) but we can often

obtain information about the moments of v. The nth moment of v is defined[2]
(v‘) = 2WD“). (3.2.1)
1
In a bond percolation model the capacitance may be related to D(v) as[3]
2 x Energy = CV02 = N [ cbv2D(v)dv, (3.2.2)

where C is the efl‘ective capacitance of the system, V0 is the applied voltage, c, is the

capacitance of an insulating bond, v is the voltage across a bond and N is the total number
of bonds in the lattice. In models where c, is a constant, C may be found from
C = No, (13)/V,2 . (3.2.3)

Thus the efi‘ective capacitance is related to the second moment of D(v).

On the other hand from Eq.(3.2.1) the maximum bond voltage in a system may be
written,
v ~|im,_,,,(v")""'. (3.2.4)

the extreme moment.

We call (#0:: 00

In terms of v" Eq.(2.2.5) can be rewritten as

 

40

 

 

 

 

 

 

 

 

Fig.3.]: The capacitance ratio C(0)/C( f ) , (El and A) and the dielectric breakdown field
ratio E, (f )/E, (f) , (+). (a) The results calculated using the square lattice dielectric

breakdown model. Each point is an average over at least 50 configurations of 100 x 100
lattices[3]. (b) Experimental results on samples of parafin wax containing stainless steel
balls(U) and copper particles(A). The full curves are a guide to the eye.

 

41

Eb (f )/ Eb (0) ~ 15,, (0)/Eap (f)
~ Vo/(Lvm). (3.2.5)
From the Eqs.(3.2.2) through (3.2.5) the dielectric constant is related to the second
moment of the local electric field and the dielectric breakdown field is related to the very
high moment of the local electric field.

One of the possible technological applications of these measurements concerns the non
destructive testing of a component for dielectric breakdown by measuring capacitance.
Since the capacitance and dielectric breakdown depend on difi‘erent moment of the local
field distribution, the dependence on particle concentration will be different. Once the
results of Fig.3.l are established, then for a given sample size a measurement of the

capacitance can be used to predict what the corresponding breakdown voltage would be.

3.3 Statistics

The statistics applicable to a given series of measurements depends on the moments
associated with the measurement. Capacitance depends on the low moments of the local
field distribution. For such a quantity sample to sample variability obeys the central limit
form[2]. The central limit theorem tells us that if we have a large number of experiments
which measure some random variable, X, then the probability distribution of the average of
all the measurements approaches a Gaussian regardless of the form of the distribution for
X itself. Hence in the case of the dielectric constant

P(aN)=exp(-N[aN -a,,]2/oz), (3.3.1)
where 3” is the dielectric constant of a sample of size N, 3,, is the dielectric constant of an
infinite sample and 0 depends on disorder and for example diverges near the percolation
point.

Since the dielectric breakdown field depends on the highest moments of the electric

field, it does not obey the central limit theorem. The statistics of the dielectric breakdown

42

field are described by the extreme distributions[3-7]. In extreme statistics the cumulative
distribution is more convenient than the differential distribution. This experimental
distribution is shown in Fig.3.2. C (E) is the probability that a sample will fail when an
electric field E is applied to it.The Weibull distribution is most often used in fitting
breakdown distributions in engineering applications of materials.[8] It is given by

C(E) = 1 - exp(—N(E/E,,)"'), (3 .3 .2)
where E0 and m are constants. This expression for C(E) gives a straight line of gradient
m if 1n[—1n(1 - C(E))/N] is plotted against In E.

More generally we assume that there are unspecified defects in a sample. The larger
the defect appearing in a sample, the lower the breakdown voltage will be. The failure
statistics is therefore based on the probability of the larger defects occuring i.e., on the
distribution at the extreme or large defects. For an example which does not directly apply
to the pure paraffin samples, we consider the case where steel particles form the defects.
In this case the smallest breakdown voltage is associated with the largest clusters and the
statistical distribution of breakdown voltage will depend on the distribution of clusters at
the large cluster extreme. If the distribution at extreme sizes is exponential one gets the
Gumbel form[9]

C(E) = l- exp(—cN exp(—lrE0 /E)), (3.3.3)
where c and k are constants weakly dependent on the steel particle concentration. This
type of statistic gives a straight line on a plot of ln[-ln(1— C(E )) / N ] against l/E.

For the purpose of comparison of these statistics 300 pure paraflin discs were tested.
Fig.3.2 shows the cumulative failure distribution. Plots of -ln[—ln(l — C(E)) / N ] against
l/E and In E are shown in Fig.3.3 to test the two statistical forms. It is immediately clear
that neither form fits the whole range of values. The plots are therefore divided into two
regions, high and low breakdown fields. In the low breakdown field region 12 values are

0.088 for the Weibull form and 0.129 for the Gumbel form. In the high breakdown field
region 12 values are 1.291 for the Weibull form and 0.905 for the Gumbel form. In other

43

 

 

 

 

1 E— 8533 53
o :— ED
5 g [95“]
I! 0.8 E
“5 C- CD
g 06 L
B ' 5 m
3 9 an
9 :
a 0.4 .— E33
6} i E33
5 E ii
0.2 E a
L E]
a
OE.SQr-LELEUPBD... 111. l 1 1.
5 15 25 35 45 55 65
E,(in KV)

Fig.3 .2: A cumulative probability of failure of paramn discs against breakdown voltage.

44

 

    

 

 

 

 

 

 

 

 

l I [YYIYYYTI‘Y‘YIT‘YY]V'IVIIYWYY
' ' ‘ 4
6 - ,/ _
'1] ‘
q
.4
(a) <
u— —-1
4
.
Cl ‘
‘
4 — —1
4
q
H q
.
“—‘ -—4
g 4
4
A 4
ur <
_1
V q
0 ~
4
I -4
v- —«
V 4
C "
4
T _‘
H —‘
C ‘
4
w 4
1
4—1
-.
.(
4
q
—4
4
q
q
.4
—4
p
p
4 LALLILLLELlAIJAlAAIlll1111ILLLLLLLLJLLLLILLALIALJL
0.01 0.03 0.05 0.07 0.09 0.11
IIIIII IIIIIIIVYVIYYfiYY—rTTIYIIITIIIIITIIYIII'7
F 4
e » —
.' 4
(b D- <
u- ) ,‘ 4
r- ." 4
~ 21 «
b 1,: 4
)- t/ -d
)- .4
h '4
4 _ 2
I- 4
I- <
r- .-
H '- .
— fl
1- 4
n 4
A 4
I- '1
LU 2 .— _4
V v- -(
Q - <
b 4
- 4
F _ —4
‘ , - 4
C " 3
p .
T I- d
H o 1" ‘_‘
C " ‘
)- 4
T l- ‘
)- -r
P -‘
b d
b d
1- 4
P 4
-2 ~ —
b 4
b 4
p 4
1- 4
b- —-4
h- 4
h 1
P 4
4 PILJLLLLMLLLLJALLLLLLALULLAAlluLLLLAJllALLLLIAL‘

-4.5 -4 -3.5 -3 -2.5 -2
-ln(E)

Fig.3.3: Tests of breakdown statistics. (a) Ifthe modified Gumbel distribution (Eq.(3 3.3))
is correct the data as plotted here should be linear; (b) If the Weibull distribution
(Eq.(3.3.2)) is correct this way of plotting the data should yield a linear relationship. The
fill curves are a guide to the eye. The data are fi'om 300 samples of pure paraffin(0.3mm
thick).

 

45

words, Weibull form is more appropriate than Gumbel form in the low breakdown field
region and the converse is true in the high breakdown field region. The separate linear fits
suggest that there are two different kinds of breakdown process. A similar plot has been
presented by Fischer and Nissen for breakdown measurements in polyethylene using a
ramping voltage, and in these data there is also a clear indication of two processes.[10]
They suggest that the higher-gradient Weibull plot at high electric fields was intrinsic
breakdown and the lower-gradient tail was considered to arise from macroscopic
imperfections in the sample and termed technological breakdown.

In our experiments both distribution forms gave very good fits. 300 failures is a large
sample set for this type of test. However, in order to differentiate between the two

statistics at least 1000 failures would be required.[l l]

 

Bibliography

[1] Y.S.Li and P.M.Duxbury, Phys. Rev. B 40, 4889(1989)

[2] L.E.Reichl, A Modern Course in Statistical Physics(l980)(Austin, TX, University of
Texas Press) 1

[3] P.M.Duxbury, P.D.Beale, H.Bak and PA. Schroeder, J. Phys. D. Appl. Phys. 23,
1546(1990)

[4] P.M.Duxbury and P.L.Leath, J. Phys. A: Math. Gen. 20, L41 1(1987)

[5] R.W.Coppard, L.ADissado, S.M.Rowland and R.Rakowski, J. Phys: Condens.
Matter. 1,3401(1989)

[6] R.M.Hill and LADissado, J. Phys. C: Solid State Phys, 16, 4447(1983)

[7] S.M.Rowland, R.M.Hill and L.A.Dissado, J. Phys. C: Solid STate Phys. 19,
6263(1986)

[8] W.Weibull, J. Appl. Mech. 18, 293(1951)

[9] E.J.Gumbel, Statistics of Extremes(l958)(New York: Columbia University)

[10] P.H.H.Fischer and K.W.Nrssen, IEEE Trans. Elec. Ins. EI—11,37(1976)

[1 l] P.M.Duxbury, P.L.Leath and P.D.Beale, Phys. Rev. B 36, 367(1987)

46

Chapter 4

Dielectric properties of colloidal suspensions
- Theoretical reviews

4.1 Introduction

Large values of the dielectric constant have been observed with colloidal solutions and
biological cells at low frequencies[ 1 ]. H.P. Schwan and co-workers reported very large
dielectric constants of polystyrene spheres (size ~ .lum) suspended in KC] solution[2].
The dielectric constant they observed is about 1000 at frequencies below lKHz (See
Fig.4. 1). This is remarkable since the dielectric constant of KCl solution is about 80 and
that of the dry polystyrene is about 3.

There has been a lot of theoretical attempts to explain the dielectric properties of
colloidal suspensions[3 -9]. There are two starting points for the theoretical description
since they occupy an intermediate position between true solutions and coarse disperse
systems. The first one is to extend the theories of homogeneous mixtures and solutions
and the second is to extend the macroscopic theories on the polarization of the interface

between two media. Two theories are reviewed and discussed.
4.2 Complex permittivity

It is customary to express the dielectric properties of materials in terms of the complex

permittivity which is defined by

47

 

48

 

 

 

 

 

3.000“ axlO‘Scm"
I II
8 - 26.98
”LR
2.00% 800 ‘
. 26.88
4_ 600
i 400
100°1— _ 26.78
._ 200 "
_ M
0 111110 1 1 1 11111L 1 J 1111111 1 1 1 111111 1 1 1 111111 26-68
0.1 1 10 100 1,000

f(KHz)

Fig.4.1 : Real (8') and imaginary (8") parts of the complex dielectric constant of a
suspension of polystyrene particles as a function of frequency: particle diameter 0.188um,

particle concentration 30%.

 

49

s‘ = 6' —ie", (4.2.1)
where e' and e' are the real and imaginary part of the permittivity, and i: J: . The
electrical behavior of a material may be expressed in terms of the admittance Y, i.e.,

Y: G+i0)C, (4.2.2)
where G and C are the conductance and the capacitance. For a plane parallel capacitor

with a dielectric material the conductance and the capacitance are given by

A A A
G: — andC=ae —=e-—,
°d °'d d

where A is the area of the electrodes and d is their separation, o is the conductivity and

(4.2.3)

80 and a, are the permittivity of the free space and the relative permittivity respectively.

Therefore for a unit cube Eq.(4.2.2) becomes

Y=o+i03a (4.2.4)
The complex permittivity and the unit admittance are related by

Y=ime‘. (4.2.5)
Comparing of Eqs.(4.2.4) and (4.2.5) gives that the following relations hold,

8' = a, (4.2.6)

8" = o/(o. (4.2.7)

The quantity 6" is called the dielectric loss since it is associated with the energy loss
during the relaxation process. Many polarization processes are characterized by a finite
relaxation time. At sufficiently high fi'equencies, when the period of the field is
commensurable with the relaxation time, the phase shift between the polarization and the

field becomes perceptible and manifests itselfin dielectric loss.

4.2a Debye Equations

The Debye theory is concerned with a gas of non interacting permanent dipoles. An

applied electric field tends to aline them. At the same time thermal motion acts to restore

 

 

50

the original disorder. In the absence of an external field a group of independent permanent
dipoles in a liquid or solid are distributed randomly in angle. When they are subject to an
external field each dipole rotates to line up in the direction of the field. They will be
completely polarized after a long time. As soon as the field is turned off the polarization
decays exponentially as the angular distribution becomes random, i.e.,

P = P,e‘”‘, (4.2.8)
where t is called the relaxation time and is independent of time. From this assumption the

dielectric constant of the substance can be derived as[ l 0]
8 - 8,,

 

8'00 =8,,+ ’ . 4.2.9
( ) 1+it0t ( )
Separating the real and imaginary parts we find
8 -8
8'=8 + 3 w ’ 4.2103
°° l+c0212 ( )
8 -8 t
a" = L—g’k} (4.2. 10b)

where 8, and 8,, are the dielectric constants at very low frequency and high frequency,

respectively. These equations are called Debye equations since they were established by
Debye and subsequently have been applied to many substances. Their behavior is very
similar to that illustrated in Fig.4.1 though the mechanism behind the Schwan et al.

measurements of Fig.4.] may be quite difi'erent.

4.3 Maxwell-Wagner theory

The dielectric dispersion observed in heterogeneous systems was explained by
Maxwell from the point of impedance theory[l l]. The basis of this model is illustrated in
Fig. 4.2(a) which depicts a parallel plate capacitor filled by layers of two materials a and b.

a, and 8, are the real part of the dielectric constants of material a and b, k, and k, are

 

“I" . _.

51

 

 

 

 

(b)

Fig.4.2 : (a) The model of a binary heterogeneous system. (b) The equivalent circuit of the

system.

 

52

their respective conductivities and f is the volume fraction of the material b. The total

impedance of the equivalent circuit (Fig.4.2(b)) is
1 1 l

 

 

 

 

 

 

. = . + . . (4.3.1)
G+rcoC G, +zc0C, G, +103C,
Substituting Eqs.(4.2.3), (4.2.4) and (4.2.5) into Eq.(4.3. 1) gives
1 l 1
—.-= .(l-f)+ .f. (43.2)
8 8, 8,
where
8; =3, +—k—, (4.3.3a)
la)
. k
e, = s, +ij. (4.3.3b)
Using Eqs.(4.3.3), Eq.(4.3.2) can be rewritten by
e‘ = e, +£54.54, (4.3.4)
1+1t0t 10)
where
8 8,
8 = a , 4.3.521
h 8, +f(8. ’56) ( )
8 k2 + 8 k2 -8 k2
= a D f( b a 025), (4,,35b)
[k, +f(k, — k,)]
I = 8" +f(‘°’° ‘8”), (4.3.56)
kb +f(ka -kb)
k, - k‘k“ (4.3.5d)

 

- 1‘. +f(ka —k,)'

The frequency dependent behavior of Eq.(4.3.4) is illustrated in Fig.4.3. The dielectric
phenomena of this system are associated with the polarization at the interface between the
dielectric layers. The polarization is due to charges which are piled up at the boundary of
the layer in the external electric field. The theory predicts a high value of 8' (>300 in this
calculation) which decreases rapidly at frequencies where the charge transfer cannot keep

up with the applied frequency.

 

53

 

 

 

 

 

350 1 120
E 1
E; DDCEIEIIEIII] 13cm BBQ: 7
300 T '1 100
i 1
250 E a 41
P j 80
: D i
200 ._— j
'0: i D . 60 m:
150 :— D 3
" 1:1
1 MA 40
100 '— 8'=80 A 1% .
_ 1' - a ‘
: DE313131111111 DDCEKKIID
50 - £59 20
_ A 4
A A
" A
. A A ,
0 WW ? ‘ 1111111 1 1 1111111 1 1 1m AAAJM4 0
1° 10° 1.000 10.000 100.000 1E+006 1E+oo7
f(Hz)

Fig.4.3: Real part(E]) and imaginary part(A) of dielectric behavior of Eq.(4.3.4). In this
plot a, '= 8080, e, = 28,, k, = 1x10'5, k, =1.5x10'9, and f=3/500 are used.

54

Wagner developed a dielectric theory pertaining to this kind of polarization for dilute
suspension of spherical particles[12]. Now let's consider an infinite disperse system with
complex dielectric constant 8;. The spherical particles which have uniform size are
distributed evenly and randomly. The distance between them is much larger than the size
of each particle so that the interactions can be neglected. Thus the system is polarized in
the external electric field only. The induced field due to this polarization at very large
distance from the particles can be represented by two methods as illustrated in Fig. 4.4.

