Transcript The Problems of Electric Polarization

Tutorial lecture 1 in Kazan Federal University
The Problems of Electric
Polarization
Dielectrics in Static Field
Yuri Feldman
1
Ancient times
1745 first condenser constructed by Cunaeus and Musschenbroek
And is known under name of Leyden jar
1837 Faraday studied the insulation material,which he called the dielectric
Middle of 1860s Maxwell’s unified theory of electromagnetic phenomena
1847 Ottaviano-Fabrizio Mossotti
 = n2
1879 Clausius
2
1887 Hertz
1897 Drude
Lorenz
1912 Debye
Internal field
Dipole moment
Robert Cole
3
Dielectric response on mesoscale
Dielectric spectroscopy is sensitive to relaxation processes
in an extremely wide range of characteristic times ( 10 5 - 10 -12 s)
Broadband Dielectric Spectroscopy
Time Domain Dielectric Spectroscopy; Time Domain Reflectometry
10-6
10-4
10-2
102
0
104
106
108
1010
1012
f (Hz)
Porous materials
and colloids
Macromolecules Glass forming
liquids
Clusters Single droplets
and pores
Water
ice
4
Dielectric response in biological systems
Dielectric spectroscopy is sensitive to relaxation processes
in an extremely wide range of characteristic times ( 10 5 - 10 -11 s)
Broadband Dielectric Spectroscopy
Time Domain Dielectric Spectroscopy
10-1
0
101
102
103
104
105
106
107
108
ice
Cells
Proteins
109
1010
H

H3N+ — C — COO
R
Amino
acidsAsn
Ala
Asp Arg
Cys Glu Gln His
Ile Leu Lys Met
Phe Ser Thr Trp
Tyr Val
1011
1012
f (Hz)
Water
DNA, RNA
Lipids
Tissues
P
N+
-Dispersion
 - Dispersion
-
Head group
region
 - Dispersion
 - Dispersion
5
Electric dipole - definition
The electric moment of a point charge relative to a fixed point is defined
as er, where r is the radius vector from the fixed point to e.
Consequently, the total dipole moment of a whole system of charges ei
relative to a fixed origin is defined as:
m   ei ri
i
A dielectric substance can be considered as consisting of elementary

charges ei , and
ei

 eri 0

i
i

if it contains no net charge.






