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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.
1
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:
H
   0e kT
(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.
2
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 (DR = 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:
  8a 3
(9.2)
where  is the coefficient of viscosity.
3
Let us start with the diffusion equation:
C (r , u, t ) / t  DT  2r C  DRu2C
(9.3)
where DT and DR are, respectively, the transnational and rotational
diffusion coefficients,  r is the gradient operator on the space (x,y,z)
and  u is the rotation operator  u  u   /  u. In this equation
C(r,u,t)d2ud3r is the number of molecules with orientation u in the
spheroid angle d2u and center of mass in the neighborhood d3r of the
point r at time t. The microscopic definition of C is
N
C ( r , u, t )    (r  ri ) (u  ui (t ))
(9.4)
i 1
Here ri(t) and ui(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  u  u   /  u
is related to 4

I  i (u   /  u)
the dimensional angular momentum operator of


2
quantum mechanics; that is  u  i I and    I
2
u
It should be recalled

2
that the spherical harmonics Ylm(u) are eigenfunctions of I
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 {Ylm(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:
4a 3
0 
kT
(9.5)
5
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:
  (0)  (t ) 
C(t ) 
 e t / 0
  (0)  (0) 
(9.6)
This result was generalized to the case of prolate and oblate
ellipsoids by Perrin and Koenig:
6
a) Prolate ellipsoid:
=b/a <1
b
a
8 a 3
a 
3 kT 2   2
1  4
 1  1  2 

 1
ln


1   2 

16 a 3
b 
3 kT 1  1 2
(9.7)
1  4
 1  1  2  1
 2
ln
 
2

1 


(9.8)
b) Oblate ellipsoid: >1
7
8 a 3
a 
3 kT 2   2
1  4
 1
2
16 a 3
b 
3 kT 1  2  2
 1
2
tan 1

tan


2  1  1
1  4
1
(9.9)

 1 
2
1
(9.10)
2
In the case of ellipsoid of revolution the dipole correlation function can
be written in the following way:
C(t)  A1exp(-t/ a )  Αexpt /  b )
(9.11)
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.
8
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:
1
1
 C( r, u, t ) /  t = - D T  r  ( FC) 
DR  u  ( NC)  D T  2r C  DR  2u C (9.12)
kT
kT
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)
F (r , t )   dsZ ( s) E (r  su)
N (r , t )   dsZ ( s) su  E (r  su)
(9.14)
Here linear molecule centered at r with orientation u is considered.
(r+su) is the position of a distance s from the molecular center along
the molecular axis. Then E(r+su) 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+su) is the electric force exerted on this charge by the
surrounding fluid. Likewise sudsZ(s)E(r+su) is the corresponding
torque.
9
To make the equations (9.12-9.14) self-consistent the Poisson equation
has to be used:
 r  E ( r, t )  2r  ( r, t )  4( r, t )
(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:
1
(9.16)
1 
2 DR
2 
1
2(1   )DR
(9.17)
10
where DR is the rotational diffusion coefficient, and
4 2

 0
3kT
(9.18)
Correlation function can be written in the following way:
1
 ( t )  ( 2 e  t /  1  e  t / 2 )
3
(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:
kT
D( ) 
 ( )
(9.20)
11
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 (t)
and the force is replaced by a torque N(t). The frictional torque is
proportional to the angular velocity:
t
N (t )     (t  t ') (t ')dt '
(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 0 independent on
frequency. The other part is due to dielectric friction and is denoted by
D(). The sum is
 (  )  0   D (  )
(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:
12
D ( ) 
2kT (s   )[ ( )  s ]
i
s [2 ( )  ]
(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 (t) can be found from knowledge of the distribution
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:
L[ 
dC (t )
 ( ) 1
]  {1  i [
]}
dt
2 kT
(9.25)
It is convenient to introduce in this case the frequency dependent
relaxation time () defined by
13
 ( ) 
 ( )
(9.26)
kT
One can now write for molecular DCF the following relation:
dC(t ) 
(s   )[ ( )  s ]

L[ 
]  1  i 0 
dt
s [2 ( )   ] 

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:
M
2 s  

0
 s  2
(9.28)
which is different from the relationship obtained by Bordewijk for the
same molecular DCF
 M  0 k 1/( 2 k 1)
where k=s/
(9.29)
14
Non-exponential relaxation
2
10
''
1
empirical
Cole-Cole
law