For a single particle in an electric field the induced complex dipole potential is given by

w. =d' cos0 (4.3.6)

1’;

 

Here d ,' is the complex dipole moment of a spherical particle of radius a. First let's look
at the spherical volume of radius b containing a number of particles (See F ig.4.4(a)). Each
particle is polarized only by the external field since the interaction of the particles are
negligible. Therefore the induced potential at sumciently large distance from the sphere is

just the sum of the potentials induced by each particle and'given by
cpl, =§nb3NcpZ (4.3.7)

where N is the number of particles per unit volume. Furthermore the sphere of radius b
can be treated as a homogeneous spherical body which is characterized by the effective
complex dielectric constant 8—. (See Fig.4.4(b)). The potential due to the polarization of

this sphere is given by

(pl. =4; “:39- (4.3.8)

 

Here d; is the complex dipole moment of a sphere of radius b. When a sphere of
dielectric constant 81 is embedded in a medium with dielectric constant 8, in an external

electric field Eo solving Laplace’s equation gives the potential outside the sphere, i.e.,

<p(r,0) = -l.5‘,,rcose+-El—:—$i’—a315'0 90—32 (4.3.9)

55

 

(C) (b)

Fig.4.4: Wagner‘s model for a dilute suspension of spherical particles.

56

The first term describes the potential of the applied uniform field, while the second term is
associated with the change in potential due to the polarization of the sphere. The potential
generated by a permanent dipole oriented along an axis 0:0 in a medium with dielectric

constant 8, is
cosG

 

 

(pde r2 ' (4.3.10)
Comparing the Eqs.(4.3.9) and (4.3.10) we obtain
81 '80 3
= ——a E . 4.3.11
e 220 +81 0 ( )
In analogy with this d: in Eq.(4.3.6) and d; in Eq.(4.3.8) can be represented by
(1;: 32—75765, (43.1221)
28, +8,
and _
11;: 8 :32; a3E,. (4.3.12b)
28, +8

From Eqs.(4.3.7), (4.3.8) and (4.3.9) we deduce

d; =§1tb3Nd:. (4.3.13)

Combining Eqs. (4.3.12) and (4.3.13) we obtain

 

 

8. ‘8;— = 31's; (4.3.14)
28:J +8' 230 +31

where f = g-na3 N is the volume fi'action of the dispersed particles. By solving

Eq.(4.3. 14) for ; we derive the Wagner formula, i.e.,
— 28, +8: +2fi;(8; -8;)

 

'=e; . . . . . (4.3.15)
28, +8, —f(8l -8,)
After substituting the expressions
8‘=8 —i—°-,
0 0 0)
. .Itl
8,=8,-r—,
00
weobtain _ _ __
e_‘='s._,+e'—’8—*-ifl, (4.3.16)

57

 

 

 

 

where
28,+8l +f(8,+8,)
= , 4.3.17
2k,+kI +f(k, —k,) ( a)
— 28,+8l -2f(8,—8,)
z , 4.3.1
8" ° 28,+8l +f(8,-8,) ( 7b)
_ ._ 9 - 2 1—
81%“: (80") k,e,) f( f) 2, (4.31%)
[28, +8, +f(e, -e,)12ko +k. +f(ko - k. )]
‘7: 2k0 +kr '2f(k0-kl) (4.3.l7d)

° 2k,+k,+f(k,-k,) '
These results are very similar to Maxwell theory, and yield behaviour similar to that in

Fig.4.3.

4.4 Thin double layer approximation

The polystyrene spheres have terminating COOH or SO, groups in their surface. In
solution these groups ionize(hydrolize), leaving usually a negatively charged sphere(radius
a) surrounded by positive(counter) ions which create the electrical double layer(thickness
8) as illustrated in Fig. 4.5(a). Whereas the Maxwell-Wagner effect arises principally from
induced charges on the surface, several people have suggested that the polarization of the
double layer is responsible for the high values of the real part of the dielectric constant at
low fi'equency[7,9, 13,14]. Of these we describe the theory of Chew and Sen. The details
of their theory are given in Appendix 2. Here we emphasize the physics of what is
happening rather than the mathematics.

A consequence of the Chew-Sen theory is that for r > a +6 , the system is
electrically neutral but there is a difi‘usion cloud in which both charge densities vary
exponentially with distance. The application of an alternating electric field results in

oscillatory circumferential conduction currents within the double layer plus radial diffirsion

58

 
    

 

 

Polarization
,_ — Char es
+ \1 Neutral
Diffusion
\ Cloud
!
_ '1 Double
. Layer
_ \ \ _+ ’ I + E
+ 1 ‘ : [E0
Inner Region 1 1 1
(Double Layer) ‘_"

I

(a) (b)

F ig.4. 5: (a) A negatively charged particle in ionic solution acquires positively charged
counterions in the absence of an external electric field. (b) In the presence of an external

electric field E, the charge distribution in the double layer changes and polarization

charges develop within the particle.

59

currents from the diffusion cloud(Fig.4.5(b)). The circumferential currents are conduction
currents which do not give rise to increased dielectric constant. On the other hand the
radial current has the phase relations associated with a displacement current and is
responsible for the dielectric enhancement.

The mathematical steps are as follows. The potential in the electrolyte solution
surrounding a colloid particle is governed by the Poisson-Boltzmann equation. Chew and
Sen set up equations for the inner a < r < a +6 and outer r > a +6 regions where 6 is the
width of the double layer. The solution of the potential equations is obtained by matching
at the boundaries order by order in (6/a). The dielectric constant becomes large when the
induced dipole moments and potential develop an out of phase component. This does not
occur in the zeroth and first order approximations. Solving the equation by perturbation
method gives the potential, i.e.,

w = —e,r cosO +(Pe,/r2)a3 cos0, (4.4.1)
which is similar to Eq.(4.3.9). Here 8, = E,e/k,T. The second term describes the effect

of the induced dipole moment of a particle. The zeroth order solution is
WW) = -e,rcos0—§e, (a3/r2)cos0. (4.4.2)

Therefore the induced dipole moment is PM = -—-1- which is real, resulting in an induced

current which is purely resistive and there is no enhancement in the dielectric constant.

In the first order of (6/a), a radial current jix appears in the inner region. It is given

by
r—a

. 4t
11‘} ~ i3eoDNo 0089[X fig], X = —5— (4.4.3)

where t = tanh(‘1’, /4), ‘1’, is the potential at the surface of the particle, and D and N, are
the difliision coefiicient and the the equilibrium ionic densities in the absence of the
charged particles. The + and - signs refer to + and - ions which have different

distributions across the inner region.

60

The wide difference in amplitudes of jf‘} and jf‘} implies that these ionic currents cannot

be carried away from the inner region into the outer region by conduction current alone.
since conduction gives rise to ionic currents which are equal and opposite. Hence, we
expect a diffusion cloud of ions in the outer region which is induced by the inequality in
amplitudes of jg}.

The charge density outside the double layer determines the diffusion process. The
charge density of this diffusion cloud is given by

n, .. e-MH) (4.4.4)

9

where k =(1—i),/c0/2D. The diffusion cloud is neutral and has a size ~l/l. Fig.4.5

illustrates the change of charge distribution in the presence of an external electric field.

This neutral diflirsion cloud gives the normal component of the current and induces a

dipole moment. The induced dipole moment is pure real up to 1st order in (6/a). The

first out of phase term for P appears in the second order of (6/a) and is given by
2
. ~ +0 .. 2

2 a l-r

2 . 2
+(EJ "2:35—02 +6ln(l-—12)-——24;7(t2 +1) . (4.4.5)
4 D 8 )

(l—t2 0‘
V0.14).

We may write the Eq.(4.3.14) in terms of P for small f(the volume fraction of

 

 

2 2
Herea=[l+ka+x:

dispersed particle), when this is done the effective dielectric constant of an ensemble of
spherical particles randomly distributed in a background medium of 8, becomes

1? = 23(1 +3 fP). (4.4.6)
From Eq.(4.4.5) the enhancement comes mostly fiom the last terms. Therefore Eq.(4.4.6)

can be approximated by
e—‘ ~(e'+13)(1- 311:1), (4.4.7)
0) or

where

61

2 24t2
F z (E) 2, (4.4.8)

 

Hence the condition for enhancement is t-—> l or ‘1’, >> 1. This implies that a particle
must be highly charged. Furthermore the enhancement comes from the induced out of
phase currents which are generated from the diffusion cloud outside the double layer. The
bigger the size of the diffusion cloud the larger the enhancement. At high frequency the
rapid oscillation of the circumferential current in the double layer does not give time for a
diffusion cloud to form outside the double layer. Therefore the induced out of phase
current becomes small and the enhancement disappears. The effective dielectric constant
is given by

8—. ~ -3fF(8' +igj—l-

(001

= _3jF—-1———2[8'+28'Jc0'c -O"t +i{8'0)r +or +E(1+ cot H, (4.4.9)
(1+ 031: 0)
2

where t = 215. Fig.4.6 shows the behavior of the effective dielectric constant vs.
frequency. In this plot the following values are-used.

‘1', = 3.85, N, = 2.08 x 102’m'3

D = 1.97 x10” cmz/sec,

8,, =808,, 8, = 38,,

6/a = 1/60, a = 0.09411 andf=0.3.
In this figure the maximum in 8" occurs at the high frequency ~10’Hz. The maximum
occurs at frequency VI. The frequency of the maximum therefore is expected to be

eta-2.

62

 

 

I‘Delta

 

8160
al70
8/80
aISO
al100

 

 

 

+9000

 

 

   
 
 

3,000

 

00m

1
g

 

 

2,000

1,000

 

 

100 1,000 10,000 100,000 1E+006 1E+007 lE+008

f(Hz)

O
l r r 1 rr TI r
: 1
e i

 

fl
0

2,000

 

 

1,800

 

 

also
WC
1180

1,600

 

 

 

.— ..........

1,400

 

 

l ,200

 

 

 

 

 

1,000

 

8"

 

 

 

 

 

 

 

200

IIII'IIII IIII IIII IIII TTTI ITII IWIIIITI IIII

 

 

1 1 1111411 1 1 111111 1 1 1111111 1 11111111 1 11111111 ‘1“

100 1,000 10,000 100,000 1134006 113-+007 lE+008

f(Hz)

 

0

 

Fig.4.6: Dielectric behavior of Eq.(4.4.9).

Bibliography

1. H.P.Schwan, Advan. Biol. Med. Phys. 5, 147(1957)

2. H.P.Schwan, G. Schwarz, J.Maczuk and H.Pauly, J .Phys. Chem. 66, 2626(1962)

3. G. Schwarz, J. Phys. Chem. 66,2636(l962)

4. K.W.Wagner, Ann. Phys. 5, 817(1913)

5. J.Lyklema, S.S.Dukhin and V.N.Shilov, J. Electroanal. Chem. 143, 1(1983)

6. W.C.Chew and P.N.Sen, J. Chem. Phys. 77, 4683(1982)

7. M.Fixman, J. Chem. Phys. 72, 5177(1980)

8. J.M.Schurr, J. Phys. Chem. 68, 2407(1964)

. 9. E.H.B.Delacey and L.R.White, J. Chem. Soc. Faraday Trans. 2, 77, 2007(1981)

10. V.V.Daniel, Dielectric Relaxation, Academic Press(l967)

ll. J.C.Maxwell, Treatise on Electricity and Magnetism, 1873(Dover, New York 1954)

12. K.W.Wagner, Arch. Elect. 2, 37(1914)

l3. W.C.Chew and P.N.Sen, J. Chem. Phys. 77, 2042( 1982)

14. S.S.Dukhin and V.N. Shilov, Dielectric Phenomena and the Double Layer in Disperse
Systems and Polyelectrolytes(Halsted, NY, 1974)

63

 

Chapter 5

Dielectric properties of colloidal suspensions
- Experimental attempts

5.1 Introduction ’

Since Schwan and his co-workers presented their results for the dielectric dispersion of
dilute suspension of polystyrene microspheres[ 1], the frequency dependence of the
effective dielectric constant has been of great experimental theoretical interest. Many
experimental attempts associated with this matter have been reported[2-4]. None have
been completely sucessfirl.

In this chapter we review and criticize Schwan’s experiment which used the long cell
arrangement, and introduce other method using a small cell. We compare the pros and
cons of both setups using several difi‘erent electrodes. We point out that the use of
platinum black electrodes, which is the conventionally recommended method for reducing
electrode efi'ects, is not a good solution to reduce electrode polarization. It turns out that
platinum black acts as an insulating layer which gives rise to the Maxwell-Wagner effect.
To support this we performed measurements using electrodes covered with insulating

scotch tape and obtained similar results.

5.2 Electrode polarization

 

65

Electrode polarization is a major source of error in determination of the dielectric
properties of ionic solution[5-6]. Fig. 5.1 shows how the polarization of a sample can
take place in a parallel plate capacitor[7]. The dielectric material can be polarized by the
reorientations of the dipoles along the external field. If the material is nonconducting then
the measured dielectric constant gives a measure of this orientation process. If the
material has finite conductivity then the ions in the electrolyte accumulate at the electrodes
forming an electrical double layer. This phenomenon is called electrode polarization. In a
parallel plate capacitor configuration the voltage drop divided by the applied current yields
the complex impedance. However, this includes the additional voltage drop at the
electrode—electrolyte interface caused by electrode polarization. This is very difficult to
measure and predict. The electrode polarization efi‘ect becomes large as the frequency is
lowered and as the conductivity of the solution is raised. Our measurements in Fig 5.2
illustrate the efi'ect of electrode polarization in materials ranging from very high resistivity
transformer oil to a highly conducting 2mM KCl solution. From these results we see that
electrode polarization is large in the region l-lOOKHz where we anticipate the important
results for microsphere suspensions. Eq.(5.3 .3) below is qualitatively verified by these
results. The various experimental methods outlined below are largely designed to

eliminate errors due to electrode polarization.

5 .3 Review of Schwan's experiment

In the investigation of the impedance of materials which have high dielectric losses

substantial errors due to electrode polarization can occur. Now consider the equivalent

circuit in Fig.5.3 in which C, and R, are the capacitance and resistance of the sample,

C, and R. are the measured capacitance and resistance and C, and R, are the capacitance

and resistance due to electrode polarization[8]. The total impedance is

 

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RANDOM RANDOM
DIPOLES IONS
63:; .7 ‘1’??- I ( if: ‘9 O I? O
‘3‘; e2 .. +2 <— , :2;
; 3:, 8 9:“ (-I (+2 (5)) EFT-E2 {-2
5:) 1?,3’30‘ 5 O (:; :(_’.'_va;‘ (I:
c—fiI Q L. {:2 W — (a) I e) 1.1-1“) 1‘ i)
1‘) '3‘ 1E3! \ 64- O '1‘! x-
2: x3 (2 “is C) +) (; .__.<+)
£2 7 S 6'7 081(71):”? 3
\§ 2? {5O (2; («B (3) It“ .1): :2
3;) :\ I Q ('I'I (:7‘ (+) - \‘i‘I
ORIENTATION CONDUCTION
STARTS STARTS
('9 gfi).€) if)“.
69 059 95‘?
0&2 «£2
6269 _3'32 62,9
_ e O “I" ._..
(b) I 636 .64) (9
$9.. 9" O
O @, :é'Oe—i (5)
£9 “*3 0* ‘9
$6 6‘64: (:1
I 1+
I l
POLARIZATION POLARIZATION
«£3 O O @419 O
O O 0 e @ 99 ‘9 690 (L1 <9
GO 0 O O O O O O
045300 @QL O (+).—.19
_.. £30 1— (C) _ @9393) <—-)O _—
{3000 6363 OOOO
:0 o : O O O< O O O
00:88 88:32.83
06 O {-13 OH" O O

 

 

 

 

 

 

 

 

 

 

_L_—
‘71-
.1.

Fig.5. 1: Idealized view of the polarization due to dipoles and ions: (8) unpolarized; (b)
orientation and conduction begins; (c) fully polarized. In actual materials, both processes

can occur simultaneously.

 

67

 

 

 

 

 

 

 

 

164-006
a:
cE D
I 0 ° 0 Distilled Water
100.000: 0 0% 1mM KCl
“44 2mM KCI
: '00 Oil
1o.000_ '0
E 0
3 0
" c1
- 10005 o no
u ' 00 CI}:
E “ 000-
h A AA ”0 El 0
1w i- AAA 0 D
E M aaaaam A gflggb 0 0000A
;
10F
E
4: + + ++¢++++ + ++++++¢ + + +++~M++ + + +++¢+++
1 L i LLJILII 1 1111111 1 Allllll L 1 111111
0.1 1 10 100 1.000
f(KHz)

Fig 5.2. The apparent dielectric constants of several liquids which are different in

conductivity.

 

68

(b)

Fig.5.3: (a) Equivalent circuit of dielectric cell. (b) Equivalent circuit of the total measured

addmittance.

69

l l l

 

 

_———l . +RP+ . C = 1/ . , (5.31)
/R, + 1a) C3 10) P /R.. +10) Cm
From Eq.(5.3. 1) we can derive that
R_ = R, +RP +R‘(mCmRm)2, (5.3.2)
Cm=Q+-—2—1—7. (5.3.3)
03 CPR,

Many assumptions have been used in these equations, namely l/(o C5 >> R5, Rm << l/co Cm.
and R, >> RP.