If the net charge of the system is zero, the electric moment is
independent of the choice of the origin: when the origin is displaced
over a distance ro, the change in m is given by:
6
6
m   ei ro  ro  ei
i
i
Thus m equals zero when the net charge is zero.
Then m is independent of the choice of the origin. In this case equation
(1.1) can be written in another way by the introduction of the electric
centers of gravity of the positive and the negative charges.
These centers are defined by the equations:
e r  r e  r Q
i i
and
p
positive
i
positive
e r  r e
i i
negative
p
n
i
negative
 rn Q
in which the radius vectors from the origin to the centers are
represented by rp and rn respectively and the total positive charge is
called Q.
m  (rp  rn )Q
7
The difference rp-rn is equal to the vector distance between the
centers of gravity, represented by a vector a, pointing from the
negative to the positive center ( Fig.1).
+Q
a
rp
- Q
rn
Thus we have:
m  aQ
Therefore the electric moment of a
system of charges with zero net
charge is generally called the electric
dipole moment of the system.
A simple case is a system consisting of only two point charges + e and
- e at a distance a.
Such a system is called a (physical) electric dipole, its moment is
equal to ea, the vector a pointing from the negative to the positive
charge.
Under the influence of the external electrical field, the positive and
negative charges in the particle are moved apart: the particle is
polarized. In general, these induced dipoles can be treated as ideal;
permanent dipoles, however, may generally not be treated as ideal
8
when the field at molecular distances is to be calculated.
The values of molecular dipole moments are usually expressed in
Debye units. The Debye unit, abbreviated as D, equals 10-18
electrostatic units (e.s.u.).
The permanent dipole moments of non-symmetrical molecules
generally lie between 0.5 and 5D. It is come from the value of the
elementary charge eo that is 4.410-10 e.s.u. and the distance s of the
charge centers in the molecules amount to about 10-9-10-8 cm.
In the case of polymers and biopolymers one can meet much higher
values of dipole moments ~ hundreds or even thousands of Debye units.
To transfer these units to SI system one have to take into account that
1D=3.310-30 coulombsm.
9
Types of polarization
Deformation polarization
a. Electron polarization - the displacement of nuclear and electrons in the
atom under the influence of external electric field. As electrons are very
light they have a rapid response to the field changes; they may even follow
the field at optical frequencies.
-
+ +
+
Electric Field
b. Atomic polarization - the displacement of atoms or atom groups in the
molecule under the influence of external electric field.
10
Orientation polarization:
The electric field tends to direct the permanent dipoles.
Electric field
+e
-e
11
Ionic Polarization
In an ionic lattice, the positive ions are displaced in the direction of the applied
electric field whilst the negative ions are displaced in the opposite direction,
giving a resultant dipole moment to the whole body.
+
-
+
+
-
-
-
- +
+
+
-+
-
+
--
+
+ -
+
+
-
+-
+
- +
- +
+
+
-++
-
+
-
++
+
Electric field
12
The vector fields E and D.
For measurement inside matter, the definition of E in vacuum, cannot be
used.
There are two different approaches to the solution of the problem how to
measure E inside matter. They are:
1. The matter can be considered as a continuum in which, by a sort of
thought experiment, virtual cavities were made. (Kelvin, Maxwell).
Inside these cavities the vacuum definition of E can be used.
2. The molecular structure of matter considered as a collection of point
charges in vacuum forming clusters of various types. The application here
of the vacuum definition of E leads to a so-called microscopic field
(Lorentz, Rosenfeld, Mazur, de Groot). If this microscopic field is
averaged, one obtains the macroscopic or Maxwell field E .
13
For the solution of this problem of how to determine the electric field
The
main problem of physics of dielectrics is
inside matter, it is also possible first to introduce a new vector field D in
passing
from
a
phenomenological
macroscopic
such a way that for this field the source equation will be valid.
linear dielectric response to the microscopic
divDof electrons,
4
structure in terms
nuclei, atoms,
molecules
and ions.
Inisgeneral
case
the problem
According to Maxwell,
matter
regarded as
a continuum.
To use theis
still
unresolved
definition
of the field completely.
vector E, a cavity has to be made around the point
where the field is to be determined.
However, the force acting upon a test point charge in this cavity will
generally depend on the shape of the cavity, since this force is at least
partly determined by effects due to the walls of the cavity. This is the
reason that two vector fields defined in physics of dielectrics:
The electric field strength E satisfying curlE=0, and the dielectric
displacement D, satisfying div D=4.
14
The Maxwell continuum can be treated as a dipole density of matter.
Difference between the values of the field vectors arises from
differences in their sources. Both the external charges and the dipole
density of the piece of matter act as sources of these vectors.
The external charges contribute to D and to E in the same manner.
Because of the different cavities in which the field vectors are
measured, the contribution of dipole density to D and E are not the
same. It can be shown that
D  E  4P
where P called the POLARIZATION.
Generally, the polarization P depends on the electric strength E. The
electric field polarizes the dielectric.
The dependence of P on E can take several forms:
P  E
15
The polarization proportional to the field strength. The proportional
factor  is called the dielectric susceptibility.
D  E  4P  ( 1  4 )E  E
in which  is called the dielectric permittivity. It is also called the
dielectric constant, because it is independent of the field strength. It is,
however, dependent on the frequency of applied field, the temperature,
the density (or the pressure) and the chemical composition of the
system.
E
D
Dielectric sample
D E
16
Polar and Non-polar Dielectrics
To investigate the dependence of the polarization on molecular
composition, it is convenient to assume the total polarization P to be
divided into two parts: the induced polarization P caused by the
translation effects, and the dipole polarization P caused by the
orientation of the permanent dipoles.
 1
E  P  P
4
A non-polar dielectric is one whose molecules possess no
permanent dipole moment.
A polar dielectric is one in which the individual molecules possess a
dipole moment even in the absence of any applied field (i.e. the center
of positive charge is displaced from the center of negative charge).
17
Induced and orientation polarizations
Induced
polarization
Orientation
polarization
PP
 NkN k (Eμi )k
α 

