Terminal relaxation

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... 

The experiments were conducted in a cell (Fig. 4.19) at residual gas pressure of less then 10" Torr kept constant during the measurements. The surface coverage in these experiments was only lO" - 10 %. In this case, after the atomic beam was terminated, relaxation of electric conductivity has not been observed even at elevated temperatures (100 -180 C), when surface mobility of adatoms increased considerably. At larger coverages of the target surface with adatoms, or at higher surface temperatures electric conductivity relaxed to its initial value (before... 

In particular it has been conjectured that the terminal relaxation of star polymers might be the most sensitive test of the dilution exponent P in Go theta solvents suggest a mean value of nearer 2.3 [32]. A physically reasonable scahng assumption for the density of topological entanglements in a melt of Gaussian chains leads to a value of 7/3 [31]. 

Tfr is transition zone relaxation time, rte is terminal relaxation time, is reptation time... 

Various factors govern autohesive tack, such as relaxation times (x) and monomer friction coefficient (Co) and have been estimated from the different crossover frequencies in the DMA frequency sweep master curves (as shown in Fig. 22a, b). The self-diffusion coefficient (D) of the samples has been calculated from the terminal relaxation time, xte, which is also called as the reptation time, xrep The D value has been calculated using the following equation ... 

Highly entangled systems, especially those of narrow molecular weight distribution, are characterized by a set of relaxations at long times (terminal relaxations) which are more or less isolated from the more rapid processes. The modulus associated with the terminal processes is called the plateau modulus G°,. Because t]0 and depend on weighted averages over H(x), their values are controlled almost completely by the terminal processes. These experimental... 

The mean times t and tw will be called the number-average and weight-average relaxation times of the terminal region, and tw/t can be regarded as a measure of the breadth of the terminal relaxation time distribution. It should be emphasized that these relationships are merely consequences of linear viscoelastic behavior and depend in no way on assumptions about molecular behavior. The observed relationships between properties such as rj0, J°, and G and molecular parameters provides the primary evidence for judging molecular theories of the long relaxation times in concentrated systems. 

First, for M > 1WC the terminal relaxations are fairly well separated from the transition relaxations. In view of Eqs. (3.24) and (3.25), the values of rj0 and, /e° for M > Mc are properties of the terminal relaxations alone. The mean relaxation times of the terminal distribution are therefore given by Eqs. (3.26)—(3.28), combined with the experimental results in Section 5. 

The terminal spectrum is furnished by cooperative motions which extend beyond slow points on chain in the equivalent system. The modulus associated with the terminal relaxations is vEkT, which is smaller by a factor of two than the value from a shifted Rouse spectrum. It is consistent with a front factor g = j given by some recent theories of rubber elasticity (Part 7). The terminal spectrum for E 1 has the Rouse spacings for all practical purposes, shifted along the time axis by an undetermined multiplying factor (essentially the slow point friction coefficient). Thus, the model does not predict the terminal spectrum narrowing which is observed experimentally. 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

The expansion determines the terminal quantities the viscosity coefficient 7] and the elasticity coefficient u which, in their turn, determine the terminal relaxation time and steady-state compliance, correspondingly,... 

Now, we can try to relate the above results to the experimental data on the viscoelasticity of concentrated solutions of polymers. For the systems of long macromolecules, the estimated values of parameter are small. Having used expressions (6.40) for this case, one can evaluate the terminal relaxation time of the system... 

In the alternative case of large values of one can use the upper line of equation (6.40) to calculate the terminal relaxation time of the system, which coincides with the given relaxation time in order of magnitude... 

While the law with index 3.4 for viscosity is valid in the whole region above Mc, the dependence of terminal relaxation time is different for weakly and strongly entangled systems (Ferry 1980) and determines the second critical point M ... 

The difference in the molecular-weight dependence of the terminal relaxation time can be attributed to the change of the mechanisms (diffusive and repta-tion, correspondingly) of conformational relaxation in these systems. Further on in this section, we shall calculate dynamic modulus and discuss characteristic quantities both for weakly and strongly entangled systems. 

The empirical result (6.80) does not correspond to the reliable results for monodisperse (Mo = M) system well. Indeed, taking result (6.80) into account, the terminal relaxation time (6.58) can be written as... 

To provide the validity of empirical dependencies of viscosity and terminal relaxation time on the molecular length (relations (6.43) and (6.44)), the sum... 

Fatkullin NF, Kimmich R, Kroutieva M (2000) The twice-renormalised Rouse formalism of polymer dynamics Segment diffusion, terminal relaxation, and nuclear spin-lattice relaxation. J Exp Theor Phys 91(1) 150-166 Ferry JD (1980) Viscoelastic properties of polymers, 3rd edn. Wiley, London Ferry JD (1990) Some reflections on the early development of polymer dynamics Viscoelasticity, dielectric dispersion, and self-diffusion. Macromolecules 24 5237-5245 Ferry JD, Landel RF, Williams ML (1955) Extensions of the Rouse theory of viscoelastic properties to undilute linear polymers. J Appl Phys 26 359-362 Fikhman VD, Radushkevich BV, Vinogradov GV (1970) Reological properties of polymers under extension at constant deformation rate and at constant extension rate. In Vinogradov GV (ed) Uspekhi reologii polimerov (Advances in polymer rheology, in Russian). Khimija, Moscow, pp 9-23... 

The results of Van der Vegt suggest that for polydisperse polymer melts the terminal relaxation time might be predicted by... 

In principle, extrudate swell or die swell is dependent on the terminal relaxation time and on the time of residence in a capillary. The shorter the time of residence in the capillary or the longer the relaxation time the higher the die swell. This leads to (see, e.g. Te Nijenhuis, General References, 2007, Chap. 9.4)... 

One convenient strategy to interpret these results is to review the molecular characteristics of binary blends as extracted from polymer melt rheology [40]. The influence of short chains (M < Me) is to effectively decrease the plateau modulus and the terminal relaxation times as compared to the pure polymer. Consequently, the molecular weight between entanglements... 

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