Thermodynamics of Thermodynamics of Dielectric Dielectric

Relaxations in Relaxations in Complex SystemsComplex Systems

TUTORIAL 3

Static dipolesStatic dipoles It is necessary to found the Relation between It is necessary to found the Relation between

microscopic polarizabilitymicroscopic polarizability and and macroscopic macroscopic permittivity.permittivity.

From the phenomenological point of view, it is From the phenomenological point of view, it is necessary to know the kinetic of the Polarization.necessary to know the kinetic of the Polarization.

From molecular one it’s required the knowledge of the From molecular one it’s required the knowledge of the effective Electric field at which the dipole is subjected.effective Electric field at which the dipole is subjected.

4 different ways are proposed to evaluate the 4 different ways are proposed to evaluate the molecular: molecular: – Claussius – Mossotti Claussius – Mossotti – Debye Debye – OnsagerOnsager– Fouss – Kirkwood Fouss – Kirkwood

The basic idea is to consider a a spherical zone containing the dipole under study, The basic idea is to consider a a spherical zone containing the dipole under study,

immersed in the dielectric. immersed in the dielectric.

The sphere is small in comparison with the dimension of the condenser, but large The sphere is small in comparison with the dimension of the condenser, but large

compared with the molecular dimensions. compared with the molecular dimensions.

We treat the properties of the sphere at the microscopic level as containing many We treat the properties of the sphere at the microscopic level as containing many

molecules, but the material outside of the sphere is considered a continuum. molecules, but the material outside of the sphere is considered a continuum.

The field acting at the center of the sphere where the dipole is placed arises from the The field acting at the center of the sphere where the dipole is placed arises from the

field due to field due to

(1) the charges on the condenser plates (1) the charges on the condenser plates

(2) the polarization charges on the spherical surface, and (2) the polarization charges on the spherical surface, and

(3) the molecular dipoles in the spherical region. (3) the molecular dipoles in the spherical region.

Lorentz local fieldLorentz local field

Eo

+ +

+

++

+

- -- -

- -

d

Lorentz local field

Lorentz local Lorentz local fieldfield

Claussius – MossottiClaussius – Mossotti equation equation

valid for nonpolar gases at low pressure. valid for nonpolar gases at low pressure.

This expression is also valid for high This expression is also valid for high

frequency limit.frequency limit.

The remaining problem to be solved is The remaining problem to be solved is

the calculation of the dipolar contribution to the calculation of the dipolar contribution to

the polarizability. the polarizability.

Debye, extended the Claussius – Mossotti equation adding a new term in the polarization (orientational polarization).

By this way the dipolar contribution it’s taking into account

Debye equation for the static permittivity

Onsager, generalize the Debye Onsager, generalize the Debye

equation taking into account the equation taking into account the

effect of the if the permanent effect of the if the permanent

dipole moment of a molecule by dipole moment of a molecule by

the polarization of the the polarization of the

environment.environment.

1 – The cavity field, G, (the 1 – The cavity field, G, (the

field produced in the empty cavity field produced in the empty cavity

by the external field.)by the external field.)

2 - The reaction field, R (the 2 - The reaction field, R (the

field produced in the cavity by the field produced in the cavity by the

polarization induced by the polarization induced by the

surrounding dipoles).surrounding dipoles).

Onsager treatment of the cavity differs from Lorentz’s because Onsager treatment of the cavity differs from Lorentz’s because

the cavity is assumed to be filled with a dielectric material the cavity is assumed to be filled with a dielectric material

having a macroscopic dielectric permittivity. having a macroscopic dielectric permittivity.

Also Onsager studies the dipolar reorientation polarizability on Also Onsager studies the dipolar reorientation polarizability on

statistical grounds as Debye does. statistical grounds as Debye does.

The remaining problem is to take into account the interaction The remaining problem is to take into account the interaction

between dipolesbetween dipoles

Kirkwood and Fröhlich develop a fully statistical argument to Kirkwood and Fröhlich develop a fully statistical argument to

determine the short – range dipole – dipole interaction. determine the short – range dipole – dipole interaction.

gg will be different fromwill be different from 11 when there is correlation between the when there is correlation between the orientations of neighboring molecules.orientations of neighboring molecules.

When the molecules tend to direct themselves withWhen the molecules tend to direct themselves with parallel dipole parallel dipole momentsmoments, , will be positive and g>1 will be positive and g>1..