k

k
k
k
Nk
is the number of particles per volume unit;
 1 3


a
 is the scalar polarizability of a particle ;
 2
Ei is the Internal Field, the average field strength acting upon
that
It isvalue
definedofasthe
thepermanent
total electric field
at the
position
is the
dipole
vector
μk particle.
of the particle
minusover
the field
due to the particle itself.
averaged
all orientations.
k is the index referred to the k-th kind of particle.
18
Orientation polarization, Average dipole moment
The energy of the random oriented permanent dipole  in the electric field
dependent on the part of the electric field tending to direct the permanent
dipoles. This part of the field is called the directing field Ed.
Wk  - μk  Ed  Ed cosk
Averaging
W  Ed cos


E

The relative probabilities of the various orientations of dipole depend on
this energy according to Boltzmann’s distribution law:
 Ed cos  1
p d ~ exp
 sin d
kT

2
19

cos 
 cose
0

e
Ed cos
kT
Ed cos
1
sin d
2
1
sin d
2
kT
0
a

1
a
x
e
 xdx
a
a

x
e
 dx
a
1 [ xe x  e x ]aa e a  e  a 1
1




cot
anh
(
a
)

 L( a ),
a [ e x ]aa
e a  e a a
a
E cos
where
x
d
kT
L(a) is called Langeven function
E
d a
kT
and
In Fig. the Langeven function
y
L(a)
is plotted against a. L(a) has a
limiting value 1, which was to be
expected since this is the maximum
y=1/3 a
1.0
y=1
of cos. For small values of
a,
<cos> is linear in Ed:
0.8
y=L( a )
0.6
0.4
0.2
0.0
0
1
2
3
4
5
6
7
8
E
1
cos  a  d
3kT
3
if 0  a  1
a
20
The approximation of cos may be used as long as
E
0.1kT
a  d  0.1 or E 
.
kT
d

At room temperature (T=300o K) this gives for a dipole of 4D:
E 
d
0.1kT

= 3 105 v/cm
For a value of  smaller than the large value of 4D, the value calculated for Ed is
even larger. In usual dielectric measurements, Ed is much smaller than 105 v/cm
and the use of cos is allowed.
From the linear response approximation it follows that:
2
μ   cos  
Ed
3kT
Substituting this into the main relationship for the orientation polarization, we get:
2
P   N k
( Ed ) k
k

21
Fundamental equation
 1
E = P  P
4


 k2
 1
E =  N k  k ( Ei ) k 
( Ed ) k 
4
3kT
k


This is the fundamental equation is the starting point
for expressing Ei and Ed as functions of the Maxwell
field E and the dielectric constant .
22
Dipole moments
and electrostatic problems
Eo
Z
1
a
2
Let us put a dielectric sphere of radius a and
dielectric constant 2, in a dielectric extending
to infinity (continuum), with dielectric constant
1, to which an external electric field is applied.
Outside the sphere the potential satisfies
Laplace's equation =0, since no charges are
present except the charges at a great distance
required to maintain the external field. On the
surface of the sphere Laplace's equation is not
valid, since there is an apparent surface
charge.
Inside the sphere, however, Laplace's equation can be used again. Therefore,
for the description of , we use two different functions, 1 and 2, outside and
inside the sphere, respectively.
23
Let us consider the center of the sphere as the origin of the coordinate system,
we choose z-axis in the direction of the uniform field. Following relation in the
terms of Legendre polynomial represents the general solution of Laplace’s
equation:

B 

1    An r n  nn1 Pn (cos )
r 
n 0 

D 

 2    Cn r n  nn1 Pn (cos )
r 
n 0 
Eo
Z
1
a
2
The boundary conditions are:
1.
2.
3.
4.
1 r  E0 z  E0r cos
1 r a  2 r a
Since  is continuous across a boundary
 d1 
 d2 
1 
  2 

