# Lecture 9

Mechanics

Oct 29, 2013 (4 years and 6 months ago)

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1

Lecture 9

Models 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
.

2

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

0
=
1
/

(

>>

0
)
.

And

the

relationship

between

them

can

be

written

in

the

following

way
:

(9.1)

where

H

is the
energy of activation

that is required for changing the
angle of orientation. The small molecules can be rotated on
comparatively big angles. The
real Brownian rotational motion can be
valid only for comparatively big molecules 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.

3

Debye was the first who applied the Einstein theory of
rotational Brownian motion to the polarization of 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

velocity

=P/

,

where

is

the

rotational

coefficient

of

friction

that

can

be

connected

with

Einstein

rotational

diffusion

coefficient

(D
R

=

kT

/

)

and

P

is

the

moment

of

molecule

rotation
.

In

the

case

of

small

macroscopic

sphere

with

a
,

the

coefficient

of

rotational

motion

according

to

Stokes

equation

can

be

defined

as
:

(9.2)

where

is

the

coefficient

of

viscosity
.

4

Let

us

start

with

the

diffusion

equation
:

(9.3)

where
D
T

and
D
R

are, respectively, the transnational and rotational
diffusion coefficients, is the gradient operator on the space
(x,y,z)

and is the rotation operator . In this equation
C(
r,u,
t)d
2
ud
3
r

is the number of molecules with orientation
u

in the
spheroid angle
d
2
u

and center of mass in the neighborhood
d
3
r

of the
point
r

at time
t
. The microscopic definition of
C

is

(9.4)

Here
r
i
(t)

and
u
i
(t)

are, respectively, the position and orientation of
molecule

i

at time
t

and the sum goes over all the molecules. The
average value of
C

is
(1/4

)

0
,

where

0

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

5

the dimensional angular momentum operator of

quantum mechanics; that is

that the spherical harmonics
Y
lm
(u)

are eigenfunctions of

It should be recalled

corresponding

to

eigenvalue

of

l(l+
1
)
.

The

solution

of

the

equation

(
9
.
3
)

can

be

done

by

expanding

of

C(
r,u,
t)

in

the

spherical

harmonics

{Y
lm
(u)}
.

In

the

case

of

dipole

moment

rank

l

is

equal

to

one
.

In

the

case

of

magnetic

moment

l=
2
.

For

the

spherical

dipole

moment

in

viscous

media

the

result

of

equation

(
9
.
3
)

can

be

obtained

in

the

following

way
:

(9.5)

6

This

is

Debye’s

expression

for

the

molecular

dielectric

relaxation

time
.

According

to

Debye,

this

formula

valid

if
:

(a)

There

is

an

absence

of

interaction

between

dipoles
.

(b)

Only

one

process

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
:

(9.6)

This

result

was

generalized

to

the

case

of

prolate

and

oblate

ellipsoids

by

Perrin

and

Koenig
:

7

a) Prolate ellipsoid
:

=b/a <1

b

a

(9.8)

(9.7)

b) Oblate ellipsoid:

>1

8

(9.9)

(9.10)

In

the

case

of

ellipsoid

of

revolution

the

dipole

correlation

function

can

be

written

in

the

following

way
:

Let us now consider the influence of
long
-
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)

9

One

of

the

ways

to

include

these

interactions

into

Debye

theory

is

to

forces

and

torque’s

in

a

generalized

diffusion

equation

and

to

solve

this

equation

self
-
consistently

with

the

Poisson

equation
.

In

this

case

the

generalized

diffusion

equation

can

be

written

as

a

following
:

(9.12)

where

F(
r
,t)

and

N(
r
.
t)

are

the

force

and

torque

respectively

that

acting

on

a

molecule

at

(
r
,t)
.

They

are

arise

from

the

Coulomb

interactions

between

molecules

and

can

be

expressed

as
:

(9.13)

(9.14)

Here

linear

molecule

centered

at

r

with

orientation

u

is

considered
.

(
r
+s
u)

is

the

position

of

a

distance

s

from

the

molecular

center

along

the

molecular

axis
.

Then

E(r
+s
u
)

is

the

electric

field

at

the

point

due

to

all

charges

in

the

system
.

Z(s)

is

the

linear

charge

density

and

dsZ(s)
E(r
+s
u
)

is

the

electric

force

exerted

on

this

charge

by

the

surrounding

fluid
.

