Download Equations

The Problems of Electric
Polarization
Dielectrics in Time Dependent Fields
Yuri Feldman
Tutorial lecture2 in Kazan Federal University
1
PHENOMENOLOGICAL THEORY OF LEANER
DIELECTRIC IN TIME-DEPENDENT FIELDS
The dielectric response functions. Superposition principle.
A leaner dielectric is a dielectric for which the superposition principle is
valid, i.e. the polarization at a time to due to an a electric field with a
time-dependence that can be written as a sum E(t)+E’(t), is given by
P2
the sum of the polarization’s
P(to) and P’(to) due to the fields E(t) and
E’(t) separately. Most dielectrics are linear when the field strength is
not too high.
P
P1
The superposition principle makes it possible to describe the
0
polarization due to an electric field with arbitrary time dependence,
with the help of so-called response functions (t) .
E2
(t) is called the
step-response function or decay function of the
E
polarization.
E1
0
t'
t
2
For a linear dielectric this response that the total polarization is given
by:
P(t)  P(0) ( t )  E( 0 ) ( t )
At t=0
( 0 )  1
In principle, both a monotonously decreasing and oscillating behavior
of (t-t’) are possible. For high values of t, P will approximate the
equilibrium value of the polarization connected with the static field E.
From this it follows that
(  )  0
P (t )  
=
   (t  t' ) 
E
(
t
'
)
dt' =
 


 t


t
t
 E (t ' ) (t  t ' )dt'

  p (t  t ' )   p (t  t ' )
where,
called pulse-response function of
polarization. This equation gives the general expression for the
polarization in the case of a time-dependent Maxwell field.
3
Let us consider now the time dependence of the dielectric displacement
D for a time dependent electric field E.
D(t)  E(t)  4P( t )
For the linear dielectrics the dielectric displacement is a linear function
of the electric field strength and the polarization, and for those
dielectrics where the superposition principle holds for P, it will also hold
for D.
Thus, we can write for D analogously to P(t)
t
D(t )    E(t ')D (t  t ')dt'

with   D (t  t ' )  D (t  t ' )
The relation between p and D is the following:
 D (t  t ') 
1  S (t  t ') 

1
4

 p (t  t ')
4
And the relation between p and D is:
D (t  t ') 
 (t  t ') 4

p (t  t ')


The unit step function in implies that there is an instantaneous
decrease of the function D (t-t’), from the value D (0)=1 to a limit
value given by:
lim D ( t  t' ) 
t  t'
(   1)

In contrast the step-response function of the polarization cannot show,
in principle, such an instantaneous decrease, since any change of the
polarization is connected with the motion of any kind of microscopic
particles, that cannot be infinitely fast.
5
The complex dielectric permittivity.
Laplace and Fourier Transforms.
Let us consider the time dependence of the dielectric displacement D
for a time dependent Electric field:
t
D(t )    E(t ')D (t  t ')dt'

Applying to the left and right parts the Laplace transform and taking
into account the theorem of deconvolution we can obtain:
D * ( s )   * ( s )E * ( s ),
where
(2.18)

 * (s)   s   D (t ) exp( st )dt   s L[ D (t )]
(2.19)
0
(s=+i; 0 and we’ll write instead of s in all Laplace transforms i).
6
Taking into account the relation (2.20) we can rewrite (2.22) in the
following way:
(2.20)
 * (i )     ( s    ) L[ por (t )]
From another side complex dielectric permittivity can be written in the
following form:
 * (  )  ' (  )  i" (  )
(2.21)
The equation (2.19) justifies the use of the symbol  for the dielectric
constant of induced polarization, since for infinite frequency the
Laplace transform vanishes, and the expression becomes equal to .
The real part of complex dielectric permittivity ’() is associated with
real part of Laplace transform of orientation pulse-response function:

' (  )       por ( t ) cos t dt
(2.22)
0
and the imaginary part of complex dielectric permittivity ’’() is
associated with the negative imaginary part of the Laplace transform
of orientation pulse-response function:
7