 dr  r a
 dr  r a
since the normal component of D must be
continuous at the surface of the sphere
At the center of the sphere (r=0) 2 must not have a singularity.
24
The total field E2 inside the sphere is accordingly is given by:
E2 
31
E0
21   2
A spherical cavity in dielectric
In the special case of a spherical cavity in dielectric (1=; 2=1), equation is
reduced to:
3
EC 
E0
2  1
2=1
1 = 
This field is called the "cavity field". The lines of dielectric
displacement given by Dc=3Do/(2+1) are more dens in the
surrounding dielectric, since D is larger in the dielectric
than in the cavity
25
A dielectric sphere in vacuum
For a dielectric sphere in a vacuum (1=1; 2=), the equation is reduced to:
3
E
E0
2  1
where E is the field inside the sphere.
The density of the lines of dielectric displacement Ds is higher in the sphere
than in the surrounding vacuum, since inside the sphere Ds=3Eo/(+2).
Consequently, it is larger than Eo.
2 = 
1=1
The field outside the sphere due to the
apparent surface charges is the same as the
field that would be caused by a dipole m at
the center of the sphere, surrounded by a
vacuum, and given by:
 1 3
m
a E0
2
26
Type of interactions
Two types of interaction forces:
-Short range forces- interaction between nearest neighbors:
• Chemical bonds,
• Van der Waals attraction,
• Repulsion forces,
etc.
Long rang dipolar interaction forces
Dipole-dipole interaction
Dipole -charge interaction
Due to the long range of the dipolar forces an accurate calculation
of the interaction of a particular dipole with all other dipoles of a
specimen would be very complicated.
The different approaches where developed for solving this
problem.
27
Lorentz’s method
28
Non-polar dielectrics. Lorentz's field.
Clausius-Massotti formula.
The apparent
surface
For a non-polar system the fundamental equation for the dielectric
permittivity
charges
 1
(2.49) is simplified to:
E = N  (E )


4
k
k
i
k
E
k

_
_ __
E
In this case, only the relation between the internal field and the Maxwell field has to
_
be determined. Let us use the Lorentz approach in this case.
_ He calculated _the
internal field in homogeneously polarized matter as the field in a virtual
spherical cavity.
EC 
3
E
2  1
+
Lorentz’s field
Real cavity
EL 
 2
E
3
Lines of dielectric displacement
+
+
+
++
Virtual cavity
29
 1
E =  N k  k ( Ei ) k
k
4
For a pure compound (k=1)
 2
Ei  E L 
E
3
Clausius-Massotti formula
  1 4

N
 2 3
30
Debye theory; Gases and polar
molecules in non-polar solvent


k2
 1
E =  N k  k (Ei ) k 
(E d ) k 
4
3kT
k


k  1;
Ed  Ei  E L 
 2
3
E
  1 4 
2 


N  
 2 3 
3kT 
This
is generally
called thethe
Debye
equation.
It is
was
the first relationship that
But
in many
cases, however,
Debye
equation
in considerable
made the connection
between the
molecular
parameter
substance
disagreement
with the experiment.
It works
very nice
for gasesofatthe
normal
being tested
and
theone
phenomenological
(macroscopic)
pressures.
In this
case
has -1<<1 and equation
can beparameter
written as:that can be
experimentally measured.

2
  1 = 4N  
3kT





31
The reaction field and Onsager’s approach
When a molecule with permanent dipole strength  is surrounded by other
particles, the inhomogeneous field of the permanent dipole polarizes its
environment. In the surrounding particles moments proportional to the
polarizability are induced, and if these particles have a permanent dipole
moment their orientation is influenced. To calculate this effect one can use
a simple model: an ideal dipole in a center of a spherical cavity.
The inhomogeneous field of the permanent dipole
Electric dipole field lines
Line of force in the dipole field
32
The reaction field of a non-polarizable point dipole
Let us assume that only one kind of molecule is presented and a is value
approximately equal to what is generally considered to be the “molecular
radius”
Solving the Laplace equation with slightly different boundary conditions :
z


a
1 r  0
1 r a  2 r a
1.
2.
 d 
 d 
 1    2 
 dr  r a  dr  r a
3.
We can calculate:
The field in the cavity is a superposition of the dipole field in vacuum
and a uniform field R, given by:
1 2(  1)
R

2  1
1 2(  1)
and the factor of the reaction field is equal to f 
a 3 2  1
a
3
33
33
Formally, the field of dielectric can be described as the field of a virtual
dipole c at the center of the cavity, given by:
3
c 

