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Microwave dielectric relaxation spectroscopy of liquids

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O rder N um ber 9403030
M icrow ave d ielectric r e la x a tio n sp ectro sco p y o f liq u id s
Wei, Yanzhen, Ph.D .
Northeastern University, 1993
UMI
300 N. ZeebRd.
Ann Aibor, MI 48106
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MICROWAVE DIELECTRIC RELAXATION
SPECTROSCOPY OF LIQUIDS
A dissertation presented
by
Yanzhen W ei
B. S. Peking University, P. R. China, 1983
M.S. Academia Sinica, P. R. China, 1986
to
The Department of Physics
In partial f u lfillm en t of the requirements
for the degree of
Doctor of Philosophy
in the field of
Physics
Northeastern University
Boston, Massachusetts
December 1992
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Copyright 1992
Yanzhen W ei
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MICROWAVE DIELECTRIC RELAXATION
SPECTROSCOPY OF LIQUIDS
by
Yanzhen W ei
ABSTRACT OF DISSERTATION
Submitted in partial fulfillm ent of the requirements
for the degree of Doctor of Philosophy in Physics
in the Graduate School of Arts and Sciences of
Northeastern University, December 1992
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In this thesis, a new technique, which is non-invasive, fast and accurate, was
developed for measuring broadband continuous dielectric spectra of liquids and
solids. It was tested from 45MHz to 20GHz bu t is applicable to at least 40GHz.
T he technique was used to study a variety of liquid samples in which water was a
prim ary component.
A graphical representation for the complex dielectric data was discovered. It
is specially useful for conducting dielectrics, and is sensitive to high frequency
features. This makes it possible to graphically visualize m ultiple relaxation
processes with small amplitudes.
The notion of “microwave dielectric excluded volume” was introduced for
ionic and biological macromolecular aqueous solutions. This is the volume
surrounding the solute molecule which is not able to reorient at microwave
frequencies. The num ber of hydration water molecules around each solute molecule
can be extracted from the dielectric excluded volume. Results were obtained on
AC1 (A =Li, Rb and Cs) and heme protein aqueous solutions. In protein solutions,
the partially-opened conformation of m et myoglobin obtained by lowering the pH
value of the solution seems to have smaller microwave dielectric excluded volume.
The dynamics of alkali-halides aqueous solutions, was extensively studied
via dielectric spectroscopy. It was found th a t the existing kinetic polarization
theory does not explain the experimentally observed linear decrease of the solution
relaxation
tim e
with
increasing
dc
solution
conductivity
for
moderate
concentrations, which can however be explained by an empirical viscosity formula
and cation size effects. The tem perature dependence of the relaxation tim e and
conductivity in concentrated LiC l/H 20 solutions were studied and found to be in
agreem ent with the predictions of the mode-coupling theory. The vitrifying ability
of L iC l/H 20 was related to the space available to a free water molecule in solution,
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which can be estim ated from the dielectric excluded volume.
It was shown, by the dielectric investigation on the concentration
dependence of alcohol/water solutions, th at alcohol molecules strongly interact with
w ater molecules. The addition of a small am ount of alcohol,
1 -propanol,
ethylene
glycol or glycerol to water, or similarly water to alcohol, substantially changes the
dielectric behavior of the solvent, which may be the evidence of the “fusion of
relaxation channels” .
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ACKNOW LEDGEMENT
This thesis was carried out under the guidance of Prof. S. Sridhar, to whom
I would like to express m y great appreciation for his support, constant
encouragement, and especially his understanding during good and rough times. I
thank Prof. Paul Champion for his advice and his group for th e kindly help they
provided for part of this work. I also would like to acknowledge the helpful
cooperation of Prof. Diane Rigos. Invaluable technical assistance provided by R.
Ahlquist, T. Hussey and S. DiCiaccio, and both understanding and help from my
lab colleagues are sincerely appreciated.
I thank my parents and husband for their support, especially m y son Cha,
for his love and tolerance of my carelessness.
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PUBLICATIONS RELATED TO THE THESIS WORK
( 1 ) Yj. Wei and S. Sridhar.
“Technique for measuring the frequency-
dependent complex dielectric constants of liquids up to 20GHz” . Rev. Sci. Instrum .
60, 3041(1989).
(2)
Yj Wei and S.
Sridhar.
“Dielectric spectroscopy up to 20GHz of
L iC l/H 20 solutions” . J. Chem. Phys. 92, 923(1990).
(3)
W
Wei and S. Sridhar.
“Radiation-corrected open-ended coax line
technique for dielectric measurements of liquids up to 20GHz” . IEEE Trans. MTT.
39, 526(1991).
(4) Y. Wei. P. Chiang and S. Sridhar. “Ion size effects on the dynamic and
static dielectric properties of aqueous alkali solutions” . J. Chem. Phys. 96.
4569(1992).
(5) Yj. Wei and S. Sridhar. “Biological applications of a technique for
broadband complex perm ittivity measurements” . IEEE MTT-S digest, GG-4
(1992).
(6 ) P. Liu, C. M. R appaport, Y^ Wei and S. Sridhar. “Sim ulated biological
m aterials at microwave frequencies for the study of electrom agnetic hypertherm ia”.
IEEE EMBS digest (1992).
(7) Yj. Wei and S. Sridhar. “Relationships between the dielectric and
structural properties of supercooled LiCl:RH20 solutions”. J. Mol. Liquids, (to
appear).
(8 ) Y. Wei and S. Sridhar. “A new graphic representation of dielectric for
conducting media and studying high frequency dynamics” , subm itted to J. Chem.
Phys.
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viii
(9) Y Wei, T. Sage, W. Tian, P. Champion and S. Sridhar,“Hydration
shells of protein solutions via dielectric m easurem ents” . Subm itted to Biophys. J.
(10) A. Rigos, Yj. Wei and S. Sridhar. “Fusion of relaxation channels in the
dielectric properties of alcohol aqueous solutions” . In preparation.
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CONFERENCE PRESENTATIONS
(1) Y* Wei and S. Sridhar, “High frequency dielectric response from 400KHz
to 20GHz of polymers and bio-polymers”. Bull. Am. Phys. Soc., 33* No. 3, F22
6
(1988).
(2) Y* Wei and S. Sridhar, “Dielectric spectroscopy up to 20GHz of aqueous
solutions of biophysical systems”. Bull. Am. Phys. Soc., 34* No. 3, G16 3 (1989).
(3) F. Li, Yj, Wei and S. Sridhar, “Dielectric relaxation at frequencies up to
20GHz”. Bull. Am. Phys. Soc., 34* No. 3, B9 15 (1989).
(4) Y Wei and S. Sridhar, “Dielectric spectroscopy of complex liquids and
solutions from KHz to 20GHz” . M aterial Research Society. V10. 10 (1989).
(5) P. Chiang, Y Wei and S. Sridhar, “Probe of local environm ent in
supercooled liquids via GHz dielectric spectroscopy” . Bull. Am. Phys. Soc., 35*
No. 3, F28 4 (1990).
( 6 ) Y Wei. W. Tian, P. M. Champion and S. Sridhar, “Gigahertz dielectric
spectroscopy: A probe of biomolecular dynamics in solutions” . Bull. Am. Phys.
Soc., 35* No. 3, F29 1 (1990).
(7) S. Sridhar and Y. Wei. “Dielectric spectroscopy of liquids and solids.
Gordon Research Conferences: Dielectric Phenomena. July, 1990.
( 8 ) Y. Wei and S. Sridhar, “Applications of an open ended coax line
technique
for
broadband
perm ittivity
measurem ents” .
IEEE
International
Microwave Symposium. GG-4. 1992.
(9) Y W ei. A. A. Rigos and S. Sridhar, “Dielectric study of alcohol/water
m ixtures up to 20GHz” . Bull. Am. Phys. Soc., 38* No.
1,
B20 3 (1993).
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CONTENTS
ABSTRACT
iii
ACKNOW LEDGEMENT
iv
PUBLICATIONS RELATED T O T H E THESIS W O RK
vii
CONFERENCE PRESENTATIONS
ix
I
INTRODUCTION
1
1.1 OBSERVATIONS ON AQUEOUS SOLUTIONS USING DIELECTRIC
SPECTROSCOPY
1.1.1 Ionic Aqueous Solutions
4
1.1.2 Heme Protein Solutions
5
1.1.3 Alcohol/W ater Mixtures
5
1.2 CONCEPTUAL BASIS OF DIELECTRIC RELAXATION
1.2.1.
H
4
Dielectric Spectrum and Response Function
6
6
1.2.2 Connection to Microscopic Properties
8
1.2.3 Theoretical Development
9
DEVELOPM ENT OF A TECHNIQUE FO R DIELECTRIC SPECTROSCOPY
15
11.1 MEASUREMENT CONFIGURATION AND PROCEDURE
15
11.2 ANALYSIS: DE-EMBEDDING OF SAMPLE IMPEDANCE
17
11.3 MODELING OF THE COAX TERMINATION
22
11.3.1 Simple Capacitance Approximation
22
11.3.2 Full Radiation Correction
25
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m
II.4 ERROR ESTIMATION
29
11.5 DISCUSSION
33
11.6 DIELECTRIC SPECTRA OF SOLIDS UP TO 20 GHz
35
11.7 IN SITU DIFFERENCE MEASUREMENTS IN DIELECTRIC SPECTRA
35
DIELECTRIC SPECTRA AND MICROSCOPIC PROCESSES
111.1 DEBYE RELAXATION
50
III. 1.1 Debye Model
50
III. 1.2. Fitting D ata to the Debye Form
52
111.2 NON-DEBYE PROCESSES
IV.
49
56
111.2.1 Empirical Functions for e(u>)
56
111.2.2 KW W Decay Function
58
111.2.3 Fitting D ata to the Cole-Cole Form
59
111.2.4 General Fit to a Complex Dielectric Function
62
111.3 POW ER LAW BEHAVIOR
62
111.4 A NEW GRAPHICAL DIELECTRIC DATA REPRESENTATION
65
111.4.1 Single Characteristic Relaxation Processes
65
111.4.2 Enhancement in Higher Frequency Features for Multi-relaxation Spectra
69
111.4.3 Dielectric Relaxation in LiCl/ 1-propanol Solutions
71
III.4.5. Discussion
72
DESCRIPTION OF RELAXATION SPECTRA FO R AQUEOUS SOLUTIONS
85
IV.
85
1 INTERACTION WITH SOLVENT - W ATER
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IV.2 NUMBER OF BOUND W ATER AROUND EACH SOLUTE MOLECULE
V
87
IV.2.1 Microwave Dielectric Excluded Volume
88
IV.2.2 Number of Hydration W ater Molecules Around Each Solute
89
IV.3 ERROR ANALYSIS
91
IV.4 ERROR IN DIFFERENCE DIELECTRIC SPECTROSCOPY
92
STATICS AND DYNAMICS OF AQUEOUS ALKALI-CHLORIDE SOLUTIONS
94
V .l SAMPLE PREPARATION AND CONCENTRATION CONVERSION
95
V.2 A CL/H20 (A=Li, Rb, Cs) ROOM TEM PERATURE RESULTS AND ANALYSES
97
V.2.1 Static Polarization - Hydration
100
V.2.2 Kinetic Polarization
102
V.2.3 Dynamic Time Scale: r D
103
V.2.4. Ionic, Dielectric, Stokes and Hubbard-Onsager Radii in a Ionic Solution
106
V.3 RELATIONSHIPS BETWEEN DIELECTRIC AND STRUCTURAL
PRO PERTIES OF SUPERCOOLED LiCl: RH20
VI
108
V.3.1 Choice of the Fitting Function
110
V.3.2 Experiments and Results
111
V.3.3 Tem perature Dependence of LiCl: RH20 (R < 7)
114
V.3.4. Connection to Supercooling Properties of LiC l/H 20
116
HYDRATION IN BIOLOGICAL MATERIALS
131
V I.l SAMPLE PREPARATION AND CHARACTERIZATION
132
VI.2 HYDRATION SHELLS
134
VI.2.1 Dielectric Spectra and Number of Hydration W ater Molecules.
107
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VH
VI.2.2 Discussion and Dynamics of the Hydration Shell.
138
VI.2.3 pH Dependence of metMB
141
VI.3 IN SITU OBSERVATION OF THE INFLUENCE OF OXYGENATION
142
VI.3. APPLICATIONS TO MICROWAVE HYPERTHERMIA
145
VI.3.1 Results and Analysis on the Phantom and Human Blood up to 20GHz.
146
VI.3.2 Sugar Solutions for Substitutes at Desired Frequencies.
147
ALCOHOL AQUEOUS SOLUTIONS
156
V II.l DIELECTRIC SPECTRA OF ALCOHOL AQUEOUS SOLUTIONS
156
VII. 1.1 l-propanol/H 20
156
VII. 1.2 Ethylene Glycol/H20
157
V II.l.3 Glycerol/H20
158
VII.1.3 Non-Debye Behavior at x ~ 0 Concentrations
159
VII.2 DISCUSSION
REFERENCES
160
165
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CHAPTERI
INTRODUCTION
Dielectric spectroscopy is a powerful tool for studying the interactions of
molecules, the effective dipole moments, the dynamics of dipolar reorientation,
structural characteristics, collective properties of many-body systems, and many
other phenomena in condensed m atter. The dielectric region of the spectrum covers
the sub-optical frequencies and extends over 14 decades as shown in the schematic
representation in Figure 1.1. At frequencies lower than 109 Hz, the applications of a
variety of dielectric measurement methods have revealed aspects of significance
regarding phenomena such as phase transitions in liquid and liquid crystals, interand intra- molecular dipolar relaxation modes, solute-solvent interaction in
polymetric m aterials, energy dissipation in solids, and dielectric relaxation of
biomolecules and bound water. For reasons inherent to the short tim e scales and
the wavelength characteristics, the
109
- 1011 Hz range has been previously explored
at spot frequencies only. For electrolyte and biological solutions with added salt,
extensive dielectric studies were precluded by the lim itation of measurement
methods, since a GHz broadband frequency domain dielectric measurement
technique is essential to cover the characteristic relaxation frequency of water
which is about 18.5 GHz at room tem perature.
W ater is the solvent which plays a unique and im portant role in chemical
and biological processes. Despite the effort which has been devoted to the study of
w ater over a hundred years or so, there are still m any interesting aspects of its
behavior which have not been understood. An approach (which is employed in this
thesis), is to study the m anner in which solute molecules interact w ith w ater and
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-2 -
with one another in aqueous solutions, which can provide insight to understanding
the exact nature of the moleculax interaction in the liquid. Some progress in this
field has been made in the past few years. There is, however, an urgent need for
reliable experim ental d ata on aqueous solutions of a large range of solutes, which
again challenged previous measurement techniques.
This thesis attem pts to address the above issues both experim entally and
theoretically. Recognizing the lim itations of available m easurement technique when
this work was initiated in 1988, a new, fast and accurate technique was developed
which is particularly suitable up to 20GHz and possibly higher (see C hapter II).
The technique enables extensive measurements on a wide variety of liquid systems,
particularly water-based solutions and mixtures, under a variety of conditions:
varying pH, concentration or tem perature. Some of the systems studied include
ionic solutions (Chapter V), protein solutions (Chapter VI) and alcohol-liquid
m ixtures (Chapter VII). In the course of this work, a new graphical representation
was discovered, th at is particularly useful at high frequencies and in conducting
media (Chapter III). The measurements and the analyses yield im portant
information on the solute, solvent and microscopic processes including the solutesolvent interaction. I show th a t common them e underlying the dielectric response
at GHz frequencies is the notion of a “microwave dielectric excluded volume” ,
which includes the solute molecule and a “hydration” shell (C hapter IV). It is a
measure of the solute-solvent interaction. A complimentary aspect is the dynamical
process in the solution, represented by the relaxation tim e r. A feature somewhat
unique to w ater is the essentially Debye nature of the relaxation w ith
— ~ 18.5
Z ttT d
GHz at 25C. W hile this is only slightly affected by small am ounts of protein
additives, larger concentrations or other hydrogen bonded liquids cause dram atic
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-3 -
changes, signaling the disruption of the tenuous H-bond network. An interesting
connection between the dielectric and structural properties is pointed out (Chapter
V), in which the presence of a non-Debye relaxation at room tem perature appears
to signal the low tem perature glass forming capability, and is also related to the Hbond network.
Somewhat surprisingly, protein solutions axe similar to dilute ionic solutions,
except of course for the larger molecular size. The dielectric excluded volume
enables one to extract the hydration shell. The experiments show th a t dielectric
measurem ents m ay form an alternative probe of structural properties such as the
folding-unfolding transition in myoglobin, and moderate improvement in sensitivity
are required to study the change in hydration shell through the R-T transition in
hemoglobin and the effect of oxygenation. An interesting by-product of the protein
solution work was the surprising result th a t dissolved oxygen m ay affect the
dielectric properties of water. This potentially im portant result is deserving of
future study.
The work discussed here is of im portance for practical applications also. In
particular the improved accuracy of the measurem ents has been shown to yield
more precise values for blood and substitutes for hum an tissues, enabling accurate
design for microwave hypertherm ia treatm ents. A variety of other applications are
also feasible, such as monitoring epoxy curing, etc.
In the rem ained part of this chapter, I first discuss the specific systems th at
were studied, the conceptual basis of the dielectric relaxation and relevant
theoretical concepts.
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1.1. Observations on aqueous solutions using dielectric spectroscopy
1.1.1 Ionic aqueous solutions
Alkali ionic solutions, which may be regarded as the “hydrogen atom ” of
solutions, exhibit rich and fascinating phenomena. Experim ental means, such as
therm odynam ic
and
transport
measurements,
neutron
and
x-ray
scattering
experiments, have been used to probe the nature of the solvent in the vicinity of
ions, but have lim ited applicability. A more interpretable and easier m ethod to
apply is the determ ination of an apparent solution dielectric constant through
measurem ents of the electric response of an electrolyte solution to a periodic field.
A comprehensive study of concentration and cation size dependence of alkali
halides (LiCl, RbCl and CsCl) aqueous solutions was carried out. The results
verified the existing kinetic polarization deficiency theory by H ubbard and Onsager
as regards es0, but are inconsistent with it as regards the correlation between the
solution relaxation tim e and the “dc” conductivity from dilute to moderate
concentrations.
At concentrations located in the molten salt range, alkali aqueous solutions
are good glass forming liquids. A dielectric study of th e tem perature dependence of
LiC l/H 20 solutions, was carried out in order to address the issue of the nature of
the glass transition. The supercooling capability of a concentrated solution was
shown to be related to the available volume for a free w ater molecule to form
hydrogen bonds w ith others. The tem perature dependence of the relaxation tim e of
glass forming LiC l/H 20 solutions reveals an intim ate connection between the
solution relaxation tim e, conductivity and viscosity. The num ber of bound water
molecules per molecules was extracted through a static model.
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-5 -
1.1.2. Heme protein solutions
The study of the properties and functions of w ater in biological systems is
complicated by th e nature of the medium. Interesting aspects of study include the
origin of resistance toward freezing and dehydration shown by most animal and
plant tissues. This raises the question of the nature of “bound” w ater which is
currently attracting increasing amount of attention and can be characterized by
dielectric spectroscopy. It is expected th at a fraction of the w ater molecules find
themselves in a local environment interacting strongly with the biomolecule in
aqueous solutions. This w ater is believed to adsorb on the protein surface and
activate conformational fluctuations of the biopolymer. Recently, it has been
proposed th a t this “hydration shell” may contribute to the energetics of the
transition from the fully deoxygenated tense (T) to the fully oxygenated relaxed
(R) state in hemoglobin (Hb). Also, the structural aspects and dynamics in
conformational changes, for instance, the folding-unfolding transition induced by
varying the pH of a myoglobin (Mb) solution or helix to coil transition in poly- 7 benzy 1-1-glutam ate by varying solvent content or tem perature, are of great interest.
In this work, the statics and dynamics of the hydration shell around a
macromolecule such as myoglobin and hemoglobin were investigated through the
microwave dielectric spectra of the aqueous solutions. W ithin experimental error,
no detectable differences were observed between m etM b and MbCO. Changes in
microwave dielectric excluded volume induced by the conformational change in
m etM b when varying the pH value of the solution were detected. An in-situ
procedure was developed to measure small changes in the shell of hydration water.
1.1.3. Alcohol/water mixtures
Many im portant chemical and biological processes occur in the liquid phase
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-6 -
in m ulticom ponent polar solvents. A detailed knowledge of the dynamic response of
a m ulticom ponent dipolar liquid to a time-dependent external electric field is
necessary in order to understand the effects of polar solvent dynamics on these
processes.
The dielectric spectra of a series of w ater/organic liquid, such as glycerol, 1propanol and ethylene glycol, m ixture were studied systematically. All pure organic
liquids show two relaxation processes up to 20GHz. In contrast to the ionic and
protein solutions, the solution dielectric spectra seemed to become non-Debye upon
addition of small am ount of organic molecules to water. There is strong evidence of
“fusion of two relaxation channels” as described in the molecular theory of Chandra
and Bachi (1991).
1.2 Conceptual basis of dielectric relaxation
Dielectric investigations are directly relevant to questions of molecular
structure.
More specifically,
dielectric spectroscopy is a study of relaxation
processes. Relaxation is a fundam ental feature of m any processes in condensed
phases, such as dipolar, magnetic, viscous and mechanical responses to various
external fields.
1.2.1 Dielectric spectrum and response function
Upon application of an electrical field E (t) on a dielectric m aterial, its
response is described by the electrical polarization P (t) (dipole moment density).
P (t) is the resultant consequence of all the effects th at the external (relative to the
object under consideration) electric field produced on the dielectric prior to the
observation tim e t, for a homogeneous and isotropic medium, i. e.
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-7 -
p (t) = J x(t-u) E(u) du
(1.2.1)
-oo
where x (t) is called the response function. For lineax dielectrics, P is assumed to be
linear in E.
Suppose a macroscopic and uniform external electric field Eg is present in a
liquid. Moreover, suppose th a t the field is switched off instantaneously at the time
t= 0 and remains zero during the tim e interval
0
> t > oo. At times t < 0 the
electric field m aintains a constant total dielectric polarization P tot(t < 0) w ithin the
liquid. This polarization consists of several contributions P ; of different kinds
which, after the field has been switched off, decay w ith different rates. In general,
the origins of various types of polarization are as follows (Pottel, 1973).
(a) The distortion of the electronic structure of the molecules, the ions and
spatial ion distribution in the liquid.
(b) The small reversible shifts ( <
small reversible rotations
atomic diam eter) of the atoms and the
( < 2ir) of the
molecules
away from
tem porary
equilibrium positions around which oscillations are possible.
(c) The irreversible rotations, varying with local environment, of the polar
molecules, or structures with perm anent dipole moments.
The orientational response in case (c) to switching off at t = 0 a formerly
constant applied electric field E 0 can be described by
PorW = Por(0 ) * (t) = XO E 0 * (t)
(1 .2 .2 )
where Xo is the m axim um orientational polarization and 'S(t) is the decay function.
W hen a tim e-dependent electric field E (t) is applied to the liquid, the response is
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-8 -
obtained from a synthesis of E (t) by superposing step functions (Vaughan, 1979).
For an electrical field alternating with frequency w, E(t) = E 0 elU;t,
P or(t) = x H E 0 eiwt
(1.2.3)
where x (w) is known as susceptibility,
xM
= Xo T [ -
] e ~ iU}t
(1.2.4)
and is the dipolar orientational contribution to the dielectric constant or
p erm ittivity e(w). The electronic contribution in (a) to the perm ittivity ee ~ 2, with
decay tim e r e < 10' 14 sec. The tim e scale in (b) is less than 10' 12 sec and its
contribution to the dielectric constant is
- ee, in which e^, known as the
“infinite-frequency” perm ittivity, is the plateau value of the dielectric spectra when
the alternating electric field is too fast to be followed by the reorientational
rotation of dipoles in dielectrics. Therefore, the measured dielectric spectrum e(u>)
is related to x (w) through
e (w) = e'(u>) - ie"(w) =
+ x (w)
(1.2.5)
1.1.2. Connection to microscopic properties
Upon turning off the constant external field, E=0 at t= 0 , the liquid may be
thought to be in a state corresponding to a very infrequently occurring, large
spontaneous therm al fluctuation of zero-field equilibrium polarization. The function
describing the therm al polarization fluctuation at equilibrium is the tim e auto­
correlation function of the polarization fluctuation vector M (t). The decay of the
nonequilibrium polarization in Eqn. (1.2.2) can be expressed as
¥ (t)= < M (t) • M (0 )> /< M (0 ) • M (0)>
(1.2.6)
where the average is taken over an ensemble of equal systems in therm al
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-9 -
equilibrium without an applied field. The decay of a dipole m om ent in a dielectric
follows the correlation function of molecular dipole m om ent /z,
V>(t) = < /x(t) •ii(0)>/< n(0) ■fi(0)>
(1.2.7)
Various physical models (Vaughan, 1979; Bagchi and Chandra, 1991b) have been
established to relate the microscopic to the macroscopic tim e correlation function
>?(t), and hence the dielectric spectrum e(u>), through Eqn.(I.2.4) and E qn.(1.2.5).
The effects of intermolecular motion, dielectric friction due to the long-range
dipole-dipole interaction and restricted orientation caused by the short-range
repulsive forces, on the dielectric spectra of liquids can be incorporated depending
on the dielectric m aterial.
The relaxation times found by dielectric methods arise from a particular
molecular species in a liquid being unable to keep pace w ith reversals
direction of applied field. This period is related to
in the
the m axim um reorientation rate
of the species in the particular environment. Nee and Zwanzig’s (1970) theory of
dielectric relaxation in polar liquids based on a model suggested by Onsager,
brought forward the earliest relationship between the macroscopic relaxation r D
tim e in c(u) and the molecular dielectric relaxation tim e r moi for w ater
TD
~ [
1
+ (ewoo/ ewo)
] r mol
r D appears to be a direct microscopic measurement.
( 1 .2 .8 )
Changes
in the
relaxation tim e can be measured against changes in the microscopic environment.
Therefore dielectric spectroscopy is a powerful measure of molecular properties of
th e species under consideration and their environment.
1.2.3. Theoretical development
Significant advances were made in both theory and experim ent since
Debye’s (1928) pioneering work in the early part of this century.