Since the second part of the total capacitance expression increases rapidly as the
frequency decreases, it is often impossible to get a correct sample data C, from the
measured capacitance C”. One way to minimize electrode polarization effects is to use

large electrode separation to make R. large and thereby make C”' and Q comparable. A

cell with large electrode separation is very hard to handle because it is accompanied by
large stray field efl‘ects. One method of dealing with the electrode polarization problem is
as follows[9]. Fig.5.4 shows a schematic diagram of a cylindrical capacitance cell which
has very large electrode separation. The teflon cylindrical cell is filled with the sample to
be measured. The upper electrode is grounded through the outer metallic cylinder, while
the lower electrode is connected in our case to the high terminal of our impedance

analyzer. This arrangement is considered as a transmission line with the admittance Y

between the points A and B given by
_ 1 1+6” 5
21";— ‘ 3°“)
R+ ' L
z2 =( 1‘” yjmc' (5.3.5a)
2 _ (R +ij)

where the characteristic impedance is Z, the resistance per unit length of inner conductor
R; total length of transmission line I; inductance per unit length L; capacity between inner
and outer cylinder per unit length C; and propagation constant y. At low frequencies (9L is

very small compared to R. Consequently Eq.(5.3.4) may be rewritten

7O

 

 

 

 

 

 

 

Fig.5.4: Cylindrical teflon sample cell S with metallic coaxial shield M.

 

71

 

jm C y; 1 + exp[—21(jcoCR) 1/2]
n+4

 

 

~ -. (5.3.6)
R l — exp[—21(ijR) % ]
Sincelg=Go+jcoC,
V; a_ -a .
GO:G(RCDC] ea e-a +231na (5.3.72!)
2 e +e -2cosa
and
C C e -e —251na (5.3.7b)

° = (2,5030% e“ +e‘“ —2cosa'
Here G = )4? and a = 1(2RcoC)%. If the sample is highly conducting and the low
frequency range is considered, then a is much smaller than 1. Hence the conductance G0

and the capacitance Co can be expressed by
4 8
a a
_ G ”57";
o -_ 4 s
zuLLs
2! 6! 10!
4 8

l a a
+—+—+...

C0=C13i3 7’ 1“ . (5.3.8b)

4 8
—+—+—+...
2! 6! 10!

+...
(5.3.88)

 

In the limit a <<1 Eqs.(5.3.8) become
60 = Cy, and C0 =%CI. (5.3.9)

These are the conductance and the capacitance contributions due to the stray field

between sample and shield. Therefore the total capacitance is obtained from Eqs.(5.3.3)

and (5.3.9), i.e.,
A CI 1
C = —+—+—. 5.3.10
T 88°! 3 (102sz ( )

The apparent dielectric constant is
2
C 12+ '2“ 1. (5.3.11)
380A (0 Clueo I

 

 

e.=e+

72

We have used R = 1:2, where K is the conductivity of the sample. The three components
1c

of 8. plotted in a logarithmic scale are shown in F ig.5.5. From this analysis there are two
ways to proceed.
(I) Use this technique to separate a from 54- The real dielectric constant of the sample
can be obtained by subtracting the contribution of the stray and the electrode polarization
terms which are extrapolated from measurements at very high and low I-values
respectively.
(2) From measurements on H20, put a = 78 and use large I measurements to calculate C
which is assumed to be independent of sample.
The details of the method used by Schwan and his coworkers are vague but apparently
they used a method like (1) above. A typical result of their work is shown in Fig.4.] and it
is this behaviour we have tried to emulate. The graphs are similar to what would be
obtained fi'om the alignment of permanent dipoles but current interpretations are different.

Some of the results Schwan and his co-workers found are
i) the dielectric constant is proportional to the particle size,
ii) the characteristic time(i.e., the period corresponding to the maximum in e " in Fig.4. 1)
is proportional to the square of the particle radius,
iii) both the dielectric constant and the characteristic time are independent of salinity at
high salinity.

We note that Schwan et al. used concentrations as high as 30 volume % of
microspheres. Present day manufacturers rate 4% to 10 % as maximum values for

stability against coagulation.

5.3a Review of other experiments

SW2]

 

73

 

I IIIIIII

8(90!)
swim!)

I I I TIIIII

loger

I I IIIIIII

 

P4 lLlJlll L LJI Ill

 

 

Iogl

Fig.5.5: Components of the apparent dielectric constant as function of electrode distance.

74

They used a small cell with variable electrode spacing. Their cell had a guard
electrode which was used to control stray electric fields. The electrodes were sandblasted
platinum, a "well known" method of reducing the electrode effects. In their results a
dielectric loss peak appears. The experimental results shown in Fig.5.6(a) are

measurements of the sample of 222nm latex(2.2% solids) in 0.878mM KCl solutions.

Lim and F ranses[3]

They employed a cell with variable electrode spacing which is similar to Springer's but
they used a platinum black electrode instead of a sandblasted one. They also used a
microprocessor controlled bridge circuit which controlled the experiment and records data
so that the speed of measurements was increased significantly. They only give the real
part of the dielectric constant and don‘t mention the imaginary part. In Fig.5.6(b) the data

are for 450nm polystyrene latex(4.4% solids) in aqueous NaCl.

Myers and §aville[4]

A four electrodes method was used in these experiments and all electrodes were
platinum blacked. They assume that separating the current source and sink electrodes and
voltage measuring electrodes eliminates the electrode efi‘ects. The voltage drops at the
source and sink electrodes don't affect the current across the entire cell. Even though the
voltage drop at the voltage measuring electrodes is large using a high input impedance
measuring device minimizes current flow through the electrodes. Therefore the voltage
drop at the electrodes is negligible compared to the voltage drop between the electrodes.
They also used a computerized measurement system to reduce the measurement time.
Their results show a dielectric behavior similar to Schwan's. In Fig.5.6(c) the sample is

191nm latex in 0.1mM HCl solutions. The volume fraction of polystyrene balls is 4.1%.

75

 

 

 

 

 

 

 

 

 

 

 

000 a
.a
.3
700 _ (a) 3
‘1
t: a D A ._..'
_ .
000 :_ ‘3 C3 .2. a 3|
)—
500 '- —1
1— .4
1- ;. [:1 ‘3 4
*- «i
b ‘00 :_ a. _j
: D 1
r- A "
b _1
C .1
200 L _
a 1: J,
C D -1
"n r A _. 3!.
,- LA
”23
o P- L AAAA L A AAAAL A J. A 1 a AA A wAA A ; éA Ajj
100 1.000 10.000 100.000 119000 19007
f(Hz)
1.200
CB
* :1
1.000 _‘__ (b)
: 1:1
000 :_
'1‘» 000 __
)—
L—
4” '-
1.
L. 1::
+-
200 __ D
P
I D at:
o h A A A A L4 AL A 4 A J A A A A A A L A A L A A A
100 1.000 10.000 100.000
f(Hz)
000 a
I A a
- 1:1 _
700 __ (C) 3
r— A ”_"‘
000 : :3 :
I A _
500 L A I
b- —4
1. t —
400 ;_ 2
E D L
:00 ; a
.. A :3 I
P 4
I” + LA AAA A 1* A A . A a M
1111 1.000 10.000 100.000
f(Hz)

Fig. 5.6: Dielectric pr0perties(real part(D) and imaginary part(A)) fi'om (a) Springer and

Lyklema and (b) Lim and Franses (c) Myers and Saville.

250

200

50

76
In table 5.1 we compare all the measurements that we are aware of at this time. It is
clear that there are wide divergences and there are many variables whose effects have not

been studied in detail. The experimental state cannot be described as satisfactory.

Table 5.1 Comparison of the previous experimental results.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

diameter(nm) f o (KHZ) 80 solid % solution
Schwan 88 80 540 <30 1mM KCl
188 15 2450
566 1.8 3000
l 170 0.6 10,000
Lim 450 1160 4.4 NaCl
Springer 222 43.5 750 2 0.878mM KCI
Myers 191 3 800 4. l 1mM HCl
5 .4 Experimental setup

5.4.1 Capacitance cell

The experimental determination of dielectric properties is usually accomplished by
measuring the capacitance of a material using a plain parallel capacitor. If the capacitor is
ideal the capacitance is given by

A
C = e e —. 5.4.1
r 0 d ( )
But the electric field between the plates of an ordinary plain parallel capacitor is not

uniform due to edge effects. These can be reduced by using the guard ring arrangement

discussed in section 1.3. Ifthe material is a highly conducting liquid, such as an aqueous

 

77

electrolyte solution, the electrode polarization effect due to the impedance of the diffuse
double layer at the electrodes makes the determination of the sample capacitance difficult.
According to the literature this effect can be reduced by using bare platinum or platinum
black electrodes[10,l 1].

From the second part of Eq.(5.3.3) we deduce that increasing the resistance of the
sample reduces the electrode polarization effect. This implies that the electrode effect is
very small for a nonconducting sample so that it may be neglected. But if the
conductivity of a sample is high enough then it makes a considerable contribution to the
total capacitance. One way to increase the total resistance of the sample is to make a
longer sample. To do so a large, especially long, cell is needed. Fig.5.7 shows the
schematic diagram of the long cylindrical capacitance cell we used. The capacitance cell
consists of two parts. The lower cylinder contains the liquid sample and holds the lower
electrode E. The upper part supports the micrometer M and the upper electrode EU.
The upper electrode can be moved by hand in the vertical direction through the tubes
containing two guiding O-rings. It is fixed coaxially to a glass tube so that it is electrically
isolated fi'om the outer cylinder. The lower electrode is separated by epoxy from the
shield. In this arrangement we anticipated using Schwan's analysis method to compensate
the stray capacitance. The long cylindrical arrangement can reduce the electrode

polarization efi‘ect but cannot avoid the stray field efl‘ect. However in a high a, sample the

field lines will tend to concentrate in the sample.

5.4.2 Electrodes

Earlier measurements were made with brass electrodes. Difiiculties with these
experiments led us to use platinum and platinum black electrodes. The face of each
electrode was covered with platinum sheet using conducting silver epoxy. Both platinum
electrodes were electroplated with platinum black to reduce the electrode polarization

effect[12]. The plating apparatus consisted of a SOmL beaker, the platinum electrode to

 

78

 

 

 

 

O-ring
\

\\\

" ‘ J
77

/ . _ Gloss Tube

Aluminium
Suppofl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:: Teflon
’3 c Epoxy

 

 

/-'7 EU

.l/

/"

Brass/i
Cylinder

 

 

 

[EL

 

 

 

 

 

 

 

Fig.5.7: Long cylindrical dielectric cell. H and L are the high and low terminal. G is

ground. EU and EL are the upper and low electrodes. M is the micrometer.

79

be plated, a counter electrode and a DC power supply. The platinic acid solution (1.4%

RC1? and 0.02% Pb” in dilute aqua regia) was prepared fi'om 1.58g of the hydrogen
hexachloroplatinate hydrate (from Aldrich Chem. Co.) in 10 mL of aqua regia (one part
HNO, three part HCl). It was diluted to 50g of solution with deionized water. Finally
0.0lg Pb(NO,)2 was added. The lead nitrate is a powerful oxidizing agent essential to

the production of the platinum black matrix. The counter electrode used was a braided
platinum wire connected to the positive pole of a DC power supply. The electrode to be
plated was immersed in the platinic acid solution and connected to the negative pole. The
electroplating was achieved in 5 minutes with the power supply adjusted to 2V. After that
both electrodes were rinsed several times with deionized water.

Springer and Lyklema have recommended sandblasted Pt electrodes for diminishing
the effect of electrode polarization. We have also performed measurements with these

electrodes(See section 5.5.2).

5 .4.3 Microsphere samples

Initially samples were bought from Interfacial Dynamics Corporation and Duke
Scientific. The data in this thesis are for samples made by Mr. Grant Shouldice a graduate
student in the Department of Chemistry at the University of Waterloo, London Ontario.
They are generated via emulsifier-free emulsion polymerization of styrene[ l 3].

Microspheres are stabilized against coagulation though the electrostatic charge on the
particles. In the past the polymerization took place in the presence of a surfactant usually
alkyl sulfonates or alkyl carbonates. These materials adsorb at the particle surface,
enhancing the negative charge on the microspheres and thereby increasing colloidal
stability and resistance to coagulation by electrolyte ions. The use of surfactants, however
leads to a product with ill defined surface properties, unless they pass through complex

processes to remove the surfactant.

80

Now the microspheres are prepared without the use of surfactants. Instead in a
surfactant free process high concentrations of an "initiatior" are used. Charge previously
supplied by the surfactant is produced from the generation of intrinsic charges at the
particle surface, thereby negating the necessity of having charge from adsorbed surfactant
molecules.

Polymer microspheres made of polystyrene comprise chains of polystyrene held
together by van der Waals forces to form non porous hydrophobic particles. The
hydrophobic nature of these particles would encourage aggregation. Surfactant free
microspheres are stabilized against aggregation by surface charges introduced at the
particle surface by various polymer chain terminators introduced in the polymerization
process, rather than by the introduction of ionic surfactants or detergents.

Hydrophobic polymers always tend to aggregate. This tendency is prevented by
electrostatic repulsion from the presence of surface charge groups on the particle.
Aggregation can be avoided by keeping the suspensions dilute. Typically sulfate or
carboxylate lattices are sold at < 10 vol% solid. Depending on the particles' surface group
and concentration, the recommended maximum vol% solid can be considerably smaller
than that. The samples we receive from University of Waterloo have 4 vol% solid.

The latex microspheres have a pK associated with the surface group which is
analogous to the pH for hydrogen ion concentration. For stability the pH of the
suspension should be selected so that the surface charge groups on the microspheres
remain fully ionized, i.e., pH > pK+1. For sulfate latexes pK<2 and consequently the
particles are stable in acid media. For the carboxyl group pK is ~ 5 and consequenctly
these particles are not suitable for work in acidic media.

The samples have all the same vol% but different surface charge densities.
Determining the charge density is done by the conductometric titrations using 1mM NaOH
solutions. Fig.5.8 are the titration data for all solutions. In each graph the minimum is the

equivalent point which implies that the moles of base added is equal to the moles of acid

81

 

 

Q

.a

.
A

p
3

 

9
d
3
i

(J

Conductanee(mS)

9
r

0

Conductanoqu)

p

, - -

<1

1

9
3

0

d

N

---rfi--,_fis-,fi--

1 s :
l

1

<1

O

Q

I

3 v . ‘- 0.12

 

 

 

> 1 ~
L 0.115--rr11 .1.. ma 121

o
.o
.—
..

p

 

8
8
3
3
8
8
8
8
8
a
8
8

No. ddropoUmM NaOH) NO. ddfOOSUmM NaOH)

 

 

 

l
l
l

 

Conductance(m5)
2 o
3 3
<1
0
<1
<1
<1
I
l
ConductancepS)
8 8
Pfi'rfifi'r ‘7 r r
1 Q!
<1
<1
G

I
1
4.

 

’ - v .. .:.. va.. .. .... .......-A : , ; v
* v ‘ v E ; 3‘ 7 - V 9...... - . . .

0““ L.-.-....,,._.._.i_.v.._......v g. - i-"""“’f""" .. .1........... \. .. .. .. 1...._

. .. ,. ................ ._. .. .. ”nun..." .4 _ 1 , {
; a C 3 ‘ V V
. 1 vv 1 : (c) 1 : v 1* .ld)
o.152’.-..1.r..i..--ir.rri. ... 20 “1-1..-.14m4- 1.1.1.1 Ami--1....1 .
0° “ a a n a 0 2 0 C . 1° ‘2 ‘4 1. 1.

N0. dM‘lmM NIOH) No. ofdrope(1mM NaOH)

 

 

 

 

 

 

Fig.5.8: Titration data of (a) 213.8nm (b) 364.1nm (c) 367 .7nm (d) 624.3nm suspensions.

82

groups. Assuming a point charge for each acid group we may obtain the surface charge
density by dividing the number of charges by total surface area. The surface charge
densities of213.8nm, 364.1nm, 367.7nm and 624.3nm balls are 3.34, 3.61, 4.01, and 1.03

in 1.1C/cm2 , respectively. The combination of low charge density and large mass, resulted

in visible sedimentation of the 624.3nm balls in the course of several days.

5 .4 .4 Measurement setup

The HP 4192A impedance analyzer was used to measure the admittance at each
frequency. The 4192A employs the four terminal configuration measurement illustrated
in Fig.5.9[14]. To compose a measurement circuit loop in a four terminal pair
configuration, the HCUR and HpoT , LPOT and LCUR terminals must be respectively
connected together and, in addition, the shields of all conductors must be connected
together. The distinctive feature of the four terminal pair configuration is that the outer
shield conductor works as the return path for the measurement signal current. The same
current flows through both the center conductors and the outer shield conductors in
opposite directions. Thus no external magnetic fields are generated around the
conductors, i.e., the magnetic fields produced by the inner and outer currents completely
cancel each other. Because the measurement signal current does not develop an external
inductive magnetic field, the test leads do not contribute additional measurement errors
due to mutual-inductance between the individual leads.

