Relaxation time behaviour

Relaxation Time Behaviour in Ideally Dilute and Concentrated Solutions... 

Relaxation Time Behaviour of Moderately Concentrated Polymeric Solutions... 

In semi-dilute solutions, the Rouse theory fails to predict the relaxation time behaviour of the polymeric fluids. This fact is shown in Fig. 11 where the reduced viscosity is plotted against the product (y-AR). For correctly calculated values of A0 a satisfactory standardisation should be obtained independently of the molar mass and concentration of the sample. 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

The results of the fit are shown in Table I where it is seen that at most temperatures the 95% confidence limit for (X overlaps zero and hence there is little deviation from single relaxation time behaviour according to this analysis. Another inference is that the value of deduced from this procedure probably decreases with increasing temperature but the most important conclusion is that the relaxation time T is not particularly sensitive to the model chosen, the values obtained agreeing to within experimental error irrespective of whether the value of was allowed to be fitted as an unknown parameter, or whether it was clamped at zero (Debye case). 

A widespread view is that the feature comes from some crossover in the electronic density of states (DOS). The main result of the present paper is that after a proper re-arrangement of the experimental data no PG feature exists in the 63 Cu nuclear spin relaxation time behaviour. Instead, the data show two independent parallel relaxation mechanisms a temperature independent one that we attribute to stripes caused by the presence of external dopants and an "universal temperature dependent term which turns out to be exactly the same as in the stoichiometric compound YBCO 124. 

A mechanism behind the drag reduction and drag increase may be that polymer molecules or possibly entanglements of polymer molecules stretch in the flow field and become elongated. This stretching is quantified by measuring on a relaxation time behaviour for the polymer solution leaving the filter. 

M continually decreases under the influence of spin-spin relaxation which destroys the initial phase coherence of the spin motion within they z-plane. In solid-state TREPR, where large inliomogeneous EPR linewidths due to anisotropic magnetic interactions persist, the long-time behaviour of the spectrometer output, S(t), is given by... 

Figure B2.5.4. Periodic displacement from equilibrium through a sound wave. The frill curve represents the temporal behaviour of pressure, temperature, and concentrations in die case of a very fast relaxation. The other lines illustrate various situations, with 03Xj according to table B2.5.1. 03 is the angular frequency of the sound wave and x is the chemical relaxation time. Adapted from [110].

Before discussing tire complex mechanical behaviour of polymers, consider a simple system whose mechanical response is characterized by a single relaxation time x, due to tire transition between two states. For such a system, tire dynamical shear compliance is [42]... 

On increasing the moisture of cellulose from 0.5 to 16% the principal signals of cellulose shift a few ppm to higer fields. A similar, but much smaller shift is observed in cellulose acetate. The relaxation times T1 for Cl, C2, C3 and C4 diminish with increased moisture content. However, in the case of C6 there is no significant change. In the case of cellulose acetate, a similar general behaviour is observed. 

Unfortunately, this group Db depends on the assignment of a single characteristic time to the fluid (perhaps a relaxation time). While this has led to some success, it appears to be inadequate for many viscoelastic materials which show different relaxation behaviour under differing conditions. 

To prove this let us make more precise the short-time behaviour of the orientational relaxation, estimating it in the next order of tfg. The estimate of U given in (2.65b) involves terms of first and second order in Jtfg but the accuracy of the latter was not guaranteed by the simplest perturbation theory. The exact value of I4 presented in Eq. (2.66) involves numerical coefficient which is correct only in the next level of approximation. The latter keeps in Eq. (2.86) the terms quadratic to emerging from the expansion of M(Jf ). Taking into account this correction calculated in Appendix 2, one may readily reproduce the exact... 

Fig. 6.3. Quasi-static behaviour of relaxation times tgj (upper curves) and r ,i in the case of strong (1,2) and weak (3,4) collisions. The straight lines are the asymptotics of the curves after Q-branch collapse.

As can be seen, the difference in behaviour of orientational relaxation times Te,2 in models of weak and strong collisions is manifested more strongly than in the case of isotropic scattering. Relation (6.26) is... 

It is worth emphasizing, however, that in both cases C2H4 oxidation exhibits electrophobic behaviour, that the relaxation time constants x can be estimated from similar formulae (equations (4.32) and (9.3)) and that the enhancement factors A can again be estimated from the same formula1 (equation 4.20). 

Contrary to the phase separation curve, the sol/gel transition is very sensitive to the temperature more cations are required to get a gel phase when the temperature increases and thus the extension of the gel phase decreases [8]. The sol/gel transition as determined above is well reproducible but overestimates the real amount of cation at the transition. Gelation is a transition from liquid to solid during which the polymeric systems suffers dramatic modifications on their macroscopic viscoelastic behavior. The whole phenomenon can be thus followed by the evolution of the mechanical properties through dynamic experiments. The behaviour of the complex shear modulus G (o)) reflects the distribution of the relaxation time of the growing clusters. At the gel point the broad distribution of... 

Figure 10.6 Cole-Cole (or Argand) plot of /" versus / recorded on a powder sample of 3 at 4.5 K at an external dc field of 500 Oe. The dashed line indicates the expected behaviour for a single relaxation time, while...

As for the derivation of Eqs. 122,123 and 124 only the transitions 1—>2 have been counted, these equations do not describe recovery processes, where the transitions 2 —>1 are important as well. These approximations have been made for convenience s sake, but neither imply a limitation for the model, nor are they essential to the results of the calculations. Equation 124 is the well-known formula for the relaxation time of an Eyring process. In Fig. 65 the relaxation time for this plastic shear transition has been plotted versus the stress for two temperature values. It can be observed from this figure that in the limit of low temperatures, the relaxation time changes very abruptly at the shear yield stress Ty = U0/Q.. Below this stress the relaxation time is very long, which corresponds with an approximation of elastic behaviour. 

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