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Lecture 9Lecture 9Models of dielectric relaxationModels of dielectric relaxation

 

i. Rotational diffusion; Dielectric friction.

 

ii. Forced diffusion of molecules with internal rotation

 

iii. Reorientation by discrete jumps

 

iv. Memory-Function Formalism

v. The fractal nature of dielectric behavior.

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According to Frenkel the molecular rotational motion is usually only the rotational rocking near one of the equilibrium orientation. They are depending on the interactions with neighbors and by jumping in time they are changing there orientation.

In this case the life time of one equilibrium orientation have to be much more then the period of oscillation 00=1/=1/ ( ( >>>>00).). And the

relationship between them can be written in the following way:

kT

H

e

0 (9.1)

where HH is the energy of activationenergy of activation that is required for changing the angle of orientation. The small molecules can be rotated on comparatively big angles. The real Brownian real Brownian

rotational motion can be valid only for comparatively big rotational motion can be valid only for comparatively big

molecules with the slow changing of orientation anglesmolecules with the slow changing of orientation angles. In this case the differential character of rotational motion is valid and the rotational diffusion equation can be written.

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Debye was the first who applied the Einstein theory Debye was the first who applied the Einstein theory of rotational Brownian motion to the polarization of of rotational Brownian motion to the polarization of dipole liquids in time dependent fields.dipole liquids in time dependent fields.

According to Debye the interaction of molecules between each other can be considered as the friction foresees with the moment proportional to the angle velocityangle velocity =P/=P/,, where is the rotational coefficient of friction that can be connected with Einstein rotational diffusion coefficientEinstein rotational diffusion coefficient (D(DRR = kT / = kT /)) and PP is the

moment of molecule rotation. In the case of small macroscopic

sphere with radius aa, the coefficient of rotational motion according to Stokes equation can be defined as:

38 a (9.2)

where is the coefficient of viscosity.

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Let us start with the diffusion diffusion equationequation::

CDCDttuC uRrT22/),,( r (9.3)

where DDTT and DDRR are, respectively, the transnational and

rotational diffusion coefficients, is the gradient operator on the space (x,y,z)(x,y,z) and is the rotation operator .

In this equation C(C(r,u,r,u,t)dt)d22udud33rr is the number of molecules with orientation uu in the spheroid angle dd22uu and center of mass in the neighborhood dd33rr of the point rr at time tt. The microscopic definition of CC is

r

u u u u /

C u t r r u u ti ii

N

( , , ) ( ) ( ( ))r

1

(9.4)

Here rrii(t)(t) and uuii(t)(t) are, respectively, the position and orientation

of molecule ii at time tt and the sum goes over all the molecules. The average value of CC is (1/4(1/4))00,, where 00 is the

number density of the fluid. In this equation the operator is related to

u u u /

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I i u ( / u) the dimensional angular momentum operator

ofquantum mechanics; that is

u ui I I and 2 2

that the spherical harmonics YYlmlm(u)(u) are eigenfunctions of

It should be recalled

I 2

corresponding to eigenvalue of l(l+1).l(l+1).

The solution of the equation (9.3) can be done by expanding

of C(C(r,u,r,u,t)t) in the spherical harmonics {Y{Ylmlm(u)}.(u)}. In the case of

dipole moment rank ll is equal to one. In the case of magnetic magnetic

moment l=2moment l=2. For the spherical dipole moment in viscous media the result of equation (9.3) can be obtained in the following way:

0

34

a

kT

0

34

a

kT(9.5)

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This is Debye’s expression for the molecular dielectric Debye’s expression for the molecular dielectric relaxation timerelaxation time. According to Debye, this formula valid if:

(a) There is an absence of interaction between dipoles.

(b) Only one process leading to equilibrium(e.g. either transition over a potential barrier, or frictional rotation).

(c) All dipole can be considered as in equivalent positions, i.e. on an average they all behave in a similar way.

(a) There is an absence of interaction between dipoles.

(b) Only one process leading to equilibrium(e.g. either transition over a potential barrier, or frictional rotation).