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-1 0 -
Debye’s classical work relating the dielectric constant to polarization and
perm anent dipole m om ent of polar molecules, in which the Lorentz local field was
incorporated, opened up the whole field of dielectric studies. Onsager (1936)
pointed out th at the electric field at a polar molecule was in p art a reaction field
induced by its own dipole moment and as such could have no role in reorienting it,
as tacitly supposed in using the Lorentz field. The Kirkwood (1939) treatm ent
based on the separation of specific short-range effects and correlations from longrange Coulomb forces, with only the latter treated microscopically, led to a static
dielectric constant, ew0K= 67, comparable with the experiment value of 78.5 at 25C.
Later developments to generalize the Debye’s description include considering effects
such as stochastic (McConnell, 1980), inertial (Lobo et al., 1973), hydrodynam ic
memory (Sparling et al., 1986 and 1984) and internal rotation (Davalos-Orozco and
del Castillo, 1992). Also, molecular dynamics computations allow evaluation of the
dielectric properties of materials by approximating various kinds of inter-m olecular
potential (Pollock and Alder, 1981; Impey, et al., 1983) Despite considerable
methodological progress, the static dielectric constant of the polar liquids is not yet
known w ith acceptable certainty theoretically (Leeuw, et al., 1988).
The long wavelength dielectric constant e(uj) is the response governed
m ostly by the orientational motions of the dipolar molecules as discussed above.
For m any chemical processes such as the solvation of a newly created charge or a
dipole, the short tim e and interm ediate wave vector response of the polar liquid is
im portant. Hubbard et al. (1983) and simultaneously van der Zwan and Hynes
(1983) considered the role of translational polarization charge diffusion in dielectric
relaxation of polar liquids. Bagchi and Chandra (1991b) further derived analytic
expressions for both the transverse and the longitudinal parts of the frequency and
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-11-
wave vector dependent dielectric function, and presented a unified treatm ent of
polarization relaxation, dielectric dispersion and solvation dynamics in a dense,
dipolar liquid.
The following are three recent dielectric models which have stim ulated
extensive experim ental efforts and need further experim ental verification.
(i) Hubbard-Onsager's kinetic polarization deficiency.
The presence of free ions in solution implies the existence of strong local
electric fields, especially around a small ion such as Li+, a situation in which the
solvent in the neighborhood of the ions might be in a state quite different from th at
of the bulk solvent. In aqueous solutions, the w ater molecules near by an ion tend
to orient along the electric field of the ions. Also, the ions move along the direction
of the applied external field.
Traditionally, the observed decrement of static dielectric constant in ionic
solutions was directly related to the static properties of the solvent around the ion.
A more complete dynamic model was developed by H ubbard, et al. (1977-1979).
The reduction of the static dielectric constant of polar solvent, resulting from the
presence of the ions, is term ed the kinetic polarization deficiency. The effect arises
from two related phenomena. As an ion migrates the surrounding volume elements
tend to rotate according to the laws of hydrodynamics and dielectric relaxation
does not restore completely an equilibrium polarization appropriate to the local
electric field. In addition, the force th a t an external field exerts on an ion does not
develop its full strength instantly because p art of the force arises from a
polarization field th at develops by orientation of the
solvent dipoles. By
incorporating the electrical dissipation function into the hydrodynamic equation, it
is shown th a t the static perm ittivity of a solutions is reduced from th a t of the pure
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-12-
solvent by an amount proportional to the “dc” conductivity of the solution at low
concentrations.
It is also asserted th a t the ion size, relative to the characteristic
Hubbard-Onsager radius, plays an im portant role in dielectric friction as a
consequence of the self-consistent kinetic model.
(ii) Mode coupling theory
On the basis of a detailed kinetic theory for the hard sphere fluid,
Leutheusser (Leutheusser,) proposed a nonlinear feedback mechanism by which
density fluctuation could lead to the ultim ate liquid-glass transition of a
supercooled liquid. This mode-coupling transition is not associated w ith a
divergence in any static susceptibility, but corresponds to a kinetic singularity in
which small density perturbations from m etastable equilibrium cannot relax in a
finite am ount of time. The tim e decay of the correlation function is nonexponetial
over certain tim e windows. In such systems, the mode-coupling theory predicted
(Kim and Mazenko, 1992) th at there is a special tem perature T Mc (well above the
phenomenological glass transition tem perature T g), above and below which the
relaxation dynamics is substantially different. Above T MC, relaxation phenomena
(e.g. dielectric, viscosity) obey a power law
r or rf ~ ( T - T m c ) ' 7
in which
7
(1.2.9)
~ 2, and when T g< T < T MC the tem perature dependence of various
relaxations have the Vogel-Fulcher-Tammann form. The experim ental evidence for
the existence of T MC is a m atter of debate and is the topic of current research.
(iii) Fusion of two relaxation channels
In the molecular theory of dielectric relaxation in a dense binary dipolar
liquid, C handra and Bagchi (1991a) related the polarization fluctuation in
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-13-
Eqn. (1.2.2) to the density fluctuation, and hence to the two-particle direct
correlation function. The theory offers a molecular explanation of the phenomena of
fusing of the two relaxation channels of neat liquids. The relaxation times predicted
by the theory are very different from th at of the non-interacting rotational diffusion
model for a binary system. The dielectric relaxation would be significantly modified
because of inter-m olecular interactions. Experim entally, dielectric studies on
mixtures of hydrogen bonded liquids such glycerol, n-propanol and w ater can
provided such inform ation and test the applicability of this and other theories of
dielectric relaxation.
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-14-
Fig. 1.1. A schematic chart of electromagnetic spectrum , corresponding molecular
charge motions and experimental methods [based on a plot of Cole (1989)].
EXPERIMENTAL METHODS
NETWORKANALYZER
FREQUENCY
DOMAIN
k
TIMEDOMAIN |<
OPTICAL
GUIDEDWAVE
LOCKINAMPLIFIER
>1
|<— TDR-
TRANSIENT-
J__________I____
J ___
MOLECULAR CHARGE MOTIONS
DIFFUSIVE
TRANSPORT
REORIENTATIONS
iELECVIBRATIONS ITRONIC
ROTATIONS
ELECTROMAGNETIC SPECTRUM
THIS WORK
POWER
AUDIO
10
MICRO
UHF
FAR
IR
IR
VISBLE
X ------
-------------------FREQUENCY
(HZ)
10
RADIO
10 J
10
10
10
10
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-15-
CH APTER II
DEVELOPM ENT OF A TECHNIQUE FO R DIELECTRIC SPECTROSCOPY
A new experimental technique for measuring the complex dielectric constant
e — e' - je" of liquids and solids, which has been tested up to 20GHz but is
applicable up to even higher frequencies, is described. From th e m easured dielectric
spectra, information on microscopic relaxation processes can be obtained. The
technique, which utilizes a coaxial probe and a unique sample d ata de-embedding
procedure, is fast, non-invasive and accurate. The effect of probe size is discussed.
A proper radiation correction model is then applied to the simple complex
capacitance model which is only applicable to low dielectric constant materials
above 10 GHz, to extract the dielectric spectra of w ater rich solutions (i.e. high e1
and e" ) up to 20GHz. Measurements on solids are also presented.
II. 1 Measurement configuration and procedure
The measurement configuration is as shown in Fig. II. 1. A semi-rigid coaxial
cable w ith an SMA connector on one end and a flat face on th e other, is dipped
into the liquid.
The connector end is m ated to an H P 8510 Network Analyzer
(m anufactured by Hewlett Packard Co., Palo Alto, CA) - since the dominant
length of the coax (0.141, 0.085 or 0.047 inch in diameter) is required to be vertical,
the m ating to the ANA was m ade either w ith a right-angle adapter, connector or
bend in the coax.
The HP 8510 system used consists of an HP8510B analyzer, an HP8513A
reflection/transm ission test set usable from 45 MHz to 26 GHz. The source is a HP
8341B synthesizer sweeper, operating between 10 MHz and 20 GHz. The sweeper
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-16-
was used in the step mode over the desired frequency range, w ith typical selections
of 51 to 801 points. A synthesized source, and its operation in a step mode,
resulting in good phase stability, is essential to obtaining reproducible d ata free of
spurious oscillations.
A personal com puter was set up to receive pairs of data [Re and Im(p) for
each frequency point] over an IEEE-488 bus, and used for subsequent numerical
analysis.
The procedure for experim ental data collection is described below step-bystep :
A. OPEN W ith the coax term inated by free space, the m easurem ent plane
of the ANA was moved to the coax end using the electrical delay provided. The
delay, which corresponds to twice the length of the coax m ultiplied by ^
(
dielectric constant of the teflon inside the coax ), was adjusted to give a cluster of
points near the R e(p )= l, lm (p)=0 point, at the middle of the right hand side of the
display. Because of connector and line m ism atches, the cluster is not a point, but
can occupy a region. The pairwise />MA d ata was read into the computer.
B. SHORT A short at the coax end was created by raising a small vessel
(about 3 cc) filled with mercury, until the coax end was well within the liquid. This
resulted in a cluster of points around the position (Re(p)= -
1,
Im(/>)=0), at the
middle of the left hand side of the display. T he pairwise d ata pMB was again read
into the computer.
C. STANDARD LIQUID The mercury cell was removed. At this point, the
display was checked to ensure th a t the data returned to the configuration for an
open. A cell with a standard liquid, typically acetone, was now inserted so th at the
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-17-
coax end was well immersed in the liquid. The d a ta pMC was again read into the
computer.
D.
SAMPLE LIQUID The standard liquid was removed, and tim e was allowed
for the acetone to evaporate completely from the end of the coax, until the display
returned to the open configuration. The procedure was repeated w ith the liquid
sample, and d a ta pM was collected.
The
above procedure
of 3
calibration
measurements
and
a sample
measurement were taken with minim al disturbance, and without disconnecting the
connectors.
The sequence of the above procedure can be interchanged depending on the
type of m easurem ents. The coax-end is usually polished by a silicon carbide coated
abrasive sheet (320
grit) and ultrasonic cleaned in methanol for about 1 m inute
before starting a set of measurement.
H.2. Analysis: de-embedding of sample impedance
The “ open” , short and standard liquid d ata are used for computerized
calibration based on the equivalent circuit given in Fig. II.2a.
By properly
modeling Z(w,e) (c is the dielectric constant of the fluid surrounding the probe end)
of the end of the coax, we can extract three param eters A ,A 12,A 23 which are related
to Z1 ,Z 2 ,Z3, and then we can get the dielectric constant of the sample liquid.
The
calibration
procedure
of
consecutively
placing
three
known
term inations, a short circuit, an open circuit and a m atched load at the reference
plane o f a network analyzer can significantly improve the accuracy of a network
analyzer m easurem ents.
In the m easurem ent configuration shown in Fig. II. 1
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-18-
which is a very convenient way ( Sridhar, 1987; Stuchly et al., 1980; Foster et al.,
1987 ) to measure liquid samples, the network analyzer calibration alone can not
eliminate the effects o f the connectors and the coax probe. This is shown in Fig. II.3
where e'
and t" for water, determined w ith the ANA calibration only (x’s) and
w ith the calibration procedure described here below(o’s), are plotted. Clearly the
ANA calibration alone is inadequate, and leads to oscillations, which we attrib u te
to the connector mismatches.
Various kinds of calibration techniques for one port m easurem ent have been
developed and used for the measurements th a t cannot be carried out at the
reference plane of a network analyzer (de Silva and M cPhun, 1978; Kraszewski, et
al., 1983). For measurements using a semi-rigid coax probe, the proper modeling of
the frequency w and sample dielectric constant e dependent impedance Z(u>,e) o f the
probe end is crucial (Stuchly, 1980; Kraszewski, et al., 1983b).
The calibration
technique we use, which looks at the problem from a new point of view, eliminates
uncertain factors such as fringe-field (complex, frequency-dependent) capacitance
and the frequency dependence of the probe param eters, as shown below.
For a TEM mode propagating along a transmission line with characteristic
resistance Z0, its output can be completely determ ined by three param eters. It can
be easily shown th a t the output of the TEM mode propagating through two
transm ission lines which have different characteristic impedances can also be
similarly determ ined. Since the complicated linear microwave transmission system
with coax and connectors can be regarded as the combination of m any pieces of
coax w ith different characteristic impedance, three param eters need to be
determ ined at each measuring frequency. The equivalent circuit in Fig. II. 2 a with
Zlf Z2, Z3 is used to characterize the transm ission system and Z(w,e) for the
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-19-
impedance of the end of the probe.
ZM(o>,e) is th e measured output impedance which can be obtained from the
complex reflection coefficient
/>M=reW
(II.2..1)
measured by the 8510 ANA, and
Zm ^Zo^
(II.2 .2 )
A "M
ZM also can be expressed from Fig. II.2a for every frequency :
n
. Z2 (Z3 +Z(w,e))
Z m- z ‘+ z ^ + z m
<IL2-3>
Equation (II.2.3) is equivalent to
A +A 12 Z(w,e)-A23 ZM(w,e)=ZM(w,e)Z(w,e)
(II.2.4)
where
A =Z 1Z2 + Z 2Z3 + Z 3 Z1
(II.2.5)
A12= Z 1+Z2
(II.2.6)
A23 = Z 2+Z3
(II.2.7)
By properly modeling the end of the probe, th at is by knowing the way in which Z
depends on frequency w and the dielectric constant e of the surrounding medium
Z(w,e)=f(u;,e)
(II.2.8)
Z(w,t) can be calculated from known e(u>) and vice- versa.
Therefore, if the first three calibration m edia with known eA,£B,ec are
measured and the reflection coefficients pA,pB,pc are recorded, ZMA ZMB,ZMC can be
calculated from (II.2.2), ZA,ZB,ZC from
(II.2.8).Equation (II.2.4) gives three
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-20-
equations which contain three unknown param eters A,A 12,A23
The param eters
frequency,
A+A 12Za-A23Zma= Z maZa
(II.2.9)
A+A 12ZB-A23ZMB= Z MBZB
(II.2.10)
A+A 12Zc-A23Zmc= Z mcZc
(II.2.11)
A, A12 and A23 can be calculated from the above equations for each
and substituted into equation (II.2.4) to get Z(w,c)
for the sample
medium:
A-A ^ZM(M «)
Zm(w>e)’
12
where ZM(w, c) is obtained from the measured reflection coefficient pM according to
equation (II.2.2). Then the dielectric constant of the sample c(w) can be solved
from the relation (II. 2 .8 ).
If f(w, c) in Eqn. (II.2.8 ) can be further expressed in the form of
where C (oj) is a function of frequency and is not necessarily the fringe field
capacitance, since it can be complex. The sample dependence of the impedance is
included in 7 (w,r). As discussed next, 7 (w, e) = c is true only when the radiation
effect can be neglected. Eqn. (II.2.9) - (II.2.11) become
A ,,
. * . , _
'MA =
4 ^
-
Z
...
^MA
(n
2
u)
e A
ZMB
(iwc a 23) z'M
mBb == m
£n
(ii.2.i5)
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-21-
(iwC A) +
(iu S A) +
C
- (ijwC A23) Zmc =
(II.2.16)
f a
- (iwC A23) ZM= ^
(II.2.17)
W e can further rearrange the equations by substituting
Z; = Z0
where px = Tj e
( i= MA, MB, MC, M )
(II.2.18)
is the reflection coefficient of the open ( i= MA), short ( i=M B
), standard liquid ( i=M C ) or of the sample liquid (i= M
) from the probe end,
into the above four equations with a new set of A's
Ax + A 2/>m a+ A3?a =
?ApMA
(II.2.19)
A x + A 2/?m b+ A3?b
?BpMB
(II.2.20)
=
A j + A 2/?MC + A 3ec = ?c
(II.2.21)
pMC
Ai + A 2Pm + A3e = ? pM
(II. 2 .2 2 )
Solving A 1? A 2 and A3 from Eqn. (II.2.19) - (II.2.21) and substituting them in Eqn.
(II. 2 .2 2 ), c’s and p1s satisfy
( g ~ £a ) ( eB ~ £c ) _ ( P m " Pm
( £ ' £b ) ( £c ' ? a )
a
) ( Pm b ~ I m c )
( Pm ‘ ^ m b ) ( Pm c ~ Pm
a
(II.2.23)
)
e (w,e) = e B + 7— — r T 7 ------ ~ 'BT 7 ^ ----------- \------I £c ~ €a ) (.Pm ~ Pm
a
) ( Im b ~ Im c )
( £b ' £c ) ( p m ‘ ^ m b ) (/’M e ' Pm
a
_
^
(II.2.24)
)
If Eqn. (II.2.13) is used for (II.2.23), we obtain the formulation of an S-param eter
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-22-
description of the microwave network,
( Y ~ Ya) ( Yb ~ Yc) _ ( Pm ~ Pma) ( Pmb ~ Pmc)
( Y ' Yb) ( Yc - Ya) ( pM - pMB) ( pMC - pMA)
/tt 9
which applies to the case of the reference plane “ sitting” between the fringe field Cf
due to the electric field inside the coaxial line, and C (w) filled w ith dielectric
medium e*. This shows the equivalence of an S- param eter description to the “
lum ped ” circuit description of this work.
H.3 Modeling of the coax termination
Generally, the coax term inated by a medium can be described by the
equivalent circuit given in Fig. II.2b or Fig. 3a,
Z
v
M
'
=
—
G (w, e) + iB(u;,
e)T
7 H —
vI 1 1 - 3 - 1 ')
which represents the e- and w-dependent complex impedance.
II.3.1. Simple capacitance approximation
W hen ^j v ^ < < 1, a simple capacitor model is a good approxim ation to the
impedance of the coax probe end, as shown in Fig. II.2c
Z (w,e) = [ iu>C£(w)+iwC(w,e) j ’ 1
(II.3.2)
C (u>,e) = C 0 (w) e
(II.3.3)
G (w, e) = 0
(II.3.4)
with
Cf represents the fringe-field effects inside the probe, and C0 the fringe-field
coupled to the sample liquid. The effect of Cf can be included into three network
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-23-
parameters,
as in Fig.
II. 2 c.
Therefore, in equations
(II.2.19)-(II.2.21) for
calibrations
= Ci(w)
( i = A, B, C )
(II.3.5)
and in equation (II. 2 .2 2 ) for the sample measurement
e' (a;, e) = e (a;)
(II.3.6)
Therefore, Eqn. (II.2.23) give the solution, when air is used as an open w ith eA= 1
and m ercury as a short w ith | eB |—*0 0 ,
£ H = 1 - ( eu.:<lu*) ( /m b - .omc) [
(k)) _ ! ]
(II 3 7)
( Pm - Pm b ) ( p m c ' Pm a )
Note that the measurement does not depend on any information of the fringing
capacitance. Also the frequency dependent parameter CQ(u).
T he procedure next requires expressions for the standard, which is usually
acetone, eqn. (II.3.7) is used with a Debye model for € :
e = €oo + (£o ' £oo)/(1+ iwT)
(II.3.8)
with param eters obtained from the Table o f Dielectric Constants o f Pure Liquids (
NBS Circular 514, National Bureau of Standards, W ashington DC, 1958 ), e0 = 21.2
,
= 1.9 and r = 3.3 x 10"12 sec.
The results of using the above procedure are shown in Fig. II.4 for w ater as
a sample liquid (o’s). Also shown in Fig. II.4 are the
d ata taken only using the
calibration at ANA reference plane, choosing Cf = 0 and Cf = 1.9 x 10' 14 F . The
im provem ent using the present procedure is clearly evident. In particular the
unw anted oscillations present using the ANA calibration only, are elim inated by
the 3-standard calibration described here. The oscillations are due to the connector
m ism atch, which is not accounted for using the machine calibration only, but
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-24-
requires an actual calibration at the measurem ent plane. In the work of Foster et
al. (1987), a time-domain gating was utilized to reduce the effect of connector
m ism atch. The present procedure does not require such a time-domain procedure.
As discussed later, the 3-standard calibration is a more accurate representation of
the tru e frequency dependence of water.
Fig. II.5 presents the data for e' and c" for m ethanol between 45 MHz and 20
GHz obtained using the technique described above. D ata taken w ith a thick 0.141
in. (x’s) and a thin 0.085 in. ( + ’s) probe are shown. Results with the two sizes of
probes agree fairly well, with the only discrepancies appearing at higher frequencies
for e". For comparison, the Debye form eqn. (II.3.8) is also shown w ith param eters
e0 = 33.64, Cqq = 5.7 and r = 53 x 10' 12 sec., taken from Table of Dielectric
Constants o f Pure Liquids ( NBS Circular 514, National Bureau of Standards,
W ashington DC, 1958 ).
This form is expected to apply with good accuracy to
methanol, and the data are in fair agreement with it.
For water, the data for e' and e" as functions of frequency are shown in Fig.
II. 6 , for a thick 0.141 in. probe ( + ’s), a thinner 0.085 in. (x’s) and a
very thin
0.047 in. probe. Here the effects of radiation are dramatically evident in the
difference in data between the three probes.
Structure at low frequency is evident in Fig. II . 6 using the 0.141 in. probe.
It is a manifestation of a “ container” effect. This arises due to the fact th at the
liquid volume in the container acts as a dielectric-filled resonant cavity with
resonant frequency less than 5 GHz for liquids with e' «
30 - 100. This was
experim entally verified this by varying the height of the liquid and noting th at the
resonance shifts accordingly. One solution to this problem is to experim entally
dam pen this resonance and reduce its Q to be much less than
1.
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-25-
To first order of the radiation effect, the conductance G <x (a 2-b 2) 2 u>4 e5^2,
where a and b are respectively the outer and inner diameters of the coax probe.
Thus the effects of radiation drastically increase with probe size, frequency and
dielectric constant of the liquid. The differences between the thick probe and thin
probe results of Fig. II .6 are entirely due to the probe size and the larger dielectric
constant of the liquid, and increase with increasing frequency. The thin probe
minimizes the effects of radiation, but does not completely eliminate it, as can be
seen from the small difference between the d ata w ithout and w ith the radiation
correction shown in Fig. II.7.
II.3.2. Full radiation correction
W hen
f «
l
is not satisfied due to high e, large probe size or higher
frequencies, Eqn. (II.3.2) - (II.3.4) are no longer valid. A more accurate formulation
for the probe-end impedance is required.
The fringe adm ittance of a vacuum-filled flanged open-ended coax probe
th a t opens to a half-free space can be represented generally as
Yv ( W, e = l ) = Gv ( W, e= l )+ i Bv (w, e = l )
(II.3.9)
M arcuvitz ( 1951 ) expressed the Gv and Bv as
C*/2)
G v
T)
v
= In ( a /b )
I
sin7 [ J o
d6
(II.3.10)
- Si [ 2 k 0 a sin ( 6/2 ) ] - Si [ 2 k 0 b sin ( 6/ 2 ) ] } d 0
(II.3.11)
vo
n In ( a /b )
(
k o
a sin
0
)
{ 2 Si [ k 0\j( a2-f b2-
-
J 0 ( k 0 b sin
2
0
) ]2
a b cos 6 ) ]
0
a and b denote the outer and inner radii of the line, k0= w/c is the wave num ber
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-26-
in vacuum and Y vo is the characteristic adm ittance of the line. Deschamps (1962)
pointed out th a t the adm ittance of the vacuum-filled probe open to a m edium of e,
H=1 as depicted in Fig. II. 8 a, can be shown to be
Ya
(
c
^
Y av
(^
!)
(II.3.12)
where Y av is the adm ittance of the probe open to the free space. Now the radiation
from a teflon-filled coax with characteristic adm ittance Y 0 is equal to th a t from a
vacuum-filled coax of the same size but with characteristic adm ittance Y 0 /^q ,
where ct is the dielectric constant of teflon. If the vacuum-filled coax in Fig. II. 8 a
is replaced by a teflon-filled coax, the adm ittance Ya/ would be
Ya/ ( w, e ) = >fe Yav ( -y[ew, 1 ) / ^
(II.3.13)
The configuration of interest here is an open-ended teflon-filled semi-rigid coax
probe dipped into a liquid as shown in Fig. II. 8 b. There are two minor differences
from the configuration Fig. II. 8 a.
different from Y v(w,
1)
First,
the free space probe-end adm ittance is
obtained from Eqns. (II.3.9) - (II.3.11), due to the absence
of the flange. Secondly, Deschamp’s theorem is only approxim ately valid for this
configuration.
Here we introduce a frequency-dependent configuration function of
probe param eters a,
b and et, viz. a(a,b,w,et), which represents the medium-
independent modifications to the adm ittance, when applying Descham p’s theorem
to M arcuvitz’s expression in our configuration Fig. II. 8 b.
Hence the probe-end
adm ittance of the teflon-filled flat-end coax surrounded by a liquid is approxim ated
as
Y (
e) =
[ G ( w, e )+ i B ( u>, t )] = <* (a, b,w,et) Yb/
(II.3.14)
where
Yb/ = ^ Yv ( * u , l)A jq
(11-3.15)
Using Eqn. (II.3.10) and (II.3.11),
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-27-
C7"/2)
G = In (“a / M r I SHllJotko^asineJ-JcllcvrbsinO]2^ (II.3.16)
H t
o
B = ,h (I/b )^
- Si
[2
I ( 2 si [ M ‘ ( »2+ b2- 2 * b “ s 0 )1
k0^e a sin (
0 /2
)] - Si
[2
k0Afe b sin (
0 /2
)]} d 0
(II.3.17)
Expanding the integrals in the above two expressions in powers of k0^f = y
G (u, e ) = cn|? ( gx7 2+ g 27 4+ g37 6+ ... + g
k7
2k+ - )/-fc
B ( w, e ) = cW7 ( b j 7 + b 27 3+ b 37 5+ ... + b k72k_1+ ... )Afq
(II.3.18)
(II.3.19)
the probe-end adm ittance
Y = G + j B = j WC («) £ (w, £ )
(II.3.20)
C («) = a b j ( c
(II.3.21)
where
is the fringe-field capacitance due to the electric field outside the probe. jwC can
be regarded as the exact coefficient of the lowest order expansion of Y(w,e) in
powers of -\fe for the semi-rigid coax probe dipped into a liquid medium. In general,
this could be different from th a t calculated from the lowest order expansion of eqn.
