Relaxation spectrum illustrations

The range of frequencies used to calculate the moduli are typically available on many instruments. The important feature that these calculations illustrate is that as the breadth of the distributions is increased the original sigmoidal and bell shaped curves of the Maxwell model are progressively lost. A distribution of Maxwell models can produce a wide range of experimental behaviour depending upon the relaxation times and the elastic responses present in the material. The relaxation spectrum can be composed of more than one peak or could contain a simple Maxwell process represented by a spike in the distribution. This results in complex forms for all the elastic moduli. 

For higher modes, the ratio xjxt becomes sensitive to the correlations. As p increases, tp/t, decreases, as shown by Eq. (38). For illustration, this ratio is plotted semilogarithmically in Figure 2 as a function of pjN for a chain with 104 beads and for P = 0, 0.2,0.5, and 0.9. It is seen that in this one-dimensional model the relaxation spectrum is broadened as the energetic preference for extended conformations (P > 0) is increased. In particular, the longest and shortest relaxation times are related by... 

We will illustrate the difficulties and the opportunities which are associated with two complementary measuring techniques Relaxation Spectrum Analysis and Electrolyte Electroreflectance. Both techniques provide information on the potential distribution at the junction of a "real" semiconductor. Due to the individual characteristics of each system, care must be taken before directly applying the results which were obtained on our samples to other, similarly prepared crystals. 

The main feature about molten high polymers (molecular weights higher than about 104) concerns the broadness of the relaxation spectrum that characterises the viscoelastic response of these systems. This broad two-dispersion spectrum may spread over a range of relaxation times going from about 10 9 up to several seconds [4]. It is well illustrated from the modulus of relaxation observed after applying a sudden stress to the polymer the resulting sudden deformation of the sample is then kept constant and the applied stress is released in order to avoid the flow of the polymer. For example, the release of the constraint oxy(t) is expressed as a function of the shear modulus of relaxation Gxy(t) ... 

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. 

E may be aetirmined from the intersection of this asymptote with the C -axis. In addition, the relaxation time T can be estimated as illustrated in Figure 9 Although the deformation model shown in Figure 9 is extremely simplified and neglects some important factors (e.g. the fact, that instead of a single relaxation time a complicated relaxation spectrum may be valid), the model may serve as a basis for the discussion of the contact deformation behaviour of polymers. 

The parameter a is an inverse measure of the breadth of the relaxation spectrum, or a direct measure of the sharpness of the knee over which the transition from Newtonian flow to shear thinning takes place. The effect of changing this parameter is illustrated in Figure 10.7. This graph shows that... 

Although the Cole-Cole plot was first introduced in the context of a dielectric relaxation spectrum, it helped discover that the molecular mechanism underlying both dielectric relaxation and stress relaxation are substantially identical (44). Figure 8.13 provides an illustration, with temperature instead of frequency. Specifically, the same molecular motions that generate a frequency dependence for the dielectric spectrum are also responsible for the relaxation of orientation in polymers above Tg. Subsequently the Cole-Cole type of plot has been applied to the linear viscoelastic mechanical properties of polymers, especially in the vicinity of the glass transition, including the dynamic compliance and dynamic viscosity functions. 

The methods for calculating the relaxation spectrum were discussed. The calculated spectra were used to determine the MWD by the Mead method as illustrated in the examples polypropylene, LDPE and propylene a-olefin and ethylene-a-olefin copolymers. The rheological method is a quick and accurate process to determine the MWD of homopolymers and copolymers. 

FIG. 12-4. Relaxation spectrum in the transition zone for the six methacrylate polymers of Fig. 12-1, similarly identified and reduced to 100 C, to illustrate calculation of fo-... 

