Diel Lecture1

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. 1

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

ε = n2 1887 Hertz

1847 Mossotti

1897 Drude

1879 Clausius

Lorentz-Lorentz 1912 Debye

Internal field

Dipole moment 2

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

100

Porous materials and colloids

102

104

106

108

1010

1012

f (Hz)

Macromolecules Glass forming liquids Clusters Single droplets and pores

Water

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

101

102

103

104

ice

105

106

107

Cells

108

Proteins

109

1010

1011

H  H3N+ — C — COO R

Amino acids

Ala Cys Ile Phe Tyr

DNA, RNA

Asp Glu Leu Ser Val

Arg Gln Lys Thr

1012

1013

f (Hz)

Water

Asn His Met Trp

Lipids

Tissues P N+

α-Dispersion

β - Dispersion

-

Head group region

δ - Dispersion

1014

γ - Dispersion

4

ii. 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

(1.1)

i

A dielectric substance can be considered as consisting of elementary charges ei , and ei = 0 (1.2)

∑ 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 according to 5 (1.1), given by:

∆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.

∑er = r ∑e = r Q

These centers are defined by the equations: i i

and

p

positive

∑e r

i i

negative

i

p

positive

= rn

∑e

i

= rn Q

negative

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: 6

m = (rp − rn )Q 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). +Q

a

rp rn Figure 1.1

- Q

Thus we have:

m = aQ

(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 7 negative to the positive charge.

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. 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.

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. 8

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 4.4⋅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 CI system one have to take into account that 1D=3.33⋅10-10 9 coulombs⋅m.

Polarizatio n 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: ' (1.4)

ee r F= 2 r r

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: 10

F E = lim e' → 0 e'

(1.5)

The field strength due to an electric charge at a distance r is then given by:

e r E= 2 r r

(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. A simple vector-analytic calculation shows that Eq. (1.6) leads to:

∫∫ E ⋅ dS = 4πe

(1.7)

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. 11

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 = 4π ∑ e

i

(1.8)

i

This relation will still hold for the case of the continuous charge distribution, represented by a volume charge density ρ or a surface charge density σ. For the case of a volume charge density we write:

∫∫ E ⋅ dS = 4π ∫∫∫ ρ dv

(1.9)

V

or , using theorem:

Gauss

divergence

divE = 4πρ

(1.10)

This equation is the first Maxwell's well-known equations for the electrostatic field in vacuum; it is generally called the 12 source equation.

The second of Maxwell's equations, necessary to derive E uniquely for a given charge distribution, is: (1.11)

curl E = 0

or using Stokes's theorem:

∫ E ⋅ ds = 0

(1.12)

in which the integration is taken along a closed curve of which ds is a line element.

Stokes' theorem:

∫∫ (curlA) ⋅ dS = ∫ A ⋅ ds, S

C

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. 13

From (1.12) it follows that E can be written as the gradient of a scalar field φ, which is called the potential of the field:

E = -gradφ

(1.13)

The combination of (1.10) and (1.13) leads to the famous Poisson's equation:

div grad φ = ∇ ⋅ ∇φ = ∇ 2 φ = ∆φ = −4πρ

(1.14)

In the charge-free parts of the field this reduces to the Laplace's equation:

∆φ = 0

(1.15)

14

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 . 15

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. (1.16)

divD = 4 πρ

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, 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=4πρ. 16

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 = 4πP

(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:

P = χE

(1.18) 17

The polarization proportional to the field strength. The proportional factor χ is called the dielectric susceptibility.

D = E + 4 πP = ( 1 + 4πχ )E = εE

(1.19)

in which ε 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 = χE + ξE 2 E

(1.20)

For very high field intensities the proportionality no longer holds.→Dielectric saturation and non-linear dielectric effects.

P = χE

(1.21) 18

For non-isotropic dielectrics, like most solids, liquid crystals, the scalar susceptibility must be replaced by a tensor. Hence, the permittivity ε must be also be replaced by a tensor:

D x = ε11E x + ε12 E y + ε13 E z D y = ε 21E x + ε 22 E y + ε 23 E z

(1.22)

D z = ε 31E x + ε 32 E y + ε 33 E z

19

Types of polarization For isotropic systems and leaner fields in the case of static electric fields

ε −1 P= E 4π

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 20 field.

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 are displaced in the opposite direction, giving a resultant (apparent) apparent dipole moment to the whole body.

+ + + - - + + + + - +

-

+

+

-

+ + + +

- + +

Electric field 23

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: the induced polarization Pα caused by the translation effects, effects and the dipole polarization Pµ caused by the orientation of the permanent dipoles.

ε −1 E = Pα + Pµ 4π

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. 24