(II.3.14) because of the differences between the two measurem ent configurations in
Fig. II. 8 . As we shall see later, the detailed functional form of a is not im portant,
as it gets eliminated. From Eqn. (II.3.20)
e (w, e) = ( G + j B ) / j w C
(II.3.22)
which is the functional form for the effective dielectric constant in Eqn. (II.2.19) (II.2.21). Therefore, the combination of Eqn. (II.3.21), (II.3.18) and (II.3.19) are
used to calculate 7 A(u>, eA), 7 B(u>, eB) and 7 c (u;, ec ) from the known dielectric
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-28-
constant of the calibration standards eA, eB and ec . Then substitute them w ith the
measured reflection coefficients of the calibration standards, pMA, pMB, pMC and
th a t of a sample pM in Eqn. (II.2.24), the effective dielectric of the sample 7 is
extracted. W e can solve for the sample dielectric spectrum also using Eqn.
(II.3.21), (II.3.18) and (II.3.19).
The algorithm for the numerical analysis of the d ata is shown as a flow
graph in Fig. II.9. The expressions (II.3.18) and (II.3.19) for G and B were
expanded to the 31st power of
7
(most term s can be found from Misra, et al., 1990;
and Xu, et al., 1986). The coefficients gj and b; were calculated using an IBM PC.
Then e( u>, e ) was found for the open, short and the standard liquid acetone, viz.
?A, eB and ec respectively. e (w, e) of a sample was then obtained from the three sets
of calibration data, the reflection coefficient pM of the sample, and ?A, ?B and ec
(see Fig. II.9). A zero finding program from Numerical Recipes (Press, et al., 1990)
was used to solve the dielectric constant c of the sample from e (w, e) as expressed
in Eqn. (II.3.22), (II.3.18) and (II.3.19).
The radiation corrected data of water, deduced from the measurements
using a 0.047 in. probe and the analysis described above are plotted in Fig. II.7
(o’s). Also shown are uncorrected results for a 0.047 in. probe. For e', it is evident
from Fig. II.7 th a t the correction is small (8a/a< 3 %) at u/2n < 12 GHz, and only
slightly larger for higher frequencies, reaching 6 a /a = 10 % for 20 GHz. The
corrections are larger for e" viz., 6eu/en< 10 % upto 12 GHz and 12 % at 20 GHz.
The corrected d ata can be fit to a Debye curve w ith e0 = 77.6,
= 5.0 and r =
7.9 ps (at the measurem ent tem perature T = 27 °C ), exhibited as the solid line in
Fig. II.7. These param eters are more accurate than w ithout the radiation
correction, which is apparently with longer relaxation time. Thus the measurement
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-29-
enables the observation of the complete relaxation spectrum of water.
The radiation correction model is adequate for the 0.047 in. probe
(Kraszewski, et al. 1983) as a result of the small correction required, in contrast to
the large errors introduced by the 0.141 in. probe for w ater, as shown by
measurements in Fig. II. 6 . Fig. 11.10 shows the radiation corrected w ater dielectric
results for 0.141(+’s), 0.085 (dots), 0.047(o’s) in. probe and the Debye fit with
e0=78.2, €^,=5.1 and r = 7.9 ps. It is appaxent th a t a narrow probe is essential for
accurate measurements, particularly at high frequencies. The correction causes
some improvement for the 0.141 in. only at frequencies lower than
8
GHz, since the
deviation due to a simple capacitance approximation is small at low frequencies.
The superiority of the 0.047 in. probe at high frequencies is apparent. The 0.047
probe d ata for w ater shown in Fig. II. 1 were taken with a flanged coax. There are
no significant differences from a simple flat-ended coax m easurem ent, but the
flange does seem to decrease small wiggles due to a “ container effect” . The 0.047
in. probe without a flange has the added convenience of measuring sample liquid
size as small as 0.5ml.
II.4. Error estim ation
The errors arise from two sources: (1) random errors introduced by the
measurements and (2) errors arising from inaccuracies in the calibration data. The
lim itations of the coax-end impedance modeling, which are at present difficult to
quantify, are discussed in the next section.
Experim entally, the largest error (called dpM) was found to be introduced in
the measurements of the reflection coefficients
pM,
is due to long term drift of the
ANA, and are of order | 6 pM/p M | ~ 1% • The actual accuracy of the machine (e.g.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-30-
for m easurem ents of an impedance at the term inal plane of the test set), as stated
by the m anufacturer is much greater, with errors of order
0 . 1 %,
hence we ignore
them and only consider the long term drift contributions.
Inaccuracies in the values assumed for the calibrations, viz. the open, short
and the acetone standard could also introduce errors in the deduced e values. The
open calibration, done without any liquid, is taken to correspond to eA =
short, carried out using liquid Hg, is also accurate, and corresponds to
eB "
1.
The
—►oo. In
the following we examine the possible effect of inaccuracies in the assumed values
for the acetone standard, i.e., a possible error of | 6cq lec'\ = | <5ec"/£c/ ,| =: 2 % at all
frequencies in ec for which Debye form has been assumed.
Taking the derivative of Eqn. (II.2.23), we get
(? -?
b)
(?-Sa)
+ ( « W )
A-
eB - 0
(<B - * c )
(pm b ~ I
B
(< a -
c)
I
?B ‘ *A
+
( ?C * h ) (?C * *A)
ma)
( P m _ Pm b ) ( Pm '
A
A~
°
( p m ' ^ m c ) d/>MA
Pm a )
( p m ' P m c ) ^ P m b ______ ( p
( Pm b ‘ P m ) ( P m b ' Pm c )
( Pm a ~ Pm ) ( P m a ' P u c )
mb
~ Pm a )
^ ^
( P m c ' Pm b ) ( P m c ' P m a )
where d/>M, d/?MA, dpMB and dpMC come from the instability of the ANA during the
measurements and
d?A, d?B and d?c are errors from the accepted dielectric
constant of calibration standards. Substituting the open and short values of | eB | -+
oo
and
above equation is simplified as
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-31-
d?=
+ ( ?
[- T
V
v Pm
f m-m
u TP m ' P„ m a )
■P
b M
^ ___( p m ~ P m c ) d P M A
(p m ~ Pm c ) <W b
( Pm a " Pm ) ( Pm a " P m c )
+ , ------ ( '’MB - W
ifUC
( Pm b ' Pm ) ( Pm b ‘ Pm c )
]
(I, 4 2)
I Pm c ' Pm b ; I Pm c * Pm a )
Table II. 1 shows the possible errors induced by the drift of the machine,
taken to be | 6pM/p M | = 1%, c o n s ta n t a t all frequencies. Also shown axe the errors
induced by an error of
| 8ec '/cc'\ = | 8ec"/cc " | =
2 %,
also constant at all
frequencies, in the assumed standard liquid ec values. The third pair of columns in
Table
II. 1 lists the standard deviation obtained from a set of 5 actual
m easurem ents, the variation being due to drifts in the machine.
One observation th at is evident from Table II. 1 is th at although the errors
are quite acceptable for frequencies above a few 100 MHz, they are unacceptably
large a t the lowest frequencies 45 MHz to about 200 MHz. The prim ary reason for
this is the small probe size, which is inappropriate for the lower frequencies due to
the following three features : (a) the fringe field capacitance of the 0.047 in. probe
is too small, so th a t at low u> the probe impedance is too big, (b) at low u> the
reflection coefficient of the acetone calibration is too close to th a t of the open, and
(c) at low u>, the cn of w ater is small. The calibration problem could be
circum vented by the use of a different standard close to 50 O, such as a saline
solution, for which however, the ec values axe not well known.
T he m ethod obviously relies on accurate and reliable calibration standards
and their modeling. The open and short calibrations are quite reliable and accurate.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-32-
(Care m ust be taken with the use of liquid mercury, since it corrodes the metallic
coax). W e have preferred acetone as a standard liquid, because it is easily available
in pure form, has a reasonable dielectric constant and exhibits little dispersion in
the present frequency range.
axe fairly accurate.
The param eters used in the Debye form for acetone
At present
the errors due to the calibration standards axe
much less than those due to the modeling of the liquid-coax interface.
Table II. 1. Calculated errors and measured standard deviation of a and at.
M achine^
A ceto n e^
Standard D eviation^
6e'/e'
Se"1e"
Se'/e'
8e"/e"
Se'/c'
6e"! e"
0.045
> 0.30
> 0.30
0 .0 2
< 0.005
0.06
>
1
0.25
0.25
0 .0 2
0 .0 1
0 .0 1
0.05
5
0.05
0.06
0 .0 2
0 .0 1
0 .0 1
0.03
10
0.04
0.04
0 .0 2
0 .0 1
0.003
0 .0 2
15
0.03
0.04
0 .0 2
0 .0 1
0 .0 1
0 .0 2
20
0 .0 2
0 .0 2
0 .0 2
0 .0 1
0 .0 1
0 .0 1
f ( GHz )
(a) Calculated assuming a
1%
1 .0 0
shift of measured reflection coefficients, constant at
all frequencies.
(b) Calculated assuming a
2%
error, constant at all frequencies, in the assumed
values for the standard .
(c) Experim ental standard deviation determ ined from 5 sets of measurements.
The above discussion points to the fact th a t in order to minimize radiation effects
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-33-
at high u, one gives up accuracy at low u. Thus the 0.047 in. probe is best used at
frequencies > 500 MHz for liquids like water. For lower frequencies, it is better to
use m uch larger probes with greater C(w) - as is well known, measurement
problems are fewer at low w anyway.
H.5. Discussion
The “ lum ped” T circuit model used here,
the microwave measurement system.
gives a complete description of
It is simply another way of defining the
linear response of the microwave network, w ith a set of param eters different from
S-paxameters. The m athem atical equivalence between them has been dem onstrated
in Eqn. (II.2.23) and (II.2.25).
Approximation to the coax-liquid impedance has gone beyond a simple
capacitance model by including radiation effects. Essential to the approach is the
experim ental minimization of such effects by the use of a very narrow probe,
enabling the treatm ent of these effects as corrections. We next examine the nature
of the radiation treatm ent considered here.
As discussed earlier, the experimental geometry does not exactly correspond
to th a t considered by Deschamps (1962). In the treatm ent the effect of these
differences is represented as Y = a Yb'. Note that the detailed functional form o f a,
which depends
on probe dimensions a and b, and frequency f is not important,
since a cancels out in the process o f analysis. This factor is included prim arily to
account for the absence of the outer screening conductor in Fig. II. 8 b compared to
Fig. II. 8 a.
The absence of this conductor should reduce the effective capacitance
from th a t calculated from the first term of Ya', and this is in agreement with
experim ents by Xu, et al. (1986) and Kraszewski and Stuchly (1983a). In general
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-34-
the applicability of the above approximation will be lim ited by increasing a, b and
f. The radiation contribution to the impedance is greatly minimized for the narrow
0.047 in. probe because at f = 20 GHz, the wavelength in the liquid A ~ 1.5
cm/Ve7, which gives the ratio of the probe aperture and the wavelength 2a/A
~ 0.06/vef. This smaller value improves the validity of the approach here. From
the results of Kraszewski and Stuchly (1983a), we estim ate th at for a 0.141 in.
probe, a < 0.845 (1 + 0.25xl0'3f2), where f is in GHz, which is between 0.845 to
0.95 between 45 MHz and 20 GHz. The correction would be much less for a 0.047
in. probe which was used in the present work.
As noted previously, the detailed form of the dependence of a on a, b, and u
is not im portant. However, our treatm ent does not account for a possible e
dependence of a.
The other approximation made in applying Deschamp’s theorem is the
neglect of higher order modes beside the TEM mode, which would be excited in the
configuration of Fig. II. 8 b. The actual contribution of such terms is difficult to
estim ate. However it may be noted that the arbitrariness of the factor a means th at
all term s linear in Yb/ are included in the present analysis.
The radiation correction is not required for all liquids in the frequency range
from 45 MHz to 20 GHz as we have already shown in section II.3. We examine the
m agnitude of the radiation correction for various values of e' and its dependence on
frequency in Fig. 11.11. The correction is evaluated as
Sf f e =
where I = l' - i
for
,
6e'/e =
(II.5.1)
hypothetical liquids with e = d = 10, 20, 40, 80, and
independent of frequency, upto 30 GHz, and for a 0.047 in. probe. A clear feature of
Fig. II .6 is the strong ^-dependence, with the correction negligible at low frequency,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and becoming increasingly im portant with increasing w. For e = 10, the expected
relative errors shown in Fig. 11.11 are less than 5 % for the real p art, and for the
imaginary part are less than 2% for u/2 t: < 20 GHz. Therefore for liquids such as
methanol, toluene and methylene chloride whose dielectric constant is less than
10
at GHz frequencies, no radiation correction is required up to 20 GHz. W ith
increasing e', the correction becomes increasingly im portant w ith frequency. (The
curve for e' = 80 also illustrates th at the first order correction considered here is
oscillatory).
H.6. Dielectric spectra of solids up to 20 GHz
The technique described above is applicable to solid samples also. The
dielectric spectrum of a piece of plexiglass is shown in Fig. 11.12. It shows the
characteristics of dielectric properties of a solid sample at microwave frequencies constant real part with low loss. The sample was measured w ith a flanged 0.085 in.
coax line, under the consideration th at solid samples has low dielectric constant
and it is more difficult to make a good contact. A piece of styrofoam was inserted
between the sample and the experiment bench in order to improve the contact.
The shape of the solid sample is arbitrary as long as a flat smooth surface w ith 1
cm 2 area and adequate thickness is available.
II.7. In situ difference measurements in dielectric spectra
Measuring changes in dielectric constant is im portant in m any cases, such as
in the epoxy curing process. In Eqn. (II.4.2), the change of e' caused by a change in
the sample reflection coefficient due a different sample condition is given by
® =<*■- ?*) , 0
l Pm
. / v
r
Pm b ) \ Pm
. .
i 4'm
Pm a )
(n-T.i)
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Here the sample dielectric constant at the reference condition needs to be known,
b ut it is only necessary to use two calibration standards. If the sample has low
dielectric constant with 'e ~ e and air is used as a calibration w ith eA=
S e = S e = (eref- 1) , ^ m a ~ P m b ) ( p m _~ P m i * )
v Pm ' P m b ) ( ^Mref' P m a )
1
(n.7.2)
The change in sample dielectric constant from eref by varying sample condition can
be obtained directly from the reflection coefficients of two calibration standards
Pm
ai
Pmb> the sample /?Mref at the reference condition and
pM
under current
condition, of interest.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. II. 1.
Measurement configuration w ith a network analyzer, SMA connectors
and a semi-rigid coaxial probe dipped in the liquid.
Co
Aeasurement
P l a n e oF
S a m p 1e
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-38-
Fig. II.2.
(a) Equivalent circuit of the measurement configuration. Z(u,e) represents
the impedance of the coax-liquid interface, and the remaining spurious impedances
are represented by the T-circuit with impedances Zv Z2 and Z3.
(b) The equivalent circuit for Z(u/,f) w ith complex fringe-field capacitances
approxim ation for the coax-liquid interface.
(c) The resultant equivalent circuit with p art of the fringe-field capacitance
incorporated into the T-circuit.
(a )
1
Z(e.u)
g (€ , u :
jB(e
■I
3
— I— I
CU.to)
m
( c )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-39-
Fig. II.3. Equivalent circuit for the measurement configuration.
(a) The T-network representation used in this work. The impedances Z; (i
=1,2,3) include line discontinuities, such as connector impedances and also the
probe capacitance. Zm is the impedance seen by the ANA at its measurem ent
plane. Z is the impedance of the probe-liquid interface. All elements of the figure
are complex quantities.
(b) Elements of the probe-liquid impedance, including radiation corrections.
The elem ent eC enclosed in the box represents a simple capacitance approxim ation
used in earlier work, while the remaining elem ents describe the radiation
correction.
Z,
Z3
- o
ztew) 11
z,
z3
— IZ D -
—
1
l
jB(eto)
G(S«»
(a )
1
JB
eCfcoi
1
0
I
,
j(B -ec)
1
G
—
—
'ec(co)
(
J
b )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-40-
Fig. II.4.
D ata for e' and e" for w ater between 45 MHz and 20 GHz, illustrating
the effects of the present de-embedding analysis (o’s) and w ithout the de­
embedding analysis but with only the machine calibration (x’s). The de-embedding
analysis removes the effect of spurious mismatches which appear as oscillations.
100
80 jjfc
^ ooooooooo 00n
-j
& *%
°°°Oo0o
; ^ +
°°°oo0o
:
1_
+ -fF +
°OOn„
60 — V
+ +
°°°oOQ
—
+ ++ ++4f++
°0o°Ooo0o
:
*+
°°°oon
:
+ + \+ +
Ooo°ooo0
V+ Y ++ +
°0O0°oooOQ^
40 +< v
+ + # -+++ +
20
+J. V ++ ++ +
++ + +< \t + + +
+++
+ «++- + ^
•H- + . +
+ +++++ • ■
:++H-
o
0
5
10
15
20
0
5
10
15
20
100
80
60
e"
40
20
0
FREQUENCY(GHZ)
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-41-
Fig. II.5 Frequency dependence of e' and e" for m ethanol taken w ith a thick 0.141
in. probe (x’s) and thin 0.085 in. probe ( + ’s). The solid line represents the Debye
form for m ethanol.
50
40
e
i
30
20
0
5
10
15
20
0
5
10
15
20
20
€
u
FREQUENCY(GHZ)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-42-
Fig. II .6 Results of e' and e" for de-ionized water taken with a thick 0.141 in. probe
( + ’s) and thin 0.085 in. probe (o’s) without the radiation correction. The difference
between the two probes is due to radiation effects which are minimized in the
thinner probe. The solid line represents the Debye form for water.
100
— ii— i—
i— i— i— i— i— i—
i— i— i— i—
i— i—
i ii— i—
i |—
j—i—i—i—i—j—i
— i—
i—i—i—|—i
— i—
i— i—r
80
<a)
60
^
+++ R8 Sgo0l
e'
40
X
20
x* < ° oa^
X
X x x ,
++++++++++-
— i—i
— i---- 1I---—i1-----1i___I---i 1i-----1I___I___I___
i i I___
i Ii
Q 64—i
----1-----1
---- 1-----1
0
5
10
I1
i i1___I___I___
i i
I___
15
20
100
80 _
60
~
40r
-
20
,
“I
$<
^ x><xxxxxxxx>;.
^ +++++++^
+++++^ x xS S* * ^*i ¥« >
xxxxxxxx;
++^ O O C O O O O = C O O ^ ¥ ? O C ^
. ■ xxXSoO1
-
-
--- 1i--- 1i--- 1i--- 1i__ Ii__ Ii__ Ii Ii Ii Ii
0 1.8 Iv .. Ii— 1i__
0
5
10
FREQUENCY (GHZ)
Ii
Ii Ii
15
Ii
Ii
Ii
I
20
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-43-
Results of e' and e" for water between 45 MHz and 20 GHz.
Fig. II.7.
o’s :
radiation corrected results for 0.047 in probe, -fs : uncorrected results for 0.047 in
probe. The solid line represents a Debye fit to the corrected 0.047 in. results.
100
80
60
e
/
40
20
0
10
5
15
20
60
50
40
. + + + + + + + + + + + + + + 'fc £ '
.++■
20
0
5
10
15
20
FREQUENCY(GHZ)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-44-
Fig. II. 8 . (a) Configuration for Deschamp’s theorem, (b) A ctual configuration used
in this work.
( b)
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-45-
Fig. II.9. Flow diagram for the numerical algorithm used to extract dielectric
coefficients e from m easured reflection coefficients pM.
Cal. S td s .
Cal. S td s .
C alcu lated
C a lcu la ted
S o lv e d
Sampli
A 's
S o lv e d
Sample £
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-46-
Fig. 11.10. Results for water illustrating probe size and radiation correction effects.
0.047 in. flanged probe with radiation correction ( o’s ), Debye fit ( e0=78.2, eoo=
5.1, r = 7.9 ps) to the 0.047 probe data (solid line), 0.141 in. probe w ith radiation
correction ( + ’s ) and 0.141 in. probe without the correction ( solid squares ).
100
80
e
i
60
+ ■^
+ a
40
20
10
50
40
€
it
30
20
Frequency (Hz)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-47-
Fig. 11.11. Calculated magnitudes of the radiation correction, using a 0.047 in.
probe, for hypothetical liquids with e = e' =
(o’s),
10
20
(squares), 40 ( + ’s), 80
(diamonds), constant upto 30 GHz.
- 1
oO .+++ r CQana^
2
U
'H o
O
O + r_n<
o , + cfP.
O + □ n<
-2
- o + □ o'n°
-3
- 4
-o ^ o
0
7 ^ 0
jOO°
(a)
"+o
leu -i i j- I i i i-i . I i i i i I i i i i 1 i i i i i i i i i
10
15
20
25
30
|aDQnnco o 5
UJ
2
U
O
'S o
o
—2
x++ _rflDnDU. ^ o o
—
d^,
— o o+.+ □DIof
: o+! o D0 o °
r ^ o
- +no
32D°
I+O
(b)
-=
1
5
10
15
20
25
30
FREQUENCY(GHZ)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-48-
Fig. 11.12. Results of e' and e" for plexiglass between 45 MHz and 20 GHz.
5
4
3
+
++++++++■•
2
1
0
0
5
10
15
20
1.0
0.8
e
it
0.6
0 .4
+++++++++++:
0.2
++++
0.0
0
5
10
15
20
Frequency (GHz)
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CHAPTER III
DIELECTRIC SPECTRA AND MICROSCOPIC PROCESSES
A standard way of analyzing complex dielectric spectra is to utilize
graphical methods. A widely used graphical representation of frequency-dependent
complex dielectric functions e(u>) = e'(u>) - ie"(u;) for various m aterials is the wellknown Cole-Cole plot, in which e", the imaginary part of the dielectric function, is
plotted on the vertical axis against e', the real part. The Cole-Cole plot is
particularly useful for m aterials which possess one or more well separated
relaxation processes with comparable magnitudes and obeying the Debye or ColeCole forms. However, when the m aterial also possesses a conductivity, the ColeCole representation becomes less useful, because the presence of a dc conductivity
leads to a divergence at low frequencies of e". A Cole-Cole plot of the dielectric
spectrum of conducting medium can not provide any information on the relaxation
processes unless the contribution of the “dc” conductivity is subtracted out
accurately. For liquids such as alcohols, the existence of known higher frequency
relaxation processes cannot be visualized graphically from the Cole-Cole plot, which
would create ambiguities in determining the microscopic processes of the medium.
A new graphic dielectric d ata representation, which is specially useful to
conducting systems and is sensitive to high frequency features, which was
discovered during the course of this work, is presented. A comparison w ith ColeCole and modulus plots is made. Also, various empirical functional forms used to
describe dielectric spectra are reviewed and their relation to microscopic processes
are examined.
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III. 1 Debye Relaxation
III. 1.1 Debve model
Consider a dipolar molecule in a uniform fluid and the rem aining part of the
fluid is a medium applying a rotational friction torque to the dipole (Debye, 1928).
W hen a small electric field varying w ith tim e is applied to the liquid along the zdirection, the dipole m om ent distribution function f(r) deviates from the therm al
equilibrium distribution in the absence of external fields. Taking account of the
rotations produced by the Brownian movement and by the inner-fictional-forcebalanced torque M, the distribution function, which is only 8 dependent, satisfies^ ^
< l = s b ^ i sin#<k T | -
M f )]
<r a -u >
where £ is a constant measuring the inner friction and M is the torque which tends
to tu rn the molecule with dipole m om ent fi in the direction of the local field E
M = ~ n E sin#
(III.1.2)
If t <0 E = E 0 and the field is switched off at t = 0, the relaxation tim e of the
polar liquid can be obtained,
r D = C/(2KT)
(III.1.3)
For a hard sphere of radius a rotating in the liquid of inner friction constant 77,
C=
8 7T 77 a 3
(III. 1.4)
Then
td
=
(III.1.5)
If the external field varies w ith frequency u /2 tt, then the local field E can be
expressed by
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E = E 0 eiwt,
(III. 1.6)
ALEo eiu>t CQS0 j
f= A [1+
1
(III.1.7)
+ 1 ZkT
is a solution of Eqn. (III.1.1). The mean dipole m oment of the molecules is, also
complex
„2
p „iL>t
a = m -r fr z K
1 2kT
<I I U -8>
Assuming the force acting on a polax molecule is the resultant force due to
the external field and the polarization force from outside a sphere surrounding the
molecule,we can relate m to the dielectric constant of the
e - 1 M w _ 4 7r N /
, J?_
e -2 P ~
3
^ 0 + 3k T ,
1
1
.
+
1
liquid and find
\
)
/In i ox
(111.1.9)
2kT
where Mw is the molecular weight, p is the density of the liquid, N is the num ber
of dipoles per unit volume and a 0 is the distortional polarizability. It is useful to
use e0 and
where e0 and
as defined by
denote the value of e(w) for high frequency and for a static field.
Hence, we obtain Debye's expression for the dielectric constant as a function of u>,
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e = eoo+ 160+ i o)rD = e' - ie"
(III.1.12)
It has been known for at least fifty years th at Eqn. (III. 1.12) gives a good
description of relaxation processes in single particle, non-interacting systems such
as gases and, surprisingly, water. The inverse Laplace transform ation of Eqn.
(III. 1.9) gives a polarization decay function for x ( u ) = e(w) - eoo
tf(t) = Ae “ t/r D = Ae ~ " p 4
(III. 1.13)
in which A is a constant. The simplest description of the dielectrically active entity
of Eqn. ( I I I .l.ll) is th at of a dipole which can reorient under the action of an
electric field between two m etastable positions. This situation can be represented
by the double minim a potential shown in Fig. III.l with Ec =0. Relaxation occurs
by the realignment of the dipole and can be represented by a particle moving from
one m inimum to the other. As the simplest reorientation process we can consider
therm al excitation, represented by the process (a) in Fig. III.l, over the potential
barrier by means of absorption of a suitable phonon. Thermal relaxation by the
emission of a similar phonon follows. This is a single particle process and the
structure around the dipole remains in its original form w ithout change or is
m om entarily elastically distorted and immediately returns to its original form. As
described the process is pure Debye in concept and gives a time development of the
relaxation as indicated in Eqn. (III.l.13). For this type of purely activated process
we expect,
wp = u>0 e
_A
kT
(III. 1.14)
III.l.2 F itting data to the Debve form
If we include the contribution from a dc conductivity cr0 of a conducting
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m aterial, Eqn. (III. 1.12) can be generalized as
= e~ + i V
6
iS -
- ^
(IILL15)
with
e' = e “ + i f r S F
n _ ( e0 ~ eoo) w‘r
“ 1+ («
f
,
(n m # )
<*o
,m i
+ " »v
1
7.
