TUTORIAL 7TUTORIAL 7

Dielectric Relaxation Dielectric Relaxation processes at temperatures processes at temperatures

above glass transition. above glass transition. Molecular chains dynamics Molecular chains dynamics

(2)(2)

SummarySummary

Polymer materials present “structural memory”.Polymer materials present “structural memory”. The glass transition:The glass transition:

universal property of condensed amorphous matter.universal property of condensed amorphous matter. it’s a dynamic phenomenon.it’s a dynamic phenomenon.

The mean relaxation time for the The mean relaxation time for the -relaxation -relaxation show a Vogel dependency with the temperatureshow a Vogel dependency with the temperature

SummarySummary

The The DD, parameter in the VFTH equation it’s known , parameter in the VFTH equation it’s known

as the strength parameter (If as the strength parameter (If D>10D>10, strong glass , strong glass

former, If former, If D<10D<10, fragile glass former), fragile glass former)

Adams – Gibss theory: assume that the Adams – Gibss theory: assume that the relaxation it’s a cooperative process.relaxation it’s a cooperative process.

Free volume theory: assume that Free volume theory: assume that TTvv is the is the temperature at which the free volume it’s zero. temperature at which the free volume it’s zero.

SummarySummary

Experimental data for Experimental data for

the the relaxation can be relaxation can be

fitted by mean of the HN fitted by mean of the HN

empirical equation.empirical equation.

decreases when decreases when

increasing temperature.increasing temperature.

From the From the we can infer about the number of entities we can infer about the number of entities

relaxing, and the mean square dipole moment.relaxing, and the mean square dipole moment.

The Arrhenius plot (The Arrhenius plot ( vs T vs T-1-1), gives information about the ), gives information about the

dynamic of the system. dynamic of the system.

From the shape parameters, we can infer information From the shape parameters, we can infer information

about the distribution of the relaxation time.about the distribution of the relaxation time.

SummarySummary

TEMPERATURE DEPENDENCE OF THE STRETCH EXPONENT FOR THE -RELAXATION

It’s not clear the dependence of the It’s not clear the dependence of the ββKK parameter with parameter with

the temperature.the temperature.

For example, the For example, the - - absorptionabsorption in Brillouin spectra of the in Brillouin spectra of the

ionic glass formed by calcium potassium nitrate in the ionic glass formed by calcium potassium nitrate in the

temperature range temperature range 120-190°C120-190°C is fitted by with is fitted by with ββKK = 0.54 = 0.54. .

Moreover, a comparative analysis of the broadband Moreover, a comparative analysis of the broadband

dielectric behavior of propylene carbonate and glycerol, dielectric behavior of propylene carbonate and glycerol,

shows a tendency for shows a tendency for ββKK to level off at a constant value, to level off at a constant value,

smaller than unity, at high temperatures.smaller than unity, at high temperatures.

Experimental studies were performed in Poly vinyl Experimental studies were performed in Poly vinyl

acetate, in bulk polymer and solutions of the polymer in acetate, in bulk polymer and solutions of the polymer in

toluene. toluene.

TEMPERATURE DEPENDENCE OF SECONDARY RELAXATIONS

The relaxation rate of The relaxation rate of

secondary relaxations secondary relaxations

obeys Arrhenius obeys Arrhenius

behavior.behavior. The frequency of the The frequency of the

peak maximum can be peak maximum can be

written aswritten as

Pre-exponential factor

Activation energy

TEMPERATURE DEPENDENCE OF THE -RELAXATION

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 .

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:

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∞

free volume fraction

Comparing Dolittle and Vogel Comparing Dolittle and Vogel equation:equation:

Free volume fraction

~1

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 According to Fröhlich, the total relaxation strength can

be written as

the correlation between two dipoles dies away very rapidly when the number of flexible bonds separating them are four or more.

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)

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)

~0

SEGMENTAL MOTIONS The glass transition temperature of polymers is related

to the molecular weight by the empirical expression:

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.

glass transition temperature of a polymer of infinite molecular weight

constant dependent of the concentration of end groups in the system

Since the Since the -relaxation-relaxation is related to the glass transition is related to the glass transition

temperature, the average relaxation time shows a temperature, the average relaxation time shows a

negligible molecular weight dependence for negligible molecular weight dependence for M>McM>Mc.. The fact that the glass transition is a cooperative The fact that the glass transition is a cooperative

phenomenon leads to the conclusion that the phenomenon leads to the conclusion that the -relaxation-relaxation

in polymers involves cooperative micro-Brownian in polymers involves cooperative micro-Brownian

segmental motions of the chains.segmental motions of the chains. Segmental motions are associated with conformational Segmental motions are associated with conformational

transitions taking place about the skeletal bonds.transitions taking place about the skeletal bonds. The independence of the relaxation The independence of the relaxation on molecular on molecular

weight suggests that some sort of cooperativity occurs in weight suggests that some sort of cooperativity occurs in

the conformational transitions taking place in the the conformational transitions taking place in the

intervening segment in order to ensure that the volume intervening segment in order to ensure that the volume

swept by the tails of the chains is negligible; otherwise swept by the tails of the chains is negligible; otherwise

the friction energy, and the relaxation times, the friction energy, and the relaxation times, would increase with molecular weight

poly(isoprene)poly(isoprene)