5.5 Experiments and discussion

5.5.1 Long cell

In this section we give experimental data obtained using the long cylindrical cell

with three difl‘erent electrodes, brass , platinum and platinum with platinum black.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

83
HCUR I J 1 l IA‘UR
Ii - ~ Capacitor - ~ A
r 7 I F r -

Q \z__ .__, e - ,— __1/ 0
Oscillator :3 Hm <3 LM ‘Ammeter
/V\

U
Voltmeter
(a)

 

 

 

 

a“
E

LPUT LCUR

(b)

Fig.5.9: (a) Four terminal pair measurement principle. (b) Actual four terminal pair
connections.

 

84

Fig.5.10(a) shows the apparent dielectric constants of distilled water. The data set at
lOOKHz does not include polarization effects because of the high frequency. But the data
at the low frequency(lKHz) indicate that the electrode effect is getting bigger. On the
other hand capacitance data for a 1mM KC! solution (Fig.5.10(b)) show the electrode
effect occuring at a much higher frequency. For example the apparent dielectric constant
of a 1mM KC! solution at l KHz gives a strong electrode polarization effect throughout
the range of electrode separations (see Fig.5.10(b)). This is very crucial because the
system of polystyrene balls suspended in water generally has a high conductivity.
Furthermore we add some electrolyte to make the double layer efl‘ect at the balls larger.
The contribution of electrode polarization in the case of distilled water is comparatively
small due to its high resistance.

The high frequency data for water in Fig.5. 1 1(a) show negligible polarization effects.
However after the subtraction of the stray capacitance obtained by extrapolation from
large I the corrected dielectric constant is not constant. The capacitance due to stray
fields is not expected to be a function of frequency but we see differences in stray
contributions at high-l values for the three fiequencies in Fig.5.10(a). There are also
serious problems with the subtraction of electrode polarization effects. Fig.5. 1 1(b)
illustrates the imprecision of subtracting the electrode polarization part from the data for
1mM KCl at lKHz. Fig.5.11(c) shows the data of 1mM KCl at 10 KHz with subtraction
of stray capacitance but there is no way to subtract the electrode polarization effect
because there are insuficient data for good extrapolation. We conclude that it is
impossible to get correct sample data fiom the capacitance measured this way. The one

hope is to firrther reduce the electrode polarization efl‘ect.

5.5.1a Effect of electrode material

It is customary to use platinum, platinum black, and sandblasted Pt electrodes in

order to make the electrode effects smaller. From Fig.5. 12 we see that use of a platinum

85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10.000 _
t 1
~ (a) l
r- T
i
1.000 _ .
.w _ .1
A A A A
c: 1??
‘°° ~ El 8 O J l
E Frequency(KHz) I!
i- A 1 ;
_ a 10 l
_ o 100
10 1 I A L A 1 L1 1 A L L A l A A 1 L I ' I #1 A A A ll
0.1 1 10 100
d(cm)
100,000
: A
E A
I A a (b)
A
AAA
10.000 _ OAK
h D
.0) 1'” E D CEUII/
: El
: D 0 01:11::
' o o 5 o°°°' Frequency(KHz)
1CD : o O A 1
E a 10
: o 100
10 l l A L L A A A A A l A l l l A L A I A A l A I L A

d(cm)

F ig.5. 10: Apparent dielectric constants of (a) distilled water and (b) 1mM KC! solution

using brass electrode.

86

1,000

 

 

 

 

 

 

 

 

 

 

 

 

*—
l1
. (a)
a
c: i
c: D D D l
'10 100 : 9 f
_ V‘ : : A 78 I
1— l
). i
10 r A 1 A A 1 r A r g A r A A A A I A m I r A . a1
0.1 1 10 100
1(1).an
.. ‘3 1
: l
: l
‘ !
10.000 __
_ C
m >-
1,0CX) E
‘ 8
" v
v
1m LJA A 4L L 4*
0.1 1 10 100
10.000 1
l:
)—
r (C)
a
~03 1 000 : 51
3 0
l— + D
’- + D at: X
.. + 1,,
1. -+- t,”’
-+-
+x'
.. /,_+_
/ +
+
1m A 4 A A A A A A A A A A L L A LAJ A g L A A A AA;
0.1 1 10 100
d(cm)

Fig.5.1l: Apparent dielectric constants(El). (a) Real dielectric constant(O) after
subtraction of stray efl‘ects of distilled water at 100KHz. (b) Real dielectric constant(V)
after subtraction of electrode efi‘ects of 1mM KC! at lKHz. (0) Real dielectric constant(+)
after subtraction of stray effects of 1mM KCl at 1010-12.

87

 

 

 

 

 

 

 

 

’ (a) l
A
1,000 _ ‘
I i
“ E
’00 C l
a C] 6
~ 1
o l
100 J l l l 1 4‘ LA 11!
0.1 100
1.000
E
i (b)
'10 10° : g
t
10 L l 1 L1 LIII I I I I I IIIL J 1 1J_l_1
0.1 1 10 100

d(cm)

Fig. 5.12: Apparent dielectric constants of distilled water at (a) lKHz and (b) 100KHz
using brass(El), platinum(O), and platinum black(A) electrodes, respectively.

88

electrode reduced the electrode effects. But it seems that the platinum black electrode
made the electrode effects worse. Experiments on sandblasted Pt electrodes were
performed in a different cell and will be described later.

At this point we examine the role of platinum black which is frequently quoted as
reducing the electrode effect. According to Eq.(5.3.10) the electrode effect can be
reduced by making the sample resistance (R) large. If the platinum black electrode by
increasing R makes the electrode effect smaller, without any additional effect. the
dielectric constant of distilled water with platinum black electrodes at small separations
should be smaller than the others. But in Fig.5. 12 the data at low separations are not
consistent with this prediction. We argue that the introduction of a high resistance
platinum black layer confuses the issue further by introducing a Maxwell-Wagner effect.
We refer to the Figs. 5. 12 and 4.3. At a low frequency like lKHz, apparent dielectric
constants using a platinum electrode are much smaller than dielectric constants using a
platinum black electrode while the reverse is true at high frequency over lOOKHz and
small electrode separation. This is consistent with the form of the Maxwell-Wagner efl‘ect
shown in Fig.4.3. The Maxwell-Wagner contribution is very large at low frequency but it
is much smaller at high frequency. Even at high frequency the dielectric constants are
smaller than the dielectric constant of the conducting medium by itself. Data using the Pt
electrode was better but we still had difliculty in making the necessary subtraction.

In addition to the above difficulties in obtaining the accurate capacitance from the

Schwan analysis there are problems associated with the analysis of Rm. Eq.(5.3.2) can be

written as
R_ = R,(1+(a;c,R_)’)+R,. (5.5.1)

The second term in the parenthesis is the square of the quality factor of the whole system
which is much less than 1 (See table 5.2). It can further be taken that R, is small

compared with R, if the size of the electrode is not too small. Thus we can approximate
R, a: R, = l/Gs so that the total conductance should be

89

Table 5.2 The quality factor of 1mM KCl.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f(KHz) Cm Gm Qm 1 411192 f(KHz) Cm Gm Qm Qm"2
0.005 5.795-06 1.025-04 2.335-01 [3015-02 10 .305-10 1.315-03 1.765-03 3.09E-06
0.01 4.62506 1.32504 2.54501 6.43E-02 20 1505-10 1.31503 2.29503 5.25E-06
0.02 .335-06 3.425-04 1.93501 3.90502 30 1.305-10 1.315-03 2.935-03 '8.88E-06
0.03 615-06 4.365-04 1.615-01 2.595-02 40 1.305-10 1.31503 3075-03 1535-05
0.04 2.07506 6.09504 1365-01 1.35502 50 1.205-10 1315-03 4535-03 105-05
0.05 1.63506 7.11504 1.13501 1395-02 [60 1.205-10 1.315-03 5505-03 3035-05
0.06 1.33506 7.945-04 1045-01 1095-02 I70 1.215-10 1,315.03 6475-03 4.19505
0.07 1.155-06 635-04 9365-02 13755-03 [[30 1205-10 1.315-03 7335-03 5335-05
0.03 9.75507 9.205-04 13435-02 7.195-03 ll9o 1.205-10 1.315-03 255-03 6.31505
0.09 35507 9635-04 7765-02 6025-03 100 1205-10 1.31503 9.17503 405-05
0.1 7215-07 1015-03 7.16502 5.12503 200 1.195-10 1.31503 1.32502 3315-04
0.2 2.405-07 1.195-03 4.025-02 1625-03 300 1,195.10 1.315-03 2735-02 7445-04
0.3 1.165-07 1255-03 2.795-02 7305-04 400 1.135-10 1.315-03 3605-02 1.30503
0.4 6315-03 1275-03 2.145-02 4535-04 500 1.135-10 1.315-03 4.505-02 2035-03
0.5 4.435-03 1235-03 1725-02 2.975-04 .600 1.135-10 1.315-03 5.405-02 2.925-03
0.6 3135-03 1295-03 1455-02 2125-04 [700 1135-10 1.315-03 6325-02 3.995-03
0.7 2325-03 1305-03 1265-02 1535-04 [300 1135-10 1.315-03 7.215-02 205-03
03 1.79503 1.30503 1.10502 1225-04 '900 1.135-10 1.315-03 .115-02 6535-03
0.9 1.42503 1.30503 325-03 9655-05 1000 1195-10 1325-03 9015-02 3125-03
1 1165-03 1.305-03 3325-03 7955-05 2000 1.195-10 1335-03 1795-01 3195-02
2 3105-09 1.315-03 4.745-03 2255-05 3000 1.205-10 1375-03 2645-01 6.99502
3 1.405-09 1.315-03 3215-03 1.035-05 4000 1225-10 1.415-03 3.46E-01 1.205-01
14 005-10 1.315-03 2445-03 5935-06 5000 1245-10 1475-03 4225-01 1.73501
|5 .005-10 1.315-03 2295-03 5255-06 16000 1.275-10 1.56503 4.91501 2415-01
|6 4005-10 1.315-03 1.335-03 3.365-06 |7000 1.315-10 1665-03 5535-01 3055-01
[7 3505-10 1.315-03 1.375-03 505-06 Isooo 1365-10 1.795-03 6055-01 3665-01
[3 3005-10 1.315-03 1.335-03 3365-06 [9000 1415-10 1.965-03 6435-01 4195-01
[9 2505-10 1.315-03 1.725-03 .955-06 [10000 1.475-10 175-03 .795-01 4615-01

 

 

90

G, =G,+Go, (5.5.2)
where the second term is the stray conductance which is represented by Eq.(5.3.9). Both
terms in the right hand side of the above equation vary linearly with III which makes
separation of the two components difficult. In Fig.5. 13 we show that the conductance vs.
l/l graph is close to linear for Pt electrodes. For the platinum black electrode there are
wide divergence from linearity. For the platinum black electrode the additional term
which is associated with Maxwell-Wagner efi‘ect should be added in Eq.(5.5.2). Then the
total resistance becomes a very complicated function of I which is almost impossible to
separate. Because of these difliculties we proceeded to use a different method using a

smaller cell.

5.5.2 Small cell

Since, despite our efi‘orts, the above configuration still has reduced but serious
electrode and stray field efi‘ects, the small cell with guard ring arrangement[15] (See
Fig.5.14) might have a merit if the electrode polarization can be removed appropriately.
The cell has two parts which are separated by epoxy to maintain electrical isolation
between them. The top part supports a precision micrometer and the bottom part contains
the fluid. The lower electrode was machined separately, then held in the correct position
relative to the guard ring and fixed there by pouring epoxy. The upper electrode was fixed
to the spindle of the micrometer.

Fig.5.15 shows the equivalent circuit suggested by Touw, Mandel, Honijk and
Verhoog[16, 17]. For the long cylindrical cell we tried to apply Schwan's method to
eliminate stray field efl'ects. When we are using a small separation between electrodes the
stray field efi‘ect (C, and G, in Fig.5.15(a)) can be reduced by the guard ring arrangement.
In the following analysis we neglect them. The resistance(R,) and inductance( L) of the
lead wires are also negligible. Hence G... and C, of the equivalent circuit in F ig.5. 15(b)

91

0.0012

 

 

   

 

 

 

 

 

 

0.001 L.
A I
O .
.: 0.0008 __
E .
v .-
0 .
0 . :
é 0.0006 _,._ -
a : (J
2 00004 ’
O t Q
0.0002 ' ;_
0 llLlillllillllilllellllilllliLllL]LL41
0 50 100 150 200 250 300 350 400
1/1

Fig.5.13: Conductance vs. 1/1 of 1mM KCl solution using Pt(A) and Pt black(0)

electrodes at 100KHz. The solid line is a guide to the eye.

92

 

 

A : Teflon
I Brass -
Epoxy

 

 

 

 

 

 

Fig.5.l4: Dielectric cell with guard ring arrangement. H and L are the high and low
terminals respectively. G is ground, and EU and EL are the upper and lower electrodes.

93

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0-
_A'L'A'A'
I 1
l r
C r
G G.
AV‘V‘VAV AVAVAVAV
‘v‘vw-m—
; 1 1 1
l r 1 1 R, L
C C .

 

 

(b)

F ig.5. 15: (a) A detailed electric circuit of the cell. (b) A measured equivalent circuit. The

subscripts r, e, m stand for residual, electrode and measure, respectively. R, is lead wire

resistance.

94

can be related to irnpedances by
1 1 1
: + ‘
G, +ij, 6+»ij G,+jcoC,

 

 

(5.5.3)

where G, and C, are measured conductance and capacitance, and G and C are those of

the sample, and G, and C, are those of the electrode effect respectively. From this

equation we can deduce that
- -1 -1
G[G2 +mZCf,] 1 =[G(1+Q2 )] +1[G( (HQ,2 ),] (5.5.4a)
C ,[61 +mzcj]"=c[G’(1+Q1)] +C[G 1+Qj)] (5.5.415)
where Q = (0C " and Q, = coC,G,‘ are the quality factors of the solution and the double

layer on the electrode respectively. Eqs.(5.5.4) can be rewritten by

G;‘ = G“(1+Q:)(1+Q2)"+G,"(l +Qj)(1+Q3)", (55511)
C, = CG,,,G'2 (1 +Q3,)(1 +Q2) +C,G,,G,’2(1+Q:)(l +Qf)", (5551:)
where Q_— - 00 CmG;l , the quality factor of the whole system. At low frequency Q2 and Q3,

become negligible with respect to unity. Since electrode effects have no influence on
G, and C”, at high frequency we have Q. a Q so that

G;‘ = G" +0," (1 +Q:)(1 +Q3)", (5.5.6a)

C, = CGjG'z +C,G,3,G;2 (1 +Q:)(1 +Qf)"'. (5.5.6b)
From the table 5.2 for 60Hz < f < 2MHz the squares of the quality factors of the measured

quantities are much less than unity so that we can neglect Q_ in the above equations.

Therefore Eqs.(S. 5.6) can be written as

a: 4 cr'+G;'(1+Q3)=1+GR) (5.5.7)
Since the ion concentration near the electrode is greater than that of the bulk solution we
may assume that if the electrode resistance R, = G,”‘(1+Qf)‘1 is much smaller than the
bulk resistance R=G", then 0436,, and to a good first approximation, Eq.(5.5.6b)
yields

C, =C+C,G’G;’(1+Q})". (5.5.3)

Using the relations as

95

Q3 _ 1
Ce zwcp andR-E— (5.5.9)

we can prove Eq.(5.5.8) is identical to Eq.(5.3.3).

Using G = o A/d and C = 8 AM, and the fact that the electrode effect is not a fimction of
d Eqs.(5.5.7) and (5.5.8) can be expressed by

l l
G zcrA——R<32A2 —,
111 d e d2

2
C, zaA21—+const.(%) , (5.5.10b)

(5.5.lOa)

where d is the separation of electrodes and A is the area of electrode. Using the above

analysis the conductivity and dielectric constant can be obtained by fitting the total

conductance and the capacitance with a polynomial of l/d up to the second order. Table
5.3 is the results of G, and G for distilled water and 1mM KCl solution. From this data
for distilled water there is about 15% error between G, and G, and about 4% error for
1mM KCl solutions above IOOHz. The approximation G 22 G, introduced at Eq.(5.5.8) is

good for a highly conducting medium.

5.5.2a Results using Pt black electrode

Fig.5. 16 and Fig.5. 17 are results using a platinum black electrode. They have loss
peaks which should not be seen in the distilled water and KCl solutions. We believe that
using a platinum black electrode gives an additional effect so that Eq.(5.5. 10b) should be
modified to

C, zaA-‘1;+const.(%)z+MW. (5.5.11)

The third term refers to the Maxwell-Wagner polarization term which is also a function of
electrode separation. Therefore fitting the measured capacitance to the inverse of the
electrode separation gives a wrong result. These results support our contention that the

platinum black acts like an insulating layer causing Maxwell-Wagner polarization.

