Fractional free volume times

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. 

Limiting flow rates are hsted in Table 23-16. The residence times of the combined fluids are figured for 50 atm (735 psi), 400°C (752°F), and a fraction free volume between particles of 0.4. In a 20-m (66-ft) depth, accordingly, the contact times range from 6.9 to 960 s in commercial units. In pilot units the packing depth is reduced to make the contact times about the same. 

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-... 

It appeared that the fractional free-volume in filled systems increased in proportion to the polymer fraction in the surface layer, determined independently, and ranging from 0.025 to 0.043. This fact was explained by the diminishing molecular packing density on the surface. There was at the same time a decrease in the temperature Tq-The findings indicate that the criterion of constancy of the free-volume fraction at T% cannot be applied to filled systems because of the influence of the filler on the polymer structure. Thus, even for one and the same polymer, the difference in its physical structure induced by physical actions capable of changing the structure causes polymer behavior to deviate from that predicted within the framework of the iso-firee-volume concept. 

We have already discussed the possibility of changes in fractional free-volume being related to the physical structure of polymers. To show this in greater detail, a special study was made102. Hie viscoelastic properties and relaxation time spectra were studied in a filled system where a powder of hardened epoxy resin was used as the filler and the same epoxy resin as the matrix. Thus the system was identical from the chemical point of view, the only difference being in die method of preparation. 

The WLF equation is an empirical expression for the shift of relaxation time with respect to the temperature change and applies to rubbery materials. It is well known (10) that the WLF equation can be written as a function of the fractional free volume only ... 

If an increase in free volume arising from stress-induced dilatation contributes to the relaxation process in the same manner as dilatation by raising the temperature, we can estimate the shift in relaxation time with Equation 2 by substituting for the fractional free volume, f,... 

This choice relies on the assumption that a constant Tg+T corresponds to a constant free volume state. Such an approximation presumes that the thermal expansion of free volume, Uf, as well as the fractional free volume, fg, at Tg, are independent of the hlend composition. In the case of binary blends with short chains, may depend on the composition of the blend, so that a normalisation of experimental times with this parameter would be misleading. 

Free-volume theories of the glass transition assume that, if conformational changes of the backbone are to take place, there must be space available for molecular segments to move into. The total amount of free space per unit volume of the polymer is called the fractional free volume Vf. As the temperature is lowered from a temperature well above Jg, the volume of the polymer falls because the molecules are able to rearrange locally to reduce the free volume. When the temperature approaches Tg the molecular motions become so slow (see e.g. fig. 5.27) that the molecules cannot rearrange within the time-scale of the experiment and the volume of the material then contracts like that of a solid, with a coefficient of expansion that is generally about half that observed above Tg. 

Let Uq be the average size of an element of free volume and v the minimum size of the element of free volume required for movement. The probability that an element of free volume permitting movement arises in unit time is then proportional to exp(—u /uq). At each opportunity to move the ion is accelerated in the direction of the applied electrical field and picks up a component of velocity qEx in the direction of the field, where x is the time taken for the move and q is the charge on the ion. The mean drift velocity in the field direction is thus proportional to qExnex —v /vQ), where n is the number of ions per unit volume. For given n the conductivity is thus proportional to exp(—u /Uq). Strictly speaking the constant of proportionality depends on T, but this dependence is small compared with the dependence on T contained in the exponential. The quantity must clearly be proportional to the total fractional free volume Ff, so that the conductivity is proportional to exp(—c/Ff), where c is a constant for the particular polymer and ion. Substitution for Ff from equation (7.33) and rearranging leads finally to the temperature dependence of the conductivity ... 

Time-temperature superpositioning was originally derived from free volume models, which assume that the rates of molecular motions are governed by the available unoccupied space. Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, f =V /(V - Vo), where V is the specific volume and Vo is the occupied volume normalized by the mass) (Doolittle and Doolittle, 1957 Cohen and Turnbull,... 

In summary, it is clear that the o-Ps lifetime determined via the PALS technique provides accurate information on the apparent mean size of the nanoholes, which comprise the free volume in amorphous polymers. It also seems well established (see Chapter 11) that provided that the noise level in the PALS spectrum is sufficiently reduced, the distribution of o-Ps lifetimes can be obtained, which generates information regarding nanohole-size distribution. Concerns have been raised about the utility of the o-Ps intensity, I3, to characterize the number density of nanoholes and hence the fractional free volume via Eq. (12.2), because the value of I3 can be influenced significantly by the presence of species that inhibit or enhance positronium formation. We feel that we can utilize I3 values to evaluate fl actional free volumes via Eq. (12.2), provided either that the sample is rejuvenated by heating above Tg prior to measurement, and/or experiment indicates that the value of I3 remains constant within experimental error, during the time of exposure to the positron source. 

Figure 11. Fractional free volume vs. milling time.

In addition to temperature time-scale shifts, which are attributed to changes in the relative or fractional free volume (62), the magnitude of the compliance or... 

As the stress-strain linearity limit of most thermoplastics and their blends is very low, nonlinear viscoelastic behavior of heterogeneous blends needs to be considered in most cases. The nonlinearity is at least partly ascribed to the fact that the strain-induced expansion of materials with Poisson s ratio smaller than 0.5 markedly enhances the fractional free volume (240). Consequently, the retardation times are perpetually shortened in the course of a tensile creep in proportion to the achieved strain. Thus, knowledge of creep behavior over appropriate intervals of time and stress is of great practical importance. The handling and storage of the compliance curves D (t,a) in a graphical form is impractical, so numerous empirical functions have been proposed (241), eg. 

These materials cover a broad permeability range from Teflon AF 2400, one of the most permeable polymers, to Cytop , the least permeable among these amorphous perfluoropolymers. However, even Cytop is about ten times more permeable than conventional glassy polymers such as polycarbonates. The relative permeability of the perfluoropolymers is well described by their fractional free volume (FFV), a common measure of the free space in a polymer matrix available for molecular transport. For instance, both gas permeability and FFV exhibit the following order Teflon AF 2400 > Teflon AF 1600 > Hyflon AD SOX Hyflon AD 60X > Cytop . As the amount of bulky perfluorodioxole increases, the permeability and diffusivity increases. 

The view that the free volume at the glass transition temperature should be a constant was first introduced by Fox and Flory. The magnitude is reasonable, but no precise significance can be attached to it because the concept of V/ as originally introduced remains operationally undefined. Nevertheless, fractional free volumes thus calculated from the dependence of relaxation times on temperature are consistent with those calculated independently from a free-volume analysis of dependence of relaxation times on pressure (Section D) and dilution with solvent (Chapter 17), and this consistency reinforces the utility of /in equation 32. It should be noted that fg as calculated from equation 39 depends on the time scale of the experiment in which Tg was determined, and the values in the table refer to the usual scale—unspecified, but probably of the order of an hour. If the time scale were varied sufficiently to lower Tg by 8° (the extreme shown in Fig. 11-8),/ would be reduced from 0.025 to 0.022. Other concepts of free volume may lead to somewhat larger numerical values but these are irrelevant to the treatment here. 

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