Viscoelasticity relaxation time

The relaxation time that we have determined may be referred to as the terminal viscoelastic relaxation time it is equal to the relaxation time which was introduced to characterise the medium surrounding the chosen macromolecule. Thus, for >—>00, the theory is self-consistent and this confirms the statement of Section 3.1.1 that chains of Brownian particles are moving independently in a liquid made of interacting Kuhn segments. 

Comparison of the dielectric and viscoelastic relaxation times, which, according to the above speculations, obey a simple relation rn = 3r, has attracted special attention of scholars (Watanabe et al. 1996 Ren et al. 2003). According to Watanabe et al. (1996), the ratio of the two longest relaxation times from alternative measurements is 2-3 for dilute solutions of polyisobu-tilene, while it is close to unity for undiluted (M 10Me) solutions. For undiluted polyisoprene and poly(d,/-lactic acid), it was found (Ren et al. 2003) that the relaxation time for the dielectric normal mode coincides approximately with the terminal viscoelastic relaxation time. This evidence is consistent with the above speculations and confirms that both dielectric and stress relaxation are closely related to motion of separate Kuhn s segments. However, there is a need in a more detailed theory experiment shows the existence of many relaxation times for both dielectric and viscoelastic relaxation, while the relaxation spectrum for the latter is much broader that for the former. 

Ferry, J. D., and R. A. Stratton The free volume interpretation of the dependence of viscosities and viscoelastic relaxation times on concentration, pressure, and tensile strain. Kolloid-Z. 171, 107 (1960). 

Table II. Maximum Viscoelastic Relaxation Times for Block Copolymers of Styrene and a-Methylstyrene...

Experiments with aqueous solutions of DNA and of other polymers suggest that polymers having high molecular weights — and hence large viscoelastic relaxation times — do not take part in the acoustic moticm and thus do not contribute to the wave dissipation [72], For operation at 5 MHz, the maximum molecular weight for which reliable viscosity measurements can be made acoustically is about 15,000 daltons. 

Comparing of (1.71) with the experimental data allows for measurements of the viscoelastic relaxation time 0 of dilute and semi-dilute polymer solutions. 

Berne and Pecora s text [16] is sometimes incorrectly cited as asserting that eq 12 is uniformly correct for light-scattering spectra. The analysis in Berne and Pecora [16] correctly obtains eq 12. However, this analysis refers the special case of a system in which particle displacements are governed by the simple Langevin equation. In these s tems, particle displacements in successive moments are uncorrelated. However, in a viscoelastic system, the polymer solution has a memory particle displacements in successive moments are no longer uncorrelated. With respect to probes in polymer solutions, Berne and Pecora s analysis is only applicable at times mtich longer than any viscoelastic relaxation times. 

If particle motion were observed over times shorter than the viscoelastic relaxation times Particle displacements in successive moments would be correlated. Equation 12 would not be correct. log[g (q, r)]/ would not be proportional to the mean-square particle displacement during r. With respect to polymer dynamics, the interest in eqs 9-12 has been the short time regime in which viscoelastic effects are apparent in D(r), but in the short time regime eq 12 does not describe probe motion in solution. 

It is useful to describe non-Newtonian flows in terms of the Deborah number, a dimensionless group describing the ratio of the viscoelastic relaxation time, X, and a characteristic timescale for the flow, tpow. 

Ilie anomalous behaviour in the linear viscoelasticity has been explained by the tube model.Figure 7.26 shows schematically how the stress relaxation takes place in star polymers. In the crude theory, it is assumed that the centre of the star is fixed during the viscoelastic relaxation time and that the relaxation takes place only by the contour length fluctuation, i.e., by the process that the polymer retracts its arm down the tube and evacuates from the deformed tube as shown in Fig. 7.26. 

Continued cross-linking leads to an increase in both resin equilibrium modulus and viscoelastic relaxation times. Thus, material behaviour becomes increasingly elastic and generated stresses begin to decay more slowly. Finally, when the resin instantaneous glass transition temperature (Tg) exceeds its local temperature (i.e., vitrification occurs), relaxation times jump sharply and mechanical behaviour becomes highly elastic. 