While borrowing from the classical models, these phenomenological approaches have also helped to clarify and refine the concepts of free-volume and configurational entropy and have focused attention on the nature of the relaxation spectrum (7-10,109,110). When a hquid in equiUbrium at a temperature Ti is suddenly cooled to a temperature, T2, its structure has no time to adjust and its properties (such as volume V, enthalpy H, or index of refraction n) exhibit instantaneous, solidlike changes characteristic of the glassy state as illustrated in Figure 9. For example, the instantaneous change in the enthalpy H may be written as follows ... 

In the condensation reaction between a trifunctional alcohol and a difunctional isocyanate, gel formation is marked by significant changes in the velocity and attenuation in the MHz region. The relaxation in the isolated monomers can be ascribed to a combination of intra- and inter-molecular processes. Intermolecular relaxation occurs in the hydrogen-bonded structure formed as a result of hydroxyl interactions. Reaction between the isocyanate and alcohol leads to the possibility of normal mode contributions to the relaxation spectrum and consequent increase in the ultrasonic attenuation. Detailed analysis of the data is difficult because of the complex topography generated by the reaction of di- and tri-functional monomers. This study does however once more illustrate the possibility of using ultrasonic techniques to monitor polymerization processes. 

Some information concerning the intramolecular relaxation of the hyperbranched polymers can be obtained from an analysis of the viscoelastic characteristics within the range between the segmental and the terminal relaxation times. In contrast to the behavior of melts with linear chains, in the case of hyperbranched polymers, the range between the distinguished local and terminal relaxations can be characterized by the values of G and G" changing nearly in parallel and by the viscosity variation having a frequency with a considerably different exponent 0. This can be considered as an indication of the extremely broad spectrum of internal relaxations in these macromolecules. To illustrate this effect, the frequency dependences of the complex viscosities for both linear... 

Fig. 4.12. Energy level scheme of donor and acceptor molecules showing the coupled transitions in the case where vibrational relaxation is faster than energy transfer (very weak coupling) and illustration of the integral overlap between the emission spectrum of the donor and the absorption of the acceptor.

Figure 2.5. Energy level diagram (top) and spectra (bottom) illustrating the two-state model of relaxation. The energy of the absorbed quantum is Av , and the energies of the emitted quanta are hvfl (unrelaxed) and hvF (relaxed). The fluorescence spectrum of the unrelaxed state (solid curve) is shifted relative to the absorption spectrum (dotted curve) due to the Stokes shift. The emission intensity from the unrelaxed state decreases and that from the relaxed state (dashed curve) increases as a result of relaxation.

Measurement of the decay kinetics /(/) in different regions of the fluorescence spectrum. If relaxation (or any reaction in the excited state) is absent, I(t) does not depend on vem, whereas in its presence, the spectral dependence illustrated in Figure 2.7 is observed. 

A variety of results obtained in studies of dipolar relaxation in the environment of the fluorescence probe 2,6-TNS are illustrated in Figure 2.10. In the model viscous medium (glycerol at 1 °C), the fluorescence spectra exhibit a marked dependence on the excitation wavelength. When 2 varies from 360 to 400 nm, the shift of the fluorescence spectrum maximum is 10 nm with a certain decrease of the half-width. In media with low viscosity, for instance, in ethanol (Figure 2.10a), this effect is never observed. 

The proposed ID TOCSY-NOESY experiment is illustrated by the assignment of NOEs from anomeric protons H-lc and H-ld of the polysaccharide 1. Because the resonances of H-lc and H-ld overlapped, this assignment was not possible from a ID NOESY spectrum as shown in fig. 3(b). Although these protons differed in their chemical shifts, it was not possible to separate them by chemical-shift-selective filtration because of the very fast spin-spin relaxation of backbone protons (20-50 ms) in this polysaccharide. Instead, a ID TOCSY-NOESY experiment was performed in which the initial TOCSY transfer from an isolated resonance of H-2c was followed by a selective NOESY transfer from H-lc. The ID TOCSY-NOESY spectrum (fig. 3(c)) clearly separated NOE signals of the H-lc proton from those originating from the H-ld proton and established the linkage Ic —> 6a. 

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