Dielectric Relaxation processes at temperatures above glass transition Dielectric Relaxation processes at temperatures above glass transition. Molecular chains dynamics (1) TUTORIAL 6

INTRODUCTION Condensed matter has a structural fading memory reflected in the velocity with which a perturbed system forgets the molecular configuration that it had in the past. In ordinary liquids, molecular reorganization occurs very rapidly and structural memory at the molecular level is very short. The relaxation time is roughly defined as the time necessary for the system to forget the configuration it had prior to the perturbation. At the other extreme the solids are characterized for having a very large structural memory at the molecular level, reflected in large relaxation times. From a strict point of view, the distinction between solids and liquids need to be expressed in a no subjective way.

Polymers are the most important viscoleastic systems The solid or liquid character of the condensed matter is expressed in terms of the Deborah number defined as For ideal liquids  → 0 and ND = 0, while for ideal solids →∞ and ND → ∞ For the so-called viscoelastic systems  and exp are comparable and their Deborah's number is of the order of unity. Polymers are the most important viscoleastic systems Time required for the relaxation process timescale of the experiment

PHENOMENOLOGICAL DIELECTRIC RESPONSE IN THE TIME DOMAIN D(t) PHENOMENOLOGICAL DIELECTRIC RESPONSE IN THE TIME DOMAIN Let us consider the response of an isotropic polar system to a perturbation electric field defined as The orientation of the dipoles by the effect of the field is reflected in a continuous increase in the dielectric displacement with time, until eventually a constant value is reached. The time dependence of the dielectric displacement can be written as

r(ω=0) is the relaxed dielectric activity Ψ(t) is a monotonous increasing function of time, the extreme values of which are Ψ(0)=0 and Ψ(∞) = 1. u (ω=∞) is the unrelaxed dielectric permittivity. It’s caused by the distortion of the electronic cloud and the positions of the nuclei of the atoms by the effect of the electric field.

The function Ψ(t) is of entropic nature. It reflects the molecular motions that take place in the system to accommodate the orientation of the dipoles to the perturbation field. Owing to the entropic nature of the response at t>0, the dielectric displacement does not vanish when the perturbation field ceases. As a result, D(t) not only depends on the actual perturbation field but also on the electric history undergone by the material in the past.

Under a linear behavior regime, the response to perturbation fields is governed by the Boltzmann superposition principle Electric perturbation in past time Electric Displacement at time t Electric Permittivity, dependent of the actual and past events. Boltzmann superposition principle for dielectric experiments in continuous form

DIELECTRIC RESPONSE IN THE FREQUENCY DOMAIN Do is the amplitude of the response. This equation indicates that the dielectric displacement is a complex quantity: component in phase with the perturbation field: Do·cos 90º out of phase: Do·sin

POLY VINYL ACETATE =r-u

DIELECTRIC RELAXATION MODULUS IN THE TIME AND FREQUENCY DOMAINS The experimental determination of the relaxed permittivity r may involve some difficulties in cases where the conductivity contribution to the dielectric loss overlaps the dipolar one. In this situation it is preferable to analyze the dielectric results in terms of the dielectric modulus M .

Let us consider the following electric history If Do is very small, the linear dielectric phenomenological theory predicts that The dielectric relaxation modulus, M(t), is given by Φ(t) is a monotonous decreasing function of time, the extreme values of which are Φ(0)=1 and Φ(∞)=0. For low dielectric displacements, the Boltzmann superposition principle holds

In continuous form If a sinusoidal dielectric displacement D(t)=Do·sinωt is imposed on the material, the electric field can be written as E(t)=Eo·sin(ωt+) =Eo·sin(ωt)·cos+Eo·cos(ωt)·sin = Do(M’sin(ωt) + M" cos(ωt)) The electric field is a complex quantity with a component in phase with the perturbation (Eocos) and another 90º out of phase (Eosin).