(III.l.17)
where ev = 8.85 x 10"12 F /m . In general, since Eqn. (III.1.15) is a nonlinear, m ulti­
param eter and complex function, a nonlinear fitting routine is usually required to
extract e0, e^, r and a0 from fitting experim ental data. Here methods of fitting
d ata to Eqn. (III.l.15) based on principles of the least square fit to a straight line
are presented.
(i) Debye form cr0 = 0.
From Eqns. (III.l.16) and (III.1.17) for e' and e" we can eliminate the
variable u>rD,
( g ' . C0 ~beoo ^2 _j_ £//2 _ ( e0
eoo )
(III. 1.18)
which represents a semi-circle with radius €° g6” , centered at (e° ^ €o°, o) in e"-e'
complex plane for physical d ata e">0. By rearranging the above equation, we find
y = e' 2 + e" 2 = ( e0 +
) e' -
(III. 1.19)
= a e' + b
which shows a linear dependence of e'2 + e " 2 on e' with slope a = e0+ eoo and yinterception b
=
- Coe^, from which the fitting param eter e0
and can be
determ ined as
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eo — I ( a "t" >J a 2 + 4 b )
(III.l.20)
€«, = ! ( a - aJ a 2 + 4 b )
(III.l.21)
Then r can be obtained from combining Eqn. (III.l.16) and (III.l.17) w ith cr0 = 0,
II
(eQ~ ^oo) ljJT
~ 1 + ( W7 ) 2
__1_
( eo - eo o )
^
_
1
[ ! + (^ r )2 - l ]
+ ( wr
)2
( e0 - e' )
—
(jJT
T=
(IIL1-22>
and averaging over all the data points. For those circumstances where d a ta points
at the low frequency end are not as well defined as those at high frequencies, the
fitting program changes the initial fitting points to achieve an accuracy of
6t
<
0 .5
ps or Of- < 5%.
An alternative way (not used in the program) is to plot cr vs. e' first, since
II
( C0
~ ^oo) ( ^ T )
= - T 1 e' + ^ e0 = ae' + b
2
(III.1.23)
we have
r
ev/ a
e0 = - b / a
Then
(III.l.24)
(III. 1.25)
can be found from the average of
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e" ( 1 + w2 r 2 )
^oo = Co - — - J t
1•
(III.1.26)
(ii) Debye fit w ith finite dc conductivity cr0
According to Eqn. (III.l.16) and (III.l.17), we have
v
, .// _ £v (Cq - Cqo) ( UT ) 2
~ T 1 + ( wr f
+
= -
t
£,+ (
t
0
£o + (Jo )
= a-^' + bj
(III.l.27)
r , <t0 and e0 are related to ax and bj through
r = - e j ax
(III.l.28)
cr0 = bj + e0 aj
(III. 1.29)
Substituting the extracted r value from the slope ax into Eqn. (III. 1.16), we have
C = eOO + (
1 “p
2 2 ) ( e0 ■ eOO ) = a2 ( T^r
u/ T
and plot e' vs. ----- ^ ----r^.e0 and e— can be
1 +
(
LOT
2 2 )
^-*2(^-1 -30)
found from
Y
e0 = a 2 + b 2
(III.1.31)
eoo = b 2
(III. 1.32)
Plugging the value of e0 back to Eqn. (III. 1.29),the dc conductivity of the m aterial
can be determined.
A possible alternative way is as follows.
( £' -
? + ( «"
- <?& ) 2
= ( *0 • *oo
Y/i
(III.1.33)
Expanding the above Eqn., we get
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e'2 + e//2 - ( e0 +
) e' + cr0 (
Through iteration, we are able to extract e0,
v
-
—2
—f ) - e0 eoo
u) e„
(HI. 1.34)
and cr0. Then r can be calculated
from Eqn. (III. 1.2) by using
(ni.i.35)
in place of e", which represents the contribution to the imaginary of the dielectric
constant only from the dipolar reorientation,
T=
*oo )
u eD
(III.1.36)
m.2 Non-Debye Processes
III.2.1 Empirical functions for e(uA
It has been known, however, th a t the Debye model is only a first order
approxim ation to the dielectric behavior of materials. In particular, although the
Debye behavior can be approached, it is never seen in solid m aterials. A num ber of
modifications, usually empirical, have been made to the Debye form and in specific
cases it has been shown th a t these fit the experimental observations of particular
m aterials.
Dielectric spectra of many m aterials give depressed circular arcs in the e"- e'
plane, which deviate from the Debye semi-circle. The behavior can be described by
the Cole-Cole function (Cole, 1989)
£ = e' - i£" = £~ + r
r
f
i ^
< Iim )
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where 0 < a < 1. The expressions for the real and imaginary parts of e(u) are
e' =
e" =
. 11 + < wr I11' " ’ sin < T a / 2) 1 (
1 + 2 ( u r )1'° ; sin ( 7ra / 2 ) + ( o j t )2'- a '
(
1+ 2
(HI.2.2)
)(1~Q) COS ( 7TQ!/2) ( €0 - Cqq)
UT
{ ojt
(III.2.3)
)1' a sin ( 7ra / 2 ) + ( lot )2^ ' Q!^
The equation relating e' and e"
(£, _
y + [£„+
gives a circle of radius r =
2
tg
(to /2 )]2
cos(7r a / 2 ) cen*ere(l
= J
j
s
( m .2 .4)
(6° '2 °°' ~ e~ ~2 °° tgi r ) iQ
complex plane of e. One of the earlier clear-cut examples of Cole-Cole function was
observed in the partially ordered crystal HBr (Cole, 1989). The slower and betterdefined absorption peak at tem peratures lower than the order-disorder transition at
89K, was a beautiful depressed circular arc in the complex plane plot e" vs. t'
which is the characteristic of Eqn. (III.2.4).
The Cole-Davison form
e — e' - ie" =
+ - e° '
R
(1 + iu >t ) P
(III.2.5)
where 0 < f3 < 1 , originated from the measurements on glycerol, which was a
skewed arc in e"-e' plane. Our results, over a wider frequency range, when plotted
in the Cole-Cole plot shown in Fig. III.2a, also seem to be a skewed arc. F itting to
the Cole-Davidson function gives
/?=0.66. However, if the d ata were plotted as
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u>6v(e/ - e 0O)
against a = ueve" in Fig. III.2b, which willbe illustrated in the
following section, the
skewed arc appears m ost likely due to the
existence of
another relaxation process which can not be seen by plotting e" vs. e'.
More recently, Havriliak and Negami (1966) found th a t none of the above
dielectric functions was successful in giving the spectral response they had
measured in a number of polymetric materials. They proposed th a t a bi-param etric
function
6
= 6' - ie" = Coo + ------ ^ J oq
o
[ 1 + ( iOJT )1-°!] P
in which 0 < a <
1
(III.2.6)
and 0 < (3 < 1 , would be of more general applicability.
The major differences take place in the absorption shape, not in the location
of th e absorption peak, between Cole-Cole, Cole-Davidson and Havriliak-Negami
functions, as shown in Fig. III.3. C-C form is more symmetric than C-D form, and
H-N form has wider spread.
III.2.2 KW W decay function
The extensive work of W illiams' group found a form for the polarization
decay function in the expression (I.I.2), known as the Kohlrausch-W illiams-W atts
(KW W ) or “stretched exponential”
$ (t) = A e ' W
w ith
0
? >7
(III.2.7)
< 7 < 1 . The above polarization decay function has been derived from a
random walk model. The model supposed th at an electric field had been applied for
some tim e to a medium containing many polar molecules (or polar groups in
complex molecules) and the direction of their dipole moments remain frozen as the
field is removed. A further assumption is th a t the medium contains mobile defects
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th a t on reaching the site of a frozen dipole relax the m edium to the degree th a t the
dipole m ay reorient itself if the diffusion of defects tow ard dipoles is executed as a
continuous-time random walk composed of an alternation of steps and pauses and
the pause-tim e distribution function has a long tail of th e form </>(t) = t ^1
Then
the relaxation function has the stretched exponential form.
Theoretical interpretation of the widespread occurrence of characteristic
patterns of behavior has lagged far behind the accumulation of experimental
evidence and empirical descriptions. Although no solution for corresponding e(u;) in
term s of known functions exists, extensive numerical evaluations have been made
and used to show th at the calculated frequency dependence corresponding to the
KW W decay function is very similar to th at of the Cole-Davidson form, except it is
less asym m etric in the e" absorption line shape for values of fi and
7
required to
produce good average agreement.
III.2.3. Fitting d a ta to the Cole-Cole form
In the case of a broad absorption w idth relative to the measurement
frequency range, the shape features are not distinct. The Cole-Cole function is a
good description of any non-Debye behavior.
(i) Cole - Cole form <r0 = 0.
First, let
Ai =
+
£00
A 2 = - («o - too) tg (W 2 )
(III.2.8)
(III.2.9)
A3 = - c0 Coo
(III.2.10)
y = e'2 + e" 2
(III.2.11)
Eqn. (III.2.4) becomes
y=
+ A 2 e" + A 3
(III. 2 . 1 2 )
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A ; ( i = l , 2 , 3 ) can be found from solving following equations,
Ax Ee/ 2 + A 2 Ee'je"; + A 3 Ee'; = E e '^
(III.2.13)
Aj Ee'je"; + A 2 Ee/ ' 2 + A3 Ee;" = Ee/'y;
(III.2.14)
Aa Ee/ 2 + A 2 Ee"i + A 3 N = E yi
(III.2.15)
where N is th e num ber of d ata points. Solving Ai(i= l,2 ,3 ) from above equations,
and according to the definitions in Eqn. (III.2.8) to Eqn. (III.2.11), we obtain
e0 = \ { Aj + \j A j 2 + 4 A3 )
(III.2.16)
eoo = i ( A, - >1 V
+ 4 A3 )
(III.2.17)
a=
0
- | t a n - 1 [A 2/ ( e
- £oo ) ]
(III.2.18)
To find the value of r , we rearrange Eqn. (III.2.1) to the form
, 1• WT
( e - 1 e)
(N(i-a)
> = Teo
-, --V
(III.2.19)
( e' - i e") - e0
M ultiplying the above equation by its complex conjugate
(III.2.20)
( i U T r (1' a ) = e° ' ^ e' + 1 ^
1
j
( 6 , + i O - e*
we have
( wr )2 ( ^
t
= eoo T e ( + e
■ n oo
L
2
-2i )
can be solved from the above equation
1
,
2
i
t /2
,
;/2
- = A ( C
f qq, I i c , 2 "r
! cU
J
' ZC
I qqC-
—I--)2 ,,' a )
<I I L 2 -2 2 >
There is another way of obtaining r. Take the ratio of e' - e ^ and e", we
will have
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£
6 or>
. 1 ((TITMOL\\ j ___________ i __________
£// °° —
- t +aT
a a ( 2 j + ( y r ) ( ^ Cog ^
(III.2.23)
Thus,
r ° A | - s-( ¥ )
(III.2.24)
(ii) Cole-Cole form with dc conductivity a 0.
For
situations
where
the
materials
have
a
frequency-independent
conductivity <x0,
e = e' - ie" = Coo + —
- i^ g -
(111-2.25)
e" diverges at low frequencies. In order to observe meaningful features in a
dielectric spectrum , we need to know the value of the dc conductivity and replace
the y-axis value e", with (e" -
By extracting the dc conductivity
contribution to e"
(e' -
)» +
.
2 ° .)
+
tg ( m / 2 ) ] 2 = j
(IIL 2'26)
gives a depressed arc when a > 0 and a semi-circle with a — 0. If we define
y = e' 2 + ( e " -
2
(III.2.27)
Eqn. (III.2.25) can be w ritten into the same form as Eqn. (III.2.12) with A;
(i= l,2,3) defined by Eqn. (III.2.8) - Eqn. (III.2.11),
y = Axe' + A 2 e" + A3
(III.2.28)
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First, the Debye fit with conductivity is used to find an approxim ate cr0, then this
trial value is used for the value of cr0 in Eqn. (III.2.26) to find a set of A; and r
following the procedure described above. The mean square root deviation of the fit
relative to the d ata can be then calculated. The optim um aQ is found by varying
the trial crt0 when the minimum deviation is achieved.
The Cole-Cole relaxation tim e r c relaxation can be calculated by replacing
e" in Eqn. (III.2.22) by e" - £ § c 2 x c'2 i
T = l { °- + . /
(
a0 \2
J
1
° ■] ^
(III.2.29)
III.2.4. General fit to a complex dielectric functions
The above fitting routines, for Debye and Cole-Cole functions w ith or
w ithout a dc conductivity, are simple and fast. Except for the case of fitting ColeCole form with a dc conductivity, no initial trial values are required for extracting
the dielectric param eters. For more complicated cases, such as HN form or
dielectric functions w ith m ulti-relaxation processes, a general non-linear fitting
program is more powerful. By inputting initial trial values of param eters in a
dielectric function, a FORTRAN program is set up for fitting dielectric spectra to
various kinds of complex dielectric functions based on approach of G rant et al.
(1978).
m.3. Power law behavior
Hill and Jonscher (1983) developed an approach to relaxation processes that
involves cooperative interactions in solids. In general the structure surrounding the
dipole is not completely rigid which is the more general case th an th a t is discussed
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in III. 1.1, and local structural distortion can allow realignment of the dipole. Such a
process is known as configurational quantum mechanical tunneling (Dissado and
Hill 1979, 1980) and the small energies involved in the configurational adjustm ents
are stored in the local structure surrounding the dipole, forming a narrow band of
configurational states whose width is designated to be 2EC as shown in Fig. III.l. In
th e case where there is a logarithmically divergent number of excitations of small
energy, which gives a tim e development of a cooperative response in second-order
perturbation theory, the response function can be w ritten as
f2 (t) = exp [ - n
(7
-fin (iEct) + E x (iEct)]
in which E x is an exponential integral,
7
(III.3.1)
is Euler’s constant and 0 < n < 1. The
function contained in the above expression is of Gaussian form for small values of
Ect, oscillates in the region of Ect « 1 and for Ect greater than about 10 the
logarithmic term dominates to give
f2 (t) * (Ect ) - n
(III.3.2)
There is another process needed to be considered which is of the same
nature but it does not generate a relaxation current. It can be best considered as a
local fluctuation in the medium. Essentially two configurational tunneling processes
occur simultaneously but the dipolar movements involved are in opposing
directions. Microscopically dipoles reorient but macroscopically there is no net
charge displacement and hence no change in polarization. The fluctuation takes
place w ithin its own band of configurational states 2EC' (say) and, similar to
equation (III.3.1), the complete tim e dependence of the fluctuation can now be
obtained in the form
f3(ti) = exp { - m [ In (iEe't,j + In (iEc'| t - 1,|)
- E ,(iE c't,) + B ,(iEc'| t - t i ! ) ] }
(III.3.3)
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in which
0
< m < l, t x is considered to be a random variable reflecting the inability
to define the tim e at which the configurational exchange takes place in the interval
tj= 0 to the tim e at which the system is observed at t 1 = t. In the particular case
for which | t - t a| > (Ec' ) _ 1 , Eqn. (III.3.3) simplifies to th e form
f 3 (t) « t “ m ( t - t ! ) 1"
(III.3.4)
The two current forming processes, one is described in (III. 1.1) and the other as
proposed in this section, are competitive and it is necessary to average the product
of these two in term s of the fluctuation to obtain the measured relaxation current
of the interactive system. Then the frequency dependent susceptibility can be
obtained by Laplace transform ation as
X M = Xo(0) F(m,n,w/u;p)
(III.3.5)
w ith
F(m ,n,x) = (l+ ix )n “ ^ F ^ l - n, 1 - m;
2
- n; j - ^ )
(III.3.6)
where x is the normalized frequency u;/wp and 2 Fj is the Gaussian hypergeometric
function. The asym ptotic behavior of the susceptibility x (w)> the same for e(w) as
shown in Eqn. (1.2.5), at high and low frequencies, w ith respect to u>p, as
u ;» wp x " ^ ) = X;(w) cot ^
k
w, n"
(
I
I
I
.
3
.
7
)
and
w
x"M
« x(0) - x 'M
oc wm
(III.3.8)
It can be seen th a t the above equations expresses the limiting form of spectral form
of spectral shape behavior contained in the dielectric functions discussed above, as
shown in table III.l and hence are capable of describing the general situation. As
shown by Hill and Jonscher (1983), the agreement between this function and th e
experim entally observed response in solids is excellent.
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The dynamic properties of the dipolar system are contained in the function
F(m ,n,o;/wp) and it has been shown th a t the entire range of observed dielectric
behavior in the sub-quantum and sub-phonon range of frequencies can be described
w ithout the necessity for arbitrary distributions of param eters.
Table III. 1 Spectral functions and their power law exponents (Hill and Jonscher,
1983).
exponent u < u>p exponent u >• u p
Process
Susceptibility function Ay'(u;)
Debye
(l-fiu;/u;p)
Cole-Cole
[l-t-(W u;p)(1_Q!)] - i 1
1
Cole-Davidsor[l-f(iw/a;p) ]~@
X »
2 .0
1 .0
1 - 0
1
2 .0
1 .0
- a
X »
-
2 .0
-
1 .0
0 -1
a
-1
~P
-P
--------- --------- ^
[1
~ a ]“
1 - 0
1 - 0
i—
*
l
-Negami
l
Havriliak
-P {\-a )
r a .4 A new graphical dielectric d a ta representation
III.4.1 Single characteristic relaxation processes
(i) Semi-circle for single Debye process w ith dc conductivity
Examples of such liquids are dilute to m oderately concentrated solutions of
the alkali halides in water. The functional form of e(ui) is ( MKSA unit system):
= e- + T t S ? - 1 ^
(IIL4‘9)
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and hence the conductivity is
a(w) = i u e v e(u>)
(III.4.10)
and
ri
_
,.,c
^v (^0
^oo)
^v (^0
eoo)
1
and
a " - weve0O= u;ev(e' - e ^ ) = 6^ e°r £°°)
i+ ^
2
.
(III.4.12)
Define
£ _ Cv(^0 COo)
° —
r
,,
°o o
_
—
_
o
(111.4.13)
i e v (e 0 — eoo) _
*
_
i c
(111.4.14)
r ---------- — <^0 i ®
<rx ' = a '
(III.4.15)
crx " = wev(e' - Cqq)
(IH.4.16)
Eqn. (III.4.3) and (III.4.4) can be w ritten into forms similar to the Debye
expressions of e' and e"
a X ~ a oo “ i - ( . ^ V
<
= r f w
«
- O'oo) 2 + <TX"
(III.4.17)
(IIIA18)
Taking
2
= Y
= - S (o j -
(III.4.19)
and rearranging the above equation, it becomes
Wx ~ (a oo ~ <5/2)
]2
+ <rx " 2 = (S/2)2
(III.4.20)
which is the equation for a circle centered at (cr0 0 -<5/2, 0) w ith radius S/2. W hen
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crx" = 0 , there axe two x-interceptions at
xi = a oo ~ S/2 ~ S/2 = cr0
x 2 = a oo
(III.4.21)
S/2 4- 8/2 = Cqq
-y(€ o ~ e°°\
= a0 +
(III.4.22)
and
r = | = €y ^ -° r €oo)
(III.4.23)
The above expressions suggest th at plotting wev(e' - e ^ ) vs. a' gives a semi­
circle for a conducting Debye liquid, which is similar to the dielectric spectrum of a
pure Debye liquid w ithout dc conductivity, presented in the Cole-Cole plot. The
low frequency x-intercept gives the solution dc conductivity aQ and the radius of
the semi-circle is proportional to the magnitude of the susceptibility and inversely
proportional to the relaxation time. This is indeed the case, as shown in Fig. III.4,
in which the d ata for several solutions of of CsC1/H20 are shown, along with the
theoretical form Eqn. (III.4.12). The improvement over a Cole-Cole plot, shown in
the inset of Fig. III.4 is apparent.
(ii) Characteristic slope for non-Debye dielectric forms
The expressions for the real and imaginary parts of e(u;) for Cole-Cole
function (Cole, 1989)
6
where
0
< a <
1,
= e '- ie" = 6 ^ + Y ^
'-r^ y - a
(111-4.16)
are
£' = £„ +
I1+ (^
)(‘-a)
(
)l' a sin (
1 4 -2
u t
( W 2) 1(
ira /2
) 4- (
u it
- <oo)
(m A 1 7 )
) 2^
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(
e" =
1
+
2
lo t
)(1-Q:) cos ( ira/2) ( e0 - e ^ )
( wr )1' Q; sin ( r a / 2 ) + ( wr
The equation relating e' and e"
(«- _ ^
)*+ (e»+
gives a circle of radius r =
50^
2 ~^os(7ro / 2 )
, tg (ra/2 )]2 = 4
;
(hi.4.19)
centere(* at (C° 'g6” , - -° p°° t a n ^ ) in the
Cole-Cole plot. In th e cr^" - cp ' plot, the curve is not so regular since u>(e' - e^) is
not zero at high frequencies. However, the curve approaches an asym ptotic slope
tan (^ p )
when the frequency is significantly higher th an th e characteristic
relaxation frequency of the spectrum . The value can be estim ated from
f c x > u ( e' - e°o) = sin( x ) (eo “ e°o) u<*
f e o o w e” = cos( ^
(III.4.20)
+ a°
“ e°°)
(III.4.21)
where ratio is ta n (^ p ). Therefore, the angle against cr^ - axisconstructed by the
asym ptotic
line is
Curves of different a value 0, 1/10 and
1/6 w ith e0 =55,
eoo= 5, t = 8.85ps and <jo=10 (fim ) ' 1 are plotted in Fig. III.5.
If the dielectric spectrum appears to be of the Cole-Davidson form
e = e' - ie" = e ^ +
- ° ' £°°
(1 + iu;r)
(III.4.22)
P
or Havriliak-Negami bi-param etric function
e = e' - ie" = eoo + ---------------[ 1 + ( iw r ) 1'a ]
a
(III.4.23)
P
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in which 0 < a < 1 and 0 < j3 < 1, or Hill-Jonscher's power law description
e(w) = eoo + (co-€oo) F(m ,n,w r)
with
F(m,n,wT) = (l+ iu rr)n “
- n, 1 - m; 2 - n; y q ^ j :)
(IH.4.24)
where 2 Fj is the Gaussian hypergeometric function, the asym ptotic behavior for e'
and e" are u ~
^ _ A 1 ~ a ) an(j wn-i respectively. Therefore the curve in cr^" - a x
plane will also approach a straight line at frequencies m uch higher than the
characteristic frequency u c = ^ of the material.
III.4.2 Enhancem ent in higher frequency feature for m ulti-relaxation spectra
Another interesting feature about this graphic representation is th at it can
enhance the characteristics of Debye-like processes of shorter relaxation times
probed by the dielectric measurements. From the expression of the radius, Eqn.
(III.4.15), we see th at if the amplitudes of the susceptibility y 10 = eio_ e ioo ~ X20
= e20 -
6200 )
the shorter the relaxation tim e is, the bigger the radius becomes. This
can be also understood from the point th a t what are plotted are the dielectric
values m ultiplied by frequency. The higher the frequency is, the bigger the
enlargement on ( e '- e ^ ) and t" is produced. As a consequence, the possible higher
frequency features suppressed in the Cole-Cole plot due to a small am plitude,
would become obvious in a a x "- cry,' plot.
It has been long speculated (Minami, et ah, 1980) th at two relaxations gave
a b etter fit to the dielectric spectrum of 1 -propanol than single Debye, Cole-Cole or
Cole-Davidson fits. But as shown in the Cole-Cole plot of our broad band
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dielectric d ata and fits in Fig. 111.6(a), one can not easily distinguish a Cole-Cole fit
from a two Debye relaxation tim e fit and there is no apparent signature of the
existence of a shorter relaxation process. There have been argum ents which imply
th a t one can extract an arbitrary num ber of relaxation processes in certain
situations, depending on the function form chosen to fit a dielectric spectrum.
However, the curve for 1-propanol d ata in a x - crx' plot shown in Fig. 111.6(b)
shows the clear evidence of the existence of only one other relaxation process in the
solution in the frequency range.
As a comparison between the shorter relaxation process and its Debye fit
through the a x '- a x plot, we found th at the m ism atch is big. As can be seen
clearly from Fig. 111.6(b), the shorter relaxation process is better described by a
Cole form w ith a=0.13 (solid line) than two Debye relaxations (dashed line).
For pure glycerol at room tem perature, it seems to be a skewed arc in the
Cole-Cole plot as already shown in Fig. 111.2(a), which was a typical example of
the Cole-Davidson form (Cole, 1989) Our data can be fitted to both C-D dispersion
w ith /?=0.66 and a two Debye relaxation form w ith same order of magnitudes in
standard deviation. There is no evidence of differences in Cole-Cole plot between
two type of fits. W hich relaxation model represents the real dipolar dynamics
better? Our a x -c rx
plot in Fig. III.7 shows th at there are at least two distinct
relaxation processes in pure glycerol which is a drastically difference in dynamics
from the picture of a single distribution of relaxation tim e given by the ColeDavidson fit. Detailed work on the aqueous solutions of glycerol and 1-propanol will
be described in Chapter VII. Here the study on LiCl/l-propanol solutions is
presented.
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III.4.3 Dielectric relaxation in LiCl/l-propanol solutions
The dielectric spectra of LiCl/ 1-Propanol solutions, with concentrations of 0,
0.5 and 1.1 mol%, were studied, and the result for 0.5 mol% solution is presented in
Fig. III.8. Also plotted are the fits to the combination of two Cole-Cole
*(") = 'oo + ■ . , Al ...a . - ■ , , A; vi-a2 - i
1 + ( lwrj
1
1 +
( i w r2)
2
Stvv
(111.4.25)
F itting param eters for two Debye and two Cole-Cole combinations for 1-propanol
and L iC l/l-propanol are listed in Table I. The static dielectric constant is
eo = eoo + A l + A 2.