2  1
The presented model involves a number of simplifications, since the
dipole is assumed to be ideal and located at the center of the
molecule, which is supposed to be spherical and surrounded by a
continuous dielectric.
The reaction field of a polarized point dipole
In this case the permanent dipole has an average polarizability , and
therefore the reaction field R induces a dipole R and satisfies the
equation:
R  f( μ  R )
Under the influence of the reaction field the dipole moment is increased
considerably, the increased moment is:
μ  μ  R
*

n2  1
3 
a
n2  2
Considering
We can obtain that
   n2

2  1 n 2  2

 2  n 2 3
In the case of polar dielectrics, the molecules have a permanent dipole
moment , and both parts of the fundamental must be taken into
account.


k2
 1
E =  N k k ( Ei ) k 
( Ed ) k 
4
3kT
k


In the case of non-polar liquids the internal field can be considered
as the sum of two parts; one being the cavity field and another the
reaction field of the dipole induced in the molecule Ei=Ec+R.
35
For polar molecules the internal field can also built up from the cavity
field and the reaction field, taking into account now the reaction field
of the total dipole moment of the molecule.
z

a

The angle between the reaction field of
the permanent part of the dipole moment
and the permanent dipole moment itself
will be constant during the movements of
the molecule.
It means that in a spherical cavity the permanent dipole moment and the
reaction field caused by it will have the same direction. Therefore, this
reaction field R does not influence the direction of the dipole moment of the
molecule under consideration, and does not contribute to the directing field
Ed.
On the other hand, the reaction field does contribute to the internal
field Ei, because it polarizes the molecule. As a result, we find a
difference between the internal field Ei and the directing field Ed.
36
Since the reaction field R belongs to one particular orientation of the
dipole moment, the difference between Ei and Ed will give by the value
of the reaction field averaged over all orientations of the polar molecule:
Ei - Ed  R
z

a


The direction field Ed can be obtained by
the following procedure:
) remove the permanent dipole of a
molecule
without
changing
its
polarizability;
) let the surrounding dielectric adapt itself to the new situation;
) then fix the charge distribution of the surroundings and remove the
central molecule.
The average field in the cavity so obtained is equal to the value of Ed
that is to be calculated, since we have eliminated the contribution of R
to Ei by removing the permanent dipole of the molecule.
37
9kT (    )(2    )
 
2
4N
 (   2)
2
This equation is generally called the Onsager equation. It makes
possible the computation of the permanent dipole moment of a
molecule from the dielectric constant of the pure dipole liquid if the
density and  are known.
38
Relationships between the different kinds of
electric fields
In this case of the internal field can be considered
In order
to describe
of matter the different kinds of
The
non-polar
liquidspolarization
as the sum of two parts; one being the cavity field
electric fields where introduced
: Maxwell’s
field Efield
field
; internal
and another
that is the reaction
of the dipole
Ei ; Lorentz field EL ; direction
field Ed reaction field R and
induced in the molecule:
Polar molecules
 2
the cavity field EC
E E E R 
For polar molecules the internal field can also built up from the cavity field and
the reaction field, Itaking into
L accountCnow the reaction field of the total dipole
Let
usof the
consider
moment
molecule. the relationships between these
3
E
fields
in thefield
different
systems.
Since
the reaction
R belongs to
one particular orientation of the dipole
moment, the difference between Ei and Ed will give by the value of the reaction
Gases and polar molecules in non-polar solvent
field averaged over all orientations of the polar molecule:
Ed  E I Ei E
- LEd EC RR 
 2
3
E
The Kirkwood –Froehlich approach
In the above approximations we considered nonpolar systems, polar dilute systems and polar
systems without short range interactions. In all
these cases the dipole-dipole interactions between
molecules where not taking into account.
A more general theory was developed by Kirkwood
and subsequently refined by Froehlich. This
approach takes into account the dipole-dipole
interactions, which appears in a more dense state
under the influence of the short range interactions
40
In this case the external field working in the sphere is the cavity field

3
E0 = EC 
E
2  1
N dipoles i
E is the Maxwell field in the material outside the
sphere
  1 = 4n E0 
1
1

2

e

A

e



M

0
0


3kTn
 E  E 0  n
Here n=N /V is the number density, and the
tensor A plays the role of a polarizability;
N-N molecules are considered to form a
continuum
N
N
N
 M 0   i  j
2
i 1 j 1
0
M =  i
For non polarizable molecules
i =1
 e  A  e 0  0
4 3  M 2  0
   1 =
V 2  1 3kT
M2
N
0

i 1
N
dX
i  M exp(U / kT )