96

Table 5.3 Comparison of conductivity values obtained from direct measurement(Gm) and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

from using analysis(G).

f(KHz) (31110120)! (30120) Gm(KCl) G(KC1)I 10012101110120) (30120) Gm(KCl) G(KCl)
0.005 2425-04 2.31504 5.33504 3.23504] 1012.63504 3.04504 6395-03 7165-03
0.01 2535-04 2.97504 9595-04 523504] 201263504 3.03504 6.39503 7.16503
0.02 2595-04 3035-04 1.30503 4.70504] 3012.63504 3.03504 6.39503 7175-03
0.03 2615-04 3.05504 2565-03 1.14504] 4012.63504 3.03504 6.39503 7.16503
0.04 2.61504 3045-04 3205-03 9.57504] 5012.63504 3.03504 6.39503 7.16503
0.05 2625-04 3.04504 3.74503 1.37503] 6012.63504 3.03504 6.39503 7.16503
0.06 2.62504 3.05504 4135-03 2.73503] 701263504 3.03504 6.39503 7.16503
0.07 2.62504 3.04504 4545-03 3.37503] 301263504 3025-04 6.39503 7.17503
0.03 2.62504 3.04504 4345-03 4.11503] 901263504 3.02504 6.39503 7.16503
0.09 2.62504 3.04504 5095-03 4.67503] 10012.63504 3015-04 6.39503 7175-03
0.1 2.62504 3.04504 5305-03 5135031 200] 2.64504 3.03504 6.39503 7.17503
0.2 2.62504 3.04504 6235-03 7.11503] 30012.64504 3.02504 6.39503 7.16503
0.3 2.63504 3.04504 6535-03 7.39503] 40012.65504 3.02504 6905-03 7.16503
0.4 2.63504 3.04504 6.70503 7.45503] 5001266504 3.05504 6.90503 7.16503
0.5 2.63504 3.04504 6765-03 7.33503] 60012.66504 3.05504 6.90503 7.16503
0.6 2.63504 3.04504 6305-03 7.39503] 700D.67E-04 304504 6.91503 7.16503
0.7 2.63504 3.03504 6325-03 7.33503] 3001263504 3065-04 6915-03 7.16503
0.3 2.63504 3.03504 6335-03 7.32503] 9001269504 3095-04 6925-03 7.17503
0.9 2.63504 3.04504 6355-03 7.32503] 100012.69504 3.07504 6.92503 7.17503
1 2.63504 3.03504 6.35503 7.27503] 200012.79504 2955-04 7025-03 7225-03

2 2.63504 3.04504 6335-03 7.21503] 300013.00504 3265-04 7135-03 7355-03

3 2.63504 3.04504 6.33503 7.20503] 4000L3.26E-04 3255-04 7435-03 7505-03

4 2.63504 3.04504 6.39503 7.175031 500013.63504 3.59504 7.75503 7.65503

5 2.63504 3.04504 6.39503 7.19503] 6000] 4.37504 2.77504 3.13503 7.73503

6 2.63504 3.04504 6.39503 7.19503] 7001 5.47504 1.16504 3.73503 7.33503

7 2.63504 3.04504 6.39503 7.16503] 30001705504 4.73504 9.43503 7.41503

3 2.63504 3.04504 6.89E-03 7.17503] 9000W53504 212503 1035-02 7.15503

9 2.63504 3.04504 6.39503 7.13503] 100001135503 -7.16503 1.14502 4.46503

 

 

 

 

 

 

 

97

 

 

 

 

 

 

 

 

67 5
55-3 : D C] :1 1:1 3" A i
_- A (a) 7. 4-5
I A j
66.6 I
- , 4
: A _. D
C C]
1; D
P A
66 L A _
- A 2.5
3 AA A
C A
65.8 _ C] a 2
: DEC]
65.6 _ Cl —« 1.5
" A 1:11:
I E] [:1 :1
fi.‘ ’- 1 1 1 1 1 1 1 1 1 1 1 L 1 1 1 1 1 1
10,“ 1CD.“ 150-006
f(Hz)
95 DA 60
: AA
90 L D 1:10 (W “ 55
F A
1- D _‘ w
: D A
85 C]
T C]
1- [j _‘ 45
*' A
0” w __ D "‘ 40 m:
,_ A
: C1
75 1— [1th
1 A
. AA
1- AA D _ w
70 ‘ A D 53D
7 A Q31:
‘ — 25
‘ A
‘ A
_ AAA 5
1 1 L1 LJ_L_1 1 1 1 1 1 1 1 111 1 1 1 1 1 1 1 1 m
1.“ 10.” 100,000 15+ooe
f(Hz)

Fig.5. 16: Real part(El) and imaginary part(A) of dielectric constants of (a) distilled water

and (b) 624.3nm suspensions with platinum black electrode.

98

 

 

 

 

 

 

 

 

24o . 450
E 151
22° _— . A (a) -<. 400
1- Q‘ A
" A A ._1
200 _— 1: @2134“ A .. 350
I A
150 " r:— ,
__ H A
160 ~__ I
b A ‘ m
b : A _l 250 ‘
1‘0 :— ‘3 A I
I A —-' 200
120 ,~_ 1: A 1
'— |
,_ I: —4 150
L— .
too ,___ DD (
: D% ‘3 9
L —( 100
E D a 31 Sam
'- 1 1 11 1 1 4 1 1 1 1 1 1 1 1 1 ‘g‘Akani 50
1.000 10.000 100.000 18006
35° A 300
_ A A I
1- A 1
' A 700
300 ' 15’ A (b) A)
: D A '4 600
250 "_
—4 500
K :1
P
. m L _‘ m m.
w __ D A
” D - 300
150 :_ A D A
; A 1:: A — 200
100 __ A b AAA»
_ 4» 100
1— A D
so t D DEC] 0
1.000 10.000 100.000 15+oos
130 180
9:.
12° _ A (C)
” ‘3 — 140
no ’_ 1:1
- d 120
Z A :1 A
o l- _‘ m
w 1” __ C1 ‘00 .
" (:1
C (:1 A _ so
- :1
N _ C] A
: A Aaflu 60
r- -1
3° _ D -—< 40
_ D D
:3
:1
m As L 41% 1 Ag L A A 9?

 

 

 

3
§
.8.
§
3'.
3

Fig.5.17: Real part(E]) and imaginary part(A) of dielectric constants of (a)367.7nm,
(b)364.1nrn and (c) 213.8nm suspensions with platinum black electrode.

99

We firrther confirm the effect of a Maxwell-Wagner term as follows. As mentioned
earlier in Sec.5.2 the large resistance of the sample gives a smaller electrode effect. We
may increase the total resistance between the electrodes, i.e., the total resistance of the
system by covering the electrode with an insulating layer. For this purpose we covered
the electrodes with scotch tape. Fig.5. 18 and 5.19 show dielectric constants of several
samples obtained using the brass electrode covered with tape. There is an enhancement of
the real part of the dielectric constant and a loss peak in the imaginary part. The Maxwell-
Wagner effect predicts such dielectric behavior for a mixture of insulating and conducting
layers (See Fig.4.3).

5.5.2b Variation of measurement with time

At this point we remember the conductivity drift with time pointed out by the previous
investigators[1,2,4]. They suggested that this drift may be caused by a temperature
change or a coagulation of the particles during a measurements. Even a small change of
conductivity can afi'ect the dielectric behavior of the suspensions. It needs to be corrected
by additional measurements to determine the time dependence. Fig. 5.20 shows the time
dependence of the conductivity of 1mM KCl. Neither the conductivity nor the
capacitance of the sample change appreciably with time. The only change in conductivity
is 0.3% during 10 minutes but the capacitance does not change at all. It is also true for
the distilled water. For the suspensions, however, for frequencies <10 KHz there was a
very substantial time variation as indicated in Figs.5.21 and 5.22. Because the change in
suspensions is much larger than that in a pure electrolyte we conclude that the time
variation is associated with the suspended microspheres, -possibly with coagulation- rather
than with a temperature change.

Figs.5.21 and 5.22 illustrates the time variations of 624.3nm suspensions at two

different electrode separations. For the capacitance the time dependence is frequency
dependent for f < 10 KHz. At 0.1 KHz it is particularly severe, and for f > 10 [(112 it is

100

 

 

 

 

 

 

 

"00° ‘ 1000
b A A i
.. A AA ‘
A A
_ AA. AA
C3 Cl C] C Baal]:
L1)
A :1
A D g
1.1 D i
Ow 1w __ DD H ~ 1m m:
E A Chit] '
_ C :J : :3
.. A; i
a: «a?
” H3?
!
’ i
i
'1
1o 1 1 1111111 1 1 1111111 1 1 1 11111110
1.“ 10.” 100.000 154'”
f(Hz)
1,000 LG!)
1 A .3
1— A A AAAQfiP
1.
" AAAA
UCJCIID D
£3 E] [:1 :1 [:1 g 8113:1113 1:1 4 100
A D
C]
A DC]
7" 10° 3 AA DEL}: 0"
1.. A
- A
A
P A -‘ 1O
" A
L.
A
’ (b)
10 4 1 1 1 11111 1 #4 L14LLL 1 P L 11111 1
1.“ 10.000 100.000 1E+ooe

f(Hz)

Fig.5. 18: Real part(El) and imaginary part(A) of dielectric constant of (a) distilled water
and (b) 1mM KCl solution with brass electrode covered with tape.

101

 

 

 

 

 

 

 

1,“ b 1 1.000
1 (a) ;
: A 21;“ '
A ‘
._. ,_. |
,\ ' --» ~ 100
1 M1133 —_,w
:3 C1 C1 CJCEEZIIIS g 1: gm 1‘4-
.0) 100 _ C] D C3 : m
L. ngl E% ‘ z
I A?“ I
- A A :1 2 _
i- A —-' _4
A i 10
I
1
10 1 1 1111111 1 1 1 111111 1 1 1 111111 1 1 1 11111 l1
1.“ 10,” 1M,” 18006 18007
f(Hz)
1.” 1.000
p (b)
A A A
A
r- AAM /\ 4 1m
C] GUIDED a A AA \ 41)
D Damn] “L131
- 100 a 1:] (*1
OJ : y D -
: A m
_ A
+- A D D DCEIInliaw
" A
,. A
10 l L L L L4 L11 1 l l l l 1 11 l J L llllll l L # L 1 l 1 1 LJ 1
1 .000 10.“ 100.000 184-008 1E+00
f(Hz)

Fig.5.19: Real part(D) and imaginary part(A) of dielectric constants of (a) 367.7 nm and

(b) 364.1nm suspensions with brass electrode covered tape.

102

0.01

 

0.001 fiiiiiiiiifiiiiiifiiififlififiifiiiiwfim9

 

0.1KI-lz
1KHz
10KHz
1MHz

Capacitance(F)

O D O D +

 

 

 

 

. g- A.-. o .A .1 .A “ ‘ ‘ ‘ ‘ t
lE‘OlO seesaw. sausage. new. at else .513 'asamoaeammamaan'aama151315

0 50 100 150 200 250 300 350
Time(sec)

0.0029

 

OOOOdOOOOOOObOOOOOOOOOOOOOO 0000000OOOOC)

”’23 ‘00ssssasssssssssmssssssssssé駧s§ssssssssss ~

4 A A
0.0027

 

0.0026

 

0.1K":
1KHz
10KHz
100KHz
1MH2

0.0025

 

0.0024

OUJD‘I’

 

 

0.0023

 

Conductancumho)

0.0022

 

 

0.0021

 

rilt'1' Ui'jljiil Ill' TUVT'UYj' II

1
I
I 0 I
I I i
. . .
5 l s
z 2 i
1 ; :
. z :
i i i
l 1 t
s x s
l l l
. .
1 .
E i
i
l I
‘ 2
g i
. .
: i
'-. E i i 1
E i E E i
; ; l I
1 E E i '.
l : : ‘ l
: . : z 1
3 3 i i Z
+ ‘ : x : ;
i i I I Z
i i E 3 5
: . 1 . . 1
a ‘ i
. .; . '. .. .' .. .
z : 1 .'
g . , . . .g .,
E Q 1 r E .
. , 1 ,

 

 

0.002 1
350

§

150

§

250

O
M
O
y—o
8

Fig.5.20: Time variation of dielectric behavior of 1mM KC] solution using small cell with
Pt electrodes.

103

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lE-OO7
(a)
f: 1124103 :J
3 :
0 .
r: .
é - o 0.1KHz
g _ e 1OKHz
a c 1MHz:
8 113.009 . . ,.
lE-OIO 11L1i11L1L111iL+L1l1111:lLLL121111111L111111114111
oroozoosoo4oosoo6m7oosoo9001.ooo
Time(sec)
115-008
E 15.009 _
g E
5 : 10,;KHz
g. _ in 1MHz
0 115-010 ‘ '
0
15011LJHiILILEIIJLiAIIIiIIIlilllliliuiullilllljllll

 

 

 

01002003004005006007008009001.000
Time(sec)
Fig.5.21: Time variation of capacitance of 624.3nm suspensions at (a) 5mm and (b)

7.6mm using Pt electrodes. For clarity only the data for 3 of the 5 frequencies studied are

shown.

Conductanctho)

Conductanctho)

0.00036
0.00034
0.00032
0.0003
.00028
.00026
.00024

COCO

.00022
0.0002
0.00018
0.00016
0.00014

0.00012

0.000115

0.00011

0.000105

0.0001

9.5E-005

E

8.5Eu005

8E-005

7. 5E-005

7E-005

104

 

 

, .  ‘ ‘ ‘0.1KH2
‘ . .......1OKHZ.., ..

 

 

D

.. ...........1 u.

 

....................

'UIUI'UIIIEUIIU'IIII

 

. . .. ..o...1..._.. .......1 .. ...a......... .. .. .

 

 

llllllllllllllllllLllllllllll.lllllllllllllllllll

100 200 300 400 500 600 700 800 900 l .000

Time(sec)

 

O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

05.1KHz . i (b) >
_ .. 1OKHz %
L- 1 0 1MH.Z. .........
E i

 

 

 

 

 

 

 

 

VIII '1'! ITII IUII

= a a : s a g 5
L4lllljljlllllllllll_l_.llJlllllllllJlLJLl11111 [All

100 200 300 400 500 600 700 800 900 l .000

Tlme(sec)

 

O

Fig.5.22: Time variation of the conductance of 624nm suspensions at (a) 5mm and (b)

7 .6mm using Pt electrodes. For clarity only the data for 3 of 5 frequencies studied are

shown.

105

negligible. In the conductance the time dependence is severe at all frequencies. The time
dependence also depends on electrode spacing. Thus it is almost impossible to make a
time correction and since the variation within a few minutes is very rapid it is difficult to

initialize the time compensation.

5.5.2c Sand blasted electrode

At this stage we remembered the recommendation of Springer and Lyklema
regarding sand-blasted electrodes, and as a last resort we performed measurements to see
whether their use would reduce the electrode polarization and time variation problem.
Since the Pt electrode on the present cell was too thin for sandblasting a mew cell with
different dimensions was made to accommadate a thick sand-blasted electrode. F ig.5.23
show the time variation of 213.8nm suspensions for a continuously applied field(line) and
for an intermittent applied field. The intermittent measurements were performed by
measuring 5 frequencies quickly(about 2 seconds) at a mean time and then repeating there
at a second mean time and so on. Afier the intermittent measurement were finished a new
sample of the same suspension was found and measured quasi-continuously. The line is a
computer drawn line through the experimental points. Initially the variation with time is
rapid and this means that the starting points of the 2 curves are different. If we shift the
curves so that the capacitance value is the same at t=0, then the conclusion is that a
substantial portion of the time dependence is associated with the application of an electric
field. That is electrophoresis is present, as well as coagulation. Electrophoresis is
diminished by the use of small fields. Thus we endeavored to do using an applied voltage
of 100 mV. Figs.5.24 and 5.25 show the time variations of 624.3nm suspensions at two
different electrode separations. Again the time variations are different at each separation
and at each frequency and even worse than that using Pt electrodes. Figs.5.26 and 5.27
show the time variation of the capacitances and conductances of four suspensions using

sand-blasted electrodes at 1.3mm electrode separation. The time variation of conductance

106

112—007 _

.1»

u -— -_' _-._. “j“

 

 

l E-008

0.1m

1 TFIIII"

 

113-009 ..... .. .. . 0.1KH2

1OOKHz
o 1MHz

I 1I11rfl'f

 

 

 

Capacitanch)

 

115-010

lE‘Oll l 1 1 1 I L L 1 1 L 1 1 1 1 1 1 1 1 1 1 J J .1 L 1 a L 1 LM
0 100 200 300 400 500 600 700

Time(aec)

 

 

 

0.00034

0.00032 .. .... ., - . 1b.). .. ,

0.0003 ° .. O-IKHZ ;,

 

 

 

 

 

 

 

0.00028 ° 1M”!

