Transcript Cole-Cole Plot and Debye Relaxation

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.
1
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)
1
Por (t )   Por (t )

with solution:
1
Por (t )   Por (0)e

 t /
(7.2)
2
It follows that in this case the step-response function of the orientation
polarization is given by an exponential decay:
or
 t /
(7.3)
 P (t )  e
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:

or
or
 t /
p
P

  (t )  e
(7.4)
Complex dielectric permittivity as it was shown in last lecture can be
written in the following way:
 * ( )    (s   ) L[por ]
Substituting (7.4) into the relation one finds the complex dielectric
permittivity:
1
 ( )     ( s    ) L[e

*
t / 
s  
]   
1  i
(7.5)
3
Splitting up the real and imaginary parts of (7.5) one obtains:
 s  
 ' ( )   
1   2 2
(s   )
 ' ' ( ) 
1   2 2
(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:
 or
p ( t )   g k exp( t /  k )
k
(7.8)
4
 (t )  
or
p
k
gk
k
exp(  t /  k )
gk
 ( )    (s   ) 
k 1  i k
*
g
with
k
1
(7.9)
(7.10)
(7.11)
k
For a continuous distribution of relaxation times:

 por (t )   g ( ) exp( t /  )d
(7.12)
0

 (t )  
or
p
0
g ( )

exp(  t /  )d
(7.13)

with
g ( )d
1  i
0
 * ( )     ( s    ) 

 g( )d  1
0
(7.14)
(7.15)
5
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 timedependent 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 t=0.
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 t=0 should be zero, which is contrast to the
behavior of Eq.(7.3; 7.8 and 7.13) for dielectric relaxation.
6
Therefore these equations cannot describe adequately the behavior of
the response function near t=0, 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 "
 ' ' ( )   Ak (   k )
k
(7.16)
7
The corresponding frequency dependence of ‘() is obtained from
Kramers-Kronig relations:
 k Ak
 ' ( )  1   2
 k k   2
2
(7.17)
As this expression gives the contribution by the induced polarization, its
value for =0 is the dielectric permittivity of induced polarization  :
  1 
Ak
2

 
k
(7.18)
k
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)
P (t )   k2 P (t )
which is the same equation as for a harmonic oscillator in the absence
of damping, to which the term resonance applies.
8
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:

G(ln  )d ln 
1  i
0
 * ( )     ( s    ) 
with
(7.20)

 G(ln  )d ln   1
0
The corresponding expressions for the response functions are:
(7.21)
9

 por (t )   G (ln  ) exp(  t /  )d ln 
(7.22)
0

 exp(  t /  
 (t )   G (ln  )
 d ln 



0
or
p
(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:
 Por (t )  e  t /

or
P
 por   (t )  e t /
10
 s  
 ( )   
1  i
*
 s  
 ' ( )   
1   2 2
(s   )
 ' ' ( ) 
1   2 2
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:
11
(s   ) 2 1
 ' ' ( ) 
 (s   ' ( ))
2 2
1  

(s   )
 ' ' ( ) /  
  ( ' ( )   )
1   2 2
(7.24)
(7.25)
It follows that both
methods yield
straight lines, with
slopes 1/  and ,
respectively, and ´
intersecting axis at s
and .
12
. The Cole-Cole equation.
The first empirical expressions for *() was given by K.S. Cole and
R.H.Cole in 1941:
( s    )
 ( )   
1  (i 0 )1

 ' ( )    (s   )
1  ( 0 )
1  2( 0 )
1
1
(7.26)
1
sin 
2
(7.27)
1
sin   ( 0 ) 2 (1 )
2
1
( 0 ) cos 
2
 ' ' ( )  ( s   )
1
1
1  2( 0 ) sin   ( 0 ) 2 (1 )
2
1
(7.28)
13
In time domain the expression for the pulse-response function cannot
be obtained directly using the inverse Laplace transform to the ColeCole 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:
 t 
1
( 1)
or
 
p (t )  
 0 n 1  n(  1)   0 

n
 n (1 )`1
,t  
(7.29)
14
. Cole-Davidson equation
In 1950 by Davidson and Cole another expression for *() was
given:
( s    )

(7.30)
 ( )    
(1  i 0 ) 
This expression reduces to the Debye equation for =1. Since
1 + i 0  e i 1   2 2  e i / cos
where =arctgo, separation of the real and imaginary parts is easy,
leading to the following expressions for  and :
 ' ( )    (s   ) cos  cos 
 ' ' ( )  (s   ) cos   sin 
(7.31)
(7.32)
15
From Cole-Davidson equation, the pulse-response function can be
obtained directly by taking the inverse Laplace transform:
1  t
or
 p (t ) 
 
 0 ( )   0 
 1
e  t / 0
(7.33)
16
. The Havriliak-Negami equation
 ( )   

( s    )
1  i  
1

(7.34)
0
It is easily seen that this equation is both a generalization of the ColeCole 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 :
cos 
 ' ( )     ( s    )
 /2
(7.35)
1


1
2 (1 )
1

2
(

)
sin


(

)


0
0
2


sin 
 "( )  (s   )
1
1
{1  2( 0 ) sin   ( 0 ) 2 (1 ) } / 2
2
(7.36)
17
where
1
arctg[( 0 ) cos 
2

1
1
1  ( 0 ) sin 
2
1
(7.37)
18
. 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:
( s    ) 0
"
 ' ' ( ) 


m sech(ln  0 )
2 2
1  0
(7.38)
where m is the maximum value of () in this case given by:
1
  (  s   )
2
"
m
(7.39)
Eqn.(7.38) can be generalized to the form:
   sec h( ln  0 )
"
"
m
(7.40)
where  is a parameter with values 0<1 and ”m is now different
1
"


from m 2 (s   ) . Applying Kramers Kroning relationship, we find that
´m in the Fuoss-Kirkwood equation is given by:
19
m"  21  ( s   )
(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.
20
. The Jonscher description
The Fuoss-Kirkwood equation can also be written in the form:
 "( ) 
m"
( /  0 )   ( /  0 ) 
(7.42)
Jonnscher suggested an expression for ”() that is a generalization of Eq.
(7.42):
 "( ) 
( / 1 )
m"
m
 ( / 2 )
1 n
(7.43)
with 0<m 1, 0 n<1. The Eq. (7.42) makes that the frequency of
maximum los is:
 m
m 1 n 
m  
1  2 
1  n

1/( m n 1)
(7.44)
21
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 lowand high frequency ranges since one then has respectively:
 
 (  )  A 
 1 
m
 
 (  )  A 
 1 
n 1
"
"
 0
(7.45)
 
(7.46)
22
23