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
radius
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
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
:
(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
add
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
made
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torcei猠replacedbya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
扥
ond
rom
knole摧d
潦
瑨e
摩獴ri扵bion
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
leads
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.
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