i' The dielectric relaxation and dielectric resonance'

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Lecture 7

• i. The dielectric relaxation and dielectric
  resonance.
• ii. The distribution functions of the relaxation
  times.
• iii. Cole-Cole distribution. Cole-Davidson
  distribution. Havriliak-Nehamy, Fuoss-Kirkwood
  and Johnsher distributions.

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Relaxation and resonance
The decreasing of the polarization in the absence
of an electric field, due to the occurrence of a
field in the past, is independent of the history
of the dielectric, and depends only on the value
of the orientation polarization at the instant,
with which it is proportional. Denoting the
proportionality constant by 1/?, since it has the
dimension of a reciprocal time, one thus obtains
the following differential equation for the
orientation polarization in the absence of an
electric field
(7.1)
with solution
(7.2)
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It follows that in this case the step-response
function of the orientation polarization is given
by an exponential decay
(7.3)
where the time constant ? is called the
relaxation time.
From (7.3) one obtains for the pulse-response
function also an exponential decay, with the same
time constant
(7.4)
Complex dielectric permittivity as it was shown
in last lecture can be written in the following
way
Substituting (7.4) into the relation one finds
the complex dielectric permittivity
(7.5)
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Splitting up the real and imaginary parts of
(7.5) one obtains
(7.6)
(7.7)
These relationships usually called the Debye
formulas.
Although the one exponential behavior in time
domain or the Debye formula in frequency domain
give an adequate description of the behavior of
the orientation polarization for a large number
of condensed systems, for many other systems
serious deviations occur. If there are more than
one relaxation peak we can assumed different
parts of the orientation polarization to decline
with different relaxation times ?k , yielding
(7.8)
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(7.9)
(7.10)
with
(7.11)
For a continuous distribution of relaxation times
(7.12)
(7.13)
(7.14)
with
(7.15)
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Equations (7.12) till ( 7.14) appear to be
sufficiently general to permit an adequate
description of the orientation polarization of
almost any condensed system in time-dependent
fields.
At a very high frequencies, however, the
deviations from Eq. (7.15) should always occur,
corresponding with deviations from (7.13) and
(7.14) at values of t that are small with respect
to the characteristic value ?o of the
distribution of relaxation times.
Physically, this is due to the behavior of the
response functions at t0. Any change of the
polarization ? is connected with a motion of mass
? under the influence of forces that depend on
the electric field. An instantaneous change of
the electric field ? yields an instantaneous
change of these forces, ? corresponding with an
instantaneous change of acceleration of the
molecular motions ? by which the polarization
changes, but not with an instantaneous change of
the velocities.
From this it follows that the derivative of the
step-response function of the polarization at t0
should be zero, which is contrast to the behavior
of Eq.(7.3 7.8 and 7.13) for dielectric
relaxation.
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Therefore these equations cannot describe
adequately the behavior of the response function
near t0, and the corresponding expressions for
?(?) do not hold at very high frequencies
(usually 1012 and higher).
The behavior of the induced polarization in time
dependent fields can be described in
phenomenological way. At frequencies
corresponding with the characteristic times of
the intermolecular motions by which the induced
polarization occurs,
there are sharp absorption lines, due to the
discrete energy levels for these motions. In a
first approximation, these absorption lines
correspond with delta functions in the frequency
dependence of ?"
(7.16)
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The corresponding frequency dependence of ?(?)
is obtained from Kramers-Kronig relations
(7.17)
As this expression gives the contribution by the
induced polarization, its value for ?0 is the
dielectric permittivity of induced polarization
??
(7.18)
It follows from (7.17) that for infinitely narrow
absorption lines, ?(?) becomes infinite at each
frequency where an absorption line is situated,
the so-called resonance catastrophe. To present
these phenomena in terms of polarization it can
be assumed that in the absence of an electric
field the time-dependent behavior of the
polarization is governed by a second-order
differential equation
(7.19)
which is the same equation as for a harmonic
