Transcript Document
Lecture 1
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.
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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
= n 2
1887 Hertz 1847 Mossotti 1897 Drude 1879 Clausius Lorentz-Lorentz 1912 Debye Internal field Dipole moment
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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 Time Domain Dielectric Spectroscopy
10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 10 8 10 10 10 12
Porous materials and colloids Macromolecules Glass forming liquids Clusters Single droplets and pores Water f (Hz)
ice
3
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 10 1
Tissues
10 2 10 3 10 4
DNA, RNA
ice
10 5 10 6
Cells
10 7 10 8
Proteins
10 9 10 10 10 11
H
H 3 N + — C — COO -
R
Amino acids Ala Asp Arg Asn Cys Glu Gln His Ile Leu Lys Met Phe Ser Thr Trp Tyr Val
Water
10 12 10 13 10 14
f (Hz)
-Dispersion
- Dispersion
Lipids
N + P Head group region
- Dispersion
- Dispersion
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ii. Electric dipole - definition
The electric moment of a point charge relative to a fixed point is defined as e
r
, where
r
is the radius vector from the fixed point to Consequently, the total dipole moment of a whole system of charges relative to a fixed origin is defined as: e .
e i
m
i
e r
i i
(1.1) A dielectric substance can be considered as consisting of elementary charges e i , and
i
e
i
0 (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
r o
, the change in
m
is according to (1.1), given by: 5
m
Thus
m
i e i
r
o
r
o
i e i
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: and
positive
e
i negative e i
r
i
r
i
r
n
e i positive
e
i negative
r
p
Q
r
n
Q
in which the radius vectors from the origin to the centers are represented by called Q
.
r
p and
r
n respectively and the total positive charge is Thus in case of a zero net charge, equation (1.1) can be written as: 6
m
(
r
p
r
n )Q The difference
r
p -
r
n is equal to the vector distance between the centers of gravity, represented by a vector negative to the positive center ( Fig.1.1).
a
, pointing from the r p + Q a Thus we have:
m
a
Q
(1.3) Figure 1.1
r n Q Therefore the
electric moment of a system of charges with zero net charge is generally called the
dipole moment
of the system.
electric
A simple case is a system consisting of only two point charges - e at a distance
a .
+ e and Such a system is called a equal to e
a
, the vector charge.
a (physical) electric dipole
, its moment is pointing from the negative to the positive 7
A mathematical abstraction derived from the finite physical dipole is the ideal or between two point charges + charge e
point dipole
. Its definition is as follows: the distance
a
e
a
/n by en .
and e i.e. replaced by and the The limit approached as the number
dipole
.
n tends to infinity is the
ideal
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
of molecules the centers of gravity of the positive and negative charge distributions do not coincide.
), since in most types The molecules that have such kind of permanent dipole molecules called
polar molecules.
Fig.2 Dipole moment of water molecule. Apart from these permanent or intrinsic dipole moments,
moment
a temporary
induced dipole
arises when a particle is brought into external electric field.
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Under the influence of this field, the the particle are moved apart: positive and negative charges in 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 e o that is 4.4
10 -10 e.s.u.
and the distance charge centers in the molecules amount to about 10 -9 -10 -8 cm .
s of the 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 account that 1D=3.33
10 -10 coulombs m .
one have to take into 9
Polarization
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:
F
ee ' r
2
r
r
(1.4) Taking one of the charges, say effect of the charge e e' , as a test charge to measure the on its surroundings, we arrive at the concept of an electric field produced by e and with a field strength or intensity defined by: 10
E
e'
lim 0
F
e'
(1.5) The field strength due to an electric charge at a distance given by:
r E
e r
r
2
r
is then (1.6)
in which
E r
is expressed in cm,
e
in electrostatic units and in dynes per charge unit, i.e. the e.s.u. of field intensity.
