Relaxations long time mode

The long time mode corresponds to the liquid-liquid transition. The relaxation time follows a Fulcher-Vogel equation, the mobility is frozen at a critical temperature, T. Such behavior is characteristic of the vitreous state since it has also been observed in inorganic glasses and even in spin glasses. The departure of from Tg is given by the empirical WLF value in polymers like polystyrene it may be very different in odier polymers like poly(cyclohexyl methacrylate). This departure is also dependent upon the thermodynamic history of the polymer. 

This result shows that the highest modes of response have the shortest relaxation times and influence the initial response of the sample. Conversely, the longest relaxation time is ti, which we can identify with the terminal behavior of the sample. For example, in Fig. 3.9 the final collapse of the modulus at long times occurs at Ti. An example will show how we can use this idea. 

Figure 13a shows the contribution of translational diffusion. The translational diffusion only describes the experimental data for the smaller momentum transfer Q = 0.037 A. Figure 13b presents S(Q,t), including the first mode. Obviously, the long-time behavior of the structure factor is now already adequately represented, whereas for shorter times the chain apparently relaxes much faster than calculated. 

The glass transition involves additional phenomena which strongly affect the rheology (1) Short-time and long-time relaxation modes were found to shift with different temperature shift factors [93]. (2) The thermally introduced glass transition leads to a non-equilibrium state of the polymer [10]. Because of these, the gelation framework might be too simple to describe the transition behavior. 

For thejth subunit in a weakly bending filament (Lbending modes have all relaxed to their equilibrium mean squared displacements, one has(109)... 

The behavior of VACF and of D in one-dimensional systems are, therefore, of special interest. The transverse current mode of course does not exist here, and the decay of the longitudinal current mode (related to the dynamic structure factor by a trivial time differentiation) is sufficiently fast to preclude the existence of any "dangerous" long-time tail. Actually, Jepsen [181] was the first to derive die closed-form expression for the VACF and the diffusion coeffident for hard rods. His study showed that in the long time VACF decays as 1/f3, in contrast to the t d 2 dependence reported for the two and three dimensions. Lebowitz and Percus [182] studied the short-time behavior of VACF and made an exponential approximation for VACF [i.e, Cv(f) = e 2 ], for the short times. Haus and Raveche [183] carried out the extensive molecular dynamic simulations to study relaxation of an initially ordered array in one dimension. This study also investigated the 1/f3 behavior of VACF. However, none of the above studies provides a physical explanation of the 1/f3 dependence of VACF at long times, of the type that exists for two and three dimensions. 

The mode coupling theory of molecular liquids could be a rich area of research because there are a large number of experimental results that are still unexplained. For example, there is still no fully self-consistent theory of orientational relaxation in dense dipolar liquids. Preliminary work in this area indicated that the long-time dynamics of the orientational time correlation functions can show highly non-exponential dynamics as a result of strong in-termolecular correlations [189, 190]. The formulation of this problem, however, poses formidable difficulties. First, we need to derive an expression for the wavevector-dependent orientational correlation functions C >m(k, t), which are defined as... 

Kokorin YuK, Pokrovskii VN (1993) New approach to the relaxation phenomena theory of the amorphous linear entangled polymers. Int J Polym Mater 20(3-4) 223-237 Kostov KS, Freed KF (1997) Mode coupling theory for calculating the memory functions of flexible chain molecules influence on the long time dynamics of oligoglycines. J Chem Phys 106(2) 771-783... 

As in the study of water dynamics, the power spectrum density is useful to detect the long-time correlation or to detect decay slow energy transfers between optical and acoustic modes in the model given in Eq. (11). The relaxation inside optical modes is much faster than that in acoustic modes, since the frequency spectrum in optical modes is sharply localized and almost resonant while the spectrum is broadly spread in acoustic modes. 

The function (t) can be analysed by the method of cumulants [57] or by inverse Laplace transformation. These methods provide the mean relaxation rate T of the distribution function G(T) (z-average). For the second analysis procedure mentioned above, the FORTRAN program CONTIN is available [97,98]. It is sometimes difficult to avoid the presence of spurious amounts of dust particles or high molecular weight impurities that give small contributions to the long time tail of the experimental correlation functions. With CONTIN it is possible to discriminate these artifacts from the relevant relaxation mode contributing to (t). 

Both the disappearance of the slow relaxation time on a very long time-scale after preparation of solutions (from a few months to one year) and the nonreappearance of this mode when cycling between high and low salt concentrations indicate that the solution, at very low salt concentration, slowly tends to equilibrium. The thermodynamic equilibrium of salt-free polyelectrolyte solution is very difficult to obtain. Strong electrostatic repulsion dominates the solution, and some electrostatic domains or clusters stay present for a long time in the fresh solution. Only with an excess of external salt or with a very long time scale can the solution be in thermodynamic equilibrium. 

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