Table III.I. Two Debye (2DB) and two Cole-Cole (2CC, 6^=2.7) fits for
L iC l/l-propanol solutions.
c(mol%) F it Type
0
0.5
1.1
e0
Aj
'''i(ps)
<*i
a2
T2(PS)
2DB
20.9
16.9
329
0
1.3
8.8
0
0
2CC
20.9
16.9
332
0
1.3
9.7
0.13
0
2DB
19.6
15.3
333
0
1.3
13.5
0
0.042
2CC
19.6
15.6
329
0.02
1.3
10.3
0.17
0.042
2DB
18.4
13.9
373
0
1.4
20.4
0
0.064
2CC
18.4
14.2
365
0.03
1.5
12.2
0.20
0.064
«2
CTo(j4l)
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III.4.4 Discussion
(i) The effects of
on the applicability of the &X ~ a X
It will be noted th at, similar to the role th at <r0 plays for a meaningful ColeCole
curve,
the
proposed
graphical
measurements requires a param eter
representation
for
d ata
of
dielectric
which has to be input separately.
A. Single Debye with a dc conductivity
For a spectrum of dielectric function expressed in Eqn. (1), the effects of cr0
on e" and e' are
w-tO
e' -* e0; e" - oo
(III.4.26)
u>—too
e' —t e^; e" - t 0
(III.4.27)
which gives the divergence at u —* 0 as shown by the dashed line in the inset of
Fig. III.9. If plotting a " = weve' vs. a ' = u>eve " of the Debye model, noticing th at
is not subtracted from e', we observe
w-tO
ueve' - t 0; ueve" - t cr0
(III.4.28)
w -oo
ueve' -
(III.4.29)
oo;
ueve" - a0 + €-i €° ~ e^
which gives rise to the divergence at w—too for the dashed line in Fig. III.9.
B. Single Cole-Cole form with dc conductivity
According to Eqn.(III.4.17) and (III.4.18), when u » u>c = ^
d
---■ f a
= <* sin(^p) (e0 - O
^ = a c o s ( ^ ) (e0 - O
wa “ 1 +
“ 1
(III.4.30)
(III.4.31)
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Therefore, the asym ptotic slope s0 in this imaginary versus real conductivity plot
would be
30 = 3(1^77 = tan(¥ > + a c o s t ^ H f o - O ^
(IIL4'32)
Since 0 < a < 1, the slope increases with increasing frequency, which would give a
tail curving upwards as shown by the dot-dashed line in Fig. III.9.
C. Other empirical dielectric functions
As
discussed
above,
the
asym ptotic
behavior
of
the
susceptibility
x(k>) = e(w) -Coo, expressed by other widely used empirical forms, approaches a
straight line as th a t of Cole-Cole form. The asym ptotic slope s0 in the complex
plane of conductivity would present a similar form as Eqn. (III.4.20) and (III.4.21)
except th a t a different constant would appear in the place of the sin(^p) and
c o s ( ^ ) term s. Therefore, the effect of not subtracting the induced polarization e^,
or assuming it is zero, would also give a tail curving upwards.
D. Existence of a second relaxation process at higher frequencies well
separated from a main single Debye process.
W hen a second u>c2 relaxation process with relatively narrow bandw idth
occurs where u>cl <C u , to certain extent, the upwards tendency caused by
would be depressed by the downward curvature contributed by the second
relaxation process. The 1.1 mol% LiCl/l-propanol d ata presented in Fig. III.9 is a
typical example. The curvature after the “kink” still manifests the existence of two
distinct relaxation processes for the L iC l/l-Propanol solutions. Such a plot for the
pure glycerol d ata also dem onstrates the existence of a second relaxation process in
th e frequency range, but its characteristics are not as clear as w hat are given in the
~ °x
pl°k
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In practice, the value of
is usually not well-known and can be varied to
produce a meaningful curve. If the data do not access high frequencies, then the
representation is insensitive to this param eter, bu t it is still able to provide insight
to the existence of possible shorter tim e relaxation processes.
(ii) About the dielectric modulus plot for conducting media
The electric modulus is defined as
M =M ' + iM " = - K
e(w)
(III.4.33)
which was used by Floriano and Angell (1989), is an alternative way of treating the
low frequency divergence in e" caused
by the existence of a dc conductivity cr0.
However, it introduces a relaxation peak due to a non-zero <r0 besides the one
related to the dielectric relaxation process.
In sum m ary, & x ' ~ a X
depicts the contribution from the dynamics of
the dipole m om ent to the conductivity, in addition to the dc conductivity. The plot
is in the wevx(w) plane. For a medium with a frequency independent dc
conductivity, similar to the Cole-Cole plot, instead of taking away the contribution
of a dc conductivity from the imaginary part of the dielectric constant as needed
for a meaningful Cole-Cole plot, subtracting a proper value of
is im portant to
observe the characteristics of a spectrum. It enhances the high frequency feature of
a spectrum. Thus, it is a useful graphical representation for dielectric spectra taken
at high frequencies relative to its m ain relaxation process for m edia w ith or without
dc conductivity. We also pointed out that data which seem to be C-C, C-D or H-N
like according to a Cole-Cole plot, may posses two or more relaxation times which
will possibly be evident when plotting wev( e '- e 00) vs. weve".
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(iii)
Kinetic polarization effect in LiCl/l-propanol solution
The dielectric constant for liquid alcohols are characterized by th e simple
Debye dispersion in the low frequency region and by strong deviation in the high
frequency region from the simple Debye type (Minami, 1980), as has become
apparent for the first tim e, to the author's knowledge, through the new d ata
representation. The low-frequency dispersion is attrib u ted to the dipole relaxation
of clusters of alcohol molecules hydrogen bonded with each other, and the
dispersion observed in the high frequency region are ascribed to th e internal
rotation of hydroxyl groups and/or the reorientation of free alcohol molecule. From
the fitting param eters listed in Table III.l, we observe th at the m ajor effects from
ion is on the m ain relaxation dispersion at lower frequencies. The decrease of its
static dielectric constant is proportional to the conductivity of the solution which is
predicted by the HO theory.
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Fig. III.l. The potential energy diagram of a m any-body two-level system which
represents the energy of a large number of individual, interacting systems. The
potential m inim a correspond to preferred orientations or positions. The shaded area
at the bottom of the minim a are the correlated states of w idth 2EC. The arrows a
and a' denote therm ally assisted transitions in which a significant am ount of energy
is exchanged w ith the phonon bath. Processes b and c are configurational tunneling
transitions of the “flip” and “flip-flop” types, respectively. The energy 2B is the
zero point splitting due to internal fields in the surrounding media [Modification of
a plot by Hill (1983)].
2B
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Fig. III.2. (a) Cole-Cole plot and (b) a x - plot with £^=4.2, of pure glycerol d ata
and Cole-Davidson fit with e0 = 43.1, £^=4.2, r= 1 .8 ns and /?=0.66, mean
deviation 6 = 0.28.
20
n
0
10
20
30
40
6'
1.0
^
8
JUJ
i
0.8
0.6
°
^ ^ ooooooooooo^
a
0.2
0.0
0
0.5
1
1.5
2
w e ve ''
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Fig. III.3. Theoretical curve for Debye (solid line), Cole-Cole (<*=0.2, dot-dashed
line), Cole-Davidson (/?=0.8, dashed line) and Havriliak-Negami (<*=0.2 and /3=0,
dotted
line)
empirical
functions
for
eo=60,
^00=5,
r=1000ps
w ithout
dc
conductivity.
60
e
1
40
20
10
10
40
30
n
20
10
10
10
10 '
12
Frequency (Hz)
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Fig.
III.4.
crx ' ~ a X
^or
c = 0.66mM, e0 = 70.5,
CsC1/H20
d ata
and
corresponding fits; o's:
= 5.2, r = 8.1 ps and <r0 = 7.6 (fim )'1; + ’s: c = l.lm M ,
e0 = 68.0,
= 4.8, r = 7.8 ps and cr0 = 12.0 (fim)"1; and diamonds: c = 2.5mM,
e0 = 57.6,
= 4.5, r = 7.0 ps and a 0 = 24.6 (fim)"1. Inset is the corresponding
Cole-Cole plot.
100
20
I
W 40
40
60
80
100
UJ
3
20
0
20
40
60
80
100
cjewe"
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Fig. III.5. a x ~ a X
f°r Cole-Cole dispersion e0 = 55,
= 5, r = 8.85ps and
<r0 = 10 (fim)-1 w ith a = 0 (solid line, pure Debye), a = 0.1 (dotdashed line) and
a —1/6 (dashed line). D otted lines represent the asym ptotic behavior of a — 0.1 and
a = 1/6, respectively. The angles formed with respect to the crx '- axis are 7t/20 and
tt/12.
120
100
80
B
c:
60
x
b
40
20
0
20
40
<7 X'
60
80
100
120
(G m ) " 1
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Fig. III.6. (a) Cole-Cole plot for pure 1-propanol d ata (o’s) and Cole-Cole fit
(dashes), two Debye relaxation tim e fit (dotdash) and DB -f-CC fit (solid line), (b)
ax" ~ ax
^or Pure 1-propanol data (o’s), C-C, DB+Cole and two Debye fit.
15.0
12.5
10.0
7 .5
5 .0
2 .5
0.0
0
10
5
20
15
25
6'
1.0
0.8
~x
0 .4
0.2
0.0
0
0 .5
oy
1
(n m )
1.5
2
1
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Fig. III.7.
~ a x Plot f°r pure glycerol d ata (o’s), Cole-Davidson fit (Dashed
line) and Debye + Cole-Cole fit (solid line).
5
4
U>
I
3
(JJ
3
2
1
0
0
0.5
1
1.5
2
co eve
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Fig. III.8. a x ' ~ <TX
^or 0.5 mol% LiCl/1-propanol d ata ( + ’s) and two Cole fit
(solid line) w ith €^=2.7, A!=15.5, ^ = 3 2 9 ps, ai= 0.02, A2=1.4, r 2=10.3 ps,
a 2=0.17 and <ro=0.042
Inset: Cole-Cole plot of d ata and the fit.
1.50
25
20
1.25
8 1.00
20
25
> 0 .7 5
0 .5 0
0 .2 5
0.00
0
0 .2 5
0 .5
0 .7 5
1
1.25
1.5
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Fig. III.9. Verification of the existence of two relaxations for the 1.1 mol% LiCl/1propanol sample, when plotting a"= weve' vs. a'= u>eyt" directly, instead of
subtracting the effect of
on the data representation. The dashed line is a Debye
curve with €^=2.7, A=14.2, r = 365ps and cr0 = 0.064
which approaches the
vertical line a'= 0.408 (ftm )'1 indicated by the arrow, when frequency is much
higher than the characteristic frequency. The dash-dotted line is a Cole curve with
€^=2.7, A=14.2,
t
and a = 0.03, which represents the low
= 365ps, <r0 = 0.064
frequency p art of the data but curves up at high frequencies. The solid line is the
two Cole-Cole fit with €^=2.7, A1=14.2, r x=365ps, a x=0.03, A2= 1.5, r 2=12.2ps,
a 2= 0.20 and cr0= 0.064
which fits the d ata set (x's). Inset: Cole-Cole plot of
the d ata and the fits, from which the differences of the two kinds of fits, Debye and
Cole-Cole, are not distinguishable.
1.0
20
0.8
0.6
3
10
20
b 0 .4
0.2
0.0
0
0.1
0.2
0 .3
0 .4
0.5
<r'(co)
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CHAPTER IV
DESCRIPTION OF RELAXATION SPECTRA FOR AQUEOUS SOLUTIONS.
In this chapter, a general picture of the dielectric relaxation spectra of
aqueous solutions are depicted. A model related to the concept of the “microwave
dielectric excluded volume” developed during the course of this thesis is presented.
In general, as described in the previous chapter, the dielectric dispersions of
solutions axe composed of several relaxation processes. Each of them can be present
in the form of any one of the empirical dielectric functions. The characteristic
relaxation times reflects microscopic information related to param eters such as
internal viscosity and dielectric friction, which will be discussed in chapter V. The
dielectric
relaxation
distribution
param eter
in
Cole-Cole,
Cole-Davidson or
Havriliak-Negami form, probes structural properties of dielectrics as will be shown
both in chapter V and VII. For aqueous solutions, the static dielectric constant of
the solute modified free w ater in the solution is used to extract the volume of a
hydrated molecule, num ber of hydration w ater around a solute molecule, etc.
Errors in these extracted physical quantities introduced by measurements and
modeling are estim ated.
IV. 1 Interactions w ith solvent - water
For a simple hydrophilic species of solute in water, the dielectric spectra
usually manifests three relaxations, related to the solute dipole (if there is any),
solute-solvent interaction and free solvent. For dilute to m oderately concentrated
alkali-halide ionic solutions, there should exist relaxations of bound w ater and free
water. Ion-pair relaxation has not been observed in this work. The dielectric
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spectra of macromolecular aqueous solutions is expected to be composed of at least
three relaxation processes, corresponding to the rotation of macromolecules, bound
w ater molecules and free water molecules, respectively, in order of decreasing
relaxation times.
M any biological macromolecules, such as proteins, fold spontaneously into
one stable conformation determined by their amino acid sequence. By treatm ent
w ith certain solvents, a protein can be unfolded, or denatured, to give a flexible
polypeptide chain th a t has lost its native conformation. W hen the denaturing
solvent is removed, the protein will usually refold spontaneously into its original
conformation. The effect of conformational change in myoglobin molecule on
solution dielectric properties by varying the pH value of the solvent will be
discussed in Chapter VI.
In water, one of the most im portant factors governing the folding of a
protein is the distribution of its polar and nonpolar side chains. The many
hydrophobic side chains in a protein tend to be pushed together in the interior of
the molecule, which enables them to avoid contact with the aqueous environment
(just as oil droplets coalesce after being mechanically dispersed in water). By
contrast; the polar side chains tend to arrange themselves near th e outside of the
protein molecule, where they can interact with other polar molecules and with
water. The motion of water molecules near a protein molecule are “hindered” to
some extent depending on how close they are to the protein molecule. An effective
hydration shell is usually drawn to describe the degree of intensity of the proteinw ater interaction.
As has been shown in Chapter II, the characteristic relaxation tim e of water
is about 8.5 ps at 25C, corresponding to a characteristic frequency of ~ 18.5 GHz.
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The distinction between the hydration and free solvent molecules is largely a
m atter of tim e scales involved, and the strength of the solute-solvent interaction.
The rotational relaxation tim e of the bound w ater is much longer (G rant, et al.
1986; Tao and Lindsay, 1988 and 1989) than th a t of the free water. Above 1GHz,
the dom inant contribution to the dielectric spectra is from the rotational relaxation
of free
w ater
molecules
in
solution.
Therefore,
the
microwave
dielectric
spectroscopy of ionic and macromolecular aqueous solutions at frequencies higher
th an 1GHz essentially probes the relaxation of solute modified free water in
solutions.
IV. 2 Number of Bound W ater Around Each Solute Molecule.
In principle, there are three ways of detecting the hydration shell for a
system displaying relaxations due to solute, solute-solvent and solute modified
solvent: (a) Derive the dielectric radius from the rotational relaxation tim e of the
hydrated polar molecule. Then the volume of the bound w ater around a molecule
can be extracted if the volume of the molecule can be estim ated ( Schlecht, et. al.
1969 ). (b) O btain the hydration water relaxation spectra from 1 MHz to about 1
GHz directly or by extrapolation (G rant, 1974). (c) Measure the solute-modified
free w ater spectrum at frequencies higher than 1 GHz, then extract the am ount of
the bound water in the solution and hence the num ber of bound w ater molecules
around each solute molecule. For simple ionic solutions, m ethod (a) is excluded
since there is no intrinsic dielectric moments. It is very difficult to observe the
dielectric relaxation of hydration w ater around ions due to the strong electrode
polarization effect (Davey, et al., 1990) below GHz frequencies. The only approach
is (c) in order to obtain information on hydration w ater and hence the interaction
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between water. For aqueous solutions of biological macromolecules, cases (a) and
(b) axe complicated or by the fact th a t there is usually not a single relaxation
process, due to the complex nature of macromolecules. Several m easurem ents, for
example, have shown th at the low frequency dielectric spectra cannot be simply
described either by multiple Debye relaxation processes or a distribution of
relaxation times (G rant et al., 1986; Hanss and Banerjee, 1967; Schlecht et al.,
1969). However, the dielectric properties of bulk w ater provide us w ith a unique
approach to determ ine the hydration of biological m aterials such as proteins,
hum an blood and tissues, by determining the free w ater content from the dielectric
spectrum in the microwave frequency range characteristic of the free water
relaxation process, i.e. by m ethod (c).
IV.2.1 Microwave dielectric excluded volume
On the basis of the idea th at at high enough frequencies, both the solute and
bound solvent molecules are not able to respond to the fast changing electricm agnetic field, a hydrated solute molecule as a whole, acts like a “void” with no
rotational dipole moment. This is the basis of the newly introduced notion of the
“ microwave dielectric excluded volume” V D w ith dielectric constant of eex due to
only the induced polarization, as illustrated in Fig. IV. 1.
T he existence of these “voids” in a bulk solution tends to lower the
dielectric constant of the solution. The dielectric spectrum of the solute modified
free solvent can be determined by fitting the dielectric d ata at high frequencies to a
dielectric function described in C hapter III. The effect of low perm ittivity voids on
bulk solvent dielectric constant may be modeled by electrostatic theory. W e define
the dielectric excluded volume fraction R v as the to tal volume of the hydrated
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solute molecules ( VD ) per liter of th e solution. By solving Laplace’s equation
subject to the relevant boundary conditions,
the perm ittivity of the solution is
found to be
r
( r• 1
-
r
fr -I 2 ( 1 ' ^
£‘ ( " 1 “ U u )
which is the well-known
W agner
+
( 1 +
2 R v ) e ex
,jV o i x
( 2 + R v ) ^ w) + ( l - R v ) ^
(IV-2-1)
solution obtained by Maxwell (1892)and generalized by
(1914)and eex is related to ex and ehyd through the same
equation
if
R «=v ,/ v d,
2 ( 1 ‘ Rex) ehyd+ ( 1 + 2 Rex) 6X
e"’d ( 2 4- R eJt ) ehyd + ( 1- Rv ) ex
W hen u - * 0 , from Eqn.
(IV.2.2)
,
( ^wO
^sO ) ( 2 ^wO + ^ex )
v ~ 7 T 7 — XX— w X
I. " e w0
■ e s0 / \ e w0
*— V
eex /
/ t i T o ox
(IV.2.3)
where ew0 and ewoo are the static and high frequency dielectric constants of bulk
water. R v = 0 if the dielectric constant of the voids is ew0 (es0 = ew0) and R v =1
when es0 = ewoo. The microwave dielectric excluded volume can be found as
where N0 is Avogadro’s constant and c is the solution concentration in Molar.
IV.2.2 Num ber of hydration w ater molecules around each solute.
In reality, there is no sharp boundary between the bound w ater and free
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w ater in solution. However, as a first order approximation, the volume estim ated in
Eqn. (IV.2.4)
can be used to represent the total volume of voids per liter of the
solution, which includes the total volume of solute molecules and the bound water.
The continuous changing of the binding strength of solute to w ater molecules is
equivalent to a shell of tightly bounded w ater molecule and the rest are free water
molecules. Thus, the amount of free w ater per liter by weight is
W fw = ( 1 - Rv ) d w ( kg )
(IV.2.5)
where dw is the density of bulk water in kg/1. The weight of the total amount of
w ater in 1 liter of the solution can be found as
W w = ds - 10'3MWc ( kg )
(IV.2.6)
where Mw is the molecular weight (grams/mole) of the solute and ds (kg/1) is the
solution density. The difference between the total amount of water and the free
w ater is the weight of the hydrated w ater per liter of the solution. Therefore, the
num ber of hydrated water molecules for each solute molecule is given by
Nhyd = ( W w - W fw ) / (0.018 c )
_ (ds - d w ) + RV- 1 0 '3MWc
0.018 c
The weight ratio of bound w ater mass to solute mass is then
w hyd = 18 N hyd/ M w
(IV .2.8)
Physical quantities of the solution needed, in order to derive the microwave
dielectric excluded volume and the number of bound w ater molecules around each
solute molecule are, fitted static dielectric constant esQ of the solute modified
solvent, concentration c, molecular weight Mw of the solute, density d of the
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solution and the void dielectric constant eex. Also the static dielectric constant es0,
density dw and molecular weight of the solvent. The volume o f the solute molecule
is not required.
IV.3. Error analysis
In expression (IV.2.7), quantities related to the measurement are R v , c and
ds. The error in the number of bound w ater around each solute molecule can be
estim ated from
ANhyd —^
d„ <5R
{ 0*018 c }2+
Nl,yd + 18* ) ScC }2 + { 0.018 c^
<5Na2 + SN22 + <5N32
(IV.3.1)
and according to Eqn. (IV.2.3),
* R V _ r r ___________ewo/es0______________ ^es0 ]2
Rv
( ewo/eso - 1) ( 2 ew0/es0 + 1) eso
r_________ ewo/ewOO____________ ^eex
12 1 1 / 2
( 2ew0/ ewoo +1) (ewo/ewoo~l) e- J 1
,r y o
(AV'd,i2)
where <5eg0 is the root mean square deviation estim ated from four or five repeated
m easurem ents performed on one sample and 6eex is the possible deviation of the
dielectric constant of the “void” from the assumption. From Eqn. (IV.2.4), the
error in the volume of a hydrated protein molecule can be estim ated as
AV d =
r ^R v
12
*cK !
i r i/
6
c
12
+ t VD — 1
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The equation for
, 0.018
,2 , ry
1
/
m—
da wN
0
A N hyd
+ 1
v d
8 c
12
t
/ •
(IV.3.3)
shows th a t at higher solution concentration,
the
error in hydration w ater molecules becomes smaller.
IV.4 Error in difference dielectric spectroscopy
The fairly rapid data acquisition tim es ( < 1 sec in certain situations ) leads
to the possibility of observing in situ changes in sample properties. It becomes
crucial
in
detecting
small
effects
such
as m easurem ents
in which
when
deoxygenated hemoglobin solutions axe oxygenated. It has been suggested th at
there is an increase about 60 in hydration w ater molecule (Colombo, et al., 1992).
If nRT is used to denote the difference in the num ber of hydration waters
around each solute molecule upon a change of m easurem ent condition from R to T
and the density is assumed to be remained undisturbed, according to Eq. (IV.2.7),
Urt =
0.018 c
dw
(IV.4.1)
Here the error due to the uncertainty in th e solution density ds is eliminated so
th a t higher sensitivity to relative changes in sample properties can be achieved
than in absolute measurements.
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Fig. IV. 1 . Illustration of the model for the microwave dielectric excluded volume
v D.
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CHAPTER V
STATICS AND DYNAMICS OF AQUEOUS ALKALI-CHLORIDE SOLUTIONS
The technique for measurement of microwave dielectric spectra described in
chapter II has made the study of aqueous ionic solutions much more accessible,
because at such high frequencies the common “electrode polarization” effect, due to
the existence of ionic species, is suppressed and the free water relaxation
range
( ~ 18.5GHz) is accessed. Aqueous solutions of AC1 (A=Li, Rb and Cs) may be
considered to be simple, at least conceptually, in contrast to more complicated
electrolytes. However they exhibit a rich behavior in regard to their electrical and
other physical properties (Angell, 1981). For instance, certain concentrations
exhibit glassy behavior at low tem peratures (Sridhar and Taborek, 1988) and in
fact are some of the classic glass forming liquids. A particularly appealing feature is
th a t the concentrations can be easily varied over a wide range from dilute to very
concentrated solutions. Partially for these reasons, it is also one of the most widely
studied by several techniques (Sridhar, 1988; Elarby-souizerat, 1988; Wei, 1990;
Taborek, 1986; Neilson, 1983; Hagemeyer, 1989; Narten, 1970.
For AC1/H20
(A=Li, Rb and Cs) solutions of dilute to moderate
concentration, detailed measurements of the perm ittivity e = d - \a" as functions of
frequency between 45 MHz and 20 GHz and concentration c, were carried out.
From the data, several parameters: the static and high frequency lim iting values e0
and Cqq, the dc conductivity <r0, the dielectric relaxation tim e r D, and the hydration
num ber nhyd, are extracted and studied as functions of ion size and concentration.
For m oderate concentrations both es0 and r D decrease linearly with solution
conductivity. W hile the behavior of es0 can be understood in terms of either static
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or Hubbard-Onsager kinetic polarization models, the experim ental results for r D are
at present not understood quantitatively in term s of these models. However a good
correlation of the r D d ata w ith empirical viscosity results suggests an alternative
explanation based upon the solution viscosity, modified by ion size effects, which
play an im portant role in the dielectric response. The various length scales relevant
to dielectric and conductivity processes in the solutions are also studied
At higher solute concentration, the relation between the dynamics of
dielectric relaxation, the glass-forming ability and properties in the supercooled
state of LiCl:RH20 solutions are investigated. A close correspondence between the
tem perature dependencies of the Cole-Cole relaxation tim e r c , the conductivity
r dc(T) and the viscosity »?(T), is observed. r c (T) is well-described by a power law
form (T-Tq)^, with T 0 ~ 210K and
~ 2, a form m otivated by mode-coupling
7
theory. The data suggest th a t non-Debye response at room tem perature is
associated with the ability to vitrify at low tem peratures. A microscopic basis for
the avoidance of crystallization is provided by obtaining the m ean radius available
o
to a w ater molecule, which is shown to approach 1.9A in the highly concentrated
solutions. This suggests a strong confinement of water molecules by the ions, which
prevents macroscopic crystallization from occurring when cooled.
V .l Sample preparation and concentration conversion.
Aqueous solutions of AC1 ( A=Li, Rb and Cs) were prepared by dissolving
AC1 powder (Fisher certified) in deionized water.
Sample concentrations are
initially in mol%
c -
*
s
l
0
0
x
s
a
S
^
j
<V
- L
1 >
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Table V.I. Solution concentrations of AC1/H20 studied
LiCl
RbCl
CsCl
-f
c (M olar)+
c (Mol%)
R
0.55
1 .0
100
1.35
2.5
40
2.14
4.0
25
2.65
5.0
20
5.11
1 0 .0
10
7.20*
14.5
6.7
9.62*
2 0 .0
5
13.38*
29.0
3.4
0.54
1 .0
100
1 .1
2 .0
50
2.5
5.0
20
2.7
5.3
18.9
5.2
1 1 .1
9.0
0.33
0 .6
167
0.49
0.91
110
0 .6 6
1 .2
81.3
1.06
2 .0
50
2.49
5.0
20
4.5
1 0 .0
10
d from 0.55 M to 5.11 M is obtained by interpolation using data from ref
(CRC handbook)
* d is obtained by extrapolation using d ata from ref (CRC handbook)
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The mole ratio in R in AC1:RH20 is given by
Cmol%
(V.1.2)
To convert concentrations in cmol% to M olarity ( cM),
°M
moles of AC1
volume of solution in liter
1.8+M wcmol%/1000
(V.1.3)
where ds is the solution density and Mw is the molecular weight of the solute in
grams. Concentrations of LiCl/H 20 vary from 0.55M to 13.38M (29.0 Mol%) ,
R b C l/H 20 from 0.54M to 5.2M and C sC l/H 20 from 0.33M tO 4.5M. Sample
concentrations studied are listed in Table V .l.
