Relaxation spectra mechanisms

The /5-transition is the closest to glass transition (a-relaxation). It may be seen in relaxation spectra (mechanical, dielectric, etc) in a form of a separate peak or an inflection on the low temperature wing of the a-peak. Sometimes, the /5-transition is identified only at low frequencies (v = 10 -10 Hz), while at high frequencies (v = 10 —10 Hz) it merges with the wing of or-peak. As a rule, /5-peak is broader and less intensive than a-peak by about an order of magnitude. For many polymers, the ratio of the temperatures of the /5- and glass transitions has been shown to be approximately constant Tp/Tg 0.75) at low frequencies of about 10 to 10 Hz (141,147). 

An advantage of having the relaxation spectrum defined by Eq. (3.63) is that it can be adapted to expressions like this to calculate mechanical behavior other than that initially measured. 

Attempts have been made to identify primitive motions from measurements of mechanical and dielectric relaxation (89) and to model the short time end of the relaxation spectrum (90). Methods have been developed recently for calculating the complete dynamical behavior of chains with idealized local structure (91,92). An apparent internal chain viscosity has been observed at high frequencies in dilute polymer solutions which is proportional to solvent viscosity (93) and which presumably appears when the external driving frequency is comparable to the frequency of the primitive rotations (94,95). The beginnings of an analysis of dynamics in the rotational isomeric model have been made (96). However, no general solution applicable for all frequency ranges has been found for chains with realistic local structure. 

Thus, the goal of mechanical characterization is to determine E,, E0, x, and (3. Good examples of the applications of this relationship for linear polymers are given by Matsuoka (1986). The parameter (3, which is linked to the width of the relaxation spectrum, varies with temperature in the same way as tan 8 it can take values of the order of 0.01-0.1 in the glassy state, and 0.3-1 at Tg. 

Increasing filler loading broadens the relaxation spectrum of the cure reaction. Broadening the relaxation spectrum by filler loading also has been found in the mechanical spectrum of cured rubber from the glass transition region to rubbery plateau region [15]. 

This chapter discusses the dynamic mechanical properties of polystyrene, styrene copolymers, rubber-modified polystyrene and rubber-modified styrene copolymers. In polystyrene, the experimental relaxation spectrum and its probable molecular origins are reviewed further the effects on the relaxations caused by polymer structure (e.g. tacticity, molecular weight, substituents and crosslinking) and additives (e.g. plasticizers, antioxidants, UV stabilizers, flame retardants and colorants) are assessed. The main relaxation behaviour of styrene copolymers is presented and some of the effects of random copolymerization on secondary mechanical relaxation processes are illustrated on styrene-co-acrylonitrile and styrene-co-methacrylic acid. Finally, in rubber-modified polystyrene and styrene copolymers, it is shown how dynamic mechanical spectroscopy can help in the characterization of rubber phase morphology through the analysis of its main relaxation loss peak. 

Helical polypeptides constitute a sinqjler biopolym syst which also displays a distinct rotational mechanism of dielectric dispersion. The a-helix structure implies an appreciable permanent dipole moment parallel to its long axis and a relaxation frequency whidi may become very low because of the considerable length of the particles. Since the latter cannot be made uniform for synthetically prepared samples a more or less extended relaxation spectrum is inevitable. 

Figure 3.10 Predictions of the temporary network model [Eq. (3-24)] (lines) compared to experimental data (symbols) for start-up of uniaxial extension of Melt 1, a long-chain branched polyethylene, using a relaxation spectrum fit to linear viscoelastic data for this melt. (From Bird et al. Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

These results for and tjo may reflect the upper limit nature of the model. Each chain in a monodisperse liquid is assumed to rearrange its conformation by reptation alone and as though it were moving in a permanent network of obstacles. Competition from other mechanisms would of course increase the rate of relaxation and thus i oduce a lower viscosity. It would also increase the breadth of the relaxation spectrum and thus produce a larger value of the Gn product ... 

