Free volume fractions, calculation

The free-volume fraction calculated from equation (10-10) (see Table 10-2) agrees very well with the values given in Table 5-6. It has been em-... 

The sizes and concentration of the free-volume cells in a polyimide film can be measured by PALS. The positrons injected into polymeric material combine with electrons to form positroniums. The lifetime (nanoseconds) of the trapped positronium in the film is related to the free-volume radius (few angstroms) and the free-volume fraction in the polyimide can be calculated.136 This technique allows a calculation of the dielectric constant in good agreement with the experimental value.137 An interesting correlation was found between the lifetime of the positronium and the diffusion coefficient of gas in polyimide.138,139 High permeabilities are associated with high intensities and long lifetime for positron annihilation. 

Dielectric constants of these materials can be further lowered by known means such as by incorporating air bubbles into the materials or by inhibiting crystallization. A difference of a couple of hundredths in the DE value may be important when one is at the low extremes. Recently Singh et al. calculated the DEs ofpolyimide films from the measured free volume fraction and found that the calculated values, are close to the experimental result.1415 ... 

The radial distance distribution in simple atomic and molecular fluids is determined essentially by the exclusion volume of the particles. Zemike and Prins [12] have used this fact to construct a one-dimensional fluid model and calculated its radial distance correlation function and its scattering function. The only interaction between the particles is given by their exclusion volume (which is, of course, an exclusion length in the one-dimensional case) making the particles impenetrable. The statistical properties of these one-dimensional fluids are completely determined by their free volume fraction which facilitates the configurational fluctuations. 

The definitions of the free-volume fraction discussed above are dependent on taking the volume as an initial marking-off state, i.e. on the manner of calculation. It is already apparent here that there is some uncertainty regarding the physical meaning of free-volume. 

In 60 the fractional free-volume was calculated according to the relationship ... 

In some reports83,84) the change in the fractional free-volume was calculated at temperatures above Tg for epoxy resin filled with polystyrene particles on the basis of the experimental value of the reduction factor aT and the universal value fg according to the equation... 

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... 

The free volume has been introduced intuitively, relating it to the gaps that allow conformational changes in the solid. It would therefore be possible to obtain / from the difference between the geometric volume of the segments and the total volume. Nevertheless, this type of calculation is not useful, as the free volume for molecular movement does not exactly coincide with the empty space in the solid. The concept of free volume is related to the occurrence of macromolecular motion rather than to the existence of gaps. For this reason, the free volume fraction is an empirical parameter whose value is determined on the basis of experimental results. 

To predict the viscosity reduction, the change in the free volume fraction caused by CO2 dissolution has to be calculated. Extending the definition of free volume fraction, the free volume fraction is given as a function of temperature, pressure, and dissolution of gas ... 

Problem 2.27 Assuming for the constant B in Eq. (2.25) a value of unity, calculate the free volume fraction at Tg and the expansion coefficient of free volume, orf. 

IPN crosslinked PBA, crosslinked uncrosslinked SAN Poly(p-hydroxy styrene), PVPh and EVAl Acrylic core-shell copolymer and either PBT or PET Poly(allyl methacrylate-co-butyl acrylate-co-butanediol dimethacrylate-co-styrene-co-methyl methacrylate) or poly (aery lonitrile-co-butyl aery late-co-tricy clodeceny 1 aery late-co- styrene) Poly(acrylate-V-cyclohexyl maleimide), PMI, and a copolymer PMMA — core, crosslinked butyl acrylate-styrene copolymer — middle layer, and PMMA shell d = 200-300 nm PEG/atactic PMMA blends were characterized by PVT at T = 20-200°C and P = 0-200 MPa. Free volume fraction was calculated from an equation of state... 

Since both V and Vh are known as a function of temperature, it is possible to plot V directly versus Vh, which yields values for At and Vo by means of a fitting procedure. With these results, the free-volume fraction can be calculated ... 