When the molecules prefer an ordering withWhen the molecules prefer an ordering with anti-parallel dipoles, anti-parallel dipoles, g<1g<1.

g =1g =1 in the case of no dipolar correlation between neighboring in the case of no dipolar correlation between neighboring molecules, or equivalently a dipole does not influence the position and molecules, or equivalently a dipole does not influence the position and orientations of the neighboring ones. orientations of the neighboring ones.

gg depends on the structure of the material, and for this reason it is a depends on the structure of the material, and for this reason it is a

parameter that fives information about the forces of local type.parameter that fives information about the forces of local type.

From Kremer – Schonhals book

OH

HH

H

OH

Claussius – Mossotti: Only valid for non polar Claussius – Mossotti: Only valid for non polar gases, at low pressuregases, at low pressure

Debye: Include the distortional polarization.Debye: Include the distortional polarization.

Onsager: Include the orientational polarization, Onsager: Include the orientational polarization, but neglected the interaction between dipoles. but neglected the interaction between dipoles. describe the dielectric behavior on non-describe the dielectric behavior on non-interacting dipolar fluidsinteracting dipolar fluids

Kirkwood: include correlation factor Kirkwood: include correlation factor (interaction dipole-dipole) (interaction dipole-dipole)

Fröhlich – Kirkwood – OnsagerFröhlich – Kirkwood – Onsager

Orientational polarization ()

Induced polarization

E(t)

s

Dynamic theory

Debye equationDebye equation

First order kinetic:First order kinetic:

Decay function:Decay function:

In frequency domainIn frequency domain

1,14

Debye equation doesn’t represent in a good Debye equation doesn’t represent in a good way the experimental data.way the experimental data.

Some modifications in the decay function Some modifications in the decay function was proposed by Williams – Watt, ussing a was proposed by Williams – Watt, ussing a previously Kolraush equation.previously Kolraush equation.

Advantages: represent better the Advantages: represent better the experimental data.experimental data.

Disadvantages: the Disadvantages: the parameter it’s an parameter it’s an artificial parameter and no molecular artificial parameter and no molecular relation for this parameter have been yet relation for this parameter have been yet foundfound

-12 -8 -4 00,0

0,2

0,4

0,6

0,8

1,0

experimental data KWW function Debye function

(t)

Log t

DISPERSION RELATIONSDISPERSION RELATIONS

The real and imaginary part of the complex permittivity are, The real and imaginary part of the complex permittivity are,

respectively, the cosine and sine Fourier transforms of the same respectively, the cosine and sine Fourier transforms of the same

function, that is, function, that is, (()). As a consequence, . As a consequence, ’’ and and "" are no independent. are no independent.

Kramer-Kronigs relationships

0 9 18

10-1

100

101

102

103

0 9 18

10-1

100

101

102

103

derivative " experimental "

"

log " '

2 lnder

ThermodynamicsThermodynamics Thermodynamics appear Thermodynamics appear

in the XIX century in the XIX century because of the necessity because of the necessity of describe the thermal of describe the thermal machines.machines.

It is based in postulates, It is based in postulates, without mathematical without mathematical demonstration, and as demonstration, and as the mechanics and the mechanics and electromagnetic electromagnetic postulates, establish the postulates, establish the basic physic laws.basic physic laws.

ThermodynamicThermodynamic

FUNCTIONSFUNCTIONS

Enthalpy (H)Enthalpy (H)

Entropy (S)Entropy (S)

Internal Energy Internal Energy (U)(U)

Free Energy (G)Free Energy (G)

VARIABLESVARIABLES

TemperaturTemperaturee

Density or Density or volumevolume

PressurePressure

Characteristic properties of materials

Calorific capacityExpansion coefficientElectric Permittivity

Thermodynamic postulatesThermodynamic postulates Thermodynamic are based in 4 Thermodynamic are based in 4

fundamentals laws:fundamentals laws:

U= Q-W U= Q-W (First law) (First law) (Energy balance)(Energy balance)

SSisoiso≥≥ 0 0 (Second law) (Second law)

Thermal equilibrium (Zero law): 2 systems in Thermal equilibrium (Zero law): 2 systems in equilibrium with one 3equilibrium with one 3rdrd are in equilibrium are in equilibrium between them.between them.

Perfect Crystals at 0 K, define 0 entropy. (Third Perfect Crystals at 0 K, define 0 entropy. (Third law)law)

Thermodynamics relates the Thermodynamics relates the properties of macroscopic systems.properties of macroscopic systems.

The macroscopic properties are The macroscopic properties are originated in the statistical average originated in the statistical average properties of microscopic properties. properties of microscopic properties.

Thermodynamic point of Thermodynamic point of viewview

Microscopicproperty Statistical average

Thermodynamic property

Thermodynamics is usually concerned with very specific systems at equilibrium.