The real component M’ and loss component M” of the complex dielectric modulus are given by Phase angle

From Kremer – Schönhals book

LOCAL AND COOPERATIVE DYNAMICS:BASIC CONCEPTS Relaxation response functions of liquids to weak external perturbations may provide information on the actual structural kinetics resulting from transitions in the configurational space describing the system. The normalized response function to an electric field can be expressed in molecular terms by N is the number of relaxing species and mi(t) is the dipole moment of the species i at time t

It can be fitted to a Vogel-Fulher-Tamman-Hesse (VFTH) equation The area of the normalized relaxation curve g(t) vs t, at temperature T, defines the mean relaxation time (T) The value of  in the liquid state increases rapidly with decreasing temperature in the vicinity of the glass transition temperature. It can be fitted to a Vogel-Fulher-Tamman-Hesse (VFTH) equation Vogel temperature D>10 STRONG glass D<10 WEAK glass strength parameter prefactor of the order of picoseconds

The importance of the dynamics is evident if a melt is cooled at high enough cooling rate to avoid its crystallization. The fast cooling leads the system in a super-cooled liquid, and then, to a glass. The glass transition temperature, at which the super-cooled liquid forms a glass, is a dynamic property. It is often defined as the temperature at which the relaxation time is 200 s, a reasonable maximum relaxation time for dynamic experiments, but obviously Tg, increases/decreases as the timescale of the experiment decreases/increases. The relaxation time reflects the time involved in the structural rearrangement (glass transition) of the molecules.  changes by many orders of magnitude from the liquid to the glass.

supercooled Molar enthalpy liquid Cpliquid glass fast Cpglass liquid DHmelting slow crystal Cpcrystal Temperature 12/1/2018

The glass transition is a universal property of condensed amorphous matter, independent of its molecular weight. Below Tg the dynamic response depends on the thermal history, and aging phenomena may occur. Owing to the latent heat of melting, a supercooled liquid has substantially higher entropy than the crystal. However, the liquid loses entropy faster than the crystal below the melting temperature (Tm)

The difference between the entropies of the liquid and the crystal becomes rather small just below the glass transition. Extrapolation of the entropy of the glass to low temperatures, indicates the existence of a temperature TK, called the Kauzmann temperature, at which the crystal and the liquid attain the same entropy.

The most prominent theoretical approaches to the glass transition are the Adams – Gibbs and the Free Volumes theories. The Adams-Gibbs theory is the first to consider the glass transition as a cooperative process. In the free volume approach Tv is considered to be the temperature at which the free volume would be zero.

In the kinetic and fluctuation model the volume of the cooperatively rearranging region is defined as the smallest volume that can relax to a new configuration independently. The cooperativity continues well above Tg, and increases with decreasing temperature. Another approach to the study of the glass transition is to determine the response of liquids deep in the liquid state

The mode coupling theory explains the glass transition in terms of a dynamic phase transition occurring at a critical temperature T, significantly above the glass transition temperature. A coupling scheme has been proposed by Ngai and coworker, in which the first-time derivative of the KWW equation is taken to be the master equation in the time domain for application to isothermal and non-isothermal experiments.

RESPONSES OF GLASS FORMERS ABOVE Tg TO PERTURBATION FIELDS The temperature dependence of the mean relaxation time of supercooled liquids obeys Vogel equation. This absorption, called the -relaxation process, may display at high temperatures conductive contributions on the low frequency side of ”(ω) -relaxation displays a KWW stretched exponential decay The wider the -relaxation, the lower the value of the stretch exponent(βKWW).

-Relaxation for Poly 2,4-Difluorbencyl methacrylate

In the vicinity of Tg, the KWW dependence with temperature is described by the VFTH equation. The mean relaxation time associated with the glass-rubber absorption is given by According to the linear phenomenological theory, the glass-liquid relaxation can be obtained in the frequency domain by means of

The -relaxation in the frequency domain is described by the empirical Havriliak-Negami equation In this expression, HN is related to the maximum of the -peak in the frequency domain, whereas HN and βHN are fractional shape parameters fulfilling the following conditions: βHN>0 and HNβHN<1. HN, and βHN parameters are related to the limiting values of the slopes in the low- and high-frequency regions of the double logarithmic plot " vs ω, by means of the expressions High f Low f

Although m and n are uncorrelated parameters, for polymeric materials 0<n<0.5 At frequency higher than the corresponding to the -relaxation some low molecular glass formers shows an excess loss or wing not accounted for by the empirical expressions commonly used to describe the -process

For many systems in which the -relaxation is described by the Davidson-Cole equation, the excess wing contribution can be described by the scaling law "~ωa with the exponent. a is lower than the value of the exponent of the Davidson-Cole expression. No accepted model exists to describe the macroscopic origin of the excess wing. Besides the -relaxation, many glass formers, especially polymers, display at high frequencies one or more relaxations (β, γ, , etc) that are believed to be associated with local intramolecular relaxations in the main chain or in side groups.

