High Relaxation spectrum

In the limit of high frequencies the integral for the loss modulus tends to zero as the denominator in Equation 4.50 tends to infinity. The storage modulus tends to G(oo) which is just the integral under the relaxation spectrum ... 

This result is very interesting because whilst we have shown that G(0) has been excluded from the relaxation spectrum H at all finite times (Section 4.4.5), it is intrinsically related to the retardation spectrum L through Jc. Thus the retardation spectrum is a convenient description of the temporal processes of a viscoelastic solid. Conversely it has little to say about the viscous processes in a viscoelastic liquid. In the high frequency limit where co->oo the relationship becomes... 

The high frequency elastic modulus does not appear in the retardation spectrum but is an intrinsic part of the relaxation spectrum. These features are reinforced when the interrelationship between the spectra are considered. 

The shape of the relaxation spectrum predicted by Eq. (22) does indeed fit rheological data on pure star melts better than the quadratic expression calculated for stars in permanent networks [27], except at high frequencies where the assumption of activated diffusion breaks down (it may easily be verified that Ugff(s)[Pg.218]

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. 

For example, Figs. 2.43 and 2.44 present the measured [55] viscosity and first normal stress difference data, respectively, for three blow molding grade high density polyethylenes along with a fit obtained from the Papanastasiou-Scriven-Macosko [59] form of the K-BKZ equation. A memory function with a relaxation spectrum of 8 relaxation times was used. 

The dependence of rf, rf, G, and G" on frequency reflects the ability of macromolecular systems to flow like Newtonian fluids if the experimental time allowed them, feXp = 1 /< , is very large compared to the time that they require to fully respond macromolecularly. This temperature-dependent, material-characteristic time is commonly called the relaxation time, X, although it is actually a relaxation spectrum (7). Conversely, when /exp is very short, that is, co is very high compared to X, the macromolecular system can only respond like an elastic solid, able only to undergo deformation and not flow. In... 

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) ... 

Figure 46. Linearized plots of the three parameters xa, vmin, and %min, determined from the MCT analyses of the relaxation spectrum in the high-temperature regime. Plotted are the scaling law amplitude (SLA) as indicated (a) from the dielectric spectra of glycerol (cf. Fig. 18a) (adapted from Ref. 136) (b) from the light scattering spectra of 2-picoline (cf. Fig. 18b) (from Ref. 183).

Before the gel point the connectivity is small and the material typically relaxes rapidly. Near the gel point the relaxation time rises sharply and at the gel point it diverges to infinity (or at least to very long times for a finite sample) in addition, the relaxation spectrum does not contain a characteristic time anymore. After the gel point, if the network has reached a high degree of development, the maximum relaxation time of the final network is also very short. 

Figure 9.14 Relaxation spectra for high (III) and low (II) molecular weight fractions of polymers (I) represents the relaxation spectrum of a dilute polymer solution. (From Ref. 1.)...

In a general case, when the asymptote is approached, the following conditions should be satisfied Ty/r > 1 and Tf > 1. However, in practice the linear dependence Y(Tlri) at high values of (Ty ) over a finite rai e of Tjr can also be observed at the values of r and comparable to Tf. In this case, the characteristic times determined from the slope of Y(Tfr]) or Y (77T )(by using Eq. (1.2.25)) are not true average values (of the type of r or r ) over the entire range of the relaxation spectrum, but have a different physical meaning. 

Thus (compare Sect. 5.5), a situation can be envis ed, in which the relaxation spectrum includes both such rapid high-frequency motions with Tf f that Tf/TfLf> 1 over the entire experimental range of T/ij (except its lowest values) and relatively slower motions with Tf > Th f for which, however, the condition jflTj) > 1 is also fulfilled over the given range of TIri-... 

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,... 

A relaxation spectrum similar to that of Fig. 4.2 is obtained for the diffusional motion of a local-jump stochastic model of IV+ 1 beads joined by N links each of length b, if a weak correlation in the direction of nearest neighbor links is taken into account for the probability of jumps (US). On the other hand, relaxation spectra similar to that of the Rouse theory (27) are obtained for the above mentioned model or for stochastic models of lattice chain type (i 14-116) without the correlation. Iwata examined the Brownian motion of more realistic models for vinyl polymers and obtained detailed spectra of relaxation times of the diffusional motion 117-119). However, this type of theory has not gone so far as to predict stationary values of the dynamic viscosity at high frequencies. 

It is assumed that all of the relaxation processes with apparent time constants of 10 m.sec. to 2 sec. derive from water protons. In the case of samples containing large amounts of liquid fat, dissolved oligosaccharides or amino acids, this may not be the case, since C-H proton relaxation rates fall within this range. Some other measurement (e.g. integration of the high resolution spectrum) may be necessary to correct for this potentially major source of error. 

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