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The Problems of Electric Polarization

Dielectrics in Time Dependent Fields

Yuri Feldman

Tutorial lecture2 in Kazan Federal University

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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 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.The superposition principle makes it possible to describe the polarization due to an electric field with arbitrary time dependence, with the help of so-called response functions (t) .(t) is called the step-response function or decay function of the polarization.

t' t

P 1

P 2

E 2

E 1

E

P

0

0

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For a linear dielectric this response that the total polarization is given by: )t()(E)t(P(0)P(t) 0

At t=010 )(

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 )(

dt'ttt=

=dt't

t'ttt

t

t

)'()'(

)()'()(

E

EP

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.

)'()'( tttt pp

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Let us consider now the time dependence of the dielectric displacement D for a time dependent electric field E.

)t(P4E(t)D(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)

D E( ) ( ' ) ( ' )t t tD

t

t dt'

( ' ) ( ' ) D Dt t t twithThe relation between p and D is the following:

D pt t S t t t t( ' ) ( ' ) ( ' ) 1

14

5

And the relation between p and D is:

D pt tt t

t t( ')( ')

( ')

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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:

t tD t t

'

lim ( ' ) ( )

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.

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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:

D E( ) ( ' ) ( ' )t t tD

t

t dt'

Applying to the left and right parts the Laplace transform and taking into account the theorem of deconvolution we can obtain:

),s(*E)s(*)s(*D (2.18)

where)]([)exp()()(

0

* tLdtstts DsDs

(2.19)

(s=+i; 0 and we’ll write instead of s in all Laplace transforms i).

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Taking into account the relation (2.20) we can rewrite (2.22) in the following way:

)]([)()(* tLi orps

From another side complex dielectric permittivity can be written in the following form:

)("i)(')(*

(2.20)

(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:

0

dt ttorp cos)()(' (2.22)

and the imaginary part of complex dielectric permittivity ’’() is associated with the negative imaginary part of the Laplace transform of orientation pulse-response function:

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0

sin)()('' dttorp (2.23)

Let us now reconsider the relationship between time dependent displacement and harmonic electric field:

tiEtEeEtE ootio sincos)( (2.24)We’ll rewrite in this case the relation (2.18) in the following form:

t sinE"t cosE)(')t(D oo (2.25)that can be presented as follows:

where

2200 "' ED tg ( ) ="( )'( )

and

D t D t D t D t( ) cos cos sin sin cos( ) 0 0 0 (2.26)

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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 . 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 with respect to the electric field gives rise to absorption of energy.

2

2

The total amount of work exerted on the dielectric during one cycle can be calculated in the following way:

20

/2

0

/2

0

20

/2

0

))((''41

sincos)(''coscos)(')(41

41

E

tdttdtE

EdDW

t t

t

(2.27)

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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:

sinED)(")E(W oo

o

88

2

(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)

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The Kramers-Kronig relationsThe Kramers-Kronig relations are ultimately a consequence of the principle of causality - the fact that the dielectric response function satisfies the condition:

0t 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:

 

' ( ) ( ) cos

por t dt

0

' ' ( ) ( ) sin

por t dt

0

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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:

])[()(

)cos(sin

])[()(

)sin(cossin

txdtx

txt

txdtx

txtdxx

xt

(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

cos1

xt

xdx t

(2.32)

This called Hilbert transform

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Let us apply this to the ´():

' ( ) ( )sin

( ) sin

1

1 1

por

por

txt

xdxdt

xt 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 :

' ( ) ( / )

"( )

1x

xdx

and similarly:

' ' ( ) ( / )' ( )

1x

xdx

(2.34)

(2.35)

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' ( ) ( / )

"( )

2 2 20

x xx

dx

' ' ( ) ( / )

' ( )

2 2 20

xx

dx

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:

(2.37)

(2.36)

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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:

)(1)( tt oror PP

with solution:

P Por ortt e( ) ( ) / 1

0

(2.39)

(2.38)

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It follows that in this case the step-response function of the orientation polarization is given by an exponential decay:

Por tt e( ) /

where the time constant is called the relaxation time.