Another type of relaxation is the so-called Johari-Goldstein β-relaxation, which seems to be a rather universal property of glass formers. This relaxation even appears in relatively simple systems in which intramolecular motions are absent. In some materials, the Johari-Goldstein β-relaxation may overlap the excess wing of the -relaxation process.

BROADBAND DIELECTRIC SPECTROSCOPY OF SUPERCOOLED POLYMERS Some polymers with flexible side groups are characterized for displaying high dielectric activity even in the glassy state. poly(5-acryloxy methyl-5-ethyl-1,3-dioxacyclohexane) (PAMED)  β

Below the glass transition temperature, the isotherms present a wide β-relaxation whose maximum shifts to higher frequency with increasing temperature Usually, the β-peak in the frequency domain is symmetric and displays half-widths of 2-5 decades. The normalized dielectric loss for this process is often described by the empirical Fuoss-Kirkwood equation The parameter m, accounts for the broadness of the β-process in such a way that, the larger the value of m, the narrower is the absorption max/2 Half width

Also, the intensity of the β-process increases with increasing temperature The width of the β-relaxation is often explained in terms of the distribution of both the activation energies and the pre-exponential factor that results from the variety of molecular environments that cause the relaxation.

At the higher temperatures, the distance separating the β- from the -peak decreases as a consequence of the high activation energy of the -relaxation process. A temperature is reached at which both relaxations coalesce into a single peak, named β-relaxation, the intensity of which seems to increase with increasing temperature

The loss curves at T > Tg can be fitted to a sum of an - plus a β-relaxation

The relaxation strength can be evaluated as: In the case of the β relaxation, the relaxation strength for a FK equation became The strength of the -relaxation decreases with increasing temperature as a consequence of the randomization of the dipolar orientation.

 reaches a maximum value at temperatures just slightly above Tg  reaches a maximum value at temperatures just slightly above Tg. Then, undergoes a steep decrease with increasing temperature Extrapolation to  = 0 (-onset) points to an onset temperature of 97°C in the case analyzed here

The strength of the β-relaxation displays a pattern by which it increases steeply in the temperature range in which the strength of the -relaxation steeply decreases. The total strength of the isotherms expressed as =+β remains nearly constant in the whole interval of temperatures. The location of the -relaxation in the frequency domain depends on the chemical structure but not on molecular weight for high molecular weight polymers

This behavior suggests some sort of cooperativity on the motions of the molecular chains, otherwise this relaxation should exhibit a strong dependence on chain length. The relaxation time associated with the -relaxation shows a stronger dependence on temperature than the secondary relaxation processes.

the β-process continues deep in the liquid state Based on the analysis of the splitting of the β-relaxation of condensed matter, including polymers, it has been speculated that five scenarios are possible type A scenario the separated -onset can be characterized by minimum cooperativity for  that cannot be continued to a local, noncooperative process. type D scenario there are two different but touching -relaxations, with a sharp crossover between them. type C scenario is similar to that of the conventional one, but here the -relaxation curve is above and below the splitting, and β is not the tangent to  type B scenario there is a locally coordinative β-precursor for the cooperative -process at high temperatures. Conventional type the β-process continues deep in the liquid state

 Loss factor and (b) Electric loss modulus as a function of the temperature at a frequency of 1Hz for PCHMA (), P4THPMA () and PDMA (). Arrows show the calorimetric glass transition temperature (Tg), measured by DSC γ β  γ β

Summary Polymer materials present “structural memory”. The glass transition: universal property of condensed amorphous matter. it’s a dynamic phenomenon. has entropic nature. 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.