N
dX
exp(U / kT )

41
In this equation the superscript N to dX to emphasize that the integration is performed
over the positions and orientations of N molecules (here dX=r2sindrdd, is the
expression for a volume element in spherical coordinates).
Since i is a function of the orientation of the i-th molecule only, the integration
over the positions and orientations of all other molecules, denoted as N -i, can
be carried out first. In this way we obtain (apart from a normalizing factor) the
average moment of the sphere in the field of the i-th dipole with fixed
orientation. The averaged moment, denoted by Mi*, can be written as:

N dipoles i
M i* 
N i
dX
M exp(U / kT )

N i
dX
exp(U / kT )

The average moment Mi* is a function of the position and orientation of the i-th
molecule only.
Denoting the positionM
andi* orientation coordinates of the i-th molecule by Xi and
using a weight factor p(Xi)
p(X i ) 
N 1
dX
exp(U / kT )

N
dX
exp(U / kT )

42
M
N
2
0
   p( X i  i  M i* dX i N  p( X i  i  M i* dX i
i 1
N
i  M i*    
N i
dX
cosij exp(U / kT )

N i
dX
exp(U / kT )

j=1
M2
N
0
 N 2   p( X i )
N -i
dX
cosij exp(U / kT )

N -i
dX
exp(U / kT )

j 1
 cosij   p( X i )
N -i
dX
cosij exp(U / kT )

 M 0  N
2
N -i
dX
exp(U / kT )

2
N
  cos
j 1
ij
dX i
dX i

43
( -1)(2 + 1)
N
N 2N
i
*
i

p( X  i  M i dX 
   cosij 

12
3kT
3kT
j 1
The left part of the Onsager equation for
the non polarized molecules
The deviations of Mi* from the value i are the result of molecular interactions
between the i-th molecule and its neighbors.
It is well known that liquids are characterized by short-range order and longrange disorder. The correlations between the orientations (and also between
positions) due to the short-range ordering will lead to values of Mi* differing
from i. This is the reason that Kirkwood introduced a correlation factor g which
accounted for the deviations of
N
i
*
i
2
p
(
X



M
dX


  cosij 
i
i

j 1
g
1

2
 p( X
from the value 2:
N
i
 i  M dX    cosij 
*
i
i
j 1
44
( - 1)(2 + 1)
N

g 2
12
3kT
When there is no more correlation between the molecular orientations
than can be accounted for with the help of the continuum method, one
has g=1 we are going to Onsager relation for the non-polarizable
case, for rigid dipoles with =1.
An approximate expression for the Kirkwood correlation factor can be
derived by taking only nearest-neighbors interactions into account. In
that case the sphere is shrunk to contain only the i-th molecule and its
z nearest neighbors. We then have:

result of the integration will be not depend on
N 1
Since after
averaging
the


dX

 i    j  exp(U / kT )
j=1
all terms
 the summation
*
with
=z+1.and we may write:
theMvalue
of
j
,
in
areN
equal

i
z
 dX
z
N 1
exp(U / kT )
gi  dX
1 Nz- icos
cos
ij exp(
ij U / kT )
g  1 +   p( X i )dX
j 1
N -i
dX
exp( U / kT )

45
g will be different from 1 when <cosij>0, i.e. when there is
correlation between the orientations of neighboring molecules.
When the molecules tend to direct themselves with parallel dipole
moments, <cosij> will be positive and g>1.
When the molecules prefer an ordering with anti-parallel dipoles, g <1.
Approximation of Fröhlich



9 kT    2   
g 
2
4
   2
2
46
Main relationships in static dielectric theory
Non-polar systems
  1 4

N
2
3
Polar diluted systems

2
  1 = 4N  
3kT

Polar systems
z
a


 Debye equation

9kT (    )(2    )
 
4N
 (   2) 2
2


Clausius-Mossotti equation
Onsager Equation
Polar systems, short range interactions




9 kT    2   
g 
2
4
   2
2
Kirkwood-Fröhlich equation
47