 ' ' ( )    por (t ) sin  dt
(2.23)
0
Let us now reconsider the relationship between time dependent
displacement and harmonic electric field:
E (t )  E o e it  E o cos t  iE o sin t
(2.24)
We’ll rewrite in this case the relation (2.18) in the following form:
D( t )  ' (  )E o cos  t  " E o sin  t
(2.25)
that can be presented as follows:
D(t )  D0 cos  cos  t  D0 sin  sin  t  D0 cos( t   )
where D  E
0
0
 '  "
2
2
(2.26)
 "( )
and tg ( ) =
 '( )
8
From the equation (2.26) it clearly appears that the dielectric
displacement can be considered as a superposition of two harmonic
fields with the same frequency, one in phase with electric field and
another with a phase difference

2
. The amplitudes of these fields
are given by ´E o and ” E o , respectively. Calculation of the energy
changes during one cycle of the electric field shows that the field with

a phase difference 2 with respect to the electric field gives rise to
absorption of energy.
The total amount of work exerted on the dielectric during one cycle
can be calculated in the following way:
1
W 
4
2 / 
 EdD
t 0
2 / 
2 / 


1
0 2

( E )  ' ( )  cos  t d cos  t   ' ' ( )  cos  t d sin  t 
4
t 0
t 0


1

 ' ' ( )( E 0 ) 2
4
(2.27)
9
Since the fields E and D have the same value at the end of the cycle as
at the beginning, the potential energy of the dielectric is also the same.
Therefore, the net amount of work exerted by the field on the dielectric
corresponds with absorption of energy. Since the dissipated energy is
proportional to ”, this quantity is called the loss factor.
From (2.27) we find the average energy dissipation per unit of time:
o 2

(
E
)
 o o

W
" (  ) 
D E sin 
8
8
(2.28)
where  called a loss angle.
According to the second law of thermodynamics, the amount of energy
dissipated per cycle must be always positive or zero. It means
that
    0
(2.29)
10
The Kramers-Kronig relations
The Kramers-Kronig relations are ultimately a consequence of the
principle of causality - the fact that the dielectric response function
satisfies the condition:
 ( t )  0 for t  0
(2.30)
It means that there should be no reaction before action.
Let us consider again the relations for real and imaginary part of
complex dielectric permittivity:

 '( )     por (t ) cos dt
0

 ''( )   por (t ) sin  dt
0
11
Both ‘() and “() are derived from the same generating function
p(t)
and that it should be possible in principle to “eliminate” this
function and to express ‘() in terms of “().
Let us consider the properties of Hilbert transform:


sin xt
sin( x   )t
 x   dx  cos t  ( x   )t d [( x   )t ]

cos( x   )t
d [( x   )t ]
( x   )t

 sin t 
(2.31)
In this integral we ignore the imaginary contributions arising from
integration through the pole at x=. The first integral is equal to ,
the second vanishes so that we can obtain:

sin xt
(1 /  ) 
dx  cost
x



This called Hilbert transform
(2.32)
12
Let us apply this to the ´():


sin xt
 ' ( )        (t ) 
dxdt
x 


1

1
1
    
x 

or
p

or

 p (t ) sin dxdt

(2.33)
In these manipulations we have extended the integration (2.33) to -
which is permissible in view of the causality principle. The second
integral in (2.33) is equal to “() in view of (2.30) so that one can
finally write :

 "( x)
 '( )    (1 /  ) 
dx
x 

and similarly:

 '( x)
 ''( )  (1 /  ) 
dx
x 

(2.34)
(2.35)
13
These are the Kramers-Kronig relations which express the value of
either “() or ‘() at a particular value of the frequency  in terms of
the integral transform of the other throughout the entire frequency
range (-, ). In view of what was mentioned above about the even
and odd character of these functions, one may change the range of
integration to (0, ) and thus obtain the one-sided Kramers-Kroning
integrals:

x "( x)
 ' ( )    (2 /  )  2
2 dx
x 
0
(2.36)

 ' ( x)
 ' ' ( )  (2 /  )  2
2 dx
x 
0
(2.37)
14
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:
1

Por (t )   Por (t )
(2.38)

with solution:
1
Por (t )   Por (0)e

 t /
(2.39)
15
It follows that in this case the step-response function of the orientation
polarization is given by an exponential decay:
or
 t /
(2.24)
 P (t )  e
where the time constant  is called the relaxation time.
From (2.24) one obtains for the pulse-response function also an
exponential decay, with the same time constant:

or
or
 t /
(2.25)
p
P

  (t )  e
Complex dielectric permittivity as it was shown in last lecture can be
written in the following way:
 * ( )    (s   ) L[por ]
Substituting (2.25) into the relation one finds the complex dielectric
permittivity:
1
 ( )     ( s    ) L[e