oscillator in the absence of damping, to which
the term resonance applies.
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The empirical description of dielectric
relaxation
The behavior of the orientation polarization of
most condensed systems in time-dependent fields
can, as a good approximation, be characterized
with a distribution of relaxation times. This
behavior is generally denoted as dielectric
relaxation. This implies that the complex
dielectric permittivity, characterizing the
behavior of the system in harmonic fields, in the
frequency range corresponding with the
characteristic times for the molecular
reorientation can be written with Eq. (7.14), or
if a logarithmic distribution function is used
(7.20)
with
(7.21)
The corresponding expressions for the response
functions are
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(7.22)
(7.23)
?. A single relaxation time  
The simplest expressions that can be used for the
description of experimental relaxation data are
those for a single relaxation time
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For the case of a single relaxation time the
points (?? , ?") lie on a semicircle with center
on the ?? axis and intersecting this axis at
???s and ?? ??. Although the Cole-Cole plot is
very useful to investigate if the experimental
values of ?? and ??? can be described with a
single relaxation time, it is preferable to
determine the values of the parameters involved
by a different graphical method that was
suggested by Cole by plotting of ????(?) and
???(?)/? against ??(?). Combining eqns (7.6) and
(7.7) one has
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(7.24)
(7.25)
It follows that both methods yield straight
lines, with slopes 1/ ? and ?, respectively, and
? intersecting axis at ?s and ??.
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?. The Cole-Cole equation.
The first empirical expressions for ?(?) was
given by K.S. Cole and R.H.Cole in 1941
(7.26)
(7.27)
(7.28)
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In time domain the expression for the
pulse-response function cannot be obtained
directly using the inverse Laplace transform to
the Cole-Cole expression. Instead, the
pulse-response function can be obtained
indirectly by developing (7.26) in series.
Taking the Laplace transform to the series one
can obtain
(7.29)
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?. Cole-Davidson equation
In 1950 by Davidson and Cole another expression
for ?(?) was given
(7.30)
This expression reduces to the Debye equation for
?1. Since
where ?arctg??o, separation of the real and
imaginary parts is easy, leading to the following
expressions for ?? and ???
(7.31)
(7.32)
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From Cole-Davidson equation, the pulse-response
function can be obtained directly by taking the
inverse Laplace transform
(7.33)
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?. The Havriliak-Negami equation
(7.34)
It is easily seen that this equation is both a
generalization of the Cole-Cole equation, to
which it reduces for ?1, and a generalization of
the Cole-Davidson equation to which it reduces
for ?0. Separation of the real and the imaginary
parts gives rather intricate expressions for ??
and ???
(7.35)
(7.36)
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where
(7.37)
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?. The Fuoss-Kirkwood description  
Fuoss and Kirkwood observed that for the case of
a single relaxation time, the loss factor ?" (?)
can be written in the following form
(7.38)
where ?m?? is the maximum value of ???(?) in this
case given by
(7.39)
Eqn.(7.38) can be generalized to the form
(7.40)
where ? is a parameter with values 0lt??1 and ?m
is now different from . Applying
Kramers Kroning relationship, we find that ?m in
the Fuoss-Kirkwood equation is given by
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(7.41)
The parameter ? introduced by (7.35) should be
distinguished from the parameter ? in the
Cole-Cole equation. An important difference
between both parameters is that the
Fuoss-Kirkwood equation changes into the
expression for a single relaxation time if ?1,
whereas Cole-Cole equation does so if ?0.
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?. The Jonscher description
The Fuoss-Kirkwood equation can also be written
in the form
(7.42)
Jonnscher suggested an expression for ?(?) that
is a generalization of Eq. (7.42)
(7.43)
with 0ltm? 1, 0? nlt1. The Eq. (7.42) makes that
the frequency of maximum los is
(7.44)
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The quantities ?1 and m and ?2 and n respectively
determine the low frequency and high frequency
behavior and can be obtained from a plot of
ln?(?) against ln?, which should yield straight
lines in the low- and high frequency ranges since
one then has respectively
(7.45)
(7.46)
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