Free volume, calculated

Later Kanig took into account the temperature dependence of the fractional free-volume, calculated according to his equations. Below Tg, as temperature decreases, sPfl increases as a result of a decrease in expansion volume at frozen hole volume. Above Tg, pn increases due to the sharp rise in hole volume. At Te the value ip i is at its minimum. From the condition d [Pg.75]

Figure 2.24 Correlation of the oxygen permeability coefficient for a family of related polysulfones with inverse fractional free volume (calculated using the Bondi method) [33]. Reprinted with permission from C.L. Aitken, W.J. Koros and D.R. Paul, Effect of Structural Symmetry on Gas Transport Properties of Polysulfones, Macromolecules 25, 3424. Copyright 1992, American Chemical Society...

The structure of an amorphous polyamide prepared from hexamethylenediamine and isophthalic/tere-phthalic acids was modified in order to determine the effect of chemical structure on the oxygen permeation properties. The greatest increase in permeation was obtained by lengthening the aliphatic chain. Placement of substituents on the polymer chain also led to increased permeation. Reversal of the amide linkage direction had no effect on the permeation properties. Free volume calculations and dielectric relaxation studies indicate that free volume is probably the dominant factor in determining the permeation properties of these polymers. 

FIGURE 11.9 Specific total, V, free Vf = hV, and occupied, Vqcc = (1 — h)V, volume of PC as a function of T at ambient P. h is the hole fraction calculated from V using the S-S equation of state. Open symbols, experimental data dots, S-S equation of state fits to the volume in the temperature range T>Tg-, stars, free volume calculated from Vf=N v,), where (Vf,) is the mean hole volume from PALS and is the specific hole density, assumed to be constant at N f = 0.67 X 10 g (corresponding to 0.81 nm at 300 K). (Adapted from Dlubek et al., [2007d].)... 

In contrast to Vf, the specific volume V in the glassy state of the polymer exhibits a larger slope than above Tg, which is due to the thermal contraction of Vocc ( occ,g O.Sag). The agreement of the slope dVld vh) fl-om the phenomenological relation Eq. (11.6) with dVf/d vh) from Eq. (11.5) above Tg we consider as evidence that o-Ps detects precisely the free volume calculated fl om S-S hole theory. The larger slope of dVld vh) below Tg supports the conclusion from the S-S equation of state calculation that Vocc shows here, as distinct from above Tg, a certain thermal expansion. 

The relative fractional free volume (/), calculated variously as the product Ij, vh)h, or C vh)h, where C is an empirical constant, did not exhibit regular behavior when plotted as a function of the blend composition, behaving essentially similar to I3. 

Another possible approach to indirectly characterize the membrane morphology is based on the investigation of the free volume within the matrix. Density measurements [119,120] and positron annihilation lifetime spectroscopy evaluation [47] are common methods. Typically, the comparison between the theoretical density or free volume (calculated by simple additivity rules) and the experimental one can reveal the presence of a good interfacial morphology or the presence of interface voids or clustering formation. Fig. 7.13 shows the influence of filler content on the morphology of poly(trimethylsilyl propyne) (PTMSP)/Ti02 NCMs in terms of the volumetric fraction of interface voids as calculated from a comparison of the expected and measured membrane density [119],... 

FIG. 12-8. Diffusion coefficient of n-hexadccane at 2S°C through random styrene-butadiene copolymers plotted against reciprocal of fractional free volume calculated from equation 8. Open circles, uncross-linked samples black circles, cross-linked. Lines drawn with slope of — 1/2.3Q3 as specified by equation 7. (Rhee and Ferry. ) Reproduced, by permission, from the Journal of Applied Polymer Science. 

Molar masses, glass transition temperatures, and fractional free volume (calculated from densities and Paul and Park s series of increments) of the polymers are summarized in Tables I and II (6-8). Fractional free volume (Vf) calculated by the method of Park and Paul (P) differs for different gases, depending on the diameters of the gas molecules. Here, the values for O2 and N2 are given as examples. As expected, the values for Vf increase with decreasing kinetic diameter of the gas molecules. 

The fractional free volume,/, calculated as a function of aging time using the Struik model can then be used in the following correlation to give gas permeability ... 

We assume that the unbinding reaction takes place on a time scale long ( ompared to the relaxation times of all other degrees of freedom of the system, so that the friction coefficient can be considered independent of time. This condition is difficult to satisfy on the time scales achievable in MD simulations. It is, however, the most favorable case for the reconstruction of energy landscapes without the assumption of thermodynamic reversibility, which is central in the majority of established methods for calculating free energies from simulations (McCammon and Harvey, 1987 Elber, 1996) (for applications and discussion of free energy calculation methods see also the chapters by Helms and McCammon, Hermans et al., and Mark et al. in this volume). 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Also, in cases where the dimensions of a regular particle vary throughout a bed of such particles or are not known, but where the fractional free volume and specific surface can be measured or calculated, the shape factor can be calculated and the equivalent diameter of the regular particle determined from Figure 2. 