V .2. AC1/H20 room temperature results and analyses.
Dielectric measurements were performed on the AC1/H20 aqueous solutions
with concentrations listed in Table V .l, using the technique described in Chapter
II. The real p art of the dielectric constant e' d ata of LiCl/H 20 , as a function of
frequency for various concentrations are shown in Fig. V .l, and e" vs. e' are plotted
in Fig. V.2 for higher concentrations. R bC l/H 20 results are plotted in Fig. V.3(a)
and (b). CsCl results were plotted in another d ata representation in Fig. III.4
(Note th a t a and e" are equivalent, since <r = wfve", where tv = 8.85 x 10' 12 F /m is
the p erm ittivity of vacuum).
The apparent low frequency divergence of e' is probably due to electrode
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polarization effects, which masks the bulk dielectric properties. For f > 1 GHz, e'
clearly displays a relaxational behavior associated w ith the bulk solution. The
conductivity has a non-zero low frequency plateau attributable to the ions. For
L iC l/H 20 solutions, X-ray and neutron studies (N arten, et al. 1973; Ederly and
Neilson, 1980) showed th at samples, with m oderate concentration, are m ixtures of
free w ater molecules, hydrated Li+ cations and Cl* anions and perhaps, ion pairs.
As discussed in the previous chapter, the m ulti-com ponent view of an ionic
solution, viz. ions, hydration shell and quasi-free solvent molecules, would suggest
the presence of at least two dielectric relaxation times, one pertaining to the
hydration shell r h and the other pertaining to the solvent molecules r D. The
presence of r h is a m ajor open question, of interest because of the perceived
im portance of the hydration layer in chemical and biological processes. Simple
m inded argum ents would lead to the conclusion th a t r h > > r D, since the hydration
molecules are under the strong influence of the ions, w ith the relative separation
between r h and r D being determined by the ion-solvent interaction strength. In
modeling the dielectric behavior, we make the following assumptions: The “ static”
conductivity <r0 is due only to hydrated single cations and anions. F urther this ionic
conductivity is assumed to be frequency independent. Then, the dielectric spectra
were fitted
to
Eqn.
(III.2.25), which is the
Cole-Cole function with the
superposition of a “dc” conductivity. Extracted dielectric param eters are listed in
Table V.II.
The solid lines in Fig. V .l, V.2, V.3 and Fig. III.4 are the corresponding fits.
For the moderate concentration, c < 5.2 M, discussed here, a = 0 was found
w ithin the error so th at the Debye like relaxation of the solvent is retained in the
solution. The dc conductivity is indicated by the plateau at the low frequency
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Table V.II. The fitting parameters for the aqueous electrolyte solutions.
a
Solution
c (M)
^sO
esOO
t 8 (ps)
LiC l/H 20
0
78.5
4.5
8.4
0
0
.55
69.1
4.6
8 .1
5.0
0
1.35
59.9
4.7
7.8
9.4
0
2.14
53.7
5.3
7.7
12.3
0
2.65
48.4
5.4
7.5
14.4
0
5.11
34.2
6.3
7.4
18.0
0
7.20
27.0
7.0
7.3
17.3
.0 2
9.62
27.7
7.3
7.6
14.8
.05
13.38
2 1 .6
7.6
8.5
9.3
0 .1 1
0.54
70.6
4.5
7.9
7.2
0
1 .1
6 6 .0
5.2
7.6
1 2 .0
0
2.5
55.0
5.0
6.5
25.5
0
2.7
53.7
5.5
6.4
27.5
0
5.2
42.4
6.5
36.0
43.9
0
0.33
74.5
4.4
8.3
4.5
0
0.49
72.0
5.2
8 .2
6 .1
0
0 .6 6
70.5
5.2
8 .1
7.6
0
1 .1
6 8 .0
4.8
7.8
1 2 .0
0
2.5
57.6
4.5
7.0
24.6
0
4.5
46.6
4.0
6 .0
40.4
0
R bC l/H 20
C sC1/H20
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end of Fig. V.3(b). The relaxation times of the solution r s axe close to th a t of pure
w ater but vary w ith concentration and conductivity as shown in Fig. V.4. The
differences in conductivity between the three electrolyte solutions are evidence of
the ion size effect. The dielectric properties, particularly a t the frequencies
considered here, axe dominantly due to the solvent dipoles, of course modified by
the presence of the ions.
V.2.1 Static dielectric constant
: static polarization
Fig. V.5 shows, the “ static” dielectric constant, which is obtained as the low
frequency plateau in the e'(uj) data, decreases monotonically w ith increasing
concentration (inset) and linearly with “dc” conductivity of the solution. As will be
apparent later, the volume effect is much larger than can be attrib u ted to the ions
alone, and the hydration molecules are essential. The hydration w ater molecules
however appear to have a relaxation frequency significantly less than 1 GHz, and
hence appear “frozen” in the present frequency range. W ith increasing ion
concentrations, as more and more water molecules form hydration shells around the
ions, e0 decreases due to the loss of free water. This is the traditional interpretation
of the decrement in dielectric constant, which is directly related to the static
properties of the solvent around the ion. The observed decrements are attrib u ted to
finite ion size and to the formation around the ion of clusters of irrotationally
bound solvent molecules. Making use of the model and carrying out the analyses
described in Chapter IV, the hydration num ber Nhyd, the microwave dielectric
excluded volume and therefore the radius of the hydrated ions rD are deduced.
Nhyd is the average number of hydration water molecules per electrolyte molecule,
and is shown in Fig. V . 6 , assuming complete dissociation. It is apparent that
Nhyd(LiCl) > Nhyd(RbCl) > Nhyd(CsCl) for a given concentration, with Nhyd of the
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latter two being quite close. Since the baxe ionic radius
o
o
(Horvath,
1985)
o
progressively increases from Li to Cs (0.78A, 1.48A, 1.69A), it is clear th a t the
hydration num ber is determined by the charge density at the ion surface, which
progressively decreases. As a consequence the ion-water interaction is progressively
weaker, w ith fewer water molecules bound by the Coulomb forces in the CsCl
solution than in the LiCl solution.
The static model discussed above also leads to a linear relation between Ae0
and cr0. This is simply because, in this picture, Ae0 is due to the to tal volume of the
hydrated ions, and hence is proportional to the number of ions in the solution, as is
the conductivity. This applies for low concentrations. This model then leads to
Ae° = ^ D [/I£ 2 ew0 H°eu (ewO“ eu)l a o
where VD = ^ZL (rAh3 -f r clh3).
o
r L ih =
The fitting radii for Li+, Rb+, Cs+and
o
o
Cl' are
o
2.6 A, rRbh= 2.45 A, rCsh= 2.4 Aand rcih= 3.25 A, respectively.
The values of rLih =
A
(V.2.1)
and 3.10
A
2 .6
A
and rCUl = 3.25
A
agree well w ith the values 2.43
for c < 6.9 M obtained from X-ray and neutron studies (N arten, et
al.,1973), and 2.55
A
and 3.34
A
for 3.57 Molal from neutron studies (Szasz, 1981;
Cummings, 1980; Neilson, 1983). The dynamic hydration num ber of 7.6 for low
concentrations, inferred from this measurement, is in agreement w ith th at
calculated by Impey (1983), which gives 7.2 (4.6 for Li+ and 2.6 for Cl') in very
dilute solution.
In contrast, most molecular dynamic simulation studies (Tanaka,
1987; Bopp, 1985; Impey, 1983) and experim ental studies (N arten, et al. 1973;
Neilson, 1983) yield coordination numbers of Li+ between 4-5 and th a t of Cl' about
6
. In making this comparison, it should be kept in m ind th a t the hydration
num ber derived in the present experiments refers to those w ater molecules whose
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average residence tim e is longer than 50 ps.
This correspondence leads to a physical definition of the dielectric radius r D
occurring in the static model : it corresponds to the radius of a sphere containing
the ions and the associated hydration w ater molecules, which responds (i.e. rotates)
rigidly at microwave frequencies to the E-M field.
V.2.2 Kinetic polarization
The above static polarization model is to be contrasted w ith the kinetic
polarization model proposed by Hubbard and Onsager (1977-1979) (HO). From a
theoretical point of view, it has recently become clear th a t the static explanations
of th e dielectric measurements axe incomplete; the measured dielectric decrements
can also arise from dynamic effects. In the HO model, the liquid is treated as a
hydrodynam ic continuum, and the kinetic ion-solvent interaction affects the
capacitive adm ittance (i.e. es0) in two closely related ways : (1) As an ion migrates,
the surrounding fluid dipoles rotate according to the laws of hydrodynamics, and
although dielectric relaxation tends to restore a polarization corresponding to the
local field, this process is not instantaneous and the average delay is equal to the
relaxation tim e r D; ( 2 ) the force which an external field exerts on an ion does not
develop its full strength immediately because the ion is driven partly by the
external field and partly by the polarization which develops in response to the
applied field, w ith the polarization field evolving w ith a tim e constant r D. It is
clear here th a t the dynamical relaxation tim e r D (and also the viscosity relaxation
tim e r vis), play an essential role. The HO model leads naturally to an intim ate
relation between the conductivity cr0 and the dielectric decrement Ae0 = ew0 - es0,
which can be expressed as
A e0
= A e ( d ip o le ) + A e ( io n )
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=
~
M
6 " ° 4
i ' ," l°
)
T D »
<v
-2 -2 >
This is similar to the Eqn. (V.2.1), with the correspondence (ew0//ie)V D —►47 rrDw,
and for a boundary condition of no slip when the viscous relaxation tim e is not
comparable to the dielectric relaxation time.
The slopes in Fig. V.5 obtained from least squares fits are -2 .1 O-m, -0 .9 0
fi-m and -0 .8 9 fi-m for LiCl, RbCl and CsCl respectively. These values m ay be
contrasted w ith a slope of -
0.88
Q-m calculated from the above equation assuming
no slip, which is the continuum lim it (Hubbard, et al., 1979). Note th a t the
continuum version of the HO theory does not predict any dependence on ion size indeed this appears to have been confirmed in weakly concentrated solutions
(H ubbard, et al. 1977) in which an apparently ion-size independent slope of - 2.7 fim was obtained for several solutions.
V.2.3 Dynamical time scale: r D
From the fitting param eters listed in Table V.II, no signature of hydration
shell relaxation can be found. It is possible th a t this occurs at such low frequencies
th a t it is masked by ionic polarization effects. Coming to effects of the ions on the
solvent dielectric relaxation tim e r D as shown in Fig. V.4, two features of the
dielectric spectrum are noteworthy:
(1) t d decreases w ith increasing ionic concentration. (However in th e case of
LiCl solutions, it reaches a minimum and then increases strongly). The d ata are
shown in Fig. 4.
(2) The pure solvent (water) and the solutions with decreasing r D are
essentially Debye-like. (In LiCl solutions, the high c solutions with increasing r D are
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-104-
non-Debye. W e believe th a t the same phenomena of increasing r D, non-Debye
relaxation and the conductivity m axim a should occur in the RbCl and CsCl
solutions also, however the observation is restricted due to the lim ited solubility).
Any static polarization model (such as the one discussed above) does not
address th e issue of the behavior of r D, which is after all a dynamical phenomenon.
We
therefore considered the
hydrodynamics
HO theory, which includes
on the dielectric properties.
the influence of
The result of the HO theory can be
w ritten in the form (Hubbard and Onsager, 1977) :
+ ■ £ ( ! - * ,)
T h at is,
<V -2'3>
as a result of dielectric friction, the dielectric spectrum of an
electrolyte solution is the free solvent spectrum modified by decrement spectra with
shorter relaxation times. However upon close graphical exam ination of the above
expression, the resultant solution spectrum has a apparent relaxation tim e which is
longer th an the pure solvent relaxation tim e r Dw. This can be seen from the
asymptotic u -* 0 behavior of Eqn. (V.2.3). We get e"(oS) = w rDw [ew0 + (ew0 - ew0Q
+ Ae0 )(l* 7 )]. Comparing with a Debye form , there is a positive shift of the
relaxation tim e of the solution :
A td =
where
7
t d w 2 cr0
7
(V.2.4)
is a constant of the solvent. Although eqn.(V.2.4) has the observed linear
dependence of
A r D « <r0,
the effect, viz.
A rD
unfortunately the HO theory predicts the wrong sign for
> 0.
Although a satisfactory microscopic explanation of the observed decrease of
r has been unable to be reached, there is an im portant correlation with empirical
results for the
solution viscosity, which
suggests an
alternative empirical
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-105-
explanation. W ithin the Debye theory, the Debye relaxation tim e is given by r D =
47 T7/a 3 /k BT,
where r] is the viscosity and a is the molecular radius. We can rewrite
the above expression as
Td = [4?n?w (a aw)3/k BT] (t)/ tjw).
(V.2.5)
where a aw = a is the effective molecular radius of the solvent molecule. For an
electrolyte solution, the Jones-Dole empirical formula is (Horvath, 1985): rj/ rjw = 1
+ A >fc + B c, where A is only related to ion-ion interactions and B represents the
ion-solvent interaction. Relevant values for B are : +0.146 (Li), -0.030 (Rb) and 0.046 (Cs), +0.14 (LiCl), -0.0360 (RbCl) and -0.062 (CsCl).
concentration dependent and can be w ritten a 3 =
1+
a should be
9(c).
Therefore the shift of the relaxation tim e of the solution from th a t of water
can be expressed as
A td = B c + A >|c + 9(c)
(V.2.6)
The above observation provides a plausible explanation for the decrease of
r D in RbCl and CsCl as being due to the decrease in the solution viscosity.
However the LiCl d ata for r D and i? appear to be in contradiction. This can be
resolved by noting th a t a size effect also enters in the form of the radius a. Since
o
o
the bare ionic radius of Li (0.78A) is much smaller than th a t of w ater (2A), an
apparent decrease of the effective radius [ 9(c) < 0 ] is likely, and m ay be
responsible for the decrease in r. This size effect is less im portant in the case of Rb
o
o
and Cs, whose sizes (1.47A and 1.67A) are comparable to th a t of water.
For c > 5M (15 Mol %), the measurements show th a t one cannot talk about
a single r s, in contrast to the case for lower c. The non-Debye behavior has to do
w ith the breakdown of the network correlation of hydrogen bound in water as will
be shown later in this chapter.
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-106-
V.2.4. Ionic. Dielectric. Stokes and Hubbard-Onsager radii in a ionic solution
In considering size effects in ionic solutions, exam ination of the several
length scales which govern the electrical behavior of solutions is worthwhile. Below,
comparison of various sizes deduced from viscosity, conductivity and dielectric data
is presented.
Stokes Radius
The limiting (c —►0) conductivity A0 leads to a Stokes
radius, which is related to the drift characteristics of the ion in the viscous solution,
and is given by :
rstokes = 6p7r°A077
(V.2.7)
The above expression ignores dielectric friction, and leads to a value m uch bigger
th an the ionic radius.
Hubbard-Onsager Radius The HO model naturally leads to a radial length
relevant to a point ion in a solution. This m ay be defined as (Hubbard and
Onsager, 1977)
_
f ewo' ewoo \ _
e2
~WO2
16^771
} Dw
Note th at this is characteristic of the solution only, and is independent of the ion
o
size. For w ater, a H 0 « 1.5 A at 25C.
aHO provides a useful measure of the im portance of dielectric friction. For
Li, R Li is much less than aH0, so the dielectric friction is im portant. For Rb and
Cs, the ionic radii are comparable to the HO radius, suggesting th at dielectric
friction is less im portant and provides a plausible explanation for the higher
lim iting conductivity.
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-107-
Inclusion of dielectric friction leads to a size-dependent HO radius, which is
related (Ibuki and Nakahara) to r 8toke8 :
rHO
— 4 n astokes
(V.2.8)
/x
for slip boundary condition, where
x
is a function of the ion size.
All the estim ated numbers for individual ions are listed in the following
table II. It is hard to give accurate HO radii for R b+ and Cs+ since their ionic radii
are close to a HO (Hubbard, et al. 1977 and 1978). The equivalent radii for molecules
are obtained from the additivity of the conductivity, e. g. the Stokes radius for slip
<r° = ^
= 1ST( £ + £ ] =
+
(V-29)
W here
^Stokes = I V
, rf
rA + r cl
-
( V - 2 .1 0 )
Table V.III. The Stokes, HO and dielectric radii for the ions and molecules.
Li+
R b+
Cs+
CT
Ionic radius
0.78
1.48
1.69
1.81
A0 [ ref.
38.7
77
77.7
76.4
115.1
rstokes (stick)
2.38
1.18
1.19
1 .2 1
.8
.6
.6
rstokes (slip)
3.57
1.77
1.78
1.82
1 .2
.9
.9
rsHO (slip)
1 .8
1 .8
2 .1
2 .2
1 .0 2
1 .0 2
.96
3.35
3.02
2.95
rD
6
]
LiCl
RbCl
153.6
CsCl
155.1
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—108 —
Dielectric radius
The notion of a dielectric radius has already been
introduced earlier. rD listed in table V.III are dielectric radii extracted from the
static model.
From the table we see th at, as regards the increase of the cation size, all the
radii of the AC1 molecules decrease, which gives strong evidence for the correlation
between the ion motion in the solvent and the dielectric relaxation process.
V.3 Relationships between the dielectric and structural properties of supercooled
LiCl:RH20 .
In this section, it is more clear to use the molar ratio as the unit of
concentration. For conversion to other concentration units, see V.1.1.
One very im portant feature of the LiCl:RH20 solutions is the ability to be
vitrified - for concentrations R < 12 (Elarby-aouizerat, et al., 1988), the solutions
do not freeze but undergo a well documented glass transition at T g ~ 140K. The
general question of how a liquid avoids crystallization and instead can be
supercooled into a glassy state remains an im portant issue (Angell, et al., 1970 and
1983).
The dynamical response of supercooled liquids to different kinds of stress
(electric, shear, etc.) has been studied extensively and some general trends have
been recognized. At very high tem peratures T > > T F> > T g, th e dielectric
relaxation tim e generally obeys an Arrhenius tem perature dependence, followed by
a Vogel-Fulcher-Tammann behavior closer to T g. Another general trend is the
tendency to exhibit a spread of relaxation times, typically described by a stretched
exponential tim e dependence to sudden application of stress. An attem p t to
provide a microscopic basis for the dynamical response of supercooled liquids has
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-109-
been m ade on the basis of a mode coupling theory of density fluctuations. The
initial theory exhibited an ergodic-nonergodic transition which shared many
features w ith the liquid-glass transition. An improved model (Kim and Mazenko,
1992) predicts th a t the transition takes place at tem perature T MC > T g. Above
T mc, relaxation phenomena (eg. dielectric,
viscosity) are predicted to obey a
power law dependence
ror»? ~ ( T - T m c )
in which
7
relaxations
(V.3.1)
- 7
~ 2, and when T g< T < T MC the tem perature dependence of various
have
the
Vogel-Tammann-Fulcher
form.
An
early
test
of
a
phenomenological version of the mode coupling theory was carried out by Sridhar
& Taborek (1988) using conductivity relaxation measurements on LiCl:7H20 at low
tem peratures, while more recently there have been studies of the dielectric
relaxation of several glass former (Kim and Mazenko, 1992; Schonhals, et al. 1991)
In measurements on LiC l/H 20 solutions at a fixed am bient tem perature of
25C presented in the previous section, a “ low frequency” ( ~
1
to 4 GHz) plateau
es0 in e'(w) was observed, which decreased with increasing concentration from the
pure w ater value of ~ 78.5 , saturating at about 20 for c = 20 to 29mol% (R=3.4).
It was found th a t r c decreased slightly from 8.5 ps of water to a m inim um ~ 7 ps
and then increased again. A remarkable feature of the LiC l/H 20 d ata is the
deviation th a t occurs in all the quantities for c > 15 mol% (R < 7).
The “ dc”
conductivity cr0 goes through a m aximum, e0 gradually saturates, r c has a
m inim um and the param eter a increases from zero. These results raise the
interesting question whether the
(low tem perature) glass-forming ability of
LiC l/H 20 solutions is related to the non-Debye behavior observed at room (i.e.
high) tem perature. All the dielectric spectra of RbCl and CsCl aqueous solutions
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-110-
appear to be Debye-like apparently up to the solubility lim its, during which the
“dc” conductivity of the solution monotonically increases w ith the concentration,
while r c (= r D) decreases. For all three ionic aqueous systems, the decrease of the
relaxation tim e of the solution r c from th at of pure w ater was found to be
proportional to the solution conductivity crQ. Earlier work by Sridhar and Taborek
(1988) showed th at the tem perature dependence of the “dc” conductivity of 15
mol% L iC l/H 20 solution tracks th at of the solution viscosity. This established an
intim ate connection between the conductivity properties and the structural
relaxation associated with the viscosity.
A similar connection has not been made w ith regard to the dielectric
relaxation time, and is addressed here. In addition the following questions are
discussed: How does the characteristic relaxation tim e of a glass forming ionic
solution scale with tem perature and relate to the “dc” conductivity? To what
extent do the ions disturb the w ater hydrogen bond network?
V.3.1 Choice of the fitting function
The polarization decay function <^(t) for the glass forming liquids is
generally described in terms of a Kohlrausch-W illiams-W atts form
<^(t) = exp [ - ( t / r )v]
(V.3.2)
in the tem perature range of interest. Numerical evaluations have shown th at in the
frequency domain, the Cole-Davidson dielectric function corresponds closest to the
KW W stretched exponential response in the tim e domain (Cole, 1989). A randomwalk model has been proposed (Bendler, 1985; Shlesinger, 1984) to interpret the
KW W behavior. The Havriliak-Nagami function (Havriliak and Negami, 1966) is
also used to fit the dielectric spectra of glass forming m aterials.
However, the main difference between Cole-Cole, Cole-Davidson and
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- Ill -
Havriliak-Nagami functions is the shape of the dielectric spectrum , not th e location
of the absorption peak as can be seen from Fig. III. 2. They are all different
manifestations of non-Debye behavior. Also from the discussion in C hapter III,
dielectric dispersion of Cole-Davidson form may appear to th e superposition of two
near by relaxation processes. W hat we are interested in here are the tem perature
dependence of the characteristic relaxation tim e and the trend of deviation from
Debye response. Therefore, the choice of a particular form among the various nonDebye functions is not im portant for the present work. The Cole-Cole dispersion
function is used to fit to dielectric spectra.
V.3.2 Experim ents and results
The m easurem ent technique is described in C hapter II. The tem perature
was varied by cooling in a jacket surrounded by liquid nitrogen. Tem perature was
controlled w ith an Omega controller. The rapid d ata acquisition (1 sec) enables
d ata to be taken on the fly, and is much quicker than the times required for
equilibration.
The tem perature dependencies of e'(u) and e"(u>) for LiCl:RH20 (R=5,7,20)
solutions were measured and fitted by the Cole-Cole expression Eqn. (III.2.25). As
the tem perature decreases, the static dielectric constant of the solution eso, the
relaxation tim e r c and the relaxation tim e distribution factor a all increase,
the
“dc” conductivity decreases. The fit param eters r c and <r0 are listed in Table V.IV
for various concentration. The expansion and contraction of the teflon filling inside
the coax, w ith respect to the probe tem perature, affected accurate determ ination of
e0 and
of the solutions a t lower tem peratures, e' and e" versus frequency at
several tem peratures and the corresponding fits are plotted in Fig. V.7(a) and (b)
for the LiCl : 5H20 sample.
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—112 —
Table V.IV. Dielectric relaxation tim e r c and Conductivity <r0 for LiCl:RH 20 , R =
20, 7 and 5.
Solution
T (K)
Tc ( Ps )
ao( n s )
a
Li Cl : 20H2O
296
8.69
13.5
0
288
10.7
1 1 .2
0
278
13.3
8.5
0 .0 1
268
19.0
6 .0
0 .0 2
262
2 2 .8
5.0
0.03
293
7.22
17.1
0 .0 1
289
7.12
16.2
0.03
286
7.67
15.1
0.05
283
8.05
13.9
0.04
278
9.39
1 2 .2
0 .0 2
273
10.4
10.5
0.03
268
11.4
9.0
0.06
263
15.6
7.2
0.07
258
18.1
6 .0
0.08
253
23.3
4.7
0.09
248
29.6
3.6
0.07
243
37.8
2 .8
0 .1
238
50.3
2 .0
0 .1 2
298
7.63
14.8
0.05
272
11.03
9.0
0.04
266
12.35
7.2
0 .1 1
253
16.84
5.0
0.19
235
49.78
2 .1
0.29
225
99.93
1 .0
0.32
223
130.7
0.9
0.33
218
217.1
0 .6
0.36
212
284.2
0.44
0.38
LiCl : 7H20
LiCl : 5H20
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—113 —
The measurement tem perature range of the LiCl : 20H2O sample was
lim ited by crystallization of the sample. For R = 5 and 7 samples, the rapid increase
of the relaxation tim e restricted the accuracy of lower tem perature range
m easurements.
The dielectric spectra of all solutions, become increasingly non-Debye upon
cooling to low tem perature. At the same time, the characteristic relaxation times of
all solutions increase rapidly and the conductivity dram atically decreases. The data
for
tc
and er0 were fitted to both a power law form :
. i T rMC
’ ’C
'o It1 “ t1 1 ■I7 ,
TMC
(V .3.3)
= <7oo I T T T<rMC f
(V.3.4)
-CTMC
and the Vogel-Fulcher-Tammann form
r c = Ar exp (T BI, )
1 ~ l T0
(V.3.5)
<r0 = k a exp (rp
)
1 ~ Lao
(V.3.6)
Table V.V. Param eters for power law fits for LiCl/H 20 solutions.