Simulations carried out in simple polymers such as polyethylene show that the conformational transitions are mostly of the following type …….g±tt ↔ ttg±……… …….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

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.

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.

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 β-Relaxation

f, Hz

Normal Mode RelaxationNormal Mode Relaxation In polymers containing dipoles type A and AB (some In polymers containing dipoles type A and AB (some

component of the dipole is parallel to the chain contour), component of the dipole is parallel to the chain contour), normal mode is observed at frequencies lower than the normal mode is observed at frequencies lower than the relaxation.relaxation.

This process is strongly dependent of the molecular This process is strongly dependent of the molecular weight.weight.

The mean relaxation time follows Vogel equation, but The mean relaxation time follows Vogel equation, but TTvNvN>T>Tvv

The dielectric strength of the normal mode is correlated The dielectric strength of the normal mode is correlated with the end-to-end distance.with the end-to-end distance.

Normal Mode

Normal m

ode

Maxwell Wagner Sillars Maxwell Wagner Sillars Charge carriers can be blocked

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.

Maxwell-Wagner polarization processes must to taken into consideration during the investigation of inhomogeneous materials: suspensions or colloids, biological materials, phase separated polymers, blends, 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.

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.

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.

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.

1, σ1

2, σ2

Maxwell – Wagner - SillarsMaxwell – Wagner - Sillars 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 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.

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 ConductionIonic Conduction There are a continuous increases of the loss factor when There are a continuous increases of the loss factor when

decreases the frequencydecreases the frequency

The real part of the permittivity is not affected by the The real part of the permittivity is not affected by the conductivityconductivity

The the log (The the log (”) vs log f, have a slope near to -1.”) vs log f, have a slope near to -1.

Generally it can be fitted by:Generally it can be fitted by:

The Arrhenius plot of The Arrhenius plot of σσdcdc vs T vs T-1-1 gives information about the gives information about the Activation energy of the conductivity.Activation energy of the conductivity.

It’s associated with ionic impurities in the polymer.It’s associated with ionic impurities in the polymer.

0·

s

dc

0,0023 0,0024 0,0025 0,0026 0,0027 0,0028-10,0

-9,5

-9,0

-8,5

-8,0

-7,5

-7,0

CH2* C *

CH3

C

O

CH2

O

OCH3

OCH3

SummarySummary Relaxation is weakly dependent of the Relaxation is weakly dependent of the

molecular weight for high molecular molecular weight for high molecular weight polymersweight polymers

The Dolittle equation allows to fit the The Dolittle equation allows to fit the relaxation time behavior of the relaxation time behavior of the relaxation as a function of the free relaxation as a function of the free volume.volume.

The comparison between the Vogel The comparison between the Vogel equation and the Dolittle permit to equation and the Dolittle permit to calculate the free volume at Tg.calculate the free volume at Tg.

SummarySummary

Chains containing A or AB type dipoles Chains containing A or AB type dipoles

present a Normal Mode of relaxation.present a Normal Mode of relaxation. Normal Mode:Normal Mode:

Strongly dependent on the Molecular weightStrongly dependent on the Molecular weight Dielectric Strength correlates to end-to-end Dielectric Strength correlates to end-to-end

distancedistance Relaxation times shows Vogel behavior.Relaxation times shows Vogel behavior.

SummarySummary

Maxwell – Wagner – Sillars effect:Maxwell – Wagner – Sillars effect: Appear in inhomogeneous materials Appear in inhomogeneous materials

(blends of polymers, semicrystalline (blends of polymers, semicrystalline polymers, biological samples, etc)polymers, biological samples, etc)

It’s associated with some mesoscopic It’s associated with some mesoscopic separation of chargeseparation of charge

The dielectric strength depends on the The dielectric strength depends on the effective surface between the phases.effective surface between the phases.

Summary

Electrode polarization:Electrode polarization: Macroscopic separation of charge in the Macroscopic separation of charge in the

boundary of the electrodesboundary of the electrodes Depends on the intensity of the field, on Depends on the intensity of the field, on

the Debye length, and on the thickness the Debye length, and on the thickness of the sampleof the sample