0.00026

 

 

0.00024

Conductance(mho)

0.00022

 

0.0002

 

 

 

‘jT‘UjT r'IVr'VUU TV.
I

 

 

VT

 

 

I . 1 ,
1L4 lJlLLLIlkLJllJJlIlllllLJllLlll

0 100 200 300 400 500 600 700

'l'ime(sec)

0.00018

 

Fig.5.23: Time variation of 213.8nm suspensions with sand blasted electrodes at 1.9mm

separation. For clarity only the data for 3 of the 5 frequencies studied are shown.

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15-007
: (a)
E lE-OO8 P...
0
u E
= .
g . 0.1KHz
a - .- 1OKHz
*3 112-009 ,,_ o 1MHz
0 :
IE‘OIOVllIIIIIlllllllillllilllLLLllllllll
0 100 200 300 400 500 600 700
Time(sec)
15.009
1 a? "D
a: r gimaéfigmfiow “boo W moodgfinogfl 3,%%
‘6’ _ 0 °
3 : 01KH *
115-010 ° .. _z. .. ........
g 3 . 10KHz
a - o 1MHz
a. - 1
a _ .
O
r
lE-Oll AlllilIllilljllllllillllillllllLLL
0 100 200 300 400 500 600 700
Time(sec)

Fig.5.24: Time variation of capacitance of 624.3nm suspensions at (a) 0.6mm and (b)

1.9mm using

sand-blasted Pt electrodes. For clarity only the data for 3 of the 5

frequencies studied are shown.

108

 

  
      

 

fr,

...0_.1KHz. (a) ..
10KHz

 

 

II'ITII'UIIYIIYIr-.'

.0 .o .o .0
§ o § §
fl hi —‘ ~
M 0‘ \l W

Conductance(mho)

 

0.00014

 

 

O.(XX)13 7 I I I I l I I I I l I l L I l I I I I l I I I I l I I I I l I J I I

 

 

  
 
  
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 100 200 300 400 500 600 700
Time(sec)
3.512005
3.413005 E " b
3312-005 1° ( )
‘5 . 5%,
.: 3.213005 :3..--
E : '~._
7 : -.
03.1E-005 raw-rm
g G
0 315-005 ..-..
a F
22.915005 :2; ............................. ..
8 5   am
5 2 i 0 “MHz
2.715005 :1“.-.- . ............... -
2.6E'ms : I I L L I I I l l I L 44 L L4 I 1
0 100 200 300 400 500 600 700
Time(sec)

Fig.5.25: Time variation of conductance of 624.3nm suspensions at (a) 0.6mm and (b)
1.9mm using sand-blasted Pt electrodes. For clarity only the data for 3 of the 5

frequencies studied are shown.

109

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I u I.“ 1.5. I.” I.” I.“ JJ‘ 4.” I ~ I.” 1.!“ m I.” I.” I.” I.”

tin-(nee) find-0°)

(a) (b)

 

 

 

e

 

 

 

M10 1.....- .....'..... . 4 .' i ' 2
p

 

 

Fig.5.26: Time variation of the capacitances of suspensions of (a) 213.8nm, (b) 364.1nm,

(c) 367.7nm and (d) 624.3nm using sand-blasted Pt electrodes at 0.05" separations.

 

u. u. use no u. an
Mac)

(a)

 

110

 

 

 

 

 

 

 

 

 

 

 

 

’ h a 1
ii
...... r ...,m-.- , . !
0.90! I
“new ll . mu... .,
L 10101: !
emu 1 i 11 . - - 1 1 - l
0 so u- ull 1.1. 1.311 no use was
Thu-0e)

(b)

ISM

 

0 ll 1.. I.” 2.. I.” J.“ 1.3. O”

Tit-(ac)

(d)

Fig.5.27: Time variation of the conductances of suspensions of (a) 213.8nm, (b) 364.1nm,

(c) 367.7nm and (d) 624.3nm using sand-blasted Pt electrodes at 0.05" separations.

lll

seems to be independent of frequency but that of capacitance is dependent on the
frequency. The dielectric behavior is illustrated in Fig.5.28. At low frequencies the
electrode effect is still dominant and the dielectric constant is large compared to the Pt
electrode results on those obtained by other investigators. The imaginary parts of the
experimental results indicate that none of suspensions have dielectric loss peaks. At

around 2 KHz they seem to have loss peaks but they are not the real loss peaks.

5.5.2d Results with Pt electrode

Finally we returned to the Pt electrodes which seemed to have the best properties.
Due to the timing dificulties the actual measurements were made after one hour when the
suspension is stabilized. Until we know the nature of the time variation, this means we do
not really know the nature of this stable state. Fig.5.29(a) shows the conductivity of
367.7nm suspensions vs. frequency. From 100 Hz to 1 MHz the conductivity increases
gradually but very slightly with frequency. The conductivity increase is more rapid in the
frequency range between 10 KHz and 100 KHz, where the dielectric loss peak is
expected. Here we are faced by the last major problem. The part of the conductivity
associated with dielectric loss is that associated with the AC conductivity. Therefore the
DC component of the conductivity is somehow to be subtracted from the measured
conductivity. The choice of DC conductivity is not easy because the conductivities below
1 KHz are not approaching a single limit value and DC measurements present an
decreasing conductivity with time. Therefore the choice of DC conductivity critically
affects the deduced imaginary part of the dielectric constant. Fig.5.29(b) shows the
imaginary part of 367 .7nm suspensions calculated from different choices of DC
conductivity. For the best result we chose the conductivity at 1 KHz as DC conductivity.
Therefore, to get the experimental imaginary part of each dielectric constant the following

relationship was used
(0(a)) - o(lKHz))/0) . Here a is conductivity.

112

154007

 

lEWOo

 

100,000 . . q . _6 ......... 21“”--.-

40 9 '- °_ ; :' mm
._ 5.. 971." i o 367nm

. . 29;; .... -4 62‘0". ........ .......

 

 

 

 

 

. ‘ AA“ . .In 3

 

10' I A AAAAAAA A AIIAAALI A AAAAAAAA L A AAAAAAA A AAAAAAAA A LAAAAIJ‘ A A AAAAA

l 10 100 1.000 l0.000 100,000 lE+006 1E*007

 

0.01

 

0M

 

0 V 21‘“ - . _ .. .. .. .. ......W _ (b)---- .. ... H: be,

0% 387nm E -0030” °°°~ 3 e

um 3‘

? mow“?- """" ...-m .. 3
' O

0000

 

 

 

0.007

 

 

 

 

0.006

 

0W

 

""'O"9'QM"“*Q"P 00m“- . .............

0W

 

0(mhdm)

0M3

 

 

0W2

 

 

' f ' ' Adan“
2 A.‘AM| AAA“ Adana“ AAAM AAA“

_r . 3 : ;

0M)!

 

 

IIII IIII IIII IIII IIII IIII'II'I IIII IIII IIII
O

 

 

 

O

I

h

D
woo

. . . . .
AAAAA A A AAAAA A A AAAAAAI A A AAAAAAA A A AAAAAAA A A AAAAIAA A A AA

10 100 1.000 10.000 100.000 rra+ooo mom
f(Hz)

 

.—

1M

 

 

(M

 

 

3.“

 

 

 

 

2.“
1.5”
I.”

 

 

 

 

{GM-01 may»

90

 

 

 

 

 

 

 

§

§ IIII IIII IIII IIII IIII I'II IIII IIII IIII IIII
O
0
0
D

 

 

 

 

rO

Fig.5.28: (a) Real parts , (b) conductivities and (c) imaginary parts of suspensions using
sand-blasted Pt electrodes.

113

0.01

 

(a).

0.009
0.008
0.007

0.006

Conductivity(mholm)

3.333)?

‘ .3. o cocoons o o o-z-coom

0.005

0

VIIIIIIIIIVTU"TTTIIYT—TIITUTI

0.004 1 1 11111111 1 1 1111111 1 11111111 1 1 1111111 1 1 1111111 1 1 IILUJL 1 1 111111

10 100 1,000 10,000 100,000 lE+006 1£+007
f(Hz)

 

 

 

H

100,000

 

('3)

10,000 .05-.-.- .. .. ,. - .. ... .. ._ ,. - _. ... -. __ ,, , ... - ,

I III

I

 

r' UTTTIT

 

 

 

 

 

o i . t
, ., g o 0.11047.
09 o 0 : o 0.5KHz
r- ‘3‘» : o E
°°. ~09” . lKHz .
k - 9
o: .
: 3 o o f "' o .
: 0 00006 D Q 000 :
3:1: 0 ' . o f
r- 00 d 033 00
r 0 c: D .
O
+ a O
3 i 1 9 O I
100 1 1 151111¢ 1 1 1111111 1 1 1111111 1 1 111111L 1 1 1 11111

 

100 1,000 10,000 100,000 lE+006 1E+007
f(Hz)

Fig.5.29: (a) The conductivity of 367.7nm suspensions vs. frequency. (b) The imaginary
part of 367. 7nm suspensions with choice of different DC conductivity. The numbers in

the legend correspond to the frequencies where the DC conductivity was taken.

114

F ig.5.30 to Fig.5.33 illustrate the dielectric behavior of four different suspensions with
the theoretical predictions using Sen's theory (solid lines). The data below 0.1 KHz and

above 1 MHz should be rejected due to the inappropriateness of the cell and the analysis

method. The fitting parameters are same as Sen used with Schwan et al.'s data except ‘i’o
and the volume fraction of balls. ‘i’o was varied to optimize the fit of experiment to theory.
‘Po=4.78, 5.38, 5.5, and 4.28 were used for 213.8nm, 364.1nm, 367.7nm, and 624.3nm
suspensions respectively.

Comparing the experimental results and the theoretical predictions the imaginary parts
are very close each other but there are large differences in the real parts. Especially the
characteristic frequencies obey the relationship given by Sen very well. In Sen's theory the
characteristic frequency depend only on the size of the particles (0 ac l/az . The
characteristic frequency against the inverse of the square of the particle size is plotted in
Fig.5.34. The linear behavior indicates that the experimental results agree with the theory.
At very low frequencies dielectric constants of the experimental results are enormously
large due to the electrode polarization effects. In this experimental method and

interpretation the electrode effect may be reduced a little.

115

l ,000

 

 

 

 

 

 

 

r- A V ~\
. A 1
A f
A AAA A
.00 100 r... ......KM...
L A _'
10 I I IIIIIII I I IIIIIII I I IIIIIII I I IIIIIII I I IIIIII
100 1,000 10,000 100,000 lE+006
f(Hz)
140
P .
2’
100 L
80 ..
'w

 

 

 

UTII'UIIU'IUTU

20

 

 

111111 1 J 1111111

0

1}

L’\ (J (:1 (:1 1:)t:ll_k)

 

I IIlIII

 

 

u—e
U
§‘IIII

10,000 100,000

f(Hz)

1E+006

Fig.5.30: Dielectric behavior of 624.3nm suspension using small cell with Pt electrodes.

 

1E+007

116

 

 

 

 

100,000.
b
h
'_' A
r- A
.- AA °
" 1,000 __
w :
:
b
b
D
F
100 ___
b
h
D
h
h
h
1-
P
10 I I IIIIIII I I I IIIIII I L ILIIIII I I IJIIIII l I I LIIII

 

 

 

100 1,000 10,000 100,000 1E+006 1E+007
RM

500

 

450

 

 

350

300

 

  

II!IIII'ITII‘IIUI UIII

 

250

 

8"

200

 

 

150

100

 

 

 

 

 

 

f(Hz)

Fig.5.31: Dielectric behavior of 367.7nm suspension using small cell with Pt electrodes.

117

100,000

 

r1 IIII.

- .
b
_
p
b
D
1-
p—
L

 
    

 

 

...“
0.) -
b
b
' A
D .
. A a! .
b
- a
100 _.
D
I
D
D
l-
I
p
D
10 J I IIIIIII L 1 lllllll l 1 IILIIII 1 L4 111111 1 I 111111

 

é

1,000 10,000 100,000 1E+006 1E+007
f(Hz)

450

 

 

 

 

 

350

 

 

 

300

 

 

IIII'IIII IIIITIIII
I

\

250

8"
§

§OIIII IIII IIII IIII III\

 

 

59‘

 

 

§

 

 

 

8

 

 

 

1 1111111; 1 1 1111111 1 1 111111

10,000 100,000 lE+006
f(Hz)

Fig.5.32: Dielectric behavior of 364. lnm suspension using small cell with Pt electrodes.

--O

.

118

100,000

 

10,000 I?

T IIIIIIII fi IIIIII

- 1,000 4. .....- ..

 

r 1 IIIIII

  

100

 

 

 

E..." ........ “.....u ..A........... .. .- .. .... . . ... . .
. - A Am A

h

p

h

D

)-

10 1 1 1111111 L 1 111111I 1_1 1111111 1 1 1111111 1 1 111111

 

100 1,000 10,000 100,000 lE+006 1E+007
f(Hz)

800

 

 

 

 

 

 

3 o
700 L-
600 '__,,
500 L. .....
s 400 L .......
w : O
E o 000
. o
P
100 L _____

   

 

 

 

f(Hz)

Fig.5.33: Dielectric behavior of 213.8nm suspension using small cell with Pt electrodes.

119

50

 

40

35

30

25

KKHZ)

20

15

10

I IIIIAIIII IIII IIII IIII IIII IIIIAIIII IIII
' l l l l l l l l

5

 

 

0 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIILIIIIIIILIIII

2E-006 613-006 113-005 1.4E-005 l .8E-005 2.2E-005

 

l/a2(nm-2)

Fig.5.34: The characteristic frequencies vs. reciprocals of the square of the particle

diameter.

Bibliography

1. H.P.Schwan, G. Schwarz, J .Maczuk and H.Pauly, J. Phys. Chem. 66, 2626(1962)

2. M.M.Springer, A.Korteweg and J. Lyklema, J. Electroanal. Chem. 153, 55(1983)

3. Kyung-hee Lim and El. Franses, J. Colloid Interface Sci. 110, 201(1986)

4. D.F.Myers and D.A.Saville, J. Colloid Interface Sci. 131, 448(1989)

5. H. Fricke and H.J.Curtis, J. Phys. Chem. 41, 729(1937)

6. T.M.Shaw, J. Chem. Phys. 10, 609(1942)

7. D.R.Day, T.J.Lewis, H.L.Lee and S.D.Senturia, J. Adhesion 18.730985)

8. H.P.Schwan, Z. Naturforsch. 6b, 121(1951)

9. H.P.Schwan and J.Maczuk, Rev. Sci. Instr. 31, 50(1960)

10. J.Blom, J. Phys. E. Sci. Instrum. 12, 889(1979)

ll. J.L.Oncley, J.Am. Chem. Soc. 60, 1115(1938)

12. C.A.Marrese, Anal. Chem. 59, 2170987)

13. G.T.D.Shouldice, GAVandezande and A.Rudin, European Polymer Journal(To be
published)

14. HP 4192A manual

15. N.C.Lockhart and J.W.Snaith, J.Phys. E. Sci. Instrum.]Ol 1(1978)

16. F. Van Der Touw, M.Mandel, D.D.Honijk and H.G.F.Verhoog, Trans. Faraday Soc.
67, 1336(1971)

17. F.Van Der Touw and M.Mandel, Trans. Faraday Soc. 67, 1343(1971)

120

Chapter 6

Summary

6.1 Capacitance and dielectric breakdown

From the experiments on paraffin wax discs containing metal particles we have
deduced the following.

First, the capacitances were measured by the cell with guard ring arrangement for the
sample of stainless steel and copper particles embedded in parafiin wax. For stainless steel
inclusions the critical behavior was not consistent with the theoretical prediction due to
their poor conductivity. 0n the other hand the random packing point is similar to other
results. To better test the theory we need a particle with very good conductivity,
uniformity in shape and ease in handling. For better conductivity we chose copper
particles. For these the critical behavior was much closer to the theoretical prediction than
stainless steel samples. But the critical point occured at a much lower concentration than
the random close packing point due to their irregularity in shape. The critical exponent a's
in Eq.(l.3.3) were -O.15 for stainless steel and -O.86 for copper. For a perfect conducting
spherical particle it should be -1. Even though the copper is a good conductor it is very
difficult to handle because it is easily oxidized.

Second, the dielectric breakdown field was determined with the samples containing
stainless steel particles. The efi‘ect of conducting inclusions was enormous. The dielectric
breakdown field was dramatically lowered very much even at very small particle

concentrations. The behavior at low loading limit is qualitatively similar to the theoretical

121

122

prediction. The dilute limit exponent or=3/2 was found in both experiment and numerical
simulation for relatively small sample size. It is higher than the analytic prediction 1/2 < or
< l.

The difference between the behaviors of capacitance and dielectric breakdown field
stems from their relationship to the local field distribution. The capacitance is related to
the low moment of the local field distribution while the dielectric breakdown field is
related to the extreme moment. Hence the capacitance cannot be a non-destructive probe
of dielectric breakdown fields.