The problem of determining the theoretical behavior of polymers in the glassy rates is treated by Curro et al. (34,35). The time dependence of the volume in the glassy state is accounted for by allowing the fraction of unoccupied volume sites to depend on time. This permits the application of the Doolittle equation to predict the shift in viscoelastic relaxation times. 

Figure 20. Dependence of the dielectric relaxation time on viscoelastic relaxation time.

This observation is well known and in general there are two explanations. The first one is based on stating that a viscoelastic measurement is a macroscopic displacement while the dielectric measurement is a microscopic displacement. For this reason, several thousand microscopic displacements (dielectric) are required to make one macroscopic displacement (viscoelastic). Another method of comparison has been suggested and that is to compare the dielectric relaxation time to the viscoelastic relaxation time calculated form the modulus related to the compliance through G (w) = It is the view of the present authors that such... 

This extrapolation is readily accomplished in a complex plane plot and represents the condition of /"(O) == 0. Also in this expression Tj is the relaxation time for the / (w) data and is the relaxation time for the S (w) data. These relaxation times for 5 (w) are plotted in Figure 38, and the agreement between the dielectric and viscoelastic relaxation times is now remarkable. 

The results of this section show quite clearly that the application of Scaife s remarks to the comparison of dielectric and viscoelastic relaxation times changed their ratio (viscoelastic/dielectric) for the a process of polymers from several thousand to less than 2. It should be emphasized that since o/e 3, a similar result would have been obtained from a comparison of e (w) and S (w) data. Once again this method would be similar to comparing dissimilar quantities. In the case of small viscoelastic dispersions, that is, when Jq/J s 3, the relaxation times are also similar. This is important because it shows that scaling depends on the Jq/J ratio and not that viscoelastic responses are always several thousand times slower than the corresponding dielectric ones. 

It is well known that under static conHning pressure the viscosities of ordinary liquids are increased. The same effect is observed for the steady flow viscosity of polymers and for viscoelastic relaxation times, though the latter have been usually studied in bulk deformation or bulk longitudinal deformation rather than in shear. Qualitatively this behavior can be understood in terms of the dependence of segmental mobility on the fractional free volume, since the free volume must decrease with increasing pressure just as it does with decreasing temperature. Quantitative relations analogous to equations 32 and 34 can be readily derived, not only for pressure dependence but also for dependence on certain other variables that affect the free volume. 

The pressure dependence of viscoelastic relaxation times is of considerable practical importance because of the hydrostatic pressures encountere4 in various extrusion and molding processes. 

Free volume also plays an important ro]e in the effects of two other variables on relaxation times. When diluent (/.e., solvent or plastizer) is added to an undiluted polymer, the relaxation times decrease rapidly. The magnitude of the effect can be successfully interpreted in terms of additional free volume introduced with the diluent molecules, as will be discussed in Chapter 17. Finally, under static tensile strain, viscoelastic relaxation times in hard glasslike polymers are decreased. This effect can be attributed to a free-volume increase accompanying the total volume increase which occurs because Poisson s ratio is less than (Chapter 1, equation 50) it will be discussed in Chapter 18. 

The dependence of the non-Newtonian viscosity jj on shear rate at relatively low shear rates is a property which can be classed with viscoelastic behavior in the terminal zone, since it reflects long-range conHgurational motions which are influenced by entanglements to the maximum degree. As pointed out in Chapter 10, the characteristic time r, which specifies the onset of non-Newtonian, behavior with increasing shear rate is closely related to the terminal viscoelastic relaxation time. (In this discussion of shear viscosity, the subscript 21 will be omitted from stress [Pg.380]

Equation (5.118) for the mean viscoelastic relaxation time may be applied for both non-entangled and entangled melts and yields diflFerent results for the two cases. For non-entangled melts, i.e. M < Me, we have M and 7 0 M, hence... 

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