Temperature structural relaxation time

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

Tk is often close to the Vogel-Fulcher temperature T0 discussed in connection with Figure 2, which is determined by fitting the Vogel-Fulcher relation5-8 to the temperature dependence of the structural relaxation time of the melt9 using Eq. [3] ... 

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

Here, tr is a dominant structural relaxation time of the liquid (e.g., viscosity p cx tr), T is temperature, Sc is the molar configurational entropy (defined... 

Table 4.1 Parameters related to the structural relaxation for the polymers investigated by NSE glass transition temperature Tg, position of the first static structure factor peak Qmax> shape parameter magnitude considered to perform the scaling of the NSE data, and temperature dependence of the structural relaxation time...

For PIB the apparent activation energy found for the structural relaxation time in the NSE window is almost twice that determined by NMR [136] (see Fig. 4.9 [125]). For aPP, the temperature dependence of NMR results [138] seems, however, to be quite compatible with that of the NSE data nevertheless, 2D exchange NMR studies on this polymer [139] reveal a steeper dependence. This can be seen in Fig. 4.11 [ 126]. 

From this comparison it follows that the observation of the structural relaxation by standard relaxation techniques in general might be hampered by contributions of other dynamic processes. It is also noteworthy that the structural relaxation time at a given temperature is slower than the characteristic time determined for the a-relaxation by spectroscopic techniques [105]. An isolation of the structural relaxation and its direct microscopic study is only possible through investigation of the dynamic structure factor at the interchain peak - and NSE is essential for this purpose. 

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

Finally we compare the temperature dependencies reported for the structural relaxation and the self-motion of hydrogens studied by NSE. For PI, the shift factors used for the construction of the master curve on Q,T) (Fig. 4.17) are identical to those observed for the structural relaxation time [8]. This temperature dependence also agrees with DS and rheological studies. The case of PIB is more complex [ 147]. The shift factors obtained from the study of Teif(Q>T) (Fig. 4.14b) reveal an apparent activation energy close to that reported from NMR results (-0.4 eV) [136]. This temperature dependence is substantially weaker than that observed for the structural relaxation time (=0.7 eV, coinciding with rheological measurements) in the same temperature range (see Fig. 4.20). 

Free Volume Versus Configurational Entropy Descriptions of Glass Formation Isothermal Compressibility, Specific Volume, Shear Modulus, and Jamming Influence of Side Group Size on Glass Formation Temperature Dependence of Structural Relaxation Times Influence of Pressure on Glass Formation... 

The AG model [48] for the dynamics of glass-forming liquids essentially postulates that the drop in S upon lowering temperature is accompanied by collective motion and that the fluid s structural relaxation times r are activated with a barrier height that is proportional [80] to the number z of polymer... 

In addition to vanishing at Tq, Fig. 5a shows that Sc T) exhibits a maximum s at a higher temperature 7a that is roughly twice Tq. As explained later, 7a is identified by us as the Arrhenius temperature, above which structural relaxation times exhibit a nearly Arrhenius temperamre dependence, X exp(pAp). Thus, 7a is the temperature below which collective motion initiates [111], that is, where z = sl/sc T) > 1. The LCTestimates of Sc T) for both the F-F and F-S polymers are displayed in Fig. 6 for relatively small and large representative M (M = 101 and M = 40001), where M is the number of united atom groups per chain, which is thus proportional to the molar mass Mmol- The entropy Sc in Fig. 6 is normalized by its maximum value and is presented as a function of the reduced temperature 6T,... 

Our estimates of typical values for Ap for both F-F and F-S high molar mass polymers (Ap/ B 2000K and 2600 K, respectively) are comparable in magnitude with Ap obtained for high molar mass alkanes by Tabor [84] (Ap/ B 2700 K). The interrelation between Ap and has implications regarding the magnitude of the structural relaxation time r at the crossover temperature Recent investigations [102, 137] indicate that r at the... 

One theory that describes the temperature dependence of relaxation time and structural recovery is the Tool-Narayanaswamy-Moynihan (TNM) model developed to describe the often nonlinear relationship between heating rate and Tg. In this model, the structural relaxation time, x, is referenced as a function of temperature (T), activation enthalpy (Ah ), universal gas constant (R), hctive temperature (7)), and nonlinearity factor (x) (Tool, 1946 Narayanaswamy, 1971 Moynihan et al., 1976) ... 

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