A simple vector-analytic calculation shows that Eq. (1.6) leads to:
E
dS
4
e
(1.7) in which the integration is taken over any closed surface around the charge e , and where outward normal.
dS
is a surface element having direction of the 11
Assuming that the electric field intensity is additively built up of the contributions
superposition)
of all the separate charges Eqn. (1.7) can be extended to:
(principle of
E
dS
4
i e i
(1.8) This relation will still hold for the case of the continuous charge distribution, represented by a charge density .
volume charge density or a surface For the case of a volume charge density we write:
E
dS
4
V
dv
(1.9) or , using Gauss divergence theorem :
div
E
4
This equation is the first Maxwell's electrostatic field
equation.
well-known equations for the in vacuum; it is generally called the
source
(1.10) 12
The second of Maxwell's equations, necessary to derive a given charge distribution, is:
E
curl
E
0
uniquely for (1.11) or using Stokes's theorem:
E
ds
0
(1.12) in which the integration is taken along a closed curve of which line element.
ds
is a
Stokes' theorem:
S
(
curl
A
)
dS
C
A
ds,
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 .
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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:
E
-grad
(1.13) The combination of (1.10) and (1.13) leads to the famous equation: Poisson's
div grad
2 4 (1.14) In the charge-free parts of the field this reduces to the Laplace's equation: 0 (1.15) 14
The vector fields
E
and
D
.
For measurement inside matter, the definition of be used.
E
in vacuum , cannot 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 thought experiment, continuum virtual cavities in which, by a sort of were made.
Inside these cavities the vacuum definition of
E
( Kelvin, Maxwell can be used.
).
2. The molecular structure of matter considered as a point charges in vacuum application here of the vacuum definition of
microscopic field Maxwell field E
.
( forming clusters of various types. The
E
collection of leads to a so-called Lorentz, Rosenfeld, Mazur, de Groot microscopic field is averaged, one obtains the )
.
If this
macroscopic or
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The
main problem of physics of dielectrics
microscopic description in terms of electrons, nuclei, atoms, molecules and ions, to a macroscopic or phenomenological description is still unresolved completely.
of passing from a 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.
divD
4
(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 dielectric displacement
D
,
E
satisfying curl satisfying div
D
= 4
E
=0 .
, and the 16
The Maxwell continuum Difference density can be treated as a between the values of the field vectors arises from differences in their sources. Both the of the piece of matter act as dipole density of matter .
external charges and the dipole sources of these vectors .
The external charges contribute to measured, same . It can be shown that
D
and to
E
in the same manner.
Because of the different cavities in which the field vectors are the contribution of dipole density to
D
and
E
are not the
D
E
4
P
(1.17) where
P
called the POLARIZATION.
Generally, the polarization
P
depends on the electric strength
electric field polarizes the dielectric.
E.
The
The dependence of
P
on
E
can take several forms: P
E
(1.18) 17
The polarization proportional to the field strength. The proportional factor is called the
dielectric susceptibility .
D
E
4
P
(
1 4
)
E
E
(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.
P
E
E
2
E
(1.20) For very high field intensities the proportionality no longer holds.
Dielectric saturation and non-linear dielectric effects.
P
E
(1.21) 18
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 x D y D z 11 E 21 E x x 12 E y 22 E y 13 E z 23 E z 31 E x 32 E y 33 E z (1.22) 19
Types of polarization
For isotropic systems and leaner fields in the case of static electric fields
P
4 1
E
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.
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Deformation polarization
+ + -
Electric Field
21
Orientation polarization:
The electric field tends to direct the permanent dipoles.
Electric field +e
-e
22
Ionic Polarization
In ionic lattice , the positive ions are displaced in the direction of an applied field while the negative ions direction, giving a resultant ( apparent ) body.
are displaced in the opposite dipole moment to the whole + + + + + + + + + + + + + + + +
Electric field
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Polar and Non-polar Dielectrics
To investigate the dependence of the polarization on molecular quantities it is convenient to assume the polarization
P
to be divided into two parts: effects , and the the induced polarization dipole polarization
P
P
caused by the caused by the translation orientation of the permanent dipoles.
4 1
E
P
P
A
non-polar dielectric
permanent dipole moment.
is one whose molecules possess no A
polar dielectric
dipole moment is one in which the individual molecules possess a even in the absence of any applied field, i.e. the center of positive charge is displaced from the center of negative charge.
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