Free volume fractions

For the free volume we again start from expression (3.27) 

5 Phase Transitions of Hard Spheres Plus Colloids 

For intermediate values of the scaling parameters it is assumed that the work W X, v) can be found from a Taylor expansion around A = v = 0, with the term beyond the quadratic being replaced by (5.18) 

The expression for the work to insert a spherocylinder with length L and diameter D is obtained by setting A = v = 1. Using for P the SPT expression for hard spheres 

Exercise Verify that the above expressions for a, b and c mateh with those below (3.36) in the limit that a spherocyhnder equals a sphere. 

---

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. 

An obvious refinement suggested by this second derivation would consist in ascribing different free volume fractions Vf to the two pure... 

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 ... 

Figure 28. Accessible free-volume fraction profile across a DPPC bilayer for solutes of different sizes (diameters indicated in nm 0.0 corresponds with the total free volume). Results of Marrink et al. [132]. Redrawn by permission of the American Institute of Physics...

The viscosity dependence of intramolecular excimer formation is complex. As in the case of molecular rotors (Section 8.2), most of the experimental observations can be interpreted in terms of free volume. However, compared to molecular rotors, the free volume fraction measured by intramolecular excimers is smaller. The volume swept out during the conformational change required for excimer formation is in fact larger, and consequently these probes do not respond in frozen media or polymers below the glass transition temperature. 

In a series of papers, Diphant has been used to probe the microviscous properties of various polymer oils, and free volume parameters have been extracted. 4,88 90) In a comparative study of Excimer and TICT probes, it could be shown that the response of these probes is frozen out at lower temperatures, as can be expected from the large reaction volume necessary, whereas the TICT reaction still shows sizable rates at these high-viscosity conditions. (26) Moreover, this study also showed that the free volume fraction measured by the TICT probes is larger than that measured by Excimer probes. 

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. 

Dependence of the Free-Volume Fraction at Glass Temperature on the... 

The theory of viscoelastic properties based on the free-volume concept was developed in18, the main idea being that some structural elements are displaced when the free-volume fraction exceeds a critical value /c. The frequency v of the determining molecular process may be obtained from the equation... 

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. 

It was shown that there is an empirical equation which connects the compressibility coefficient p with the free-volume fraction ... 

The decrease of Aa Tg must lead to a considerable decrease in free-volume with decreasing temperature. According to52, it was necessary to revise the postulate of constant free-volume at Tg. Taking into account the temperature dependence of Og and ai, Simha and Weil gave alternative expressions for the free-volume fraction/in terms of volumes and expansivities, respectively ... 

Summarizing, we can say that the free-volume fraction at Tg as defined by Eq. (12) diminishes as the chain packing in the bulk polymers loosens. This result confirms the analogous qualitative conclusion formulated earlier46,47. ... 

The conclusion that the free-volume fraction at Tg is not a universal parameter for linear polymers of differing molecular structure can be qualitatively confirmed by the following arguments71. Assume that at temperatures far below Tg polymeric chains are in a state of minimum energy of intramolecular interaction, Le. the fraction of higher-energy ( flexed ) bonds is zeroS4. On the other hand, let the equilibrium fraction of flexed bonds at T> Tg obey the Boltzmann statistics and be a function of Boltzmann s factor e/kT. Thus, the fraction of flexed bonds at Tg can be estimated from the familiar expression ... 

From the point of view of the ideas discussed above concerning the variability on the free-volume fraction at Tg, even for the same modes of molecular motion in different polymers, there is great interest in some new concepts about the free-volume distribution, in the system, first proposed in 24. The starting point is the suggestion that all molecular motions, like transfer phenomena, can take place only when the size of the voids or holes in the system exceeds a critical value v. This critical volume appears as a result of redistribution of the free-volume within the system. 

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. 

Figure 4.2 Shape of variation of the free volume fraction or the entropy with (T - Tg).

T 1 oo = temperature at which the free volume fraction or configurational entropy become zero, K... 

The free volume fraction, fT, is usually defined from the classical linear model of thermal expansion. 

---