Since the dielectric breakdown field is related to the extreme moment of the local field
distribution it does not obey the central limit theorem. The dielectric breakdown field
statistics are described by the extreme distribution. We have compared the Weibull
distribution and the modified Gumbel distribution with failure tests of 300 pure paraffin
wax discs. It turned out that the modified Gumbel form is more appropriate than the

conventional Weibull form.

6.2 Dielectric properties of polystyrene latex
suspensions

We attempted to measure the dielectric properties of suspensions of polystyrene
spheres. This we hoped was a simple system which would lead the way into more
complex systems such as clay suspensions. The subject is of intrinsic interest and we
anticipated comparing the results with the theory of Sen and Chew, as well as studying the
efi‘ects of difi‘erent surface groups on the polystyrene spheres.

The motivation was good, the extreme difficulties of the experiment were not
anticipated. The foremost initial problem was that of electrode polarization which gives
large effects in the area of interest. Traditional methods of reducing the electrode

polarization or correcting for its presence did not lead to satisfactory results.

123

We first attempted the measurement of the capacitance of a colloidal suspension using
the long cylindrical cell with Pt and Pt black electrodes similar to that used by Schwan et
al. There were two major problems to get the correct sample data.

F irst, the analysis method was not good enough to get rid of the stray field effect
which is unavoidable on using a long electrode separation. The subtraction of stray field
effects did not result in the anticipated a single value at high frequencies above 100 KHz.
At low fi'equencies below 1 KHz stray field effects were swamped by electrode
polarization efl‘ects so that the stray field compensation could not be applied. 0n the other
hand in the mid frequency range between 1 KHz and 100 KHz compensation for stray
field could be made, but there were not enough data to extrapolate the electrode
polarization effect.

Second, using the platinum black electrode for ostensibly reducing the electrode
polarization efi‘ect caused a Maxwell-Wagner polarization effect. This additional term is
dependent on the electrode separation, and therefore complicated the separation of
electrode polarization fiom the apparent dielectric constant.

Due to these dificulties we decided to use a cell with small separation and with a
guard ring setup. In this manner we could reduce the stray field effect. Bare Pt, Pt black
and sand-blasted Pt electrodes were used. Again using a Pt black electrode produced a
Maxwell-Wagner efi‘ect. A sand-blasted electrode was not a good substitute for Pt black
electrode. Both electrodes are known as methods to reduce the electrode polarization
effect but for us they were worse than Pt electrodes. Myers[l] pointed out that both
electrodes are used to increase the electrode surface area and as a result the electrode
impedance is decreased. He also said that these electrodes are difiicult to characterize and
it is impossible to predict in advance the electrode impedance which will result from their
use.

With the small cell the major problem investigating the dielectric properties of colloidal

suspensions is the electrode polarization. The second is the conductivity drift which has

124

reported by previous investigators to arise from a temperature change or a coagulation of
the particles. We found that temperature effects were minimal, and the main time change
arose from electophoresis and coagulation. In the literature only the conductivity drift has
been reported but we found that the capacitance also changed with time at low frequencies
below 01 KHz.

We had four suspension samples which have 4 vol% solid with different surface charge
densities. A conductometric titration was used to measured the surface charge density.
Their surface charge densities were 3.34, 3.61, 4.01, and 1.03 in uC/cmz for 213.8nm,
364.1nm, 367.7nm and 624.3nm suspensions, respectively.

The experimental data using Pt electrode were presented and compared with the
theory developed by Sen and Chew. There were still abig electrode polarization effects in
the experimental data below 1 KHz. The imaginary parts were very similar to each other.
The characteristic frequencies obeyed the relationship predicted by the theory, i.e., they

were proportional to the inverse square of the size of particles.

6.3 Recommendations

(a) Our results for the capacitance of metallic balls in a dielectric indicate that we need a
good conductor which has good sphericity and does not easily oxidize. Spherical silver
balls should be better. These were not available commercially at the time but they can be
produced[2]. The fiequency dependence of such a dielectric material is unexpected and
should be the subject of theoretical study.

(b) In the breakdown experiment the low loading limit exponent was not consistent with
the analytical result probably due to finite size efl‘ect. To examine the sample size effect,
various sample thickness could be used.

(c) Before we can pursue the study of dielectric properties of colloidal suspensions, the

problems with electrode polarization effects at low frequencies below 1 KHz have to be

125

overcome. The best candidate at this time is the improvement of the four probe method
used by Myers and Saville[l]. In this method current is supplied by 2 electrodes and the
voltage between two separate electrodes is measured. They assume that, provided the
voltage measuring device has a high input impedance, the data derived from the voltage
measuring probes should be independent of electrode polarization effects. While we
attempted this method, it become clear that the available equipment was not appropriate
for this type of measurement. After electrode polarization is satisfactorily dealt with, the
next priority is to study the time variation in detail.

Thereafter there are many interesting properties to be studied. The diversity in the
published results indicates that they are strongly dependent on the parameters of the
particle such as diameter, surface group, charge density and debris from the
polymerization process. A complete study could be more satisfactorily attained if the

samples were prepared in the premises.

Bibliography

1. D.F.Myers, Ph.D. thesis. Princeton University(l988)
2. D.M.Grannan, J.C.Garland and D.B.Tanner, Phys. Rev. Lett. 46, 375(1931)

126

Appendix 1

Derivation of Eqs.(2.2.1) and (2.2.2)

In two dimensions, consider a thin ellipse with its long axis parallel to the direction of

the electric field. In elliptic coordinates, 5,11,

x = csinhfisin n, z = ccoshécosn (Al . 1)
with the ellipse equation being

1:2 22

_b_2_+;2_ = 1 (A12)
and

c2 = (12 -b2.

The electric field applied in the z direction has amplitude E0 far from the ellipse.

Laplace's equation is then

2 2
vzr..z[gg._gnԤ).o

where t = C(sinh2 g + sin2 10%. This equation has elementary solutions like

(Al.3)

 

L5,, 1], e‘ cos n, cosh(n§) sin(m1) etc. Applying our boundary conditions which are

‘I’ = 0 for E = 5,, (Al.4a)

‘1’ —+ - 02 = —Eoccosh§cosn for E —-> oo (Al.4b)
yields the electric potentials,

‘1’," = 0 (A1.5a)

‘1‘“, = -—Eoccosh§cosn + Ae“ cosn (A1.Sb)
where

127

128

A_§a_(a+_b)
C

To get A we have used cosh £0 = a/c and sinhéo = b/c.

The electric fields are then given by

t
Al .6
1 6‘1’ ( )
E n = ---——
t 6n
Thus each component of the electric field is
E, = —{-(—Eocsinh§cosn — Ae“ cosn), (Al.7a)
E,I = —J;(Eoccosh§sinn-Ae“ sinn). (Al.7b)
The electric field along the z - axiscan be obtained by setting n=0.
cosh e9“
Ez(§)=Eo[l+ E“ AM§J.(A1.8)
The electric field at the tip of the cluster is found by evaluating Eq. (A1.8) at i = 5,0
15,, = E,(1+a/b) = E,(1+ Jinx/20%) (Al .9)

. __ b2
With K - /0.
Eq.(Al .8) can be written in terms of r , the distance from the tip,
a(a+b){(a+r)-(r2 +Zar +b2)y2}

cz(r2 + Zar +b2)l/2

b -

E,(r)=E, 1+ (A1.10)

 

 

 

where r =ccoshé—a.

When r is large,
(r +a)/(r2 +Zar +b2)l/2 = (r +a1(r +a)2 — c2]

lc2

—l/2

 

Thus Ez(r) asymptotes to a dipole form,
E,(r)= Eo(l+a2,/r2) (A1.ll)

129

with the electric dipole moment

(1,, =a(a+b)/2. (Al.12)
Since the inclusion is sharp, the electric field becomes large near the cluster tip. The nature
of this divergence is found from a small -r expansion of Eq. (A1.8) which shows

E:(r) ~ Ea, when r << 1c/2 (Al.l3)

and
E: (r) ~ Eo{c,, + c2,(a/2r)1/2} when a > r > K/2 (Al. 14)

with cl, = 1 and c2. =a/(a—b).Note that ck and c2, are 0(1) as a —> oo andx —> 00.

In three dimension we can use prolate spheroidal coordinate to get electric field around
finger-like inclusion. The coordinate transformations are

x = csinhésinncos¢

y=csinh§sinnsin¢ (ALIS)

z = c cosh 5 cos 1]

and the spheroid equation being
2 2 2

x z
F+:—,+a—,=1and 19:112-112. (A116)
Laplace's equation is then

 

 

1 a as! 1
VZ‘I’z _ 'nh _ __ . _
c2(sinh2 2, +31112 10311111: 66, (5' 5‘ at.) + c2(sinh2 g +sin2 n)sinn anlsmn an)
1 62‘? -o

 

 

 

 

 

+czsinh2§sin2n 6412 ' (Mm
The separated equations, for ‘I’ = X (§)H(n)<b(¢) , are

2

:4: = —m2¢,
1 d . dH m2

$24 nn;)+n(n+l)H—sin2nH—O, (Al.18)
l d . dX m2 _

sinhfi zflflnhgzj --n(n+ 1)X - sinhz g X — 0,

130

where am are zero or positive integer. The solutions of the first equation are cos(m(p) and

sin(m(p). The second and third equations can be rewritten by

 

 

 

 

" 2
%l(1—v2)%+n(n+1)H-l’_"v, H=o. (A1.19a)
d F dx‘ m2
EE(I‘HZ)EJ+N(H+1)X‘l-uzX=O. (Allgb)

We have changed variables by introducing u = coshE, and v = cos 11. These equations are

those solved by the spherical harmonics P‘" and their second solutions Qf. Thus the

potential ‘I’ may be written by
”5.11.40 = Z X...(§)H...(n)¢.(¢) (A120)

where
X...(€) = AP."(§) + BQ.’.'(§)
H.111) = A'P..'(n) + B'Q.‘.'(n)
¢,(¢) = Ccosrml) + Dsinmtb.
For a conducting prolate spheroid, we set m=0 by azimuthal symmetry. The boundary

conditions are
‘I’ = O for g = :0 (surface of spheroid) (Al .21a)

‘1’ -+ -Eoccosh§cosn for g -) oo (Al.21b)

which leads to the electric potentials

 

‘11,, = o (Al.22a)
‘12,, = —cE, cosh: cosrll-{1n(mnh(§/2))+1/coshg}/B] (A1 .22b)
where
B = 1n[mnh(§,/2)]+1/coshg,
=1n(c:a) +2. (Al.23)

The electric fields are

131

 

 

 

 

_ E, cosn _ . _ . 1
E, _ (Shh: F, + sin: 11),, bs1nhE, {smug ln[tanh(E,/2)] + tanhE, + cosh: 511mg} /3{
E1 = (5320:1313 1/2 :1— {1n[tanh(§ /2)] + l/cosh§}/B] . (A1 .24)

The electric field in the z-direction(setting n=0) as a function of distance from the defect

 

tip is then

E,(§) = E, — E, {ln[tanh(§/2)] + cosh g/sinh2 §}/B (A125)
We may write this equation in terms of r , the distance from the defect tip.

l/2
r + a - c c(r + a)
E = E - E l —— B.
,(r) ° °lnlr+a+c:l +r2 +2ra+b2}/ (A126)

From the Eq.(Al .25) the electric field at the defect tip is

15,, = E0 {1 — (1n[b/(c + a)] + ca/b2 )/B}. (A1 .27)
For large a and small b

15,, ~ E, {1 +a2/[b’ 1n(2a/b)]}. (Al.28)

The far-fieldflarge r) solution again has a dipole form
Elf) ~ 15.. wad/r3 (A129)

where the 3D electric dipole moment
01,, = 2c3/3B. (A130)
For 1: < r < a, the electric field reduces to

E, (r) ~ E, {1 + [1n(r/2a) +a/2r]/B} (A131)

and for r <1: ,
E,(r) ~ 15,, (A132)

Appendix 2

Derivation of Equations in Section 4.4

The potential in the electrolyte solution surrounding a colloid particle is governed by
the Poisson-Boltzmann equation. The interaction between particles can be neglected in
the dilute limit of colloid concentration. Thus only single particle is considered and the
Poisson equation to be solved is

W» = —%(N, - N_), (A2.1)
where e is the dielectric constant of the solution, and N , and N _ is the charge density in

the solution surrounding the sphere.

A2.l Static case

The electrical potential (D and the ion density N i are related by Boltzmann distribution
N, = Noem’”. (A22)
Here No is the charge density of either + or - ions far away from the particle. k, is

Boltzmann constant. T is a temperature. Combination of Eqs.(A2.l) and (A22) gives

the Poisson-Boltzmann equation
1 d d‘I’ 1 . .
7270’ ‘75) = Wm” “‘2'”

where 82 = 8k, T / (ZezN o) is the screening distance, and ‘I’ is a dimensionless potential

which is defined as ‘1’ = alt/Ir, T . The boundary conditions for Eq.(A2.3) are

1‘31"”

132

133

and

‘I’ = ‘1", at r = a.
Here a is the radius of the particle. If we take 6 as the thickness of the double layer, we
can obtain the solution of Eq.(A2.3) by matched asymptotic expansions. Using

perturbation expansion the potential can be expanded

‘11 ~ 91“” 4%)?“ +(%)‘1'(2>+.... (A24)
We have taken (5/a) as the small parameter. Combining Eqs.(A23) and (A24) we get
1 a 5 (0) 5 (1)
75"2 El? +(/0)‘¥ +..]
= g:—sinh[\11<°> + (%)‘P(‘)+...]

~_ (0) (1) (o) 5 2)
62511111)? +016~11 cosh‘I’ +0{(A) . (A25)

First consider the inner region solution. By a coordinate stretching transformation

r=a+x5 ,x6<<a, (AZ-6)

we can emphasize the double layer zone. The Eq.(AZ. 5) can then be rewritten by

(31.1: (“15), %][w<°>+(%)te£2+-]

 

 

1 (a) (1) (o) 0{ 5 2)

~57sinh‘l’m +016)? cosh‘I’m + (A) . (A27)
Collecting terms of the same power of 8 on both sides gives

d2

(tr—2 —\Pf,,°)_ - sinh W533, (A28)

£91“)— ~11") cosh ‘1'“) = 41‘1“”. ‘ (A2 9)

dx2 1m m m dx 1m '
The solutions of Eqs. (A2. 8) and (A29) are given by

1+ te ‘I’o
‘P_(°)(x)= 211(1—1 L), t=tanh(° 4 , (A210)
(1) _ 2 __ -2.r
‘P, ,,.(x) -—_——(l 1294:)[1 (1e )- 2x] (A211)

Therefore the potential for inner region up to 1st order is

134

~11 (x) = 84“” 49(9) ‘11)}:
a

 

 

1+ te" 521,-: _2,
= 21n{1 ted) + 00 _t2e_2,)[12(1 -e )- 2x]. (A212)
As x —> oo , Eq.(A212) gives

4,,-..-x4942-2.).-x.0{(§){ (A213)
0 a

The inner potential decreases exponentially with distance so that it is very small at large
distance. Consequently the outer solution can be obtained by linearizing sinh ‘I’ ~ ‘I’. The

outer potential is then

 

-(r-a )/6 -(r-a )/6
ajwfl,’ = Ae + 539 . (A214)

\P‘" (r) ~ W52 + (Z r ar

The above equation can be expressed in terms of inner variable substituting r = a + 6x so

that
2
‘1’,“ = fie“ +(§)(fixe“ +£e“) +0[(§) {. (A215)
0 a a a 0

Comparing this equation with Eq.(A2 14) we get
A = 4a! and B = 2a!’.

A22 With an external electric field, E = Eoe"“"

When an external electric field is introduced the charge densities and potential are

perturbed fi‘om the equilibrium values. We can express the total charge densities and

potentialas
N, = N, +n,, (A2.l6a)
‘I” =‘I’+\y. (A2.l6b)

N, and ‘1‘ are the equilibrium charge densities and potential ,and n, and \u are the

perturbed charge densities and potential respectively. Here ‘P's and w are normalized

135

potentials, related to the actual potential F through ‘1’ = (e/k, T )CD. The total ionic

currents are
1; = D,(—VN; x N;V\P‘). (A217)

The first term corresponds to diffusion current and the second term corresponds to

conduction current. From Eqs. (A216) and (A217) we obtain

j, = D,(—Vn, $N,Vw3Fn,V‘P)+0(E§), (A213)
We have assumed that the perturbing electric field is very small so that
Vz‘l" = “1%,:5A2L can be represented by the linear firnction of E0. We have used the

0

fact that the ionic current is zero in the absence of the external electric field. The current
and the electric field are related to each other by the continuity equation, i.e.,

V}, = ram, (A219)
Furthermore the total charge densities, potential, and currents have to satisfy the Poisson's

equation, i.e.,

vzqfl = -M
211/052 ’

where 6 = [1—z'lr,,7’/(e2 2N o )1”: is the Debye screening length, 8’ the real part of the

(A220)

dielectric constant of the ionic solution and N o the equilibrium ionic densities in the

absence of the charged particles. From the Eqs. (A216), (A218), (A219) and (A220)

 

we obtain
2 fight); (A221)
and
D,V - (—Vn, x N,Vw my?) = icon, (A222)
The boundary conditions to solve the above two equations are
j, -r‘|,,, = 0, (A223a)
w(a*) = 111(0’) (A223b)

and

136

e'fiv—
6r

6w

:8—

, ar (A223c)

 

 

9
a' a'

where 8’ is the dielectric constant of the ionic solution and a F that of the insulating
particle. The first boundary condition implies that the ions do not penetrate the insulating
particle. The second and the third are the continuity of potential and displacement current
at the surface of the particle respectively.