Likewise

s
u

dsZ(s)
E
(
r
+s
u
)

is

the

corresponding

torque
.

10

To

make

the

equations

(
9
.
12
-
9
.
14
)

self
-
consistent

the

Poisson

equation

has

to

be

used
:

(9.15)

where

(
r
,t)

is

the

charge

density

and

(
r
,t)

is

the

electrostatic

potential

at

r
,t
.

In

the

case

of

polarizable

molecules

4


in

Poisson

equation

have

replace

by

4

/

,

where

is

dielectric

constant

due

to

the

polarizability

[(

-
1
)/(

+
2
)=

o

]
.

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 multipole
moment that contributes to the orientation relaxation is the
dipole moment.

The solution of diffusion equation taking into account dipolar forces
gives the correlation function

(t)

that decays on two different time
scales specified by the relaxation times:

(9.16)

(9.17)

11

where
D
R

is the rotational diffusion coefficient, and

(9.18)

Correlation function can be written in the following way:

(9.19)

Two

relaxation

times

for

a

single

component

polar

fluid

was

found

also

by

Titulaer

and

Deuthch
,

Bordewijk

and

Nee
-

Zwanzig
.

If

Berne

discussed

the

two

correlation

times

as

decay

of

transverse

and

longitudinal

fluctuations
,

Nee

and

Zwanzig

considering

dielectric

friction

in

diffusion

equation
.

Considering

the

diffusion

equation

they

the

assumption

that

by

some

reasons

the

frictional

forces

on

the

particle

is

not

developed

instaneously,

but

lags

its

velocity
.

Considering

the

correlation

function

of

angular

velocities

they

came

to

the

frequency

dependent

friction

coefficient

in

diffusion

equation
:

(9.20)

12

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
u
(t).

The translational velocity is replaced by an angular velocity

(琩

a湤n瑨torcei猠replacedbya瑯rq略
N
(t).

The frictional torque is
proportional to the angular velocity:

(9.21)

or in Fourier components,

(9.22)

The total friction coefficient

(

)

consists of two parts. The first is due
to ordinary friction, e.g.
Stokes’ law friction

0

independent on
frequency. The other part is due to
dielectric friction

and is denoted by

D
(

).

The sum is

(9.23)

Using the
Onsager 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:

13

(9.24)

This expression is valid for
spherical isotropic Brownian motion 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

(琩

can

ond

rom

knole摧d

function

C(
u
,t)

of

orientations

as

a

function

of

time
.

This

distribution

function

obeys

the

diffusion

equation

for

spherically

isotropic

Brownian

motion
.

The

solution

of

this

equation

to

a

very

simple

relation

between

dielectric

friction

and

DCF
:

(9.25)

It is convenient to introduce in this case the frequency dependent
relaxation time

(

)

defined by

14

(9.26)

One can now write for molecular DCF the following relation:

(9.27)

From

comparison

of

(
9
.
27
)

with

the

Debye

behavior

we

are

coming

to

the

simple

relationship

between

macroscopic

and

molecular

correlation

times
:

(9.28)

which

is

different

from

the

relationship

obtained

by

Bordewijk

for

the

same

molecular

DCF

(9.29)

where
k=

s
/

15

Character of

interaction

Temperature

Structure

etcetera

is a phenomenological
parameter

is the relaxation time

?

Non
-
exponential relaxation

empirical

Cole
-
Cole
law

1941 year

(1
-

)

/ 2

16

The Memory function for Cole
-
Cole law

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

=
d
f

the memory function

a fractional derivation

17

Scaling relations

N,

are scaling parameters

d
G

is a geometrical fractal
dimension

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

is the constant depends on relaxation
units transport properties

is the self
-
diffusion coefficient

18

Hydrophilic

PAIA PAA PEI

are electrolyte polymers

PVA

Is a nonelectrolyte with strong interaction

between hydroxyl groups and water

Hydrophobic

PEG PVME

PVP

are nonelectrolyte polymers

N.

Shinyashiki, S.

Yagihara, I.

Arita, S.

Mashimo
,
Journal of

Physical Chemistry, B
102

(1998) p. 3249

T=Constant

Polymer water mixtures

19

Composite polymer
structure

H.

Nuriel, N.

Kozlovich, Y.

Feldman, G.

Marom

Composites: Part A
31

(2000) p. 69

The samples with Kevlar fibers

have the longer relaxation time

T
is not Constant

20

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

21

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.