WLF free volume

In addition to the WLF free volume, a series of other free volumes can be defined and discussed. The vacant volume can be obtained from the specific volume of the amorphous polymer v m measured at the temperature Tand the specific volume v dw calculated from the van der Waals radii. The vacant-volume fraction /vac is... 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

With further understanding how molecular rotors interact with their environment and with application-specific chemical modifications, a more widespread use of molecular rotors in biological and chemical studies can be expected. Ratiometric dyes and lifetime imaging will enable accurate viscosity measurements in cells where concentration gradients exist. The examination of polymerization dynamics benefits from the use of molecular rotors because of their real-time response rates. Presently, the reaction may force the reporters into specific areas of the polymer matrix, for example, into water pockets, but targeted molecular rotors that integrate with the matrix could prevent this behavior. With their relationship to free volume, the field of fluid dynamics can benefit from molecular rotors, because the applicability of viscosity models (DSE, Gierer-Wirtz, free volume, and WLF models) can be elucidated. Lastly, an important field of development is the surface-immobilization of molecular rotors, which promises new solid-state sensors for microviscosity [145]. 

In attempts to better understand dendrimer intramolecular morphology, considerable attention was devoted to the fractional free volume near the glass temperature [40, 49, 50], Because all of the studies were performed within the WLF temperature range, the data were analyzed using the equation... 

Table 14.2 Values of WLF Cj0 and C2° Parameters, Fractional Free Volumes f0 and/ and Thermal Expansion Coefficient af for PAMAM, PPI and PBzE dendrimers ... 

The configurational entropy model describes transport properties which are in agreement with VTF and WLF equations. It can, however, predict correctly the pressure dependences, for example, where the free volume models cannot. The advantages of this model over free volume interpretations of the VTF equation are numerous but it lacks the simplicity of the latter, and, bearing in mind that neither takes account of microscopic motion mechanisms, there are many arguments for using the simpler approach. 

According to the more widely used Williams, Landel, and Ferry (WLF) equations, all linear, amorphous polymers have similar viscoelastic properties at Tg and at specific temperatures above Tg, such as Tg + 25 K, and the constants Ci and C2 related to holes or free volume, the following relationship holds ... 

The WLF equation holds over the temperature range from Tg to about + 100 K. The constants in Eq. (5.76) are related to the free volume. This is a procedure analogous to the one we used to generate time-temperature-transformation (TTT) diagrams for metallic phase transformations in Section 3.1.2.2. 

Fig. 18. Viscosity (log rjinPa-s[lPa-s=10 pois]), as a function of temperature for polymethacrylate (a) PVC (b), and polyvinylacetate (c) according to Eq. (43) (WLF, theoretical curves). The diagram shows the great influence of the free volume near the glass transition temperature...

The free-volume concept was applied most widely in the theory of viscoelastic properties of polymers developed by Williams, Landel and Ferry (WLF theory), presented in detail in12. According to WLF theory, the changes in liquid viscosity with frequency and temperature from glass temperature T% to T may be plotted on a single master curve by using the reduction factor... 

It is apparent that only the average value corresponds to the universal value in WLF theory, and that the deviations from this value can be very considerable. In spite of this, we may believe that free-volume is determined mainly by the hole volume. At the same time it follows from Eq. (63) that the SB constant is a very complex value and a function of p and a. Plotting the experimental data in coordinates... 

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 97- it was also shown on the basis of dilatometric data that the free-volume of PMMA in the mixture with polyvinylacetate increases with the increase in FVA concentration. In 98) a large difference was reported in the viscoelastic behavior of block copolymer from that predicted by WLF theory. This theory is believed to be useful only near the Te of each component, not in the broad temperature interval including the transition from glassy to rubberlike state. This anomaly is thought to be connected with certain motions in the interphase regions, which should be looked upon as independent components of the mixture. 

From experimental data on viscoelastic properties the fractional free-volume was calculated according to the WLF equation for pure hardened resin, filled with different amounts of polymeric filler obtained from the same hardened resin. By using special methods for preparation of filled specimens, it was possible to obtain... 

Equation (5.57) is based on the equilibrium free volume of the material calculated with the WLF equation. This may be considered a consistent assumption for T > Tg (most of the experimental data are included in... 

The WLF relationship can be derived from the Doolittle relationship that links the mobility M to the free volume fraction ... 

In principle, these relationships open the way to a determination of fg which is found to decrease with crosslink density as well in ideal epoxy networks (Gerard et al., 1991), as in nonideal polyesters (Shibayama and Suzuki, 1965). However, it must be recognized that, in both series of data, it is impossible to have consistent values of, Cf, Cf, a, and fg except if BD varies with the structure, which can be considered as a serious argument against the free volume interpretation of WLF parameters. 

This relation holds well for semicrystalline polymers for amorphous polymers, it holds for T > Tg + 100°C. Below this region free volume effects predominate necessitating the use of the Arrhenius-WLF equation... 

---