Temperature dependence of the characteristic

Since both the temperature dependence of the characteristic ratio and that of the density are known, the prediction of the scaling model for the temperature dependence of the tube diameter can be calculated using Eq. (53) the exponent a = 2.2 is known from the measurement of the -dependence. The solid line in Fig. 30 represents this prediction. The predicted temperature coefficient 0.67 + 0.1 x 10-3 K-1 differs from the measured value of 1.2 + 0.1 x 10-3 K-1. The discrepancy between the two values appears to be beyond the error bounds. Apparently, the scaling model, which covers only geometrical relations, is not in a position to simultaneously describe the dependences of the entanglement distance on the volume fraction or the flexibility. This may suggest that collective dynamic processes could also be responsible for the formation of the localization tube in addition to the purely geometric interactions. 

Fig. 4.8 Temperature dependence of the dielectric characteristic times obtained for PB for the a-relaxation (empty triangle) for the r -relaxation (empty diamond), and for the contribution of the -relaxation modified by the presence of the a-relaxation (filled diamond). They have been obtained assuming the a- and -processes as statistically independent. The Arrhenius law shows the extrapolation of the temperature behaviour of the -relaxation. The solid line through points shows the temperature behaviour of the time-scale associated to the viscosity. The dotted line corresponds to the temperature dependence of the characteristic timescale for the main peak. (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)...

Fig. 4.9 Temperature dependence of the characteristic time of the a-relaxation in PIB as measured by dielectric spectroscopy (defined as (2nf ) ) (empty diamond) and of the shift factor obtained from the NSE spectra at Qmax=l-0 (filled square). The different lines show the temperature laws proposed by Tormala [135] from spectroscopic data (dashed-dotted), by Ferry [34] from compliance data (solid) and by Dejean de la Batie et al. from NMR data (dotted) [136]. (Reprinted with permission from [125]. Copyright 1998 American Chemical Society)...

Figure 6.5 Temperature dependence of the characteristics of sodium k-carrageenan particles dissolved in an aqueous salt solution (0.1 M NaCl). The cooling rate is 1.5 °C min-1, (a) ( ) Weight-average molar weight, Mw, and (A) second virial coefficient, A2. (b) ( ) Specific optical rotation at 436 nm, and ( ) penetration parameter, y, defined as tlie ratio of the radius of the equivalent hard sphere to the radius of gyration of the dissolved particles (see equation (5.33) in chapter 5). See the text for explanations of different regions I, II, III and IV.

In other crystals the splitting is too small for spectroscopic measurements, but the rate constants for incoherent tunneling have been found from relaxation measurements. The results obtained by Kapphan [1974] are listed in Table 9.1. The temperature dependences of the characteristic time scales for orientational relaxation of OH- dipoles in several different crystals are depicted in Figure 9.2. Below 5 K, the relaxation times are inversely proportional to T, but scale as T 4 at higher temperatures. 

It has already been mentioned that the properties of a dielectric sample are a function of many experimentally controlled parameters. In this regard, the main issue is the temperature dependence of the characteristic relaxation times—that is, relaxation kinetics. Historically, the term kinetics was introduced in the field of Chemistry for the temperature dependence of chemical reaction rates. The simplest model, which describes the dependence of reaction rate k on temperature T, is the so-called Arrhenius law [48] ... 

Pentyl-4 -cyanobiphenyl and 4-octyl-4 -cyanobiphenyl liquid crystals (LCs) confined in molecular sieves of MCM-41 and cloverite types are studied in a wide temperature range by dielectric spectroscopy, thermal analysis and in-situ FTIR spectroscopy. The phase transitions of the bulk LCs cannot be detected when confined in MCM-41 sieve. A relaxational process occurs due to the molecular motions in the layer at the pore walls the temperature dependence of the characteristic frequency obeys a Vogel-Fulcher-Tamman law associated to a glassy state. In the cloverite cages, the LCs keep the phase transitions of the bulk but shifted. Interactions between Lewis and Brdnsted sites and the LC molecules are monitored by IR spectroscopy. DTA measurements put also in evidence strong guest-host interactions. 

