Transcript Dielectric Relaxation processes at temperatures above

Dielectric Relaxation
processes at temperatures
above glass transition.
Molecular chains dynamics
(2)
TUTORIAL 7
Summary
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Polymer materials present “structural memory”.
The glass transition:
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universal property of condensed amorphous matter.
it’s a dynamic phenomenon.
The mean relaxation time for the -relaxation show a
Vogel dependency with the temperature
Summary
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The D, parameter in the VFTH equation it’s known as
the strength parameter (If D>10, strong glass former,
If D<10, fragile glass former)
Adams – Gibss theory: assume that the  relaxation
it’s a cooperative process.
Free volume theory: assume that Tv is the
temperature at which the free volume it’s zero.
Summary
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Experimental data for the 
relaxation can be fitted by
mean of the HN empirical
equation.
 decreases when
increasing temperature.
Summary
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From the  we can infer about the number of entities relaxing,
and the mean square dipole moment.
The Arrhenius plot ( vs T-1), gives information about the
dynamic of the system.
From the shape parameters, we can infer information about the
distribution of the relaxation time.
TEMPERATURE DEPENDENCE OF THE STRETCH
EXPONENT FOR THE -RELAXATION
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It’s not clear the dependence of the βK parameter with the
temperature.
For example, the - absorption in Brillouin spectra of the ionic
glass formed by calcium potassium nitrate in the temperature
range 120-190°C is fitted by with βK = 0.54.
Moreover, a comparative analysis of the broadband dielectric
behavior of propylene carbonate and glycerol, shows a
tendency for βK to level off at a constant value, smaller than
unity, at high temperatures.
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Experimental studies were performed in Poly vinyl acetate, in
bulk polymer and solutions of the polymer in toluene.
TEMPERATURE DEPENDENCE
OF SECONDARY RELAXATIONS
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The relaxation rate of
secondary relaxations obeys
Arrhenius behavior.
The frequency of the peak
maximum can be written as
Activation energy
Pre-exponential factor
TEMPERATURE DEPENDENCE OF
THE -RELAXATION
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Arrhenius plots of the relaxation display a
curvature.
the dependence of the peak
maximum of the relaxation in the frequency
domain is given by:
where T∞ = Tv .
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The evolution of the maximum of the  process can also
determined from the empirical Doolittle equation which
establishes that the relaxation time associated with the 
process depends on the free volume according to the following
expression:
free volume fraction
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According to the Cohen and Turnbull theory, the free volume is
zero at T∞ so the assumption that vf is a linear function of
temperature for T> T∞
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Comparing Dolittle and Vogel equation:
Free volume
fraction
~1
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For many systems investigated, vf /B = 0.0025
± 0.005
If B is assumed to be equal to unity, this would
mean that the free volume fraction at Tg would
have a universal value lying in the range 2,5 ±
0.5%.
DIELECTRIC STRENGTH AND POLARITY
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According to Fröhlich, the total relaxation strength can be
written as
~0
average of the cosine of the angle γ, made
between the dipole associated with the
reference unit i and that associated with j
within the same chain.
(INTERMOLECULAR)
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cosine of the angle between the dipole
associated with reference unit i and
unit j not belonging to the polymer
chain that contains reference unit i.
(INTRAMOLECULAR)
the correlation between two dipoles dies away very rapidly
when the number of flexible bonds separating them are four or
more.
SEGMENTAL MOTIONS
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The glass transition temperature of polymers is related to the
molecular weight by the empirical expression:
glass transition temperature of a polymer
of infinite molecular weight
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constant
dependent of the
concentration of
end groups in the
system
The glass transition temperature only shows a moderate
temperature dependence for molecular weights below the
critical value Mc ≈ 2Me, where Me is the molecular weight
between entanglements.
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Since the -relaxation is related to the glass transition
temperature, the average relaxation time shows a negligible
molecular weight dependence for M>Mc.
The fact that the glass transition is a cooperative phenomenon
leads to the conclusion that the -relaxation in polymers
involves cooperative micro-Brownian segmental motions of the
chains.
Segmental motions are associated with conformational
transitions taking place about the skeletal bonds.
The independence of the relaxation  on molecular weight
suggests that some sort of cooperativity occurs in the
conformational transitions taking place in the intervening
segment in order to ensure that the volume swept by the tails
of the chains is negligible; otherwise the friction energy, and
the relaxation times, would increase with molecular weight
poly(isoprene)
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Simulations carried out in simple polymers such as polyethylene
show that the conformational transitions are mostly of the
following type
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…….g±tt ↔ ttg±………
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…….ttt ↔ g±tg±……….
These transitions produce changes only in the central
segments, the extreme segments remaining in positions parallel
to the initial ones
LONG-TIME RELAXATION
DYNAMICS
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The relaxation behavior of polymer chains at long times (low
frequencies) depends on the orientation of the dipoles of
bonds, or groups of bonds, relative to the chain contour.
Stockmayer classified polymer dipoles into three types: A, B,
and C.
Dipoles of type A and B are rigidly fixed to the chain backbone
in such a way that their orientation in the force field requires
motion of the molecular skeleton.
Dipoles of type C are located in flexible side chains, and their
mobility is independent of the motions of the molecular
skeleton.
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Dipoles of type A are parallel to the chain contour, and the
vector dipole moment associated with a given conformation is
proportional to the end-to-end distance vector of that
conformation, that is
The vector sum of dipoles of type B and C is not correlated with
the end-to-end distance.
Some polymers exhibit dipoles with components of types A and
B, and these are called type AB polymers.
These latter polymers can be further classified into at least six
types.
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The curves representing the dielectric loss in the frequency
domain for type A polymers present at low frequencies the
normal mode process associated with motions of the whole
chain.
This relaxation is followed, in increasing order of frequencies,
by the -relaxation, reflecting segmental motions of the chains,
and, finally, by the β-process at very high frequencies, arising
from local motions
Normal Mode
-Relaxation
f, Hz
β-Relaxation
Normal Mode Relaxation
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In polymers containing dipoles type A and AB (some
component of the dipole is parallel to the chain contour),
normal mode is observed at frequencies lower than the 
relaxation.
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This process is strongly dependent of the molecular weight.
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The mean relaxation time follows Vogel equation, but TvN>Tv
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The dielectric strength of the normal mode is correlated with
the end-to-end distance.
Normal Mode