^PL
9
T
223
0 .0 1
1.82
215
0 .0 1
1.98
209
0 .0 1
2.04
210
0 .0 1
1.95
204
0.03
2.07
198
0.05
Solution
7
T
LiCl:20H2O
1.52
LiCl:7H20
LiCl:5H20
tm c(K
)
t m c (K
)
^PL
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-114-
Table V.VI. Power law fits for LiC l/H 20 solutions.
Solution
B t (K) T to(K)
LiCl:2H20
319
LiCl:7H20
LiCl:5H20
6 Vf t
Ba(K ) T cto(K)
171
0 .0 1
441
155
0 .0 1
370
163
< 0 .0 1
381
164
< 0 .0 1
341
159
0 .0 1
458
143
0 .0 2
VFT
T he value of the relevant fitting param eters are listed in Table V.V for the
power law and Table V.VI for V FT. Reasonably good fits were obtained to both
the above forms - this may possibly be due to the relatively narrow tem perature
range. The power law fit is m otivated by earlier observations on the T-dependence
of th e viscosity (Taborek, et al., 1986) and the conductivity (Sridhar and Taborek,
1988) and also has some justification from the mode-coupling theory. D ata and
both fits for r c are shown in Fig. V .8 and for cr0 are shown in Fig. V.9.
V.3.3 Tem perature dependence of LiCkRHoO (R < 7)
One feature of the glass forming solutions with R = 5 and 7,
is th a t the
power law fits to the relaxation time r c (T) and the “dc” conductivity cr0 (T) yield a
characteristic tem perature T MC around 205K, which is about 65K higher than the
glass transition tem perature 140K. The 65K difference is close to th a t predicted by
the mode coupling theory, which suggests th a t there is a ergodic-non-ergodic
transition at a tem perature 30-50K higher than the glass transition tem perature.
Although, we can not distinguish which type of fit is b etter for our data, we note
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-115-
th a t the characteristic tem peratures of ~ 160K extracted from the V F T fits, are
above T g, instead of 30-50K below as is th e more common experience.
The present result is consistent w ith results for the viscosity of a R ~ 7
solution by Taborek et al. (1986), which obey a power law fit w ith power ji ~
-2 .0 8 and T^0= 207K. Although the transition is not a sharp point, it appears to
be a significant tem perature which marks the boundary between two types of
viscous behavior: power law for T > T ^ 0 and approxim ately Arrhenius for T < T^0.
The
second
feature
which
emerges
from
the
results
is
the
close
correspondence between the dielectric relaxation tim e and the conductivity for the
glass-forming solutions of R =7 and 5. Combining the viscosity m easurem ents made
by Taborek et al. and the conductivity results by Sridhar and Taborek on a R ~ 7
sample, it is very appealing to speculate th at, for glass forming solutions,
r c(T) «
a j ?(T)
(V.3.7)
This is apparently true in our data for the high concentration (R < 7) solutions,
b ut is inapplicable to the water rich sample R=20.
In section V.2.3, it is shown th a t the alkali-halide AC1 additives modify the
dynam ic response of the solution by shortening the Debye dielectric relaxation tim e
w ith increasing concentration as
^"solution
Twater ' P &0
(V .3 .8 )
This was established at 25C by varying ion size of A. The fact th a t the
dielectric relaxation time of the solution is not simply th at of unperturbed water
molecules is further confirmed by the tem perature dependence reported here. By
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-116-
fitting the dielectric relaxation d ata of supercooled pure w ater from -
21C
to 32C,
provided by Bertolini et al. (1982), we obtain
T w ater(T )
=
2 .0 x l 0 5 ( T
-
2 1 1 )-226
(V.3.9)
whose exponent (2.26) is much bigger than th at of the LiCl:20H2O sample
(7
=
1.52).
V.3.4 Connection to supercooling properties of LiC l/H ?0
Fig. V.10 summarizes the dielectric param eters of L iC l/H 20 at T ~ 25C.
Shown are the dielectric relaxation tim e r D and the Cole-Cole param eter a as a
function of concentration.
Two significant features of the d ata are apparent :
(a) For c < 10 mol % (R >10), the relaxation is Debye like (a = 0), and
r c = r D decreases w ith increasing c. These concentrations also freeze, but at
depressed tem peratures.
(b) For c > 15 mol% (R<6.7), the relaxation is non-Debye (a > 0) and r c
increases w ith increasing c. Also these higher concentrations can be supercooled.
Thus there appears to be a plausible connection between the Debye-like
dielectric response of the liquid at high tem peratures, and th e ability to supercool
it. This connection is possible because the dielectric response is dom inated by the
free w ater, and we assume th at freezing is also determ ined by the ability for
extended hydrogen bond networking in water. (The concentrations are still too low
to view the solution as a molten salt, since the solubility lim it at 25 C is about 30
mol %).
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—117 —
Each ion gathers around it a cloud of w ater molecules, and the dielectric
m easurements suggest that the hydration layer is strongly bound. The solutions
m ay be regarded (as we do in our analysis), as composed of two components : a
m atrix of hydrated ions in a medium of water. If the concentration is sufficiently
low, the hydrated ions are well separated, and the w ater “ m edium ” is identical to
bulk water, except th a t it is divided into connected pieces due to the presence of
the hydrated ions. The prim ary evidence for this is the weak effect on r D, and the
continued Debye like response (a = 0) until concentrations c > 10 mol% are
attained. The disturbance of the H-H network by the existence of ions is equivalent
to raising the solution tem perature. We conclude th a t for c < 10 mol%, the
structure of the free water in the solution is basically unaffected, and the strong
w ater-water correlations, such as the H-H bonds, continue to be present.
F urther evidence for this picture comes from measurements of the dielectric
response of m ixtures of water with strongly associated liquids such as glycerol,
propanol, ethylene glycol, etc. (Rigos, to be published). Even small additions of
these strongly hydrogen bonded liquids dram atically affects both r D and a. (Details
of measurem ents on these liquids will be presented elsewhere). This is because
these molecules strongly disrupt the hydrogen bonding of the water molecules, to
the extent th at the solution cannot be regarded as a two component m ixture, but
rather as a microscopically disordered liquid. The m ain point which we wish to
utilize here is the observation, common to both the liquid m ixtures and the ionic
solutions, is th a t a non-Debye response in w ater signals disruption of the long-range
correlations, arising from the hydrogen bonding.
W hen the ionic concentration is increased, the ionic separation decreases to
the point where the local water-water interactions are affected, and the two
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-118-
component picture breaks down. This is signaled by a > 0 for c > 10 mol%.
W ithin the frame-work described above, the rigid ions act to confine the “ free”
w ater molecules, preventing communication with another “ free” w ater molecule,
thus disrupting the H-H bond network. From our m easurements of e0, we can
directly determine the water-water separation distance at which the hydrogen bond
network is disrupted.
The e0 results of in section V.2. at room tem perature for solutions of
different concentrations directly yield the microwave dielectric excluded volume in
the solution, i.e. the volume not available to the free water, and then the effective
radius of the space around an ion available to a free w ater molecule can be
estim ated by
t h = [ ^ ( l - v ) / c / N 0/ 2 ] 5
(V.3.10)
rfw is plotted vs. mol% concentration in F ig .V .ll, and is seen to decrease with
increasing concentration. As Fig.5 shows, each w ater molecule is increasingly
confined as the ionic concentration increases, due to the presence of the ions.
Around 10 mol% (R =10), the radius of the available space for a free water
o
molecule falls to about 2 A, which is comparable to the average radius of the
volume a bulk w ater molecule occupies at 4C, estim ated as [3*18/(4 ttN0)]1^3 = 1.9
o
A.
The confinement picture and the relation to the Debye relaxation suggests
th a t the interplay between the translational properties and the re-orientational
relaxation is im portant. This appears to have theoretical foundations, based upon
recent work by Bagchi and Chandra (1990). Using a hydrodynamic continuum
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-11 9 -
model, they show an interesting trend from Debye to non-Debye relaxation as the
diffusion tim e crosses the reorientational time. Thus in liquids where translational
contribution to polarization relaxation is significant, the response is Debye, whereas
in liquids where the orientational mechanism of polarization relaxation dominates,
the dielectric relaxation is non-Debye. This picture is qualitatively consistent with
the observations in this paper. Although a quantitative analysis remains to be
carried out, the approach taken in this work is novel, in th a t it provides an
interesting way of tuning diffusion times by varying additive concentration in
solutions.
The connection to crystallization becomes apparent if one recognizes th at
crystallization of water takes place through a macroscopic H-bonded network, and
is accompanied also by an increase in volume. The strong confinement of individual
w ater molecules by the ions not only disrupts a macroscopic network, but also the
strong electrostatic forces prevent needed volume expansion from occurring, thus
preventing crystallization in the high concentrations solutions.
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—120 —
Fig- V .l. e data and Debye fits for various LiCl/H^O concentrations.
100
1.09M
5
1.35M
2.14M
10
FREQUENCY (GHZ)
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—121 —
Fig. V.2 Cole-Cole plots for c = 7.2 M (diamonds), 9.62 M (+ ) and 13.38 M (o)
LiCl/H 20 solutions. The lines represent Eqn. (III.2.25) w ith param eters from Table
V .I.
20
15
10
5
0
0
5
10
20
25
30
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-122-
Fig.V.3. Frequency dependence of (a) e' and (b) the conductivity cr, between 45
MHz and 20 GHz, for R bC l/H 20 solutions, o’s (pure water), x ’s (0.54M), diamonds
(1.1 M), -f’s (2.7M) and filled squares (5.2 M). Similar results were obtained for
LiCl and CsCl solutions. Solid lines are the fits.
100
80
60
G'
40
20
0
0
5
0
5
10
15
20
15
20
100
80
I
60
I
40
0
10
FREQUENCY(GHZ)
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-123-
Fig.V.4. Dependence of the dielectric relaxation tim e r D on the dc conductivity cr0
for LiCl (diamonds), RbCl (+ ) and CsCl (o) solutions. Inset: Variation of r D with
concentration c, including d ata at high concentrations for LiCl solutions not shown
in m ain figure.
10
9
;o
8
Ox
<73
0 2 .5
3
5 7 .5
1 0 1 2 .5 15
c(M)
<=>
7
6
5
0
10
20
30
40
50
cr0 (m h o /m )
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-1 2 4 -
Fig.V.5. Behavior of the static dielectric constant eg0 w ith dc conductivity cr0 for
LiCl (diamonds), RbCl (+ ) and CsCl (o) solutions. Inset: V ariation of eg0 with
concentration for the same solutions.
120
100
8 0 ir
100
60
40
80
20
O
w
s^O.
c(M)
60
40
20
0
10
20
30
40
50
<j o (m h o /m )
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-125-
Fig.V. 6 . Hydration numbers vs. concentration for LiCl (diamonds), RbCl (+ ) and
CsCl (o) solutions.
10
8
6
o +
Nhyd
4
Oo
2
0
0
1
2
3
4
C o n c e n tr a tio n (M)
5
6
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-126-
Fig.V.7. (a) Real and (b) imaginary part of the dielectric constant versus frequency
for LiCl:5H20
solutions at T=272K (x's), 253K (o's), 235K (+'s) and 212K
(diamonds).
40
30
20
10
0
20
15
a
o
A
a
b
o
5
10
FREQUENCY(GHZ)
15
20
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-1 2 7 -
Fig. V . 8 . Cole-Cole relaxation tim e versus tem perature for LiCl:RH 20 , R = 5 (x's),
R = 7 (o's) and R =20 (+ 's) samples. Solid lines axe corresponding power law fits.
2
M
O
h
1
200
220
240
260
380
300
T(K)
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-128-
Fig. V.9. “dc” conductivity cr0 versus tem perature for LiCl:RH 20 , R = 5 (x's), R =7
(o's) and R =20 (+ 's) samples. Solid lines axe corresponding power law fits.
100
a
o
si
a
o
b
200
220
240
260
280
300
T(K)
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-1 2 9 -
Fig. V.10. Dielectric relaxation r c (o's) and Cole-Cole param eters a (+ 's) as a
function of concentration mol% ( = 1 /R ) for LiC l/H 20 at room tem perature.
10
0 .1 5
Debye
n on—Debye
9
0.10
8
0 .0 5
7
0.00
6
R=20
e.7
5
- 0 .0 5
>
10
15
20
C o n c e n tr a tio n (mol%)
25
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. V .ll. The average radius of the volume available for free w ater molecules
around an ion in LiC l/H 20 solutions, versus concentration.
,r r i
10
1 1
'
r~rnr p i r i i | i i 1 1 | 1 1t i | r
I I
a
_o
-
6
-
o
o
3
-
-
0
o
-
-
o
-
o
2
R-20
10
6.7
o
6
O4
-
I_J , 1 , i i i 1 i i j i 1 i i i_ i 1 i i 1 , 1 , 1 1 1
0
0
5
10
15
20
C o n c e n tr a tio n (mol%)
25
30
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—131 —
CH A PTER VI
HYDRATION OF BIOLOGICAL MATERIALS
Structural and dynamical aspects of heme protein hydration shell are
studied via dielectric spectroscopy. Since “bound” w ater plays an im portant role in
understanding the origin of resistance towards freezing and dehydration shown by
most anim al and plant tissues, the nature of the dielectric polarization responsible
for th e hydration water relaxation has been under discussion for a considerable
period of tim e. Using the new technique described in C hapter II, from the dielectric
response of solute modified “bulk” or free water in protein solutions, the num ber of
bound w ater molecules in the hydration shell of m etM b, MbCO and H b 0 2 were
determ ined. W ithin experimental error, no detectable differences were observed
between m etM b and MbCO. The Debye single relaxation response of the “bulk”
w ater in these aqueous solutions indicates th a t residence tim e of water molecule in
the hydration shell is much shorter than its rotational reorientation time. In
addition, an in situ experiment was performed in order to observe a change in
hydration when deoxyHb was converted to H b 0 2 upon oxygenation.
Effects of conformational change on the dielectric spectrum of m etM b were
observed. Proteins fold spontaneously into a stable conformation determ ined by
their amino acid sequence. Protein folding, or more specifically, the relations
between amino acid sequence, folding pathways, kinetics and the functional spatial
arrangem ent of a polypeptide chain, is presently the least well understood step in a
“central dogma” relating storage of genetic inform ation with its expression by
protein functions. By varying the pH of the solvent or tem perature of the solution,
a reversible conformational change can be introduced. The microwave dielectric
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-1 3 2 -
excluded volume of the denatured (or unfolded) m etM b at pH ~ 4 is found to be
substantially smaller than th at at pH ~ 7.
An application th at is relevant to the new technique of Gigahertz dielectric
spectroscopy is microwave hyperthermia. T hat interaction of microwaves with
biological m aterial produced heat has been known since the tu rn of the century.
However, the exploitation of this phenomena as a therapeutic tool has taken place
only recently. Of particular interest is the use of radio frequency and microwave
heating as an agency of tum or destruction in cancer therapy. The measured
dielectric constant of hum an blood at microwave frequencies provides im portant
information for the design of the antenna arrays used to focus the heating energy.
One of the widely used blood substitutes - a saline solution, turned out to have
different dielectric response from th at of hum an blood at microwave frequencies.
Proper dielectric substitutes were designed through the dielectric measurements.
VI. 1. Sample preparation and characterization
Horse heart myoglobin was purchased in a lyophilized form from Sigma
Chemical Company, St. Louis, MO. Human oxyhemoglobin (H b 0 2) was obtained
in a frozen and concentrated form from the University of M assachusetts at Lowell.
A. MetMb
Samples of native metM b were prepared by dissolving lyophilized Mb in
deionized water. The sample was then centrifuged and the concentration was
determined
by
measurements
of
the
absorption
spectrum .
The
low
pH
measurem ents utilized a 0.1M citric acid/ 0.2M potassium phosphate buffer.
B. MbCO
A m etM b sample was placed in a container well sealed with a rubber
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-1 3 3 -
septum and paraffin tape. The protein is degassed w ith N 2 (Med-Tech, Medford
MA or Wesco Co., Billerica, MA), using two needles inserted above the sample
surface to allow the gas to flow. An additional volume of w ater is also degassed and
a small quantity of sodium dithionite ( N a ^ C ^ ) is added through the septum cover
w ith a syringe to make a saturated solution. Then a few drops are injected in the
degassed myoglobin solution followed by equilibration with
1
atm osphere of CO
gas.
C. DeoxyHb
About
1
m l of concentrated, frozen oxyhemoglobin solution ( ~ 4 mM
protein, 16 mM heme) originally m aintained at liquid nitrogen tem perature was
thawed inside a refrigerator. Then it was transferred to a sealed container. The
protein was degassed with N 2 for about 50 minutes and a few drops of saturated
sodium dithionite solution were injected. The sample was tilted to create a larger
surface area and rotated every 5 to 10 minutes. The absorption spectrum was
checked for deoxygenation by diluting 5 jA of the concentrated deoxyHb solution
into 2 ml of degassed deionized water. The concentration was determ ined from the
absorption spectrum after the diluted sample was oxygenated.
B oth the sample and the calibration standard acetone are kept at 25°C in a
tem perature stabilizer, which utilizes w ater at a fixed tem perature from a
circulating bath (Neslab Instrum ents, Newington, NH).
Densities of solutions were measured using a DA-110 Density/Specific
G ravity M eter from M ettler Instrum ent Corporation, Hightstown, NJ. The
accuracy is ±0.001 kg/liter. W e assume the density of the solutions is independent
of ligation state.
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VT.2 H ydration shells
VI.2.1 Dielectric spectra and num ber of hydration w ater molecules
The d ata de-embedding process described in C hapter II, extracts the real
and imaginary parts of the dielectric function from the m easured reflection
coefficients at frequencies from 45 MHz to 20 GHZ.
The d ata were fitted to a
model of the dielectric response in order to extract useful param eters. Except for
the rise in e'(u;) at low frequencies below 1 GHz, due to contact polarization,
observed in Fig. VI. 1, VI.2 and VI. 3, the d ata are well fitted by a simple Debye
form plus a dc conductivity due to the ions :
e = e' - ie" = eTO +
“
1 +
1W Ip
where e0 is the static dielectric constant,
- i
w ev
(VI.2.1)
v
'
represents the plateau value to which e'
would fall at frequencies much higher than 1/(27ttd), t d is the Debye relaxation
tim e, cr0 is the dc conductivity of the solution which is essentially due to the free
ions, and ev = 8.85 x 10' 12 F /m is the perm ittivity of vacuum. The low frequency
rise in e" is caused by the finite conductivity a0. The Cole-Cole plot of the d ata for
m etM b in H20 (4.03 mM) is shown w ith a corresponding fit to Eqn. (VI.2.1) in
Fig. VI. 1. An experimental w ater spectrum ( e0 = 78.3, eTO = 5.6 and r D = 8.5 ps )
is plotted for comparison. Also shown are the dielectric spectra for M bCO in H20
(3.55 mM) in Fig. VI.2 and H b 0 2 in H 20 (4.3 mM) in Fig. VI.3. M easurements
were also performed on metM b samples at concentrations of 1.81 and 5.13 mM.
The fitting param eters for the protein solutions es0, esoo, r SD and <r0 are listed in
Table VI.I. The results show th a t the dielectric param eters of the solutions are
similar to those of bulk w ater, except of course for es0. Therefore, it is concluded
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-13 5 -
th at the higher frequency part of the spectrum is only due to the free w ater in the
solution, w ith a modified low frequency plateau due to the presence of the hydrated
w ater molecules, which act as dielectric voids of volume V D (illustrated in Fig.
IV .l). The microwave dielectric excluded volume VD is assumed to be a spherical
void w ith dielectric constant of eex = ewoo = 5.0, and is designed to describe both
electronic polarizability and dipolar polarizability of protein molecules; the latter
includes elastic deformation of bonds and angles, and rotation of polar groups
around bonds. VD can be obtained from the analysis described in C hapter IV. In
the previous study of ionic solutions, the conductivity (cr0) was found to contribute
to the decrease of the static dielectric constant by the am ount
7
cr0 at low ion
concentration. To correct for the effect of solution conductivity, es0 +
7
er0 was
used for the pure contribution due to the protein solution, in place of eg0 in Eqn.
(IV.2.3) in Chapter IV to calculate the excluded volume of proteins. The value
7
=
0.9 Q-m deduced from the previous work is assumed to be valid here. Then the
dielectric excluded volume per protein molecules V D, num ber of hydration water
molecules Nhyd, and the derived weight ratio whyd is calculated following the
procedure given in Chapter IV. Results are listed in Table IV.II. The possible
related errors were also estim ated according to the analyses in C hapter IV. For a
typical myoglobin solution with c = 4.03mM at 25C,
~ 2%,
6 ds ~
O.OOlkg/liter,
8es0 ~ 0.3, es0 = 70.2 and Nhyd= 360, ANhyd = 52. For other solutions, ANhyd
introduced by the measurement error of concentration c, density ds and es0 are
listed in Table VI.III. The pH3.9 metM b sample is prepared by diluting
1 ml
of a
pH7.7 sample in pure water with 1 ml of a 0.1M citrate acid / 0.2 M potassium
phosphate buffer using a graduated pipette. We estim ated th a t the volume error is
less th an 0.06 ml, which corresponds to a 3% error in dilution. Assuming a 3%
m easurem ent error in the absorption spectrum leads to
a total error in
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-13 6 -
concentration of ~
6%
for the low pH metM b sample. The values are listed in
Table II for metM b, MbCO and HbC^.
TABLE VI.I. Debye fit parameters for metM b, MbCO and H b 0 2 in aqueous
solutions.
metM b
MbCO
H b02
[c] (mM)
1.81
4.03
5.13
2.60
3.55
4.14
pH
6 .8
7.5
7.7
3.9
7.5
~ 7
T ( °C )
23
25
25
25
25
25
es0
75.7
70.2
68.4
74.4
69.7
52.34
^800
4.0
7.0
5.5
4.5
7.0
6.5
TsD ( PS )
8.4
9.2
9.1
7.9
9.0
1 0 .1
tr0 ( Qm) ' 1
0.19
9.07
0.18
0.4
0.76
0.47
Table VI. II. Protein
i information extracted from the microwave dielectric
measurements.
metMb
MbCO
H b02
c(mM)
1.81
4.03
5.13
2.60
3.55
4.14
pH
6 .8
7.5
7.7
3.9
7.5
~7
ds
1.008*
1.019
1 .0 2 2
1 .0 1 1 *
1.016*
1.071
T ( °C ) 23
25
25
25
25
25
V D(nm 3) 28.7** ±2.0
33.0 ±0.8
31.5 ±1.2
22.8 ±1.9
35.0 ± 1 .4
108.6 ±
whyd
0.24 ± 0.09
0.34 ±0.05
0.26 ±0.06
0 .0 1
±
0.40 ±0.10
0.29 ±0.02
Nhyd
^40 ± 90
360 ±50
270 ±60
10 ± 1 1 0
420 ±110
1030 ±70
0 .1 1
1 .2
* Estim ated by interpolation between c=0 - 4.03 mM
** At T = 23°C, ew0 = 79.2
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- 137-
Table VI.III. Possible errors in num ber of hydration w ater introduced during the
measurements.
metM b
MbCO
H b02
c (mM)
1.81
4.03
5.13
2.60
3.55
4.14
^sO
<5NX
0 .2
0.3
0.4
0.4
0 .6
0.3
59
41
44
83
95
48
*M/[c]
0.03
0 .0 2
0.03
0.06
0.03
0 .0 1
39
28
40
63
44
47
0 .0 0 2
0 .0 0 1
0 .0 0 2
0 .0 0 2
0 .0 0 2
0 .0 0 1
dN 3
61
14
22
43
31
13
ANhyd
93
52
63
112
109
68
*n2
H
(kg/i)
The equation for ANhyd shows th at at higher solution concentration,
error in hydration w ater molecules becomes smaller.
the
This is the reason why the
1.81 mM sample has a bigger error ANhyd. Also, from the estim ated error provided
in Table VI.III, it can be seen th at deviations of es0 in repeated dielectric
m easurem ents are the dominant error source in ANhyd. The present relative error
for the absolute num ber of hydration waters associated w ith Hb is 7%, which is not
quite good enough to detect the increase of 60 hydration water molecules (Colombo
et al., 1992) predicted to bind during the T-+R transition in hemoglobin.
However, from statistical measurements on a 4mM metM b sample, it is
found th a t
6 N X~ 8
for successive measurements carried out in a
20
m inute period if
the probe is kept in the solution. In contrast, an experiment in which the probe is
dipped in and out over a 20 m inute period gives £NX~ 21. Based on these
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-1 3 8 -
observations, an in situ procedure that, along with an accurate concentration
determ ination, was developed and should minimize the error in hydration number.
VI.2.2. Discussion: statics and dynamics of the hydration
The experimental quantity that is directly obtained from the analysis of the
measurem ents is the microwave dielectric excluded volume VD per protein
molecule, shown in Table VI.I. The im portant point is th at, for native proteins,
this volume is larger than the bare molecular volume in the crystal, VMb ~ 25. 3
nm 3 and V Hb ~ 85.4 nm 3 (Dickerson and Geis, 1969 ). The excess is attributed to
the hydration shell, and knowing VD, the weight ratio whyd and hydration num ber
Nhyd are easily obtained.
The weight ratios obtained agree with low frequency measurem ents ( G rant
et al., 1986; Hanss et al., 1967; Von Casimir et al.,
1968; Schlecht et al., 1969),
which yield values in the range 0.2-0.4. The num ber of hydrated w ater molecules
around m etM b seem to be reassuringly concentration independent w ithin the error.
Nhyd for oxyhemoglobin is about 3.5 times th a t of myoglobin. The average value for
Nhyd from the measurements on 3 samples of m etM b is 290 ± 60.
o
o
Using spheres with rMb = 18.2 A and rHb = 27.3 A to approxim ate Mb and
o
o
Hb molecules, the thickness of the hydration layer is 1.7 A and 2.3 A, respectively.
R am an spectroscopy ( Tao and Lindsay, 1989 ) indicated th a t DNA prim ary
hydration shells do not have an “ ice - like ” structure, which may also be true for
Mb and Hb. The average molecular size of bulk water molecules, w ith density d w
o
= 1, is 3.1 A. From the volume estimates, the hydration shell of native m etM b is
equivalent to 268 liquid w ater molecules (dw= 1), while the shell of oxyHb consists
of 806 w ater molecules.
