3. Introduction into the physics of dielectrics.
ii. Electric dipole - definition.
a) Permanent dipole moment,
b) Induced dipole moment.
iii. Polarization and dielectric constant.
iv. Types of polarization
a) electron polarization,
b) atomic polarization,
c) orientation polarization,
d) ionic polarization.
3
4. 4
Ancient times
1745 first condensor 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
= n2
1887 Hertz
1897 Drude
1847 Mossotti
1879 Clausius
Lorentz-Lorentz
Internal field
1912 Debye
Dipole moment
5. 5
The dynamic range of Dielectric Spectroscopy
Dielectric spectroscopy is sensitive to relaxation processes
in an extremely wide range of characteristic times ( 10 5 - 10 -12 s)
Broadband Dielectric Spectroscopy
Porous materials
and colloids
Clusters Single droplets
and pores
Glass forming
liquids
Macromolecules
10-2
10-4 100 102 104 106 108 1010 1012
Time Domain Dielectric Spectroscopy
f (Hz)
10-6
Water
ice
6. 6
Broadband Dielectric Spectroscopy
Cells
Time Domain Dielectric Spectroscopy
f (Hz)
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)
H
H3N+ — C — COO-
R
Ala Asp Arg Asn
Cys Glu Gln His
Ile Leu Lys Met
Phe Ser Thr Trp
Tyr Val
10-1
P
-
N+
Head group
region
Lipids
Proteins
Water
- Dispersion
DNA, RNA
-Dispersion - Dispersion - Dispersion
Tissues
ice
Amino acids
103
102
0 105 106 107 108 109 1010
104
101 1011 1012 1013
1014
7. ii. Electric dipole - definition
7
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:
i
i
i
e r
m (1.1)
A dielectric substance can be considered as consisting of elementary
charges ei , and
0
i
i
e (1.2)
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 according to (1.1), given by:
8. 8
i
i
o
o
i
i e
e r
r
m
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:
Q
e
e
positive
i
i
positive
i p
p r
r
r
Q
e
e
negative
i
i
negative
i n
n r
r
r
and
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.
Thus in case of a zero net charge, equation (1.1) can be written as:
9. 9
)Q
( n
p r
r
m
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.1.1).
rp
r n
Q
+
Q
-
a
Figure 1.1
Thus we have:
Q
a
m (1.3)
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.
10. 10
The limit approached as the number n tends to infinity is the ideal
dipole. The formulae derived for ideal dipoles are much simpler than
those obtained for finite dipoles.
Many natural molecules are examples of systems with a finite electric
dipole moment (permanent dipole moment), since in most types
of molecules the centers of gravity of the positive and negative
charge distributions do not coincide.
A mathematical abstraction derived from the finite physical dipole is the
ideal or point dipole. Its definition is as follows: the distance a
between two point charges +e and -e i.e. replaced by a/n and the
charge e by en.
Fig.2 Dipole moment
of water molecule.
The molecules that have such kind of permanent
dipole molecules called polar molecules.
Apart from these permanent or intrinsic dipole
moments, a temporary induced dipole
moment arises when a particle is brought into
external electric field.
11. 11
Under the influence of this 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 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 4.410-10 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 CI system one have to take into
account that 1D=3.3310-10 coulombsm.
12. Polarization
12
Some electrostatic theorems.
a) Potentials and fields due to electric charges.
According to Coulomb's experimental inverse square law, the force
between two charges e and e' with distance r is given by:
r
r
ee'
r
F 2
(1.4)
Taking one of the charges, say e', as a test charge to measure the
effect of the charge e on its surroundings, we arrive at the concept of
an electric field produced by e and with a field strength or intensity
defined by:
13. 13
e'
e'
F
E
0
lim
(1.5)
The field strength due to an electric charge at a distance r is then
given by:
r
r
e r
E 2
(1.6)
in which r is expressed in cm, e in electrostatic units and
E in dynes per charge unit, i.e. the e.s.u. of field intensity.
E
dS 4e (1.7)
A simple vector-analytic calculation shows that Eq. (1.6) leads to:
in which the integration is taken over any closed surface around the
charge e, and where dS is a surface element having direction of the
outward normal.
14. 14
Assuming that the electric field intensity is additively built up of the
contributions of all the separate charges (principle of
superposition) Eqn. (1.7) can be extended to:
E
dS 4 ei
i
(1.8)
This relation will still hold for the case of the continuous charge
distribution, represented by a volume charge density or a surface
charge density . For the case of a volume charge density we write:
E
dS 4
V
dv (1.9)
or , using Gauss divergence theorem:
4
div
E (1.10)
This equation is the first Maxwell's well-known equations for the
electrostatic field in vacuum; it is generally called the source
equation.
15. 15
The second of Maxwell's equations, necessary to derive E uniquely for
a given charge distribution, is:
0
curl
E (1.11)
or using Stokes's theorem:
0
ds
E (1.12)
in which the integration is taken along a closed curve of which ds is a
line element.
C
S
curl ds,
A
dS
A)
(
where C is the contour of the surface of integration S, and where the
contour C is followed in the clock-wise sense when looking in the
direction of dS.
Stokes' theorem:
16. 16
From (1.12) it follows that E can be written as the gradient of a scalar
field , which is called the potential of the field:
-grad
E (1.13)
The combination of (1.10) and (1.13) leads to the famous Poisson's
equation:
4
2
grad
div (1.14)
In the charge-free parts of the field this reduces to the Laplace's
equation:
0
(1.15)
17. The vector fields E and D.
17
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 .
18. 18
The main problem of physics of dielectrics of passing from a
microscopic description in terms of electrons, nuclei, atoms, molecules
and ions, to a macroscopic or phenomenological description is still
unresolved completely.
For the solution of this problem of how to determine the electric field
inside matter, it is also possible first to introduce a new vector field D
in such a way that for this field the source equation will be valid.
4
divD (1.16)
According to Maxwell matter is regarded as a continuum. To use the
definition of the field 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.
19. 19
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
P
E
D
4 (1.17)
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:
E
P
(1.18)
20. 20
The polarization proportional to the field strength. The proportional
factor is called the dielectric susceptibility.
E
E
P
E
D
)
(
4 4
1 (1.19)
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
E
P 2
E
(1.20)
For very high field intensities the proportionality no longer
holds.Dielectric saturation and non-linear dielectric effects.
E
P
(1.21)
21. 21
For non-isotropic dielectrics, like most solids, liquid crystals, the scalar
susceptibility must be replaced by a tensor. Hence, the permittivity
must be also be replaced by a tensor:
D E E E
D E E E
D E E E
x x y z
y x y z
z x y z
11 12 13
21 22 23
31 32 33
(1.22)
22. Types of polarization
22
For isotropic systems and leaner fields in the case of static electric
fields
E
P
4
1
The applied electric field gives rise to a dipole density
There can be two sources of this induced dipole moment:
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.
b. Atomic polarization - the displacement of atoms or atom groups
in the molecule under the influence of external electric field.
25. Ionic Polarization
25
In ionic lattice, the positive ions are displaced in the direction of an
applied field while the negative ions are displaced in the opposite
direction, giving a resultant (apparent) dipole moment to the whole
body.
+
-
-
-
-
+
+
+
-
-
+
+
-
-
+
+
+
-
-
-
-
+
+
+
-
-
+
+
-
-
+
+
Electric field
26. Polar and Non-polar Dielectrics
26
To investigate the dependence of the polarization on molecular
quantities it is convenient to assume the 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.
P
P
E
4
1
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.