To solve the Eqs.(A2.21) and (A222) the perturbation method is useful. The unperturbed

potential and charge densities and perturbed potential and charge densities can be

expanded
n, = "(0) 4%)"? +(%)2n§2>+..., (MM)
91 = w“) +(%)w“’ +(%)2 Wm,” (A2.24b)
N, = N5?) +(%)N§" +(%)2N§2)+..., (A224c)
q: = 14°) + (imam) +(%)’\11<2>+.... (A224d)

A2 .21 Zeroth order solution
To find the inner solution the coordinate stretching transformation is used to

emphasize the double layer region, i.e.,
_ (r — a)
’ 5

x

The gradient operator and the divergence operator can be represented in terms of the inner

 

variablexas
16 ~16 5x6
V~‘——+9——-——— f 6 0, A225
x562: we (12% °' —’ ( )
and
3-35; 1.5.), 1 {_§)_5_- 3.
V (a azx+56xx+asin91 ax aesrnOO. (A226)

From Eqs.(A2. 18) and (A225) we can see
. 8 -1 .(-1) .(o) 5 .(1) 5 2 .(2) 63
1,: — J, +1, +-},+— j, +0—,-. (A227)
0 a a

a

137

Since j, have to be bounded, as 6/a —> O,

ji")(x) = O for all x.

Using the Eqs.(A2.24) and (A225) the Eq.(A218) can be expanded
j, = D{_l_[_2n§:,, N10) fl;n:°) fl];

- 5 0x ax ax
+1. -922;N0>§‘V_m_).;N<o>iwfl;n<o>fl;n(lifl ,;
a 0x * 0x 1 6x * ax * ax

 

6x * 6x 1 6x ’ ax * at 1 6x
1 5 anO) 571(0) 0 6“,“) 0 aw“) W10) .
+—{—) — * +x-L4Ngl—1Nf’x—¥N(‘)-F +....(A2.28)

 

aa & 60

We have omitted the derivatives of ‘P‘s with respect to 6 since W's are the function of x

only. For the simplicity we have set D+ = D_ . Therefore we have

(0) (o) (o)
f" = -:3{-%— :1: N,” iii—Tn?) £617.] = 0 . (A229)

The above difi‘erential equation can be easily solved by multiplying the integrating factor

1‘1“”

e and we get
n1” = :N§°’(w‘°l + Cf” :1: C§°’), (A230)

where Cf” and C3” are the integration constants to be determined. In the limit

(6/a) —> O the Laplacian operator can be approximated by

1 a2
Combining the Eqs.(A2.2) and (A210) we obtain
.3 2
N9) = NOW” = No[:ii:§:) , t= 1mM/4). (A232)

With the Eqs. (A230), (A231) and (A232) the Poisson's equation(Eq.(A2.21)) becomes

52 (o) 1 (0) (o)
— = - n — n
ax’ "’ 2N,( * ‘ )

 

138

= cosh w‘°’(w(°’ + C5”) + C2101 sinh 811°). (A233)
Thus the zeroth order potential in the inner region is

 

1— t‘e l- t e “x
We have neglected exponentially increasing solutions. Substituting the above equation to

Eq.(A2.30) the zeroth order perturbed charge densities are

2 -x -x
"(0)__+N(0){C(°)£1_t)e_._c(0)(t(l + 2x)e i 1]]. (A235)

1— tze'z’r 2 1- tze'z‘

 

Now we have to find the solutions for outer region of the double layer. The zeroth order
solution implies that the surface potential is perfectly screened by the charge cloud around
the particle. Thus the particle looks like neutral at outside of the double layer so that the
solution of the Poisson's equation in the outer region resembles the solution for a

insulating sphere in a external electric field. They are given by

(0) _ _ _ 1 3 6

111,“, — eorcose 2eo{‘1/2)cos9, /,-)0. (A236)

for outside the particle. Here eo = EOe/k, T . The potential inside the particle is given by
v.13.) = -%eor¢089- (A237)

The continuity of displacement current at the surface of the particle implies that

 

 

e,(—; e,)cose= e g 6‘: 011°) 20 (A238)
In the limit 6 -> O the above equation cannot be satisfied. Hence
£09) = 0, 5 .3 0. (A239)
01'
—C§°){l——_ + t :J- C§°1 til—1’27) = o. (A240)
1 1 (l -tz)

To get the expression for n5,” and WM in the outer region we have to solve the

Eqs.(A2.21) and (A222). Outside of the particle the potential is exponentially small so

that we can approximate

139

N, ~No+est, ‘I’~est for r-a>>6,

where est stands for exponentially small term of order exp[—(r — a)/5]. Now the

Eqs.(A2.21) and (A222) can be written by
, (0) "(0) _ "(0)
V“ = -—“——‘— A241
‘1' 2N062 ( )

and
DV - (—Vn§,°’ : NOW/10)) = icon, . (A242)

Combining two equations we get
v2n§°1 41%).?) r {y25,)[n£°1 - 715”] = 0 . (A243)

Taking sums and difi‘erences of the above equations gives

 

V2 (113°) + n1”) + (1%anf) + n1”) = O (A244)
and

V2 (n10) — 72$”) + (1%an,” - n5”) + ()g21ng’) - 119] = O . (A245)
Finally we have

71:0) = N 061°) “39:04) 0:22) cosB , (A246)

where A = (1 - i) 0% D . By the matching criterion
11mg}, ~ limngL,
we get
No駰1acose = N,C§°’. (A247)
By applying the matching criterion on the potentials we have
C11” = —§-eoacos6 . (A248)
In terms of perturbation expansions the continuity equation in the inner region can be

rewritten by

101°) {1611" 2 J 1 a
V. ' =__£+ ___i_x_+_ '(O) +___ -(°) ' 9
1* 5 ax { a a: a1“ asinO 69(1“ 3’“ )

140

5) 16A? 2 (,,_ 2x (0,] 1 a( m '11” .
+- — - .. .. +—.— .+ e]...
1a {1(1— 61: +aj a _j‘ asrn666l1j 6 16],, )sm

2
=ic01n1°1+(5)n111+(§) 7113...]. (A249)
a a

From the above equation

a (0)
,, =0. A250
6 —J ( )

And the boundary condition Eq.(A223a) at the surface of the insulating particle implies
that
11330 = O) = 0. (A251)

Therefore j1°)(x)— - 0 for all x. In the outer region the current due to n10) and 111101
j1°)~D(-Vn1°) :1: N(°)\|I (0) ; n(°)v\y(°))

 

~ D(—Vn1°)3F Now/1”). (A252)
With Eqs. (A236) and (A2. 46) the radial current of the outer region is

112’ — 0N, cose{c<°> " 1:" .1 2 + 21’: L12 2 x {‘r’; - ej}. (A253)
By the matching criterion

limj1°1~1133= 0,

i.e.,
11m 110) ~ DNocos9C1°101=O, (A254)

where
_ 1+ M + 2202
1+ M '
Hence 571°) = 0. Consequently we can show C1” = O and C1” = 0 from Eqs.(A240) and

 

(A247). It is worth while to get the solution of the tangential current in the double layer

for the future. From Eq.(A228) the tangential component of the inner current of the
zeroth order of (5/0) is

j1°)— - ZFDN10) 6\v__1°)

141

= $DN1°131eo sine.
The zeroth order solutions are summarized below.
In the inner region

1111:} = —le0acose,

7110) = 0,

1133,: $éDN1O) ge, sin 9 .
In the outer region
111112- — -eor cosO —-1-eo(a3/r2)cose ,

(0) _
"tout - ’

3
1,) ==FDN eose{3,——1)e,.

A2 .2 .2 First order solution
From Eq. (A2. 28)

L10)1x)-_[-axi "(1);{N(0)_ ‘3 W(I) +"(1)3

615W: 6):

Multiplying it by the integrating factor ei‘P‘” we obtain

as) = emwm +C,1‘)=FC1‘1).

Furthermore the potential has to satisfy Poisson's equation, i.e.,

5 (1) _ __ (1)_ 0)
6x2 _‘V 2N0 1'" n“ ]
With Eq. (A2. 59) we get

axZ‘"

1" = cosh w(°’(w<‘> + C1‘))+ C1” sinh 114°).

(A255)

(A2563)

(A256b)
(A256c)

(A257a)
(A257b)

(A257c)

(A258)

(A259)

(A260)

(A261)

The above equation is identical to Eq.(A2.33). Thus the solutions resemble Eqs.(A234)

so that

2 —x -x
w1"= C11111-t )‘ C0)_C(1) ’(1 2’0"

l—tze‘z‘ 1-

142

1 t2 " 11 2 "
-:FN1°){C1')(——3—)f— C1"{—1—+—,x1§,—:l)]ocose. (A263)
1 1e l-t’e‘
FromEq..(A249)
5 (1) 1 5 (9)
6x 1, si—{em(sin9j,e )0 (A264)

Substituting j1°1given by the Eq. (A2 55) and solving the differential equation the first

order radial current is

 

 

11‘}- — +3e, DN, oose{xrr 1:11 i 1:11;] (A265a)

~ ..3eoDN0 003913: 313%] , x -—) co. (A267b)
From the continuity equation

v2n1‘1 + (160)11‘) =1: (%Xn1l) - n91] = 0. (A268)
This equation is identical to (A2. 43) so that the solution is

n1” = N, C“) a e w .) 1+” cos9. (A269)

r2 l + M
This result indicate that the diflirsion cloud is neutral and has a size ~ % Therefore the

Eq.(A24l) become

V211,“) = 0, (A270)
and gives the solution

1111') = P1‘)e,(azz)cose. (A271)

The radial current in the outer region is
j1',)— - D(-Vn1') 1F N,V\u1‘))
C11)“ 3e 51' "11+Ar4-23r2 j:2P111e, a
1'3 1+ M r3

 

= DN, oose{ (A272)

Therefore the outer normal currents are
~ 1131+ +(/)j1"

— -+DN, cos9[-—e°33

       

 

143

 

5 3 -A(r-a) 1+” 12 2 2P(l) 3
+(;)DNocose{C1”a er, 1,; ' at 5°" . (A273)

In the inner region r - a ~ 6:: , Eq.(A273) in terms of inner variable can be rewritten by
1,, ~ iDN, cose[3e,x x 2C1'1ot + 2P1'1e,] , r —) a . (A274)

Comparing the Eqs. (A267b) and (A274) we get
P“) = 612 CO) = _ 690’

 

1-12) (7.72).; (A2-7”
The potential in the outer region is
3 2
11’ ~ W(°)+(§_)W(1)= —e,rcosO+e—°q—,c—(£e— -..1_+ 6’ 2 (Q) . (A276)
0 r 2 1- t a

In the limit r -—-) a , Eqs.(A269) and (A276) which are the outer solutions can be written

by
n,,,, ~ (5/a)n1‘1 ~ 5N ,C1‘1cose, (A277)
1.7,, ~ “112 +(6/a)w1,,‘)~ [- age, +(6/a)P(‘1e,]acose. (A278)
In the limit 1: -) co ,the inner solutions given by Eqs.(A262) and (A263) become
1111‘) C 1'10 cosG, (A279)
"1'3... ~ No C1'109089 (A280)

By the matching criterion we can show that
C1” = P1'1e, and C111 = C1". (A281)
The unknown C11) can be obtained by the boundary condition of the displacement current,

i.e., the continuity of the displacement current at the surface of the particle.

‘52s ’(ir) 0.2)

The first order solutions are summarized below.

a t 1-312
3 _C(l)_l_tt___ c(1)L__)_ (A282)

Inner region

1- t "‘
n11L=iFN1°){:C1‘)£l———1e-:— C1‘)[il—+-21):—il]]acos6, (A283a)

(Ze-Zx 1_t2e-2x

144

 

 

 

 

1-12 6”r t 1— 2x “
W11). =[C1')(T—t,—):;-- C11) (1: re): +eP1”0]acose, (A283b)
_ “e " "
1112," = i3eoDNo cosO(x3F 2’ i 4te _ ] (A283c)
lit lite ’
Outer region
3 -k(r-a)

"1'3", - No Cma er; :IZCOSG. (A284a)
WEB, = P(l)eo(a%2)0059. (A284b)
3 "1"“)1+Ar+kz 2 2P1')e 3
111.)“: = DNo 13°39[Cm a er; 1+M r i '30“ . (A284c)

C1”, C 31') and P1” are given by Eqs.(A275) and (A282), respectively.

A223 Higher order theory
Up to first order solutions the dielectric enhancement does not occur. The first order

potential in the outer region obtained from the radial current in the double layer. Similarly
w1’can be obtained fi'om j1i). From Eq.(A249) collecting terms of the order of (8/a)

 

on both sides gives
1 5 (2) _ n1‘)— 1 (1) (o) 2 (I)
a —6xj*’ iom sine w —sin9(jte x—jte)— —jn. (A285)

11.2,) and j are givena in Eqs. (A255) and (A265a), respectively.
j1°)— - IDN 11:0) g-eo sine
and

jg? =:l:3e0DN0 cosG x=F 4’ i 419 _ .
lit lite’

 

From the continuity equation which is expanded by using perturbation expansions, i.e.,

Eq.(A228), the first order tangential current in the double layer is given by
. 1 6
112 = D—-—{-n§‘) =F[Nf)w1°) + N§°)(w1" — mm)", (A286)
0 as

where

145
(I): (o) 219“ 2 _ -2, _
NI xNI (l—rze’z")[t (1 e ) 2x], (A287)
ng”,1v§°>, W1” and (71" are given in Eqs.(A283a), (A232), (A256a) and (A283b).

respectively. Thus Eq.(A286) can be rewritten by
(I) _ (0) - 3 219" 2 _ -2x (I) (I) 3
1:9 — DNI Sln61Zeol-tze"2‘[t (l e )-2x]+C i P igeox . (A288)

Now jg) can be determined from the Eq.(A285)
1
'1” =icoN a C1‘)x$——— C10 H l:t2t C11) c039
1.: 0 { (lit)2[ 3 ( ) ]

2 1
$ DN° cos9[6e0x2 + 2x(P1')e0 1 C11) 43%)
a

 

—24eo ln(l :l: t) $1111;(P1"e0 i C1”):12e0t], x —> oo . (A288)

The second order potential and charge densities are similar to the first order solutions so

that the potential and the charge densities in the outer region up to second order are

2
w ~ w1°’+(§)w1°+(§) we)
a a

 

3 2 3
=[—r—3-l+§P1‘)+(§) P12)]%cose , (A2-39)
a 2 a a r
2
n1 ~ Eng) + (g) "9)
aze'w") (1+ ”N: 6
= 8N C1" +—C12)]. A290
° r2 1+ M a ( )

The outer radial currents can be obtained by
112,) = D(-Vn12’ * NoVw1”)
(2) 032-11”) 1+ Ar + 12,2 :t 2P12)eoa3
r3 1+ M r3 '

 

= DNo cosO[C (A291)

In terms of the inner variable the outer radial currents up to second order are

jt, ~ DN0 cos9{:tg[3eox + 2P11)e0 i 2C1')a]

2 3 3
{Q} 6e0x2 + 6,[ P1‘)eo “3(1)“: 1C1” la )
a 6 1+M

146

-2P12)e0 :F2C12)a]}, r —) a. (A292)
The inner radial currents can be obtained from
. 5 . 5 2 .
1.. ~ H192 + (J 123. (A293)
a a

where 11‘) and jg) are given by Eqs.(A265a) and (A288). Comparing Eq.(A292) and

the result of Eq.(A293) we get
1012);:0 i C12)a = 12e0 ln(l i- t) i- fit—t-[W’eo i Cm]; 6eot

 

. 2 l
- ":2 (1::)2 [C319 +t(li21)C1')]. (A294)

From the above equation we can determine P12) and C12) so that the total induced dipole

moment to second order is
2

P ~ -1 + (g) 61 2
2 a l— t

2 . 8 2
{2) ~3fl—‘3—a2 +(51n(1—t’)——34—’—-2-(12 +1) . (A295)
0 4 D 8' (1_t2) (I.

 

HICHIGQN STQTE

UNIV. LIBRRRIES
HIWNINHIIH‘ |llWNWllllllllHIHlHIHHI
0468597

 

3129301