Relaxation time theory

According to the Rouse terminal relaxation time theory [14], a polymer solution always has a corresponding characteristic relaxation frequency Ochar)- Here, the codur is the (0.79 corresponding to the point of maximum curvature of the flow curve. Therefore the calculated M is believed to be the peak MW (Mp). Mp indicates the maximum probability of... 

Fig. 9.10. Plot of g obtained to first order in t using equation (9.56) for the soft sphere liquid at p = 0.7 and y = 1.0. The curve is predicted from the relaxation time theory using equation (9.60) with TD = 0.24. The points are from the NEMD simulation of Hanley et al. (1987).

Of the adjustable parameters in the Eyring viscosity equation, kj is the most important. In Sec. 2.4 we discussed the desirability of having some sort of natural rate compared to which rates of shear could be described as large or small. This natural standard is provided by kj. The parameter kj entered our theory as the factor which described the frequency with which molecules passed from one equilibrium position to another in a flowing liquid. At this point we will find it more convenient to talk in terms of the period of this vibration rather than its frequency. We shall use r to symbolize this period and define it as the reciprocal of kj. In addition, we shall refer to this characteristic period as the relaxation time for the polymer. As its name implies, r measures the time over which the system relieves the applied stress by the relative slippage of the molecules past one another. In summary. 

Of the various parameters introduced in the Eyring theory, only r—or j3, which is directly proportional to it-will be further considered. We shall see that the concept of relaxation time plays a central role in discussing all the deformation properties of bulk polymers and thus warrants further examination, even though we have introduced this quantity through a specific model. 

In connection with a discussion of the Eyring theory, we remarked that Newtonian viscosity is proportional to the relaxation time [Eqs. (2.29) and (2.31)]. What is needed, therefore, is an examination of the nature of the proportionality between the two. At least the molecular weight dependence of that proportionality must be examined to reach a conclusion as to the prediction of the reptation model of the molecular weight dependence of viscosity. 

Not only are the creep compliance and the stress relaxation shear modulus related but in turn the shear modulus is related to the tensile modulus which itself is related to the stress relaxation time 0. It is therefore in theory possible to predict creep-temperature relationships from WLF data although in practice these are still best determined by experiment. 

Note that the term y in Eqs. 2-15 and 2-16 has a different significance than that in Eq. 2-14. In the first equation it is based on a concept of relaxation and in the others on the basis of creep. In the literature, these terms are respectively referred to as a relaxation time and a retardation time, leading for infinite elements in the deformation models to complex quantities known as relaxation and retardation functions. One of the principal accomplishments of viscoelastic theory is the correlation of these quantities analytically so that creep deformation can be predicted from relaxation data and relaxation data from creep deformation data. 

As in the case of the Rouse relaxation time, it is possible to express x2 in term of the intrinsic viscosity either by using Eq. (24) or the Kirkwood-Riseman theory [24, 47] ... 

To extract information about xj from NMR data, the transverse relaxation time Tj may be used as well as the longitudinal time T. For gaseous nitrogen it was done first with Ti in [81] and confirmed later [82] when T was measured and used for the same goal. The NMR linewidth of 15N2 is the inverse of T2, and the theory, relating to Ti to x.1, is well known [39, 83]. For the case of diatomic and linear molecules the formula is... 

This ratio of orientational relaxation times is sometimes used to identify the situation corresponding to perturbation theory [85]. 

Inequality (6.67) is the softest criterion of perturbation theory. Its physical meaning is straightforward the reorientation angle (2.30) should be small. Otherwise, a complete circle may be accomplished during the correlation time of angular momentum and the rotation may be considered to be quasi-free. Diffusional theory should not be extended to this situation. When it was nevertheless done [268], the results turned out to be qualitatively incorrect orientational relaxation time 19,2 remained finite for xj —> 00. In reality t0j2 tends to infinity in this limit [27, 269]. 

If we assume that the same discrepancy between theory and experiment in M-M enters in M +-M, we can estimate the vibrational relaxation times for the molecular ions. Table II shows the estimated vibrational relaxation times r+ at various temperatures. The values are shorter than those for the neutrals by factors given in Figure 4. 

It is an unfortunate fact that several preexisting theories have tried to explain complicated mechanical phenomena of CB-reinforced rubbery materials but they have not been so successful." " However, a recent report might have a capability of explaining them collectively," when the author accepted the existence of the component whose molecular mobility is different from that of matrix mbber component in addition to the existence of well-known bound rubber component. The report described that this new component might be the most important factor to determine the reinforcement. These mbber components have been verified by spin-spin relaxation time 2 by pulsed nuclear magnetic resonance (NMR) technique, ° while the information obtained by NMR is qualitative and averaged over the sample and, therefore, lacking in the spatial... 

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. 

Table I shows the values of the relaxation time calculated using Eqs. (249) and (250). Both the inertial time and the long time decrease with increasing density. This is in agreement with the trend of the curves in Fig. 6. Indeed, the actual estimates of the relaxation times in Table I are in semiquantitative agreement with the respective boundaries of the plateaux in Fig. 6. The estimate of Xiong, the upper limit on x that may be used in the present theory, is perhaps a little conservative.

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