In nature, the processes are mainly irreversible.

Their description requires going beyond equilibrium.

THERMODYNAMICS OF IRREVERSIBLEPROCESSES

The four main postulates of the theory are:

1 - The local and instantaneous relations between thermal and mechanical properties of a physical system are the same as for a uniform system at equilibrium. This is the so-called local equilibrium hypothesis.

2 - The internal entropy arising from irreversible phenomena inside a volume element is always a non-negative quantity. This is a local formulation of the second law of thermodynamics

3 - The internal entropy has a very simple character. It is a sum of terms, each being the product of a flux and a thermodynamic force

4 - The phenomenological description relating irreversible fluxes to thermodynamic forces are assumed to be linear.

MAXWELL EQUATIONS

where c is the velocity of light. The total charge density, , and the total current density, J are taken as the sources of the field

If the magnetization is assumed to

be zero, for a polarizable fluid, the Current will correspond to free charges and the polarization rate.

And the charge density correspond to the free and polarization density

J total current, Jf electric current of free charges, f density of free charges, p density of polarization charges, and P/t polarization current

Taking into account the relations between the electric displacement and the electric field,

we can obtain a different version of the Maxwell equation can be written as:

Conservation equationsConservation equations Mass: Mass: In the absence of chemical reactions, the rate of

change in mass within a volume V can be written as the flux through the surface dS according to:

, and v are, respectively, the mass density and velocity.

Charge: free chargesCharge: free charges

Polarization chargesPolarization charges

Conservation equationsConservation equationsLinear momentum: Linear momentum:

The equation indicates that the force exerted by the electromagnetic field on the material within the volume V is equal to the rate of decrease in electromagnetic momentum within V plus the rate at which electromagnetic momentum is transferred into V across the surface V .

is the momentum density

is the density of the Lorentz force identified with the body forces

=ED + HB - ½ (E·D + H·B)I is the Maxwell stress tensor; can be interpreted as the moment flux density.

Conservation equationsConservation equations

EnergyEnergy: :

The Poynting vector determines the density and direction of this flux at each point of the surface.

Poynting vector

electromagnetic energy flux through the surface.

work per time unit spent in production of conduction currents.

time rate of change in the field energy within the region

Internal Energy Equation

The total energy can be expressed as

Potential energy

Kinetic energy

Electromagnetic energy

ENTROPY EQUATION

For a single component system

the corresponding Gibbs equation is

Internal energy

polarization charge density

RELAXATION EQUATION

Using the entropy and energy balance equation, it is possible to express the relationship between the polarization rate and the thermodynamics functions

if To is constant and q= dJ/dt=0, Eq (3.6.1) reduces to the well-known Debye equation:

Debye equation predict instantaneous propagation of the perturbation

The Debye equations derived do not adequately represent the experimental behavior of polymers.

Instead of a symmetric semicircular arc, an asymmetric and skewed arc is observed.

To represent in a more accurate way the actual behavior, some modifications to the former theory must be made.

A more general relationship between forces and fluxes as follows

where the operator D1 represents the fractional

derivatives of order (0<<1).

Fractional derivatives were introduced in the theory of viscoelastic relaxations to give account of the deviations of the experimental data from those predicted by classical linear models, such as Maxwell and Kelvin-Voigt models, which are combinations of springs and dashpots

In terms of decay equation, the fractional derivatives In terms of decay equation, the fractional derivatives

it’s equivalent to stretch the decay function instead of it’s equivalent to stretch the decay function instead of

the exponent.the exponent.

Or which is the same, to chose a kinetic order different Or which is the same, to chose a kinetic order different

to 1(0<to 1(0<<1).<1).

t

e

P P

t

Under some considerations, the Laplace transform of the fractional derivative equation leads to the Havriliak Negammi empirical equation

This equation, contrary to the Debye equation, adequately predicts the shape of the actual dielectric data in the relaxation zones

Advantages:Advantages:

It’s possible to fit It’s possible to fit experimental data experimental data with HN equation.with HN equation.

Disadvantages:Disadvantages:

It’s based on It’s based on phenomenological phenomenological point of view. It’s not point of view. It’s not possible to relate the possible to relate the exponents with the exponents with the molecular structure. molecular structure. (no physical (no physical meaning) meaning)

Summary Summary

From thermodynamic point of view it From thermodynamic point of view it is also possible to obtain the is also possible to obtain the relaxation equations.relaxation equations.

The use of thermodynamics in The use of thermodynamics in dielectric materials relate the dielectric materials relate the macroscopic properties with macroscopic properties with microscopic ones. microscopic ones.