Maxwell Wagner Sillars
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Charge carriers can be blocked
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at inner dielectric boundary layers on a mesoscopic scale (M W S), or
at the external electrodes contacting the sample (electrode polarization)
on a macroscopic scale.
In both cases this leads to a separation of charges which gives
rise to an additional contribution to the polarization.
The charges may be separated over a considerable distance.
Therefore the contribution to the dielectric loss can be by
orders of magnitude larger than the dielectric response due to
molecular fluctuations.
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Maxwell-Wagner polarization processes must to taken into
consideration during the investigation of inhomogeneous
materials:
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suspensions or colloids,
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biological materials,
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phase separated polymers,
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blends,
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crystalline or liquid crystalline polymers.
They play also an important role in investigating the dielectric
behavior of molecules in confining space.
In the liquid crystalline state the material has a nanophase
separated structure (smectic layers) which disappears above
the phase transition.
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The charges blocked at internal phase boundaries generate the
Maxwell-Wagner polarization.
That causes a strong increase in ’ with decreasing frequency.
Above the phase transition, the phase boundaries disappear
and therefore the charges cannot be blocked anymore and ' is
reduced compared to the liquid crystalline state.
Also the slope of the conductivity contribution is influenced by
the Maxwell-Wagner process.
In the isotropic state the conductivity is nearly ohmic while in
the liquid crystalline state (with a phase separated structure)
the frequency dependence of the conductivity is weaker.
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The most simple model to describe an inhomogeneous
structure is a double layer arrangement.
Each layer is characterized by its permittivity i and by its
relative conductivity σri.
1, σ1
2, σ2
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For the complex dielectric function one gets
Maxwell, and after Wagner and Sillars have modelize the
response of an inhomogenus medium to the electric
perturbation.
Maxwell – Wagner - Sillars
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Using an system composed by spherical
particles embedded in a homogenous
medium, they found the following
expression for the phenomena:
ρ
= NR/R' is the volume fraction of the small particles.
Electrode Polarization
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Electrode polarization is an unwanted parasitic effect during a
dielectric experiment because it can mask the dielectric
response of the sample.
It occurs mainly for moderately to highly conducting samples
and influences the dielectric properties at low frequencies.
Both the magnitude and the frequency position of electrode
polarization depend on the conductivity of the sample and can
result in extremely high values of the real and imaginary part of
the complex dielectric function.
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The molecular origin of that effect is the (partial) blocking of
charge carriers at the sample electrode interface.
This leads to a separation of positive and negative charges
which gives rise to an additional polarization.
The electrode polarization effect is dependent on the electric
applied field, and the geometry of the sample.
Thickness of the sample
Debye length(L=(D·)1/2)
’’
’
Ionic Conduction
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There are a continuous increases of-7,0the loss factor when
decreases the frequency
-7,5
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The real part of the permittivity is not affected by the
log 
conductivity
-8,0
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The the log (”) vs log f, have a slope near to -1.
Generally it can be fitted by:
  dc 
 


·

-9,0 0

-8,5
*
CH2
s
CH3
C
*
C
O
O
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The Arrhenius plot of σdc vs T-1 gives
-9,5 information about the
Activation energy of the conductivity.
CH2
OCH3
OCH3
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It’s associated with ionic impurities-10,0
in0,0023
the0,0024
polymer.
0,0025 0,0026
1/T, K
-1
0,0027 0,0028
Summary
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 Relaxation is weakly dependent of the
molecular weight for high molecular weight
polymers
The Dolittle equation allows to fit the relaxation
time behavior of the  relaxation as a function
of the free volume.
The comparison between the Vogel equation
and the Dolittle permit to calculate the free
volume at Tg.
Summary
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Chains containing A or AB type dipoles
present a Normal Mode of relaxation.
Normal Mode:
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Strongly dependent on the Molecular weight
Dielectric Strength correlates to end-to-end
distance
Relaxation times shows Vogel behavior.
Summary
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Maxwell – Wagner – Sillars effect:
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Appear in inhomogeneous materials (blends
of polymers, semicrystalline polymers,
biological samples, etc)
It’s associated with some mesoscopic
separation of charge
The dielectric strength depends on the
effective surface between the phases.
Summary
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Electrode polarization:
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Macroscopic separation of charge in the
boundary of the electrodes
Depends on the intensity of the field, on the
Debye length, and on the thickness of the
sample