   

10

1  (i )
0

10
-1
10 -4 -3 -2 -1 0
1
2
3
4
10 10 10 10 10 10 10 10 10
1941 year
is the relaxation time

Character of
interaction
?

is a phenomenological
parameter
Structure
''
0.2
Temperature
0.0
etcetera
-0.2
-0.4
(1-)  / 2
0.0 0.2 0.4 0.6 0.8 1.0
'
15
The Memory function for Cole-Cole law
t
d
f (t )    M (t   ) f ( ) d
dt
0
the memory function
a fractional derivation
z f ( z)  1   M ( z) f ( z)
M ( z )  z1 
df
 1
  D0 [ f (t )]
dt
R.R. Nigmatullin, Ya.E. Ryabov,
Physics of the Solid State, 39 (1997)
Fractal set
Interection
with thermostat
L. Nivanen, R. Nigmatullin, A. LeMehaute,
Le Temps Irrevesibible a Geometry Fractale,
(Hermez, Paris, 1998)
 = df
time
16
Scaling relations
ln( N )
df 
ln(  1 )

 0

 R 
N  G  
 R0 
 Ds  R
2
dG
dG ln(   s )

2 ln(  /  0 )
N,  are scaling parameters
dG is a geometrical fractal
dimension
 0 is the limiting time of the system self-similarity in the time domain
Ds is the self-diffusion coefficient
Ds
 s  2 G d G is the constant depends on relaxation
R0
units transport properties
17
T=Constant
Polymer water mixtures
dG ln(   s )

2 ln(  /  0 )
s

1.00
PAA
PEI
PEG
PVME
0.95
dG
0
 0 s
PVA
1.7
7.210 -12
1.510 11
1.06
PAIA
1.4
6.510 -12
1.710 11
1.12
PAA
1.1
6.310 -12
2.110 11
1.32
PEI
1.3
4.910 -12
2.710 11
1.31
PEG
1.5
4.510 -12
2.81011
1.2
PVME
1.4
3.610 -12
4.21011
1.5
PVP
0.9
0.7210 -12
2401011
17.2
PVA
PAIA
PVP
0.90
0.85
0.80
0.75
-11
, [s]
10
PAIA PAA PEI
PVA
are electrolyte polymers
Hydrophilic
Is a nonelectrolyte with strong interaction
between hydroxyl groups and water
PEG PVME
PVP
Hydrophobic
are nonelectrolyte polymers
N. Shinyashiki, S. Yagihara, I. Arita, S. Mashimo, Journal of Physical Chemistry, B
18
102 (1998) p. 3249
T is not Constant
Composite polymer
structure

dG ln(   s )

2 ln(  /  0 )
dG
1.4
C rystalline
nylon
1.2
C rystalline
nylon +
Kevlar fibres
s
0.45
0.40
 0 s
0.35
0.30
Quenched
nylon
Quenched
nylon +
kevlar fibers
0
Crystalline nylon
Quenched nylon
Quenched nylon+Kevlar
Crystalline nylon+Kevlar
(transcrystalin)
0.50
7.110
-3
2 . 210 3
16
0.25
1.3
1.5
8.710
2 . 4210
1.6
-3
-1
8 . 910 3
5 . 210
2
1 . 410
2
77
0.20 -7
10
-6
10
-5
10
-4
10
, [s]
126
The samples with Kevlar fibers
216
have the longer relaxation time
H. Nuriel, N. Kozlovich, Y. Feldman, G. Marom
Composites: Part A 31 (2000) p. 69
19
T is not Constant
Water absorbed in
the porous glass

h%
0.64
A
B
II
C
I
III
0.60
0.56
II
0.63
A
1.2
0.52
0.48
0.44
B
C
1.4
3.2
0.40
0.36
-7
10
III
3.39
I
3.6
-6
10
-5
10
-4
10
, [s]
Samples are separated in two groups according
to the humidity value h.
A. Gutina, E. Axelrod, A. Puzenko, E. Rysiakiewicz-Pasek, N. Kozlovich, Yu.
Feldman, J. Non-Cryst. Solids,235-237 (1998) p. 302
20
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.
21