Thermodynamics of Dielectric Relaxations in Complex Systems

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27 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Thermodynamics of
Dielectric Relaxations
in Complex Systems


TUTORIAL 3

Static dipoles


It is necessary to found the Relation between
microscopic polarizability

and
macroscopic permittivity.


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


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



4 different ways are proposed to evaluate the
molecular:


Claussius


Mossotti


Debye


Onsager


Fouss


Kirkwood




The basic idea is to consider a a spherical zone containing the dipole under study,
immersed in the dielectric.



The sphere is small in comparison with the dimension of the condenser, but large
compared with the molecular dimensions.



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.



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


(1) the charges on the condenser plates


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


(3) the molecular dipoles in the spherical region.

Lorentz local field

E
o

+

+

+

+

+

+

-

-

-

-

-

-

d




Lorentz local field

Lorentz local field



Claussius


Mossotti

equation

valid for nonpolar gases at low pressure.



This expression is also valid for high
frequency limit.



The remaining problem to be solved is
the calculation of the dipolar contribution to
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
equation taking into account the
effect of the if the permanent
dipole moment of a molecule by
the polarization of the environment.



1


The cavity field, G, (the
field produced in the empty cavity
by the external field.)



2
-

The reaction field, R (the
field produced in the cavity by the
polarization induced by the
surrounding dipoles).



Onsager

treatment

of

the

cavity

differs

from

Lorentz’s

because

the

cavity

is

assumed

to

be

filled

with

a

dielectric

material

having

a

macroscopic

dielectric

permittivity
.



Also

Onsager

studies

the

dipolar

reorientation

polarizability

on

statistical

grounds

as

Debye

does
.



The

remaining

problem

is

to

take

into

account

the

interaction

between

dipoles


Kirkwood

and

Fröhlich

develop

a

fully

statistical

argument

to

determine

the

short



range

dipole



dipole

interaction
.




g

will

be

different

from

1

when

there

is

correlation

between

the

orientations

of

neighboring

molecules
.




When

the

molecules

tend

to

direct

themselves

with

parallel

dipole

moments
,


will

be

positive

and

g>
1
.



When

the

molecules

prefer

an

ordering

with

anti
-
parallel

dipoles,

g<
1
.



g

=
1

in

the

case

of

no

dipolar

correlation

between

neighboring

molecules,

or

equivalently

a

dipole

does

not

influence

the

position

and

orientations

of

the

neighboring

ones
.




g

depends on the structure of the material, and for this reason it is a
parameter that fives information about the forces of local type.

From Kremer


Schonhals book



Claussius


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




Debye: Include the distortional polarization.





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




Kirkwood: include correlation factor
(interaction dipole
-
dipole)




Fröhlich


Kirkwood


Onsager


Orientational


polarization (

)

Induced polarization

E(t)





s

Dynamic theory

Debye equation


First order kinetic:




Decay function:




In frequency domain

1,14


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


Some modifications in the decay function was
proposed by Williams


Watt, ussing a
previously Kolraush equation.




Advantages: represent better the experimental
data.


Disadvantages: the


parameter it’s an
artificial parameter and no molecular relation
for this parameter have been yet found

DISPERSION RELATIONS


The real and imaginary part of the complex permittivity are,
respectively, the cosine and sine Fourier transforms of the same
function, that is,

(

)
. As a consequence,



and

"

are no independent.





Kramer
-
Kronigs

relationships

Thermodynamics


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


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

Thermodynamic


FUNCTIONS



Enthalpy (H)



Entropy (S)



Internal Energy
(U)



Free Energy (G)


VARIABLES



Temperature



Density or
volume



Pressure

Characteristic properties

of materials


Calorific capacity

Expansion coefficient

Electric Permittivity


Thermodynamic postulates


Thermodynamic are based in 4 fundamentals
laws:




U= Q
-
W


(First law)
(Energy balance)




S
iso


0

(Second law)




Thermal equilibrium (Zero law): 2 systems in
equilibrium with one 3
rd

are in equilibrium between
them.


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



Thermodynamics relates the properties of
macroscopic systems.


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


Thermodynamic point of view

Microscopic

property

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 IRREVERSIBLE

PROCESSES


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,
J
f


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 equations


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 charges









Polarization charges

Conservation equations

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 equations



Energy
:







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
T
o

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
D
1


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



Or which is the same, to chose a kinetic order different to
1(0<

<1).





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:



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


Disadvantages:



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

Summary


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


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