Normal-mode relaxation process

Figure 6 Normal mode relaxation processes in linear polymers (a) motion of a polymer molecule when subjected to an oscillating shear field (b) diagrammatic representation of the motion of an ideal flexible chain when in an oscillating shear field (c) viscoelastic relaxation of a polymer in solution, ri is the high frequency limiting value, tjq is the zero frequency and...

In this study, both the normal mode relaxation of the siloxane network and the MWS processes arising from the interaction of the dispersed nanoclay platelets within the polymer network have been observed. Although it is routine practice to observe the primary alpha relaxation of a polymeric system at temperatures below Tg, in this work it is the MWS processes associated with the clay particles within the polymer matrix that are of interest. Therefore, all BDS analyses were conducted at 40°C over a frequency range of 10 to 6.5x10 Hz. At these temperatures, interfacial polarization effects dominate the dielectric response of the filled systems and although it is possible to resolve a normal mode relaxation of the polymer in the unfilled system (see Figure 2), MWS processes arising from the presence of the nanoclay mask this comparatively weak process. 

A.3. The Normal-Mode (n) Relaxation Process The term normalmode relaxation refers to the long-range motions of the end-to-end dipole moment vector along a polymer chain, and thus corresponds to the comparably slow motion of a whole chain (Adachi 1997). This relaxation mode is characteristic of polymers with dipoles fixed parallel to the mainchain (type A polymers). A representative class of such polymers are the polyethers (-CH2—CHR—O—) , with R H [e.g., poly(propylene glycol) (Hayakawa and Adachi 2001) poly(butylene oxide) (Casalini and Roland 2005)], for which, with the exception of a few members [e.g., poly(styrene oxide) (Hirose and Adachi 2005)], a strong normal-mode relaxation signal can be resolved... 

In a wide range of polymer systems it has been observed that in the frequency range 10 kHz-10 MHz, a contribution to the ultrasonic relaxation can be identified due to the dynamic shear viscosity contribution to equation (1). The dynamic shear viscosity arises from cooperative normal mode distortion of the whole polymer molecule (Figure 6). Studies of the ultrasonic relaxation in the frequency range 10 kHz-10 MHz for dilute polystyrene solutions in toluene exemplifies these processes (Figure 7). It is apparent, as predicted by theory, that as the molecular weight of the polymer increases so the frequency of the first normal mode relaxation decreases. Similiar effects to those outlined above have been observed in poly(methyl methacrylate), ... 

Furthermore, it may be seen that for all the normal modes of relaxation, including the most rapid, the freely jointed chain model and the Rouse model are identical if we set n = N + 1 that is, the relaxation time xp of the pth normal mode of a freely-jointed chain is the same as that of a Rouse marcromolecule composed of N + 1 subchains, each of mean square end-to-end length b2. Moreover, for the special choice a = 0, Eq. (10) is true for arbitrarily large departures from equilibrium. We thus seem to have confirmed analytically the discovery of Verdier24 that quite short chains executing a stochastic process described by Eqs. (1) and (3) on a simple cubic lattice display Rouse relaxation behavior. Of course, Verdier s Monte Carlo technique permits study of excluded volume effects, quite beyond the range of our present efforts. 

The processes observed in the depolarized Rayleigh spectrum correspond to internal modes of motion. Thus, they may have relaxation times which substantially exceed those obtained from the longitudinal or bulk relaxation alone. Nevertheless they are a part of the a relaxation process as it is normally observed in the creep compliance. All processes with the same shift factors make up the full a relaxation. In liquids with substantial depolarized Rayleigh scattering the slowly relaxing part of the W scattering is also dominated by the orientation fluctuations associated with the internal modes of motion. Each internal mode contributes some intensity, but it is believed that fairly short wavelength modes dominate the scattered intensity. 

The molecular relaxation process has been studied by the autocorrelation function of normal modes for a linear polymer chain [177]. The relaxation spectrum can be analyzed by the Kohlrausch-Williams-Watts function [177,178] ... 

As expected, two relaxation processes are observed for PIP in the bulk the segmental mode, related to the dynamic glass transition representing the dynamics of the polymer segments, and the normal mode, sensing the chain dynamics (Fig. 9). 

In the experiment, the normal mode is related to the fluctuations of the end-to-end vector of free (nonimmobilized) chains, while the novel relaxation process was assigned to the fluctuations of the end-to-end distance of terminal subchains (hence, the confinement-induced mode is by its nature also a normal mode, not of the whole chain but of the terminal subchains only). The dynamics of these two relaxation processes is related to the corresponding end-to-end distances, consequently, their distribution, in dependence on thickness and molecular weight, is simulated in our study. 

Similar heterogeneous model has been used to develop a relaxation function by Chamberlin and Kingsbury (1994), who consider the localized normal modes to be involved in the relaxation process. Localized (domains) regions are assumed to be present between Tg and T. They are described as dynamically correlated domains (DCD). A Gaussian distribution of the domain sizes has been assumed, with each domain characterized by a Debye relaxation time. Expressions for the dielectric susceptibility have been derived and used to fit the experimental susceptibilities of salol, glycerol and many other substances with remarkable agreement over 13 decades of frequency (even when only one adjustable parameter is employed). 

We next briefly discuss a second liquid phase chemical process, namely, the vibrational energy relaxation of high-frequency solute normal modes [33],... 

As mentioned in the introduction, the above discussion of the small-, large-, and intermediate-molecule limits of electronic relaxation processes can also be utilized with very minor modifications to discuss the phenomena of intramolecular vibrational relaxation in isolated polyatomic mole-cules. ° Figure 4 is still applicable to this situation. The basis functions are now taken to be either pure harmonic vibrational states, some local-mode vibrational eigenfunctions, or some alternative nonlinear mode-type wave-functions. In the following the nomenclature of vibrational modes is utilized, but its interpretation as normal or local can be chosen to suit the circumstances at hand. 

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