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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 PHENOMENOLOGICAL THEORY OF LEANER DIELECTRIC IN TIME-DEPENDENT FIELDSDIELECTRIC IN TIME-DEPENDENT FIELDS

The dielectric response functions. Superposition principle.

A leaner dielectric is a dielectric for which the superposition A leaner dielectric is a dielectric for which the superposition principle is valid,principle is valid, i.e. the polarization at a time ttoo due to an a electric field with a time-dependence that can be written as a sum E(t)+E’(t),E(t)+E’(t), is given by the sum of the polarization’s P(tP(too)) and P’(tP’(too)) due to the fields E(t)E(t) and E’(t)E’(t) separately. Most dielectrics are linear when the field strength is not too high.

The superposition principlesuperposition 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 response functions (t)(t) .(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 t' t

P 1

P 2

E 2

E 1

E

P

0

0

3

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’)(t-t’) are possible. For high values of tt, PP will approximate the equilibrium value of the polarization connected with the static field EE. From this it follows that

0 )(

dt'ttt=

=dt't

t'ttt

t

t

)'()'(

)()'()(

E

EP

where, called pulse-response functionpulse-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 DD for a time dependent electric field EE.

)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 P, it will also hold for DD.

Thus, we can write for DD analogously to P(t)P(t)

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

t

t dt'

( ' ) ( ' ) D Dt t t twith

The relation between pp and DD is the following:

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

14

5

And the relation between pp and DD is:

D pt tt t

t t( ')( ')

( ')

4

The unit step function in implies that there is an instantaneous decrease of the function DD (t-t’), (t-t’), from the value DD (0)=1 (0)=1 to a limit value given by:

t tD t t

'

lim ( ' )( )

1

In contrast the step-response function of the polarizationstep-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 The complex dielectric permittivity. Laplace and Fourier permittivity. Laplace and Fourier Transforms.Transforms.Let us consider the time dependence of the dielectric displacement DD for a time dependent Electric fieldtime dependent Electric field:

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

t

t dt'

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

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

where

)]([)exp()()(0

* tLdtstts DsDs

(2.19)

(s=s=+i+i; ; 00 and we’ll write instead of ss in all Laplace transforms ii).

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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)(')(* )("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 ( ) ="( )

'( )

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 superposition of two

harmonic fields with the same frequencyharmonic 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 E oo and ”” E E oo ,

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 worktotal 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

))((''4

1

sincos)(''coscos)(')(4

1

4

1

E

tdttdtE

EdDW

t t

t

(2.27)

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Since the fields E E and DD 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. loss factor. From (2.27) we find the average energy dissipation per unit of time:

sinED)(")E(

W ooo

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 0 (2.29)

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The Kramers-Kronig The Kramers-Kronig relationsrelationsThe Kramers-Kronig The Kramers-Kronig relationsrelationsThe Kramers-KronigKramers-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. 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 pp(t)(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(cos

sin

txdtx

txt

txdtx

txtdx

x

xt

])[()(

)cos(sin

])[()(

)sin(cos

sin

txdtx

txt

txdtx

txtdx

x

xt

(2.31)

In this integral we ignore the imaginary contributions arising from integration through the pole at x=x=. The first integral is equal to , the second vanishes so that we can obtain:

( / )sin

cos1

xt

xdx t

( / )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 x

xdx

' ( ) ( / )"( )

2 2 20

x x

xdx

' ' ( ) ( / )

' ( )

2 2 20

x

xdx

' ' ( ) ( / )' ( )

2 2 20

x

xdx

These are the Kramers-Kronig relationsKramers-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-KroningKramers-Kroning

integrals:

(2.37)

(2.36)

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Relaxation and resonanceRelaxation 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 1/1/, 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 relaxation time.time.

(2.24)

From (2.24) one obtains for the pulse-response functionpulse-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 formulasDebye 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 kk ,

yielding: k

kkorp )/texp(g)t(

(2.29)

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por k

kk

k

tg

t( ) exp( / )

*( ) ( ) s

k

kk

g

i1

(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 d

i10(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 massa motion of mass under the influence of influence of forcesforces that that depend on the electric field.depend on the electric field. An instantaneous An instantaneous change of the electric fieldchange of the electric field yields an instantaneous change instantaneous change of these forces,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=0t=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,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 functionsdelta 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=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 catastropheso-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 harmonic oscillator in the absence of dampingabsence of damping, to which the term resonance applies.

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Dielectric properties of Dielectric properties of waterwater

Dielectric properties of Dielectric properties of waterwater

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 s

and = = .

Although the Cole-Cole plotCole-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 21

1 (2.24)

' ' ( ) /

( )( ' ( ) )

s

1 2 2

(2.25)

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

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2. The Cole-Cole equation.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

11

2

1 21

2

01

01

02 1

(2.26)

' ' ( ) ( )

( ) cos

( ) sin ( ) ( )

s

01

01

02 1

1

2

1 21

2

(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 Laplace transformtransform to the Cole-Cole expressionCole-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 1

10 1 0

1 1

(2.29)

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3. Cole-Davidson equation 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.=1. Since

where =arctg=arctgoo, 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 equationCole-Davidson equation, the pulse-response function can be obtained directly by taking the inverse Laplace inverse Laplace transformtransform:

por tt

te( )

( )/

1

0 0

1

0

(2.33)

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4. The Havriliak-Negami equation4. The Havriliak-Negami equation

( )

( )s

i1 0

1 (2.34)

It is easily seen that this equation is both a generalization of the Cole-Cole equation, to which it reduces for =1=1, and a generalization of the Cole-Davidson equation to which it reduces for =0=0. Separation of the real and the imaginary parts gives rather intricate expressions for and :

' ( ) ( )

cos

( ) sin ( ) ( )/

s

1 21

201

02 1

2

"( ) ( )

sin

{ ( ) sin ( ) }( ) /

s

1 21

201

02 1 2

(2.35)

(2.36)

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arctg[( ) cos

( ) sin

01

01

1

2

11

2

where

(2.37)

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5. The Fuoss-Kirkwood description  5. The Fuoss-Kirkwood description  Fuoss and KirkwoodFuoss 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 mm is the maximum value of (()) in this case given by:

m s" ( )

1

2Eqn.(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<0<11 and ””mm is now

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

m s" ( )

1

2

32

)(21'

sm

The parameter introduced by (2.35) should be distinguished from the parameter in the Cole-Cole equationCole-Cole equation. An important difference between both parameters is that the Fuoss-Fuoss-

Kirkwood equationKirkwood equation changes into the expression for a single relaxation time if =1=1, whereas Cole-Cole equationCole-Cole equation does so if

=0=0.

(2.41)

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6. The Jonscher description 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<m0<m 1, 0 1, 0 n<1. n<1. The Eq. (2.42) makes that the frequency of maximum los is:

mm n

m nm

n

1 1 21

1 1/( )

(2.44)

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The quantities 11 and mm and 22 and nn respectively determine the low frequency and high frequency behavior and can be obtained from a plot of lnln””(()) against lnln, 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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New Fitting Software-”DATAMA”New Fitting Software-”DATAMA”

The ProblemComplex 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?