In order to see whether the results are sensitive to the exact shape of the potential field, some calculations have been made in which the field w r) was replaced by a square well. The depth of the well was taken equal to the value (Eq. 31) of w(o) for an L-J-D- field, while the radius was taken equal to the value (at— a) valid for hard spheres. In this approximation the free volume is equal to m (a —or)3, and hence in formula 38... 

For the calculation of free volume inside the cell, which is essential for the design of a ZEBRA cell to keep internal pressure low for safety reasons, the density of molten NaAlCl4 over the full temperature range between 160 and 600 °C should be known. Berg et al. [33] have compared these values with the literature. The densities are compiled in Table 9. 

The number of the constraints to chain mobility shown in Fig. 16 decreases with increasing temperature, reflecting the increase of the free volume. From a comparison of the spectra in Fig. 15 with line shapes calculated for flexible chains on a diamond lattice 23 (one can infer that the average length of the flexible unit increases from 3-5 bonds at room temperyture to about 10-15 bonds at 380 K. Our model thus can quantitatively explain the gradual increase of the number of conformations accessible to the chain motion. The earlier XH wide line data 72 are also in accord with our findings. 

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. 

Network properties and microscopic structures of various epoxy resins cross-linked by phenolic novolacs were investigated by Suzuki et al.97 Positron annihilation spectroscopy (PAS) was utilized to characterize intermolecular spacing of networks and the results were compared to bulk polymer properties. The lifetimes (t3) and intensities (/3) of the active species (positronium ions) correspond to volume and number of holes which constitute the free volume in the network. Networks cured with flexible epoxies had more holes throughout the temperature range, and the space increased with temperature increases. Glass transition temperatures and thermal expansion coefficients (a) were calculated from plots of t3 versus temperature. The Tgs and thermal expansion coefficients obtained from PAS were lower titan those obtained from thermomechanical analysis. These differences were attributed to micro-Brownian motions determined by PAS versus macroscopic polymer properties determined by thermomechanical analysis. 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. This independence means that, if we want to calculate the entropy difference between a pair of states joined by an irreversible path, we can look for a reversible path between the same two states and then use Eq. 1 for that path. For example, suppose an ideal gas undergoes free (irreversible) expansion at constant temperature. To calculate the change in entropy, we allow the gas to undergo reversible, isothermal expansion between the same initial and final volumes, calculate the heat absorbed in this process, and use it in Eq.l. Because entropy is a state function, the change in entropy calculated for this reversible path is also the change in entropy for the free expansion between the same two states. 

The free volume is calculated using equations (8) and (9) with the following parameters. 

Activation Volume and Free Energy Calculated from Experimental Data... 

Fig. 4.3. (A) Composite multispecies benthic foraminiferal Mg/Ca records from three deep-sea sites DSDP Site 573, ODP Site 926, and ODP Site 689. (B) Species-adjusted Mg/Ca data. Error bars represent standard deviations of the means where more than one species was present in a sample. The smoothed curve through the data represents a 15% weighted average. (C) Mg temperature record obtained by applying a Mg calibration to the record in (B). Broken line indicates temperatures calculated from the record assuming an ice-free world. Blue areas indicate periods of substantial ice-sheet growth determined from the S 0 record in conjunction with the Mg temperature. (D) Cenozoic composite benthic foraminiferal S 0 record based on Atlantic cores and normalized to Cibicidoides spp. Vertical dashed line indicates probable existence of ice sheets as estimated by (2). 3w, seawater S 0. (E) Estimated variation in 8 0 composition of seawater, a measure of global ice volume, calculated by substituting Mg temperatures and benthic 8 0 data into the 8 0 paleotemperature equation (Lear et al., 2000).

Equation (23) was found to be obeyed by a number of systems such as poly(ethyl acrylate)-benzene, rubber-benzene, and poly(methyl acrylate)-ethyl acetate [12], According to the equation, a plot of [In(DT/D0)] l versus ([)[ will yield a straight line, and from its slope the free volume parameter p can be determined. To construct this plot,/( / , 0) is first calculated as [27]... 

Miller, M. A. Reinhardt, W. P., Efficient free energy calculations by variationally optimized metric scaling concepts and applications to the volume dependence of cluster free energies and to solid-solid phase transitions, J. Chem. Phys. 2000,113, 7035-7046... 

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. 

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