(2.24)

From (2.24) one obtains for the pulse-response function also an exponential decay, with the same time constant:

por

Por tt e

( ) / (2.25)

Complex dielectric permittivity as it was shown in last lecture can be written in the following way:

*( ) ( ) [ ] s porL

(2.26)

Substituting (2.25) into the relation one finds the complex dielectric permittivity:

ieL st

s

1

][1)()( /*

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Splitting up the real and imaginary parts of (2.26) one obtains:

' ( )

s

1 2 2

' ' ( )

( )

s

1 2 2

(2.27)

(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:

kkk

orp )/texp(g)t(

(2.29)

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por k

kk

kt

gt( ) exp( / )

*( ) ( ) s

k

kk

gi1

(2.30)

(2.31)

with gkk

1 (2.32)

For a continuous distribution of relaxation times:

por t g t d( ) ( ) exp( / )

0

(2.33)

por t

gt d( )

( )exp( / )

0

(2.34)

* ( ) ( )( )

s

g di10

(2.35)

g d( )

10

(2.36)with

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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 time-dependent 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.

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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).

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 "

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,

k

kkA )()('' (2.16)

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The corresponding frequency dependence of ‘() is obtained from Kramers-Kronig relations:

' ( ) 1

22 2

k k

kk

A (2.17)

As this expression gives the contribution by the induced polarization, its value for =0 is the dielectric permittivity of induced polarization :

12 Ak

kk(2.18)

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 tt k PP (2.19)

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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Dielectric properties of water

109 1010 1011

0

20

40

60

80

Water at 20oC

',''

Frequency MHz

2~ s

s

23

*( )

s

i1

' ( )

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:

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' ' ( )

( )( ' ( ))

ss

2

2 211 (2.24)

' ' ( ) /

( )( ' ( ) )

s

1 2 2

(2.25)

It follows that both methods yield straight lines, with slopes 1/ and , respectively, and ´ intersecting axis at s and .

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2. The Cole-Cole equation.

The first empirical expressions for *() was given by K.S. Cole and R.H.Cole in 1941:

( )( )

( )s

i1 01

' ( ) ( )

( ) sin

( ) sin ( ) ( )

s

112

1 212

01

01

02 1

(2.26)

' ' ( ) ( )

( ) cos

( ) sin ( ) ( )

s

01

01

02 1

12

1 212

(2.27)

(2.28)

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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 (2.26) in series. Taking the Laplace transform to the series one can obtain:

por

n

n

n

tn

tt( )

( )( )

,( )`

1 110 1 0

1 1

(2.29)

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3. Cole-Davidson equation In 1950 by Davidson and Cole another expression for *() was given:

( )( )

( )s

i1 0

1+ i 0

2 21 e ei i / cos

This expression reduces to the Debye equation for =1. Since

where =arctgo, separation of the real and imaginary parts is easy, leading to the following expressions for and :

' ( ) ( ) cos cos s

' ' ( ) ( ) cos sin s

(2.30)

(2.31)

(2.32)

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From Cole-Davidson equation, the pulse-response function can be obtained directly by taking the inverse Laplace transform:

por tt

te( )

( )/

1

0 0

1

0

(2.33)

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4. The Havriliak-Negami equation

( )

( )s

i1 01 (2.34)

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 :

' ( ) ( )

cos

( ) sin ( ) ( )/

s

1 2120

10

2 12

"( ) ( )

sin

{ ( ) sin ( ) }( ) /

s

1 2120

10

2 1 2

(2.35)

(2.36)

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arctg[( ) cos

( ) sin

01

01

12

112

where(2.37)

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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:

)1

)()('' 0

"20

20

sech(lnms

where m is the maximum value of () in this case given by:

m s" ( )

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Eqn.(2.38) can be generalized to the form:

(2.38)

(2.39)

" " sec ( ln ) m h 0(2.40)

where is a parameter with values 0<1 and ”m is now

different from . Applying Kramers Kroning relationship, we find that ´m in the Fuoss-Kirkwood equation is given by:

m s" ( )

12

32

)(21'

sm

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.

(2.41)

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6. The Jonscher description

The Fuoss-Kirkwood equation can also be written in the form:

"( )( / ) ( / )

"

m

0 0

Jonnscher suggested an expression for ”() that is a generalization of Eq. (2.42):

(2.42)

"( )

( / ) ( / )

"

mm n

1 21 (2.43

)with 0<m 1, 0 n<1. The Eq. (2.42) makes that the frequency of maximum los is:

mm n

m nmn

1 1 21

1 1/( )(2.44)

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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:

A)(

A)(

n"

m"

1

1

1

0

(2.46)

(2.45)

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