Fractional free volume/was then calculated according to the previous definition [Eq. (10.15)]. Its variation with temperature is shown in Figure 10.10, together with the theoretical free-volume fraction fi values of/(obtained assuming spherical holes, plotted as circles in Figure 10.10) are systematically lower than h for all the structures. Furthermore, the expansion coefficients of/are higher than the corresponding values deduced from the theory. 

For a further comparison of the free volume of polymers with different Tg values and different chemical structures, we show in Figure 11.11 the fractional hole free volume h calculated from PVT experiments employing the S-S equation of state. Polymers denoted as conventional (hydrogen polymers and others) have hole fractions ranging up to 0.15, while at 606 K Teflon AF2400 has a value of A = 0.374. 

The simplified procedure starts with computation of the characteristic P, 1, V parameters from the PVT data at T> Tg. Next, from Eqs. (14.2) and (14.3), the fictitious hole fraction in the glassy state at T < Tg and P (a prime indicates an independent variable in the vitreous state) is calculated as /lextrapoi = ( P )-Subsequently, from the PVT data at T < Tg, using Eq. (14.2), the hole fraction in the glassy state, hgu = h(T, P ), is computed. Thus, for the same set of T, P, the hole fractions that the melt would have, /lextrapoi, and the factual one, /igiass > extrapoi, are determined. From the isobaric values of h versus T, the frozen fraction of free volume is calculated as [McKinney and Simha, 1974]... 

Solid body. In amorphous systems, calculate the hole fraction in the vitreous state, h = h P and then the pressure and composition dependencies of the frozen free-volume fraction, FF = FF(P, w). The semicrystalline systems must be treated as supercooled liquids (described by the S-S equation of state) comprising dispersed crystals, described by the Midha-Nanda-Simha-Jain equation of state [see Eqs. (6.32) to (6.34)]. 

At a certain polymer concentration, however, the viscosity of the mixture becomes so great that the polymer segments can no longer move freely. Because of this freezing-in process the specific volume of the amorphous polymer v is larger than the specific volume of the liquid polymer v° would be at the same temperature. Conversely, the density of the liquid polymer is higher than the density of the solid The solid polymer has vacant sites or what is called free volume. These vacant sites should be seen as having diameters of the order of those of atoms. The free-volume fraction/w, p is calculated as... 

Table 5-7. Various Free-Volume Fractions of Amorphous Polymers Calculated from Crystalline Density at 0°C°...

The free-volume fraction /exp calculated from Equation (10-51) agrees very well with the values given in Table 5-7 (Table 10-6). According to the empirical Boyer-Simha rule, the product (a/ - aam) Tg generally has a value of about 0.11 for a large number of polymers. Partially crystalline polymers and polymers with relaxation mechanisms lying below the glass transition temperature are exceptions to this rule. 

These values are considered as the universal WLF constants, giving the widest range of agreement over many prominent amorphous polymers. From these choices one calculates that the free-volume fraction /g at the glass-transition temperature is 0.025 and the dilatational thermal coefficient of expansion of a sub-cooled polymer melt at the glass-transition temperature is a = 2.8 x 10 ... 

PEG/atactic PMMA blends were characterized by PVT at T = 20-200 °C and P = 0-200 MPa. Free-volume fraction was calculated from an equation of state Schmidt and Maurer 1998... 

An expression for the work of insertion W can be obtained from scaled particle theory (SPT) [31]. SPT was developed to derive expressions for the chemical potential and pressure of hard sphere fluids by relating them to the reversible work needed to insert an additional particle in the system. This work W is calculated is by expanding (scaling) the size of the sphere to be inserted from zero to its final size the size of the scaled particle is Act, with X running from 0 to 1. In the limit 2 0, the inserted sphere approaches a point particle. In this limiting case it is very unlikely that the depletion layers overlap. The free volume fraction in this limit can therefore be written as... 

In Fig. 4.10 we compare the free volume fraction a calculated from (4.6), in the good solvency limit with Ss/R in (4.7) from (4.33), with Monte Carlo simulation results of Fortini et al. [51] for q = 1.05 along the binodal gas-liquid curve. Except for some deviation at large colloid volume fractions the agreement is excellent. 

---