Dielectric Relaxation processes at temperatures above glass transition Dielectric Relaxation processes at temperatures above glass transition. Molecular chains dynamics (2) TUTORIAL 7

Summary Polymer materials present “structural memory”. The glass transition: 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 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 Experimental data for the  relaxation can be fitted by mean of the HN empirical equation.  decreases when increasing temperature.

Summary 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 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.

Experimental studies were performed in Poly vinyl acetate, in bulk polymer and solutions of the polymer in toluene.

TEMPERATURE DEPENDENCE OF SECONDARY RELAXATIONS 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 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 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. ~0 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) 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)

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. constant dependent of the concentration of end groups in the system glass transition temperature of a polymer of infinite molecular weight

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)

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.

Normal Mode -Relaxation β-Relaxation 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 Relaxation 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. This process is strongly dependent of the molecular weight. The mean relaxation time follows Vogel equation, but TvN>Tv The dielectric strength of the normal mode is correlated with the end-to-end distance.

Normal Mode 

 Normal mode

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.

For the complex dielectric function one gets 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 - 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 Conduction There are a continuous increases of the loss factor when decreases the frequency The real part of the permittivity is not affected by the conductivity The the log (”) vs log f, have a slope near to -1. Generally it can be fitted by: The Arrhenius plot of σdc vs T-1 gives information about the Activation energy of the conductivity. It’s associated with ionic impurities in the polymer.

Summary  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 Chains containing A or AB type dipoles present a Normal Mode of relaxation. Normal Mode: Strongly dependent on the Molecular weight Dielectric Strength correlates to end-to-end distance Relaxation times shows Vogel behavior.

Summary Maxwell – Wagner – Sillars effect: 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 Electrode polarization: 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