From Fig. 22, the temperature dependence of the characteristic time x evaluated from the aging dynamics is quite similar to that evaluated from the dynamics of the Ki-process, but is completely different from that evaluated from the dynamics of the... 

In most supramolecular structures, the temperature dependence of the characteristic dielectric relaxation time follows the Arrhenius equation, r = Toexp(A dip/ T). where tq is the preexponential factor that is often of the magnitude of the vibrational time scale and A dip is the activation energy of the dipolar process.The dipolar process of the host lattice and the trapped molecules follows this behavior, but A trapped molecules is less than that for the host lattice molecules. In ice ciathrates, the dipolar processes of the water molecules that form the host lattice and the guest molecules inside the cages of this lattice occur at widely different time scales. This allows for a reliable attribution of the dielectric spectra features to water molecules and to the guest molecules. As an example of the magnitude of the dielectric properties of supiainolecular structures, the data on selected ice clathrates and other inclusion compounds are summarized in Tables 1 and 2. 

Other important experimental methods for elucidating the nature of chemical bonds in crystals are the study of the elastic and thermal properties and the determination of phonon spectra. Many of these properties can be regarded as thermodynamic stability criteria and are quantitative measures of the second derivatives of the bond energies with respect to the atomic spacing. Moreover, such investigations yield temperature dependences of the characteristic themodynamic functions. 

Early it was obtained some other equation for the temperature dependence of the characteristic time of segmental movement for diluted solution of polystyrene in toluene ... 

The experimental dependence of the self-diffraction intensity of the extraordinary polarized radiation on the normalized temperature r = (T— 7V)/r is shown in Fig. 1. The same figure shows the temperature dependence of the characteristic hologram-erasure time tff, which is equal to the writing time as well as to half the characteristic relaxation time of the nonlinearity. The value of, which is proportional to (J ), increases as the phase-transition point is approached, even though in this case the hologram-erasure time, which determines the stationary value of (Ref. 2), decreases. The dependence of the refraction intensity of the ordinary polarized radiation is similar in form. 

Figwe 29 Top TEM micrograph of PET melt-crystallized at 200 °C. Bottom Temperature dependence of the characteristic lengths / and of semicrystalline PET, derived from the ID autocorrelation (CF) and interface distribution functions (IDF). The major component I2 is obtained from the difference of the computed Lb and fy The crossed line corresponds to an evaluation of /i CFfrom TEM data. With permission from Haubmge, H. G. Jonas, A. M. Legras, R. Macromolecules 20M, 37,126." ... 

The temperature dependence of the characteristic time x of molecular motions responsible for the glass transition is strongly non-Arrhenius. Above Tg it is described by the Vogel-Fulcher law... 

Fig. 9. Temperature dependence of the characteristic frequency of the AFM spin fluctuations in the CuOj planes in Lai 55Euo.24GdooiSro,Cu4. o>sc(1 ) ha t>een calculated from the spin-lattice relaxation rate of Gd which is shown in the inset. From Kataev et al. (1997).

The measurements were used to calculate the temperature dependence of the characteristic molar volume, Fo(/10 m mol ) as... 

To conclude the application of the scheme to single liquids, and in addition to the groups of liquids considered above, some simple molecular fluids were examined by Assael et al. (1992b). These are CS2, CeHja, CCI4, CH3CN and CH3CI, for which self-diffusion, viscosity and thermal conductivity measurements were all available to provide a more critical test of the correlation. As before, the data are used to calculate the temperature dependence of the characteristic molar volumes, Fb(/10 m mol ), as... 

The temperature dependence of the characteristic frequency of the Goldstone process in the C phase depends on the material studied. For example, for polymer 43 it appears to be temperature insensitive, but at the transition to the J phase the Goldstone mode becomes rapidly shifted to much lower frequencies. In systems with the glassy transi-... 

Figure 19 Temperature dependence of the characteristic tan(5 peak frequency for iocai and buik PVAc measurements. Lines are fits to the VF equation. From Crider P. S. Majewski, M. R. Zhang, J. Oukris, H. israeioff, N. E. J. Chem. Phys. 2008, 128, 044908. °...

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