The Dielectric Normal Mode

We return once again to the frequency-dependent and temperature-dependent measurements of the dielectric function of polyisoprene (PI) presented in Sect. 6.3.2. As shown in Figs. 6.20 and 6.21, two relaxation processes exist. The low frequency process, the normal mode, is the one of interest here. As has already been mentioned, it reflects the movements of the end-to-end distance vector R of the chain. The Rouse model enables these movements to be treated in the case of melts that are not entangled. Earlier, we learned that the motion of the end-to-end distance vector is to a large part due to the superposition of the three lowest order Rouse modes, polarized in the x, y and -directions. Therefore, the dielectric normal mode, when measured for samples with a molar mass below the entanglement limit, may be identified with these primary modes. 

For a Rouse chain built up of Nr polar sequences, each one carrying a dipole moment with a longitudinal component p, the total dipole moment Pp is given by 

The fluctuation-dissipation theorem provides an exact description of the step Aenm(t) associated with the normal mode. Emplo dng Eq. (8.6) in combination with Eq. (8.2) and using the relation 

Equation (8.93) relates the variance of the dipole moment of the polymer to the mean squared end-to-end distance of the chain. We therefore substitute in Eq. (8.89) by R, thus obtaining 

Now we employ the Rouse model. As the end-to-end distance vector is essentially determined by the lowest order Rouse modes, we can also represent the time correlation function in good approximation by 

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Fig. 4.7 Temperature dependence of the mean relaxation time (r) divided by the rheological shift factor for the dielectric normal mode (plus) the dielectric segmental mode (cross) and NSE at Qinax=l-44 A (empty circle) and Q=1.92 A (empty square) [7] (Reprinted with permission from [8]. Copyright 1992 Elsevier)...

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. 

Imanishi Y, Adachi K, Kotaka T (1988) Further investigation of the dielectric normal mode process in undiluted cis-polyisoprene with narrow distribution of molecular weight. J Chem Phys 89(12) 7685-7592... 

Recently the analysis was extended to the polymer chain dynamics (i.e., the dielectric normal mode) for polymers that have dipole moment parallel as well as normal to the backbone. They are polypropylene glycol (PPG), 1,4-polyisoprene (PI) [186], and polyoxybutylene (POB) [187]. The normal mode relaxation times (strictly speaking, the longest normal mode relaxation times, t ) taken at various combinations of temperature and pressure superpose to a single master curve when plotted against using the... 

The dielectric normal mode reflects the fluctuations of the end-to-end vector and is dominated by the slowest chain normal mode. As higher modes are scaled and shifted to higher frequencies, the timescale of the normal mode peak, (where is the frequency... 

We have already met this particular molecular weight dependence, in Eqs. (5.119) and (5.120), when formulating the average viscoelastic relaxation time f of polymer melts. Roughly speaking, r gives the time required by a chain for a complete conformational reorganisation. This also implies a full reorientation of the end-to-end distance vector of the chain. It is exactly this motion which shows up in the dielectric normal mode. 

Fig. 5.21. Molecular weight dependence of the relaxation time of the dielectric normal mode in cis-PIP. Data from Boese and Kremer [58]...

Fig. 6.7. General shape of the time dependent dielectric function e(t) of PIP showing the a-process and the dielectric normal mode (schematic drawing)...

K. Adachi, H. Okazaki, and T. Kotaka. Application of scaling laws to the dielectric normal mode process of di-polyisoprene in solutions of inflnite dilution to the bulk. Macromolecules, 18(1985), 1687-1692. 

The peculiar name normal mode needs a comment. As will be explained in detail in the next chapter, chain dynamics in melts may be described with the aid of two theoretical models known as the Rouse model and the reptation model. In the framework of these treatments chain kinetics is represented as a superposition of statistically independent relaxatory normal modes. As it turns out, the dielectric normal mode is associated with the mode with the longest relaxation time. For non-entangled melts this is the lowest order Rouse mode for entangled melts, it is the lowest order reptation mode. 

Hence, the effect of the entanglements is two-fold, since both the elastic and the viscous properties are concerned. The observations all indicate the existence of a critical molar mass, introduced earlier as the critical molar mass at the entanglement limit, denoted by Me. Polymers with low molar masses, M < Me, exhibit no entanglement effects, but for M > Me they show up and become dominant. All properties that are founded on motions on length scales corresponding to a molar mass above Me are affected. This holds, in particular, for the viscosity and the dielectric normal mode since these include the whole polymer chain. On the other hand. Rouse dynamics is maintained within the sequences between the entanglement points, as has already been mentioned. 

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