Longest viscoelastic relaxation time

Due to the presence of entanglements, a semidilute solution has an elastic behavior over short times and a viscous behavior over long times. The characteristic time which separates these two behaviors is the longest viscoelastic relaxation time T. ... 

The puzzling quantity is the longest viscoelastic relaxation time Tr. If the friction involved in the relaxation is purely hydrodynamic, it has been shown that where L is the length of the tube in which the... 

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. 

Dielectric properties reflect different averages of chain configuration and motion than viscoelastic properties and can thus be used to track features of chain dynamics that are different from those to which the stresses respond [27]. For example, the longest dielectric relaxation time is twice the longest Rouse stress relaxation time. Thus, this technique is useful for evaluating molecular models for relaxation processes, particularly constraint release mechanisms in the tube model as shown in Section 9.5.3.1. Cis-polyisoprene is particularly well suited for dielectric relaxation studies [28]. 

The parallel between po and JeR has been noted elsewhere (208,213,328,329), and is not in fact fortuitous. It follows rather directly from the empirical observation that departures of t (y) and r/ (co) from t]0 are governed by the longest relaxation times of the system, combined with slight extensions of a reduced variables argument suggested by Markovitz for linear viscoelastic behavior (329). Suppose one wants to compare the forms of the dynamic moduli on... 

No exact general criterion is available when it is necessary to include the relaxation terms in the equations of change however, relaxation terms are necessary for viscoelastic fluids, dispersed systems, rarefied gases, capillary porous mediums, and helium, in which the frequency of the fast variable transients may be comparable to the reciprocal of the longest relaxation time. 

The temporary network model predicts many qualitative features of viscoelastic stresses, including a positive first normal stress difference in shear, gradual stress relaxation after cessation of flow, and elastic recovery of strain after removal of stress. It predicts that the time-dependent extensional viscosity rj rises steeply whenever the elongation rate, s, exceeds 1/2ti, where x is the longest relaxation time. This prediction is accurate for some melts, namely ones with multiple long side branches (see Fig. 3-10). (For melts composed of unbranched molecules, the rise in rj is much less dramatic, as shown in Fig. 3-39.) However, even for branched melts, the temporary network model is unrealistic in that it predicts that rj rises to infinity, whereas the data must level eventually off. A hint of this leveling off can be seen in the data of Fig. 3-10. A more realistic version of the temporary network model... 

Intrinsic Viscosity. The intrinsic viscosity is the most important viscoelastic quantity in the longest relaxation time t0i of Eq. (1.21). The factor M rjJJl T is determinal if the equilibrium properties of the solution and the viscosity of the solvent is given. The quantity Sj seems... 

The reptation model is more powerful than you might think. You can get much more out of it than just the simplest basic laws for the viscosity, the longest relaxation time, and the diffusion coefficient of a chain in a polymer melt. This model allows you to describe, for instance, the relaxation of a pol mier after a stress has been released, or the response to a periodic force. As a result, you gain a fairly complete picture of the dynamics of polymer liquids, and of their viscoelasticity in particular. 

Some of the manifestations of viscoelasticity are delayed relaxation of stress after cessation of flow phase shift between stress and strain rate in oscillatory shear flow shear thinning (decrease of viscosity) at shear rates exceeding the reciprocal of the longest relaxation time and normal stress differences in shear flow, whose magnitudes are related to the relaxation time spectrum. A very convenient observation for experimentalists is that there is a close similarity between the shear viscosity and first normal stress difference as functions of shear rate and the corresponding parameters, complex viscosity and storage modulus, as functions of frequency in a small amplitude oscillatory shear. 

Analysis of dynamics at the gel point and theory of viscoelasticity provide a method to determine the static scaling exponents. Indeed, scaling arguments allow to show that the viscoelastic functions, G and G , at different stages of the network formation, can be superimposed into a master curve, provided that frequency and complex modulus are renormalized by appropriate reaction time (tr) dependent factors. The theory shows that the renormalisation factors for the frequency and the complex modulus are the longest relaxation time (iz) and the steady-state creep... 

The polymer is a branched polyethylene melt with = 1.55 x 10 and Mw/M = 11.9, flowing at 170 °C in a 3.3 1 planar contraction. The hnear viscoelastic properties and the nonlinear parameters for a four-mode PTT equation are shown in Table 10.1. Different values of e were used for each mode, but with a constant value of f = 0.08. These parameters provide a reasonable fit to the transient and steady-state shear and extensional data, although the nonlinear parameters for the two longest relaxation times cause small oscillations in startup of simple shear that are not observed experimentally using parameters that ehmi-nate the shear oscillations causes the calculated extensional stresses to be too low, and the contraction flow results are sensitive to the extensional stresses. The mean relaxation time was 1.74 s, the average velocity in the downstream channel was 7.47 mm/s, and the downstream channel half-width (the characteristic length) was 0.775 mm. The Weissenberg number based on downstream channel properties was therefore 16.8. 

Even at very low concentrations, the viscoelastic properties are affected by in-termolecular interactions as evidenced by the extrapolation plots of Fig. 9-10. A simple analysis of the initial concentration dependence has suggested that the first effect is an increase in the longest relaxation time. Recent theoretical calculations of Muthukumar and Freed, based on the Freed-Edwards theory of concentration dependence of viscosity in polymer solutions, show that the pth.relaxation time has a concentration dependence of the form... 

In the terminal zone where G and G" are proportional to co and co respectively, the viscoelastic properties are dominated by the longest relaxation times and these are determined by long-range motions in which a molecule of high molecular weight... 

The viscoelastic properties in the terminal zone are dominated by the characteristic constants rjo, J% Ac, and i,o (equation 74 of Chapter 3) and if the molecular weight distribution is sharp, additional constants are the terminal relaxation time r I (or or tj) and a related characteristic time constant r, which sets the scale for the shear dependence of non-Newtonian viscosity. Here the entanglements have their maximum effect in influencing the properties which reflect the longest-range molecular motions. 

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