(c) All dipole can be considered as in equivalent positions, i.e. on an average they all behave in a similar way. The molecular dipole correlation function in this case will be the simplest exponent:

C tt

e t( )( ) ( )

( ) ( )/

0

0 00 (9.6)

This result was generalized to the case of prolate and oblate ellipsoids by Perrin and KoenigPerrin and Koenig:

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a) Prolate ellipsoid: =b/a <1=b/a <1 b

a

a

a

kT

8

3

1

2

1

1 11

3 4

2

2

2

ln

b

a

kT

16

3

1

1 1

1

1 1 1

3 4

2

2

2

2ln(9.8)

(9.7)

b) Oblate ellipsoid: >1>1

8

a

a

kT

8

3

12

11 1

3 4

2

2

1 2tan

b

a

kT

16

3

11 2

11

1

3 4

2

2

1 22tan

(9.9)

(9.10)

In the case of ellipsoid of revolution the dipole correlation function can be written in the following way:

)/txpeΑ)exp(-t/AC(t) ba1

Let us now consider the influence of long-range forceslong-range forces such as Coilomb, or dipolar forces on the results of the Debye theory. In this case each molecule not only experiences the usual frictional forces which give rise to a diffusion equation, but also must respond to the local electric field which arises from the permanent multiple moments on the neighboring molecules.

(9.11)

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One of the ways to include these interactions into Debye Debye theorytheory is to add forces and torque’s in a generalized diffusion equation and to solve this equation self-consistently with the equation self-consistently with the Poisson equationPoisson equation. In this case the generalized diffusion equation can be written as a following: t = -

1

kTD

1

kTDT TC u t FC D NC C D Cr R u r R u( , , ) / ( ) ( )r 2 2 (9.12)

where F(F(rr,t),t) and N(N(rr.t).t) are the force and torque respectively that acting on a molecule at ((rr,t).,t). They are arise from the Coulomb interactions between molecules and can be expressed as:

F r t dsZ s E r su( , ) ( ) ( ) (9.13)

N r t dsZ s su E r su( , ) ( ) ( ) (9.14)

Here linear molecule centered at rr with orientation uu is considered. ((rr+s+su)u) is the position of a distance s s from the molecular center along the molecular axis. Then E(rE(r+s+suu)) is the electric field at the point due to all charges in the system. Z(s)Z(s) is the linear charge density and dsZ(s)dsZ(s)E(rE(r+s+suu)) is the electric force exerted on this charge by the surrounding fluid. Likewise ssuudsZ(s)dsZ(s)EE((rr+s+suu)) is the corresponding torque.

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To make the equations (9.12-9.14) self-consistent the Poisson equation has to be used:

r rr r rE t t t( , ) ( , ) ( , )2 4 (9.15)

where ((rr,t),t) is the charge density and ((rr,t),t) is the electrostatic potential at rr,t,t. In the case of polarizable molecules 4 in Poisson equation have replace by 44//,, where is dielectric constant due to the polarizability [([(-1)/( -1)/( +2)=+2)=oo].]. Also the dipole moment of the linear molecules might be taken as an effective dipole moment.In the absence of net molecular charges, the only In the absence of net molecular charges, the only multipole moment that contributes to the orientation multipole moment that contributes to the orientation relaxation is the dipole moment. relaxation is the dipole moment.

The solution of diffusion equation taking into account dipolar forces gives the correlation function (t)(t) that decays on two different time scales specified by the relaxation times:

1

1

2

DR

2

1

2 1

( )DR

(9.16)

(9.17)

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where DDRR is the rotational diffusion coefficient, and

4

32

0kT(9.18)

Correlation function can be written in the following way:

( ) ( )/ /t e et t 1

32 1 2 (9.19)

Two relaxation times for a single component polar fluid was found also by Titulaer and DeuthchTitulaer and Deuthch, Bordewijk and Nee- Bordewijk and Nee- ZwanzigZwanzig. If BerneBerne discussed the two correlation times as decay of transverse and longitudinal fluctuations, Nee Nee and Zwanzigand Zwanzig considering dielectric frictiondielectric friction in diffusion equation. Considering the diffusion equation they made the assumption that by some reasons the frictional forces on the particle is not developed instaneously, but lagslags its velocity its velocity. Considering the correlation function of angular velocities they came to the frequency dependent friction coefficient in diffusion equation: D

kT( )

( )

(9.20)

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In this case in the theory of rotational Brownian motion, the position of the particle is replaced by its orientation, specified by the unit vector uu(t).(t). The translational velocity is replaced by an angular velocity (t)(t) and the force is replaced by a torque NN(t).(t). The frictional torque is proportional to the angular velocity:

N t t t t dtt

( ) ( ') ( ') ' (9.21)

or in Fourier components,N ( ) ( ) ( ) (9.22)