Another very significant characteristic of multiple mechanism relaxation is the pronounced change of the shape of the spectrum of relaxation times with temperature with increasing temperature the relaxation spectrum not only shifts to shorter time values, but a dip appears and deepens indicating increased separation with temperature of those parts of the spectrum due to the different mechanisms. (This is shown in Fig. 21.) The high temperature or long-time part of the spectrum consists,... 

In most liquids and to a good approximation, y0= (1-n). Therefore n can be evaluated by evaluating p. P itself is given by the ratio WJW, where and W are the Debye and actual widths of the relaxation spectrum. Even when loss curves (dielectric, mechanical or any other) are fitted to other standard analytical functions such as Cole-Cole (CC), Davidson-Cole (DC) or Haveriliac-Negami expressions, (see earlier section) one can determine p using the empirical relations... 

The treatment presented thus far applies to systems where only one independent variable is subject to relaxation. Frequently, however, m > 1) such variables are needed to describe the relaxation properties of interest. Under these circumstances, a set of m relaxation equations of the type given in Eq. [4] can be established. Accordingly, m relaxation times are determined and in a specific relaxation process each relaxation time will contribute its share to the overall effect in proportion to a corresponding amplitude. The ensemble of relaxation times and amplitudes is called the relaxation spectrum of the process under consideration. It reflects the underlying molecular rate mechanism. Thus, in principle, experimental relaxation spectrometry offers a way to elucidate kinetic mechanisms. 

The relaxation spectrum may include a term corresponding to x = 0. This may be expressed as t) on the first line of Eq. (1.10) and gives rise to an additional term icot] x in Eq. (1.11). Ordinary solvents, usually of low molecular weights, are Newtonian fluids which have only one mechanism corresponding to t = 0 with rj m = t]s, where % is the viscosity. It is customary to describe the viscoelastic behavior of dilute polymer solutions with G ((o) — icoris instead of G (w) This corresponds to the exclusion of a term tjs d (t) from the definition of H (t) which is assumed to be the contribution of the solvent with viscosity ris. 

Dynamic Properties. The location of the mechanical properties on the time scale is determined by the longest relaxation time and so by the intrinsic viscosity as seen above. Now we will consider details of the theoretical discrete relaxation spectrum which is determinal by... 

Mechanism (a) responsible for libration band near the edge of far IR spectrum and for nonresonance Debye relaxation spectrum... 

The nature (e.g., the softening range) of a hard occlusion in the interior of a rubber particle can never be determined by dynamic mechanical methods. Any dispersed phase can only be characterized in the region of higher modulus. Its low modulus or higher temperature properties are completely lost. Thus, the relaxation spectrum of a composite is generally not a superposition of the component spectra. 

To gain fmther insight into the crystallization mechanism of PDS, the shape and breadth of the relaxation spectrum were examined before and after crystallization. The comparison is made in Figure 9.11, where a normalized plot of dielectric loss as a function of frequency was constructed. Before the onset of crystallization (temperatures below 8°C), dipole loss curves superimpose quite well suggesting thermodielectric simplicity in the applied frequency interval between 100 Hz and 1 MHz.17 As crystalli-... 

As a particular example of a coupled system, we shall now calculate the relaxation spectrum for the mechanism... 

The fimction (A) is referred to as the distribution of relaxation times or the relaxation spectrum. In principle, once (A) is known, the result of any other type of mechanical experiment can be predicted. In practice (A) is determined from experimental data on E t). Since the distribution of relaxation times is so broad, it is more convenient to consider In A. Hence we introduce the function H(ln A), where the parenthesis denotes functional dependence, to replace (A) as... 

The (3 relaxation in polyethylene, which is most prominent in the low-crystallinity LDPE, is associated with the amorphous regions and almost certainly corresponds to what would be a glass transition in an amorphous polymer a diiference in its position in mechanical and dielectric spectra is therefore not surprising. The a relaxation, as discussed in section 7.6.3, is associated with helical jumps in the crystalline regions and, provided that the lamellar thickness is reasonably uniform, might be expected to correspond to a fairly well-defined relaxation time and to a narrow peak in the relaxation spectrum. The dielectric peak is indeed quite narrow, because the rotation of the dipoles in the crystalline regions is the major contribu-... 

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