Relaxation free-volume theories

From the family of AG (P, T) curves the projection on the (P, T) plane of the critical lines corresponding to the UCFT for these latexes can be calculated and this is shown plotted in Figure 4. It can be seen that the UCFT curve is linear over the pressure range studied. The slope of the theoretical projection is 0.38 which is smaller than the experimental data line. Agreement between theory and experiment could be improved by relaxing the condition that v = it = 0 in Equation 6 and/or by allowing x to be an adjustable parameter. However, since the main features of the experimental data can be qualitatively predicted by theory, this option is not pursued here. It is apparent from the data presented that the free volume dissimilarity between the steric stabilizer and the dispersion medium plays an important role in the colloidal stabilization of sterically stabilized nonaqueous dispersions. 

However, because measurements are kinetically determined, this is a less accurate form of the equation. Very often it is observed that the measured shift factors, defined for different properties, are independent of the measured property. In addition, if for every polymer system, a different reference temperature is chosen, and ap is expressed as a function of T — rj, then ap turns out to be nearly universal for all polymers. Williams, Landel and Ferry believed that the universality of the shift factor was due to a dependence of relaxation rates on free volume. Although the relationship has no free volume basis, the constants and may be given significance in terms of free volume theory (Ratner, 1987). Measurements of shift factors have been carried out on crosslinked polymer electrolyte networks by measuring mechanical loss tangents (Cheradame and Le Nest, 1987). Fig. 6.3 shows values of log ap for... 

The idea that the fractional free-volume at glass temperature as found experimentally depends on the mode of molecular motions was put forward in 196746 47 as a result of calculating/g from data obtained from isothermal volume relaxation for some polymer systems. By estimating average relaxation time at different temperatures it was possible to find the fractional free-volume/g at Te according to WLF theory. If we accept the validity of the theory as regards the universal dependence of the reduction factor aT on (T - Tg), then on the basis of data on Aa and theoretical values aT calculated from universal values of the coefficients C and C, it is possible to make an estimate of/g. In this case the value found corresponds to the universal one. If, however, we use the experimental values aT, the fractional free-... 

In reality, the data on isothermal contraction for many polymers6 treated according to the free-volume theory show that quantitatively the kinetics of the process does not correspond to the simplified model of a polymer with one average relaxation time. It is therefore necessary to consider the relaxation spectra and relaxation time distribution. Kastner72 made an attempt to link this distribution with the distribution of free-volume. Covacs6 concluded in this connection that, when considering the macroscopic properties of polymers (complex moduli, volume, etc.), the free-volume concept has to be coordinated with changes in molecular mobility and the different types of molecular motion. These processes include the broad distribution of the retardation times, which may be associated with the local distribution of the holes. 

We therefore believe one of the most pressing problems in the field of free-volume theory and glass-transition theory to be the development of new concepts and obtaining of new parameters for the corresponding states. Miller37 as early as 1968 introduced the idea that the glass transition corresponds to the iso-relaxation state, which at molecular weights of polymers below the critical ones may be replaced by an iso-viscous state. 

Recent work by Zhang and LeBoeuf (in review) examined the effects of the presence of three solvents—water, acetone, and benzene—on the molecular mobility and structural relaxation of a humic acid through DSC analysis combined with molecular dynamics. Again, antiplasticization behavior was observed in two of the three systems (i.e., HA-water and HA-acetone) where solvents present in relatively low concentrations exhibited potential to form hydrogen bonds with the humic acid. Antiplasticization and plasticization behaviors were further interpreted from the perspective of hydrogen bonding analysis and free volume theory. 

In this system the a relaxation can be analyzed by the symmetric equation of Fuoss-Kikwood and a new model which is similar to Havriliak- Negami equation used in the analysis of dielectric spectroscopy. According to the Tg values calculated for these systems, the free volume can be appropriately described by the free volume theory. The analysis of these families of poly(methacrylate)s allow to understand in a good way the effect of the structure and nature of the side chain on the viscoleastic behavior of polymers [33],... 

On the other hand, as in the analysis of the previous systems the temperature dependence of the a relaxation follow the Vogel-Fulcher-Tamman-Hesse (VFTH) [88-90], The 7% values obtained are 337 5 and 274 5 for PBCHM and PBCHMM respectively. By this way and using equation (2.31) the m parameter obtained are 1579 and 1804 and the relative free volumes at Tg are 3.2 and 2.6% for PCBuM and PCHBMM, respectively, in good agreement with the free volume theory. 

The Kauzmann temperature plays an important role in the most widely applied phenomenological theories, namely the configurational entropy [100] and the free-volume theories [101,102]. In the entropy theory, the excess entropy ASex obtained from thermodynamic studies is related to the temperature dependence of the structural relaxation time xa. A similar relation is derived in the free-volume theory, connecting xa with the excess free volume AVex. In both cases, the excess quantity becomes zero at a distinguished temperature where, as a consequence, xa(T) diverges. Although consistent data analyses are sometimes possible, the predictive power of these phenomenological theories is limited. In particular, no predictions about the evolution of relaxation spectra are made. Essentially, they are theories for the temperature dependence of x.-jT) and r (T). 

Kawamura, J. and Shimoji, M. Ionic conductivity and glass transition in superionic conducting glasses II. Structural relaxation and excess free volume theory. J. Non-Cryst. Solids 88 295, 1986. 

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (see Table 1, WW) provided a rather good description of dielectric relaxation experiments in polymer melts, but they are not of considerable help in understanding molecular phenomena since they are not associated with a molecular model. In the same way, the glass transition theories account well for macroscopic properties such as viscosity, but they are based on general thermodynamic concepts as the free volume or the configurational entropy and they completely ignore the nature of molecular motions. 

The temperature dependency of the apparent energy of activation of the bleeching process as exhibited by the curvature in the Arrhenius plot (Fig. 3) is typically found for, e. g., dynamic mechanical relaxation processes (17), which leads to the connection whith the free volume theory. The latter processes are best described by the WLF-equation (18), log a =Cj(T-Tg)/(C +T-Tg), i. e., a master plot is obtained when plotting the logarithm of... 

The free-volume model proposed by Vrentas and Duda (67-69) is based on the models of Cohen and Turnbull and of Fujita, while utilizing Bearman s (7j0) relation between the mutual diffusion coefficient and the friction coefficient as well as the entanglement theory of Bueche (71) and Flory s (72) thermodynamic theory. The formulation of Vrentas and Duda relaxes the assumptions deemed responsible for the deficiencies of Fujita s model. Among the latter is the assumption that the molecular weight of that part of the polymer chain involved in a unit "jump" of a penetrant molecule is equal to the... 

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