The total friction coefficient (()) consists of two parts. The first is due to ordinary friction, e.g. Stokes’ law friction Stokes’ law friction 00 independent on frequency. The other part is due to dielectric dielectric frictionfriction and is denoted by DD(().). The sum is

)()( D 0 (9.23)

Using the Onsager reactive fieldOnsager reactive field and calculating the transverse angular velocity and torque in terms of time dependent permanent dipole moment, they obtained an explicit expression for the dielectric friction coefficient:dielectric friction coefficient:

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D

s s

s

kT

i( )

( )[ ( ) ]

[ ( ) ]

2

2 (9.24)

This expression is valid for spherical isotropic Brownian spherical isotropic Brownian motion of a dipole in an Onsager cavitymotion of a dipole in an Onsager cavity. To obtain the molecular DCF it is necessary to average over distribution of orientations at time t, for a given initial orientation and then to average over an equilibrium distribution of initial orientations. The average of (t)(t) can be found from knowledge of the distribution function C(C(uu,t),t) of orientations as a function of time. This distribution function obeys the diffusion equation diffusion equation for spherically isotropic Brownian motionfor spherically isotropic Brownian motion. The solution of this equation leads to a very simple relation between dielectric friction and DCF:

LdC t

dti

kT[

( )] { [

( )]} 1

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(9.25)

It is convenient to introduce in this case the frequency dependent relaxation time (()) defined by

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( )( )

kT

(9.26)

One can now write for molecular DCF the following relation:

LdC t

dti s s

s

[( )

]( )[ ( ) ]

[ ( ) ]

120

1

(9.27)

From comparison of (9.27) with the Debye behavior we are coming to the simple relationship between macroscopic and molecular correlation times:

Ms

s

2

2 0(9.28)

which is different from the relationship obtained by Bordewijk for the same molecular DCF

Mkk

01 2 1/( ) (9.29)

where k=k=ss//

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Character of interaction

Temperature

Structure

etceterais a phenomenological

parameter

is the relaxation time

?

Non-exponential relaxationempirical

Cole-Cole law

1941 year

)(1 i

(1-) / 2

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The Memory function for Cole-Cole law

)()(1)( zfzMzfz dftMtfdt

d t )()()(

0

L. Nivanen, R. Nigmatullin, A. LeMehaute, Le Temps Irrevesibible a Geometry Fractale, (Hermez, Paris, 1998)

R. R. Nigmatullin, Ya. E. Ryabov, Physics of the Solid State, 39 (1997)

Fractal set

)]([10 tfD

dt

df

1)( zzM = dfthe memory function

a fractional derivation

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Scaling relations

N, are scaling parameters

dG is a geometrical fractal dimension

)/ln(

) ln(

2 0 sGd

)ln(

)ln(1

N

d f

0

Gd

R

RGN

0

0 is the limiting time of the system self-similarity in the time domain

is the constant depends on relaxation units transport properties

2 RDs

Gdss G

R

D20

sD is the self-diffusion coefficient

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HydrophilicPAIA PAA PEI are electrolyte polymers PVA Is a nonelectrolyte with strong interaction

between hydroxyl groups and water HydrophobicPEG PVME

PVP are nonelectrolyte polymers

N. Shinyashiki, S. Yagihara, I. Arita, S. Mashimo, Journal of  Physical Chemistry, B 102 (1998) p. 3249

T=Constant

Gd 0 s s0

Polymer water mixtures

)/ln(

) ln(

2 0 sGd

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Gd 0 s s0

Quenchednylon +kevlar fibers

1 . 3 2 . 4210 -1 5 . 210 2 126

C rystallinenylon +Kevlar fibres

1 . 5 1 . 6 1 . 410 2 216

Composite polymer structure

H. Nuriel, N. Kozlovich, Y. Feldman, G. MaromComposites: Part A 31 (2000) p. 69

The samples with Kevlar fibers

have the longer relaxation time

)/ln(

) ln(

2 0 sGd

T is not Constant

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Water absorbed in the porous glass

A. Gutina, E. Axelrod, A. Puzenko, E. Rysiakiewicz-Pasek, N. Kozlovich, Yu. Feldman, J. Non-Cryst. Solids,235-237 (1998) p. 302

Samples are separated in two groups according to the humidity value h.

T is not Constant

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Conclusions

I The Cole-Cole scaling parameter depends on the features of interaction between the system and the thermostat.

II The Cole-Cole scaling parameter and the relaxation time are directly connected to each other.

III From the dependence of the parameter on the relaxation time, the structural parameters can be defined.