Transcript Tutorial 3 – Thermodynamics of Dielectric Relaxations in Complex

Thermodynamics of
Dielectric Relaxations
in Complex Systems
TUTORIAL 3
Static dipoles
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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
Lorentz local field
 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
-
-
-
Eo
d
-

-
+
+
+
+
+ +
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
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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.
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1 – The cavity field, G, (the
field produced in the empty cavity
by the external field.)
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2 - The reaction field, R (the
field produced in the cavity by the
polarization induced by the
surrounding dipoles).
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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.
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Also Onsager studies the dipolar reorientation polarizability on
statistical grounds as Debye does.
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The remaining problem is to take into account the interaction
between dipoles
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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
OH
H
H
OH
H
 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
Dynamic theory
E(t)
s
Orientational
polarization ()

Induced polarization
Debye equation
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First order kinetic:
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Decay function:
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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.
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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
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1,0
0,8
(t)
0,6
0,4
experimental data
KWW function
Debye function
0,2
0,0
-12
-8
-4
Log t
0
DISPERSION RELATIONS
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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
0
9
18
3
3
10
10
derivative "
experimental "
2
"
10
1
10
1
0
10
-1
10
10
0
10
-1
10
0

"
der
2
10
  '

2  ln 
9
log 
18
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.
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Thermodynamic
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FUNCTIONS
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Enthalpy (H)
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Entropy (S)
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Internal Energy
(U)
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Free Energy (G)
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VARIABLES
Characteristic properties
of materials
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Temperature
Calorific capacity
Expansion coefficient
Electric Permittivity
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Density or
volume
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Pressure
Thermodynamic postulates
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Thermodynamic are based in 4 fundamentals
laws:
 U= Q-W
 Siso≥ 0
(First law) (Energy balance)
(Second law)
 Thermal equilibrium (Zero law): 2 systems in
equilibrium with one 3rd are in equilibrium between
them.
 Perfect Crystals at 0 K, define 0 entropy. (Third law)
Thermodynamic point of view
Thermodynamics relates the properties of
macroscopic systems.
 The macroscopic properties are originated
in the statistical average properties of
microscopic properties.
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Microscopic
property
Statistical average
Thermodynamic
property
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Thermodynamics is usually concerned with
very specific systems at equilibrium.
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In nature, the processes are mainly
irreversible.
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Their description requires going beyond
equilibrium.
THERMODYNAMICS OF IRREVERSIBLE
PROCESSES
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The four main postulates of the theory are:
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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.
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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
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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
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4 - The phenomenological description relating irreversible fluxes to
thermodynamic forces are assumed to be linear.
MAXWELL EQUATIONS
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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
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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
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 total current, Jf electric current of free charges, f density of free
charges, p density of polarization charges, and P/t  polarization current
J
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
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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.
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Charge: free charges
Polarization charges
Conservation equations
Linear momentum:
is the momentum density
=ED + HB - ½ (E·D + H·B)I
is the Maxwell stress tensor;
 can be interpreted as the
moment flux density.
is the density of the Lorentz force identified with the body forces
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 .
Conservation equations
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Energy:
work
per time unit
spent in production
of conduction
currents.
electromagnetic
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Poynting vector
energy flux through the surface.
time
rate of
change in the field
energy within the
region
The Poynting vector determines the density and direction
of this flux at each point of the surface.
Internal Energy Equation
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The total energy can be expressed as
Electromagnetic energy
Kinetic energy
Potential energy
ENTROPY EQUATION
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For a single component system
the corresponding Gibbs equation is
polarization charge density
Internal energy
RELAXATION EQUATION
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Using the entropy and energy balance equation, it is possible to
express the relationship between the polarization rate and the
thermodynamics functions
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if To is constant and q= dJ/dt=0, Eq (3.6.1) reduces to the wellknown Debye equation:
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Debye equation predict instantaneous propagation of the
perturbation
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The Debye equations derived do not adequately represent the
experimental behavior of polymers.
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Instead of a symmetric semicircular arc, an asymmetric and
skewed arc is observed.
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To represent in a more accurate way the actual behavior, some
modifications to the former theory must be made.
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A more general relationship between forces and fluxes as
follows
where the operator D1 represents the fractional derivatives of
order  (0<<1).
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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
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In terms of decay equation, the fractional derivatives it’s
equivalent to stretch the decay function instead of the
t
exponent.

  e 
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Or which is the same, to chose a kinetic order different to

1(0<<1).
P
P

 t


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Under some considerations, the Laplace transform of the
fractional derivative equation leads to the Havriliak Negammi
empirical equation
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This equation, contrary to the Debye equation, adequately
predicts the shape of the actual dielectric data in the relaxation
zones
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Advantages:
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Disadvantages:
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It’s possible to fit
experimental data
with HN equation.
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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.
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