*
t / 
s  
]   
1  i
(2.26)
16
Splitting up the real and imaginary parts of (2.26) one obtains:
 s  
 ' ( )   
1   2 2
(2.27)
(s   )
 ' ' ( ) 
1   2 2
(2.28)
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:
(2.29)
 or
p ( t )   g k exp( t /  k )
k
17
 (t )  
or
p
k
gk
k
exp( t /  k )
gk
 ( )    (s   ) 
k 1  i k
*
g
with
k
1
(2.30)
(2.31)
(2.32)
k
For a continuous distribution of relaxation times:

 por (t )   g( ) exp( t /  )d
(2.33)
0

 (t )  
or
p
0
g ( )

exp( t /  )d
(2.34)

with
g ( )d
1  i
0
 * ( )     ( s    ) 

 g( )d  1
0
(2.35)
(2.36)
18
Equations (2.33) till ( 2.35) 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 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.(2.3; 2.8 and 2.13) for dielectric relaxation.
19
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
(2.16)
20
The corresponding frequency dependence of ‘() is obtained from
Kramers-Kronig relations:
k Ak
 ' ( )  1   2
 k k   2
2
(2.17)
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
(2.18)
k
It follows from (2.17) 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:

(2.19)
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.
21
Dielectric properties of water
W ater at 20 o C
s
80
 s      ~  2
 ', ''
60
40

20
0
10 9
10 10
Frequency M Hz
10 11
22
 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 (2.6)
and (2.7) one has:
23
(s   ) 2 1
 ' ' ( ) 
 (s   ' ( ))
2 2
1  

(s   )
 ' ' ( ) /  
  ( ' ( )   )
1   2 2
(2.24)
(2.25)
It follows that both
methods yield
straight lines, with
slopes 1/  and ,
respectively, and ´
intersecting axis at s
and .
24
2. 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
(2.26)
1
sin 
2
(2.27)
1
sin   ( 0 ) 2 (1 )
2
1
( 0 ) cos 
2
 ''( )  ( s   )
1
1
1  2( 0 ) sin   ( 0 ) 2 (1 )
2
1
(2.28)
25
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 (2.26) 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  
(2.29)
26
3. Cole-Davidson equation
In 1950 by Davidson and Cole another expression for *() was
given:
( s    )

(2.30)
 ( )    
(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 
(2.31)
(2.32)
27
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
(2.33)
28
4. The Havriliak-Negami equation
 ( )   

( s    )
1  i  
1

(2.34)
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
(2.35)
1


1
2 (1 )
1

2
(

)
sin


(

)


0
0
2


sin 
 "( )  (s   )
1
1
{1  2( 0 ) sin   ( 0 ) 2 (1 ) } / 2
2
(2.36)
29
where
1
arctg[( 0 ) cos 
2

1
1
1  ( 0 ) sin 
2
1
(2.37)
30
5. 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
(2.38)
where m is the maximum value of () in this case given by:
1
  ( s   )
2
"
m
(2.39)
Eqn.(2.38) can be generalized to the form:
   sec h( ln  0 )
"
"
m
(2.40)
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:
31
 m'  12  ( s    )
(2.41)
The parameter  introduced by (2.35) 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.
32
6. The Jonscher description
The Fuoss-Kirkwood equation can also be written in the form:
 "( ) 
m"
( /  0 )   ( /  0 ) 
(2.42)
Jonnscher suggested an expression for ”() that is a generalization of
Eq. (2.42):
 "( ) 
( / 1 )
m"
m
 ( / 2 )
1 n
(2.43)
with 0<m 1, 0 n<1. The Eq. (2.42) makes that the frequency of
maximum los is:
 m
m 1 n 
m  
1  2 
1  n

1/( m n 1)
(2.44)
33
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
(2.45)
 
(2.46)
34
New Fitting Software-”DATAMA”
35
The Problem
Complex systems necessarily entail
complex dielectric behaviour. How do we
“mine” the nuggets from all the dirt??
Most Fitting routines consider one
variable, frequency, and a parameter set.
Yet the energy landscape is governed by
temperature. How do we fit over 2
variables and a parameter set?