These values for Nhyd are smaller than the hydration
num bers derived from the direct dielectric measurements. The above estimates
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-139-
suggest th a t the hydration shell corresponds roughly to a single atomic layer of
w ater surrounding the protein molecule and th at the hydration w ater is more
tightly packed than bulk w ater at 4°C.
From
previous studies on ionic solutions (Chap. V ), it is clear th a t buffer
ions will also bind w ater and lower es0, although to a m uch smaller degree.
Therefore, w ith the exception of the low pH solutions, only pure deionized water
(w ithout buffer) was used as the solvent for the measurements.
As is evident from the previous discussion, the relaxation tim e observed is
th at of free water, since it is close to th at measured in pure water. Thus, the
hydrated w ater does not exhibit any dynamical signature, but appears “ frozen”,
between 1 and 20GHz. There are varying estim ates of the relaxation tim e of bound
w ater in biological m aterials ( Tao and Lindsay, 1988; Gestblom, 1991 ), but our
results imply th at such a relaxation occurs at frequencies <
1
GHz for the proteins
studied here.
Similarly, the protein itself does not offer any significant dielectric resonance
in the region between 1 and 20 GHz. If we assume th a t the protein can be modeled
as a dam ped harmonic oscillator with an associated charge, according to Eqn.
(7.51) in Jackson’s E&M (1975), the dielectric response of a group of Z electrons, if
they move together w ith the same binding frequency w0 and subject to similar
damping forces characterized by
7
, is
47 rNZ 2 e 2
1
u)2 - u 02 - i
7
w
(VI.2.2)
Therefore,
47 rNZ 2e 2
(VI.2.3)
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-140-
at u; = w0, e" reaches m axim um
e»
c max
4?~NZ2e 2 __1
m
u;07
fVI o 4 '\
Iv
and
e" (u) __
f"
~
e max
72
( , I2
( W
-
u} u Q
, , 2 \2
CUq j
_L
+
r j l , ,2
7
(VI.2.5)
^
«" (wi/2)
1
at w1/2 where - „
= ±,
^ max
"
( ^02 - ^ l/2 2 )2
7 =
W02
”
(VI.2.6 )
( W 0 - U71 / 2 ) 2
I f w l/2 — wO + A
\j U 0
~
~
LOXj 2 )
If w1/2 = w0 - A
UJq
- bJ-i/n
T w0r - /( w0 - w1/2 ).2
'
i - =
<VI-2-8>
Assume f0 = 10 GHz and the resonance peak w idth is 1 GHz, we get
So th a t
7
7
+=
6.
45
7
_ = 6.13
is about 27r ( 1GHz ). Now we estim ate e"max w ith the above
7
value.
Suppose the mass in e"max is a fraction of the weight of a whole myoglobin molecule
m = ^
where Mw = 18.8 kg and if [c] is the
(VI.2.9)
concentration in mM.
N = cN 0 = N 0 c (m oles/m 3) = N 0 10' 3 c ( moles/1 )
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—141 —
[ c ] N0
Therefore
„
where ix>0 = 20 7r,
max
7
_
1 4tt N 02 e2
101 8 M w 7 W ( )
Z2 [c]
(VI2 10)
(V i.Z.lU j
Q
~ 27r ( frequency in GHz ). W e have
e"max = 1-57 x IQ' 10
(VI.2.11)
[c] —4, if Z is taken to be 100, and 6 is 10'3, e"max ~
6
x 10' 3 which is not visible.
Finally the spherical approximation to the hydrated volume used to derive
Eqn. (IV.2.3) may not be unreasonable, even though it ignores details of the
structures involved. This is partly because any anisotropies are averaged over in
the m easurement, which is inherently isotropic.
VI.2.3 pH dependence of metM b
New insights can be anticipated from structural characterization of both the
unfolded and the folded polypeptide chain under the conditions of the folding
milieu. Because water is excluded almost entirely from the interior of globular
proteins, different solvation of the polypeptide chain in
the unfolded and folded
forms m ust be an im portant factor. Myoglobin undergoes a global conformational
change
at
low
pH
which
leads
to
a
partially
unfolded
structure.
This
conformational transition is accompanied by a blue shift and broadening of the
Soret absorption band, which appears at 409.5nm in the native protein. The
population of native horse heart metM b as a function of pH is determ ined from
analysis of the optical absorption spectrum (Wei, et al.). The transition at pH =
4.7 is attributed to a transition from a folded to a partially unfolded structure, as
previously observed for sperm whale metM b at pH4.3 ( Sage et al. 1991 ). An
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-142-
im portant question is whether this is accompanied by a change in the hydration
num ber, and we addressed this issue w ith high frequency dielectric experiments.
O ur m easurem ent on a pH = 3.9 metM b sample revealed a decrease, within error,
Q
of the dielectric radius by about 2 A, using the analysis discussed previously. The
resulting excluded volume is close to th at calculated for an unhydrated native
protein.
The large error in the determ ination of Nhyd in the low pH m etM b sample is
prim arily due to the difficulty in obtaining an accurate value for the protein
concentration.
The tim e constant for the conformational transition appears to be
on the order of a m inute or less, and it was thus difficult to determ ine a reliable
value for the isosbetic point between the native and partially unfolded protein,
which would allow direct determ ination of the protein concentration at low pH.
T he concentration was therefore calculated from the measured sample volumes of
the unbuffered protein solution and the low pH buffer.
Unfortunately, measurements near the midpoint of the transition at pH4.7
were not able to be carried out, due to sample precipitation at the millimolar
concentrations required for the dielectric measurements. Under the more acidic
conditions of the experiment the sample is completely soluble.
V I.3. In situ observation of the influence of oxygenation on the hydration proteins
The fairly rapid data acquisition times ( < 1 sec in certain situations ) leads
to the possibility of observing in situ changes in sample properties. This can be
achieved in measurements in which deoxygenated hemoglobin solutions are
oxygenated, and which axe carried out using special cells not requiring sample
substitution.
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-143-
Here nRT is used to denote the difference in the number of hydration waters
axound each protein molecule upon oxygenation. For in situ m easurem ents, if the
density is assumed to be the same in the oxygenated and deoxygenated solutions,
then according to Eqn. (IV.4.1),
nRT =
(h018 [c] dw
(VI.3.1)
Here the error due to the uncertainty in the solution density ds is elim inated so
th a t higher sensitivity to relative changes in sample properties can be achieved
th an in absolute measurements.
The experiments were initiated when the freshly prepared deoxyHb solution
was placed in a cell filled w ith nitrogen gas. The dielectric probe was positioned at
a depth of 0.5 cm in the solution. D ata were taken about once every m inute. After
10
m inutes, pure
0 2
was introduced into the cell and measurements were recorded
at one m inute intervals up to 50 mins. Fig. VI.4(a) shows the real p art of the
dielectric constant e' at
1
GHz vs. tim e for a 4 mM deoxyHb solution. At 1GHz, e'
is a good measure of relative changes in es0. We note the decrease with tim e for
the first
10
m inutes in panels b and c, which is probably due to tem perature drift
in the instrum ent or to small changes in the probe-liquid interface. A fter the
introduction of
0 2
in (a), a downward trend begins between
1 0 -2 0
m inutes,
at
which tim e e' begins to increase, and ultim ately levels off. During the measurem ent
a color change was observed and an aliquot of sample, diluted in degassed water,
revealed an absorption spectrum of completely oxygenated Hb at the end of the
measurem ent sequence.
A t other frequencies, plots of c' vs. tim e show the same trend, but with
smaller am plitude. The long tim e lim it of d for oxyHb is higher than th a t of
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-144-
deoxyHb up to 20 GHz.
If there is no contribution from the solvent and no
changes in Hb volume take place during the oxygenation process, these results
would indicate th at the dielectric excluded volume of deoxyHb is laxger th an th at
of oxyHb. This suggests th at th e number of bound w ater molecules m ight decrease
upon oxygenation, contrary to the proposal (Colombo et al., 1992) th a t an
additional 60 w ater molecules bind upon oxygenation.
To exclude any possible effects due to the presence of dithionite and
degassed w ater, we repeated the same measurem ents on a
1m l
dithionite solution
(Fig. VI.4b) and on a 2ml sample of deionized water deoxygenated w ith N 2 gas
(Fig. VI.4c). The response of the dithionite/H 20 solution to the introduction of
oxygen is similar to th at of the deoxyHb solution, but is smaller and saturates
faster. W hat is most surprising is the response of the degassed water to 0 2. The
result shown in Fig.VI.4c is the same in situ measurem ent except th at the
dielectric probe was positioned about 1 cm below the liquid surface.
The above results contrast with measurements on deionized w ater performed
for 50 mins in air, which show only a small monotonic decrease in the real p art of
th e dielectric constant (Ae' ~ 0.4), due to the instrum ental drift.
The oxygen diffusion into a solution in our configuration can be modeled
w ith a one-dimensional diffusion equation. W hen this is done, we find th at the
predicted tim e scale for approaching the saturating oxygen concentration is much
longer than th at over which the dielectric signal changes in Fig. VI.4.
This
suggests th a t convection rather than diffusion may be the dom inant mode for
oxygen transport into the solution. The analysis of the tim e dependent response is
complicated not only by the convection, but also by the reactions of
0 2
w ith
dithionite and Hb. Nevertheless, if we assume th a t the increment in e' is ~ 1, the
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—145 —
upward trend of the d ata in Fig. VI.4a between 20-50 m inutes could indicate an
increase of ~
200
w ater molecules in the hydration sphere of H b 0 2. On the other
hand, we cannot exclude the possibility th at this is due to the interaction of
0 2
directly w ith the structure network of liquid water as suggested by Fig. VI.4c.
A lternatively, the small downward inflection th at takes place between 10-20
m inutes in Fig. VI.4a, could be taken as an indication of a T —>R transition th at
decreases the
protein hydration as predicted by Colombo et al. (1992).
Further
study is clearly needed in order to separate the different mechanisms affecting the
dielectric m easurem ents in the presence of oxygen.
The above results strongly suggest the surprising conclusion th a t the
dielectric properties of w ater are affected (at a level of 3%) by dissolved 0 2. This is
a hitherto unknown property of water. A dm ittedly, our results are preliminary, and
more complete experiments are underway to further study this new and surprising
result. Nevertheless, the present experiments indicate th a t the influence of 0 2 on
w ater cannot be ignored when studying oxygenation effects in biological samples.
V I.4 Applications to microwave hyperthermia
Accurate measurements of dielectric properties of biological substances are
essential for both fundam ental studies and biomedical applications. Microwave
energy can be effectively used in hypertherm ia treatm ent of tumors, therm al
angioplasty for rechannelizaton of arterial occlusion and other disorders. Dielectric
properties
of tissue,
which
determine
the
absorption
and
propagation
of
electrom agnetic energy through the tissue, is crucial for these kind of applications.
Since it is very difficult to obtain the distribution of actual power density within
the body directly, it is necessary to simulate hum an biological m aterials in
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—146 —
microwave frequencies. Using the new technique for the precise measurement of the
dielectric properties of hum an blood, phantom s and other substitutes were tested
up to 20 GHz. A previously sim ulated m aterial for high water content tissue at the
frequency range from 100MHz to 1 GHz was modified to m atch high frequency
dielectric data of hum an tissues.
VI.4.1 Results and analysis on the phantom and hum an blood up to 20
GHz.
The dielectric spectra of blood, llg /1 of N aC l/H 20 and the phantom axe
shown in Fig. VI.5.
The fit for the muscle phantom (Allen, et al., 1988) e0 = 74, eOQ= 4.6 and r
= 8.3 ps, gives the dielectric excluded volume 0.04 liter in 1 liter of solution which
indicates there is 96 wt.% of free water.
Compare to the w ater content given by
th e recipe in Ref. [7], about 0.5% of water is bound to other ingredients.
The
dielectric spectrum
of hum an blood seems to
be much more
complicated. It can not be well fitted into single Debye behavior
Cole-Cole
expression or the form with two Debye relaxation tim e for the whole frequency
range,, due to the complexity of the ingredients and the high viscosity. Human
blood contains about 45% of red cells, 1% of platelets and white cells, and the rest
is the plasm a which has 90% of water content.
However, (i) the dipole moments of big biological molecules are not able to
respond to the microwave frequencies; (ii) water molecules bound to the proteins,
etc., in blood, apparently have about one order of m agnitude longer relaxation
times due to stronger interaction with the biological molecules; they become
“ dielectric voids” . Therefore the high frequency end of the spectrum should only
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-1 4 7 -
represent the behavior of the free w ater dipole moment.
The fit to the blood spectrum from about 8 GHz to 20 GHz gives the values
in Debye form as e0 = 55.6, eQO= 4.5 and r= 8.4 ps, which indicates th a t it is only
the free w ater left for the high frequency end.
Using eex = 5 in Eqn. (IV.2.3) results in an estim ation of 76% of w ater in
blood, which disagrees w ith the blood water content of 50%. eu = 37 would give
50% of water content in blood, which indicates the proteins in blood have very
large polarization dipole moments.
The dielectric spectra in Fig.VI.5, show th at the salt w ater solution, which
im itates the conductivity of the hum an blood, is not a good dielectric substitute for
blood at microwave frequencies.
The microwave dielectric measurements on a muscle phantom and hum an
blood suggest th at besides the conductivity, the dielectric excluded volume, which
is not the real volume of ingredients, plays an im portant role in substitutes and
phantom s for microwave hypertherm ia (Brown). Large molecules with strong
polarizability may be necessary for the composition of such materials.
VI.4.2 Sugar solutions for substitutes at desired frequencies.
As shown in the above discussion, it is difficult and expensive to sim ulate
the dielectric spectra of hum an tissues in the whole microwave frequency range.
However,
the operating frequency range is usually much narrower for a specific
instrum ent. A sucrose/NaCl aqueous solution formula (A) for S-band frequency
range (lOO-lOOOMHz) is modified to obtain the desired frequencies properties as
shown in Fig. V I.6. Table VI.IV gives the proportions by weight for samples
corresponding to various compositions of simulated muscle m aterial.
Table VI.V
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-1 4 8 -
gives the measured results of the mixtures and hum an blood a t 8 and 10 GHz and
compares them w ith the published dielectric constant and conductivity.
Table VI.IV. Composition of simulated materials
NaCl (wt.%)
Sucrose (wt.%)
W ater (wt.%)
A
1.4
46.2
52.4
B
1.4
27.5
71.1
C
2.0
27.5
70.5
D
2.0
25.0
73.0
Table VI.V.
Comparison of various compositions.
Dielectric Constant
Conductivity
8GHz
10GHz
8GHz
10GHz
Ref.*
40
39.39
8.33
10.0
Blood
46
43
9
13
A
26
23
9
11
B
41.5
38
11
15
C
40.5
37
11
15
D
42
37.5
12
16
* Hartsgrove, et al. 1987.
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-1 4 9 -
It can be seen th a t increasing sucrose concentration decreases th e dielectric
constant and conductivity. The dielectric properties of sim ulated m aterials at high
frequencies axe dom inated by the am ount of sucrose.
Based on th e measured
dielectric properties, composition (B) has been chosen for microwave frequency
range studies around 8 GHz. In general the concentration of sucrose is the
determ ining factor in establishing phantom characteristics. This is because sucrose
molecules have significant effects on the w ater structure, which determ ines the
microwave dielectric response. The effect seems to be similar to alcohols in w ater as
will be discussed in the following chapter.
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-150-
Fig. VI. 1. Cole-Cole plot of metM b ( c=4.03 mM ) d ata ( o’s ) and the Debye fit
(—): e0 = 70.2,
= 5.5, r sD = 9.2 and cr0= 0.07 ( fim )_1. Also plotted are d ata (
+ ’s ) and Debye fit ( solid line ): e0 = 78.3, £^=5.5 and r wD = 8.5 ps for deionized
w ater at 25°C.
80
60
20
0
20
40
60
80
e'
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—151 —
Fig. VI. 2. e' and e" vs. frequency for MbCO ( c=3.55 mM ) d ata ( o’s ) and the
Deby fit ( - ’s ): e0 = 69.7,
= 7.0, r sD = 9.0 and a0= 0.76 ( fim J*1.
Also
plotted are the water d ata ( + ’s ) and fit for comparison.
100
80
e/
60
40
20
0
5
10
15
20
0
5
10
15
20
60
40
20
FREQUENCY(GHZ)
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—152-
Fig. VI. 3. e' and e" vs. frequency for H b02 ( c=4.14 mM ) data ( o’s ) and the
Debye fit ( - ’s ): e0 = 52.34, e^ = 6.5, rsD = 10.1 and cr0= 0.47 ( fim )*x. Also
plotted are the water data ( + ’s ) and fit for comparison.
100
80
e
i
60
40
20
0
5
10
15
20
0
5
10
15
20
60
40
20
FREQUENCY(GHZ)
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—153 —
Fig. VI. 4. e' at 1 GHz vs. tim e for (a) Hb in dithionite/w ater L ~ 1 cm, d ~ 0 .5
cm. (The gap is due to a com puter d ata transfer error).
55
e
54
53
52
51
50
10
0
20
30
40
50
30
40
50
(a)
7 9 .0
7 8 .5
ei
7 8 .0
7 7 .5
7 7 .0
10
0
20
(b)
80
LI
1—I—1~I I I I I I 1 I I I I T T T I
78 E € '
76
l l l l l'
02
?+H+b
74
72
70
10
20
30
40
50
(c)
Tim e ( m in . )
(b) dithionite/w ater L ~ 1 cm, d ~ 0.5 cm and (c) degassed w ater L ~ 2 cm, d
cm. Oxygen was let in at t = 10 min.
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—154 -
Fig. VI. 5. Dielectric spectra of human blood (o's) at 25C, and fit for high
frequencies (e0=55.6, €^=8.4, r=8.4ps, dashed line), muscle phantom (+'s) and fit
(e0=73.6, £^=4.6, r=8.3ps, solid line), llg /1 NaCl/H20 solution (solid squares).
100
80
e
I
60
C °< IO o 0 B W »
“ “ Oooooo
40
20
I
?
10
Frequency (Hz)
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—155 —
Fig. VI. 6. Real part of the dielectric constant e' and conductivity a vs. frequency
above 5 GHz for published data (o's), blood sample (solid squares), Formula (A)
(+'s) and modified formula (B) (x's).
80
60
20
50
20
+ +
Frequency (Hz)
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—156 —
CHAPTER VII
ALCOHOL AQUEOUS SOLUTIONS
Alcohol aqueous solutions form another class of m aterial th at has not been
extensively studied at high frequencies due to the instrum entation lim itation. In
order to further observe how solutes interact in the presence of the hydrogen
bonding network of water, concentrations from pure alcohol to pure w ater for
mono-alcohol (1-propanol and ethylene glycol), poly-alcohol (glycerol) aqueous
solutions were investigated systematically. The dielectric characteristics due to
interactions between different species of polar molecules, are distinctly different
from th a t of ionic and biomolecular solutions.
VII. 1 Dielectric spectra of alcohol aqueous solutions.
For pure mono-alcohols, up to three relaxation ranges have been observed,
among which the low-frequency one characterized by a single relaxation tim e is
dom inant for those alcohols where the Kirkwood correlation factor is large. Two of
the relaxations are in the frequency range below 20GHz. For poly-alcohols, one
relaxation range characterized by the Cole-Davidson equation described in Chapter
III have been found to occur frequently. However, due to the lim itation of the ColeCole or frequency graphic method, as shown in Chapter III, the Cole-Davidson
behavior is most likely superposed by several relaxation processes with close tim e
scales.
VII. 1.1 1-propanol/HoO
The
real and imaginary
parts
of the dielectric spectra for a few
concentrations vs. frequency are plotted in Fig. VII. 1.
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-157-
The dielectric behavior of the l-propanolx(H20 ) 1.x m ixture is clearly nonDebye. As is evident from the cr^-plot as shown in Fig. III.6 in chapter III, ColeCole form is not good fits of pure propanol. The form w ith two Debye relaxation
times:
1 + i U)T1
+
(VII. 1.1)
1 -f- iwT2
gives a b etter fit in general for the alcohol-rich solutions. The corresponding fitting
param eters are listed in Table VII.I.
Table VII.I. F itting param eters to two Debye relaxation form Eqn. (VII. 1.1) for
l-propanolx(H20 ) 1.x.
Concentration
Low / relaxation
High / relaxation
RMS+
Error S
X
^OO
Aj
r i(ps)
A2
t
1.00
2.7
16.9
326
1.3
8.8
0.11
0.80
3.0
17.5
196
2.0
13.1
0.17
0.61
3.2
18.5
129
3.5
15.6
0.20
0.42
3.4
16.9
86
5.8
17.1
0.18
0.19
4.2
22.5
46
20.0
15.3
0.30
0.074
3.6
39.7
40
21.5
9.2
0.30
2
VII. 1.2 Ethvlene Glvcol/HoO
Fig. VII.2 shows the real and imaginary parts of the dielectric spectra for a
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—158 —
few concentrations vs. frequency for the ethylene glycol aqueous solutions. A t large
x, the two Debye relaxation model in Eqn. VII. 1.1 gives a better fit as shown by
solid lines. The corresponding fitting param eters are listed in Table VII.II.
Table VII.II. F itting param eters to two Debye relaxation form in Eqn. (V II.I.1) for
(Ethylene glycol)x(H20 ) 1.x
Concentration
Low / relaxation
High / relaxation
RMS+
X
eoo
Ai
T j(p s)
A2
t2
Error
1.0
4.2
33.0
123.1
3.4
11.9
0.12
0.72
4.2
37.1
85.2
4.7
13.3
0.17
0.62
4.2
40.7
75.5
5.1
12.0
0.32
0.46
4.2
41.9
55.7
6.4
12.2
0.16
0.22
4.4
44.0
31.0
15.1
11.8
0.22
0.088
3.8
35.4
19.7
32.6
9.2
0.25
V II.I.3 Glvcerol/H20
The
real and
imaginary
parts
of the
dielectric
spectra for a few
concentrations of glycerol aqueous solutions vs. frequency are plotted in Fig. VII.3.
Glycerol is a poly-alcohol. The dielectric behavior of pure glycerol is a
typical example of Cole-Davidson dispersion. The Glycerolx(H20 ) 1.x m ixture is
clearly non-Debye. However, according to Cole-Cole and frequency plot as shown in
Fig. VII.3 for the m ixtures, and Fig. IV.7 for pure glycerol, Cole-Davidson form is
not a good fit. The form with two Debye relaxation times in Eqn. (V II.1.1) gives a
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-1 5 9 -
b etter fit. The corresponding fitting param eters are listed in Table VII.III.
Table VII.III. Fitting param eters to two Debye relaxation form in Eqn. (VII.1.1)
for glycerolx(H20 ) 1.x
Concentration
Low / relaxation
High / relaxation
RMS+
Error
X
eoo
Aj
ti (p s)
A2
t
1.0
4.8
31.9
1189
4.9
104
0.56
0.82
4.9
33.1
867
5.8
71.6
0.54
0.53
5.4
36.8
312
8.4
41.2
0.59
0.42
5.3
41.5
145
8.1
21.2
0.43
0.20
5.4
40.8
55.1
18.4
15.6
0.29
0.076
4.4
32.1
27.2
36.5
10.3
0.31
0.048
3.9
30.0
24.2
40.4
9.6
0.24
VII.1.4. Non-Debve behavior at x
2
0 concentrations
As can be seen from above tables, the two relaxation tim es get closer and
closer and finally they are expected to converge to one relaxation tim e of water.
Therefore, at very low alcohol concentrations, it is more difficult to fit the data
accurately to the two Debye relaxation model. Also, the two Debye fit in Fig. VII. 1
for the very dilute solution x = 0.046 is not as good as the fits for other more
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-160 -
concentrated ones. Alternatively, the x ~ 0 concentrations were examined in term s
of the Cole-Cole form
+
i +
(V IL 1 ' 2)
Values of A , £«, and r c axe listed in Table. VII.IV.
Table VII.IV. Cole-Cole fit for various water-rich alcohol aqueous solutions.
X
Coo
A
tc
a
6
0.032
3.5
69.8
12.1
0.05
0.1
0.016
2.2
74.1
9.8
0.05
0.1
Ethylene Glycol
0.031
2.5
74.0
9.8
0.04
0.1
Glycerol
0.021
1.2
75.8
9.5
0.07
0.1
0.010
1.2
75.8
9.2
0.06
0.1
Solute
1-Propanol
VH.2. Discussion
The propanol results agree on other m easurements w ith alcohol-rich
concentrations (Barthel, et al. 1990; Perl, et al. 1983). For glycerol/H 20 , results
agree w ith measurements at spot frequencies and on aqueous solutions x ~ 1. No
dielectric measurements on Ethylene glycol aqueous solutions have been reported to
our knowledge. As can be seen from table VII.IV, addition of a small am ount of
alcohol molecules to w ater courses the dielectric dispersion of solutions to deviate
from Debye single relaxation behavior. The addition of a small am ount of water
also dram atically change the dielectric relaxation pattern of pure alcohols as shown
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-161 -
by Table V II.I to Table VII.III. Alcohol molecules strongly disturb the w ater
hydrogen bonding network.
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-1 6 2 -
Fig. VII. 1. Real and imaginary part of the dielectric spectra vs. frequency
for (l-propanol)x(H20 ) 1.x solutions: pure 1-propanol (solid squares), x=0.61 (x's),
x=0.19 (diamonds), x=0.074 (+'s) and water (o's).
100
80
e
i
60
40
20
50
40
u
30
20
10
10
Frequency (Hz)
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Fig. VII.2. Real and imaginary part of the dielectric spectra vs. frequency for
(Ethylene glycol)x(H20 ) 1.x solutions: pure ethylene glycol (solid squares), x=0.72
(x's), x=0.22 (diamonds), x=0.088 (+'s) and water (o's).
I
°
°
°
° ooooo
+ M 1—Mill tHlHI I—
I —
t- ,
1-1- I I I
1 0 10
.//
Frequency (Hz)
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—164-
Fig. VII.3. Real and imaginary part of the dielectric spectra vs. frequency for
(Glycerol)x(H20 ) 1.x solutions: pure glycerol (solid squares), x=0.53 (x's), x=0.20
(diamonds), x=0.076 (+'s) and water (o's).
100
80
e
i
60
40
20
101
10
50
40
30
.//
20 v
Frequency (Hz)
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-165 -
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