Polymers free volume fraction

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. 

Dependence of the Free-Volume Fraction at Glass Temperature on the Molecular Parameters of Linear Polymers... 

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. 

Different approaches were used to describe the yielding of polymers quantitatively. Some theories took into account the free volume fraction. Eyring considered thermally activated mechanisms, and Robertson s model was based on changes of chain conformations. Argon s and Bowden s models were based on a metallurgical approach and a dislocation theory. A brief summary of the existing yielding theories is presented. 

Figure 5 Density profiles of grafted (solid curves) and free polymer chains (dotted curves) as a function of the spatial coordinate z, calculated using iSAFT (gray curves) and SOFT (black curves) methods, (a) Free polymer bulk volume fraction tpf = 0.75 separation between the platelets H = 50, and (b) free polymer bulk volume fraction (pf = 0.2 separation between the platelets H = 80. Other parameters Ng —101, pg — 0.1, and Nf= 100.

For a fixed strain rate, a comparison of Eq. (74) and experimental data [51, 52] of miscible blends is shown in Fig. 32. Curves 1 and 2 represent, respectively, the PPO/PS blends in compression, and the PPO/PS-pCIS blends in tension.Table 2 lists the three parameters fjf2, CK, and A/f2 used in curves 1 and 2. The unique feature here is the presence of a maximum yield (or strength) for 0 <

nonequilibrium interaction (A < 0). Such phenomenon does not occur in incompatible blends or composite systems. Table 2 also reveals that the frozen-in free volume fractions which are equal to 0.0243 and 0.0211 for polystyrene and for PPO, respectively. These are reasonable values for polymers in the glassy state. In the search for strong blends, we prefer to have —A/f2 > 1, and a larger difference between the yield stresses of blending polymers. 

Polymer name o-Ps Life- time x3(ns) Free- Volume Radius (A) o-Ps Inten- sity %) Free- Volume Fraction (%) Longer o-Ps Lifetime X4 (ns) Longer o-Ps Intensity I4 (%) Ref Comments... 

The constants C and C2 are nearly universal and, within a reasonable degree of approximation, are valid for many polymers, taking the values — 17.4 and 51.6 K, respectively. From the value of Cj it can be deduced that the free volume fraction, fg, at the glass transition is approximately 0.025. When the free volume fraction falls to this low value it seems that the conformational changes in the solid cease to occur. The value of C2, together with fg = 0.025, allows the thermal expansion coefficient of the free volume to be known, which turns out to be ay 4.8 x 10 K valid for the great majority of polymers. 

The influence of factors such as chemical structure, molecular weight, cross-linking and plasticizers in the glass transition of polymers can be related to the changes that they provoke on the free volume fraction, which, as we already know, reaches a critical value at the glass transition temperature. The factors affecting the glass transition can be classified into two types (1) molecular factors, i.e., those related to the chemical structure of the polymer chain, and (2) external or controllable factors. 

If the different tactic configurations of a single polymer, for example, poly(methyl methacrylate), are considered the lowest value of Tg corresponds to the isotactic polymer. At T < Tg the specific volume of the isotactic polymer is lower than that of the atactic one, and the free volume fraction is the same for both polymers therefore the volume occupied will be less in the isotactic polymer. Nevertheless, at T > Tg, both tactic configurations have similar specific volume consequently the temperature at which the free volume is equal to 0.025 of the total volume is lower in the isotactic form than in the atactic one. 

Let us consider a polymer of density p and molecular weight M . In this case, the number of chains per unit volume is p(iV /M ) thus the number of chain terminals per unit volume is 2p(jV /M ), where is Avogadro s number. If the contribution of one chain terminal to the free volume is represented by 0, the total free volume fraction due to the chain terminals, f is given by... 

The macroscopic free-volume is an important parameter closely related to the rheological behavior of an amorphous material. It is generally accepted that the glass transition is regarded as an iso-free-volume state, and the free-volume fraction (fg) at the Tg is around 0.025 for many monomers and polymers. The WLF method has been widely used for determining the macroscopic fg value. On the other hand, the microscopic analysis of the free-volume has also been uti-... 

Assuming that Ci, C2, and C3 are not affected by temperature, pressure, and dissolution of CO2, they can be determined from a viscosity-shear rate curve of the neat polymer. Namely, the coefficient, Ci, which is equivalent to n - 1, can be determined by the slope of the viscosity and shear rate curve. The values of C2 and C3 can be determined from data of viscosity vs. free volume fraction of the neat polymer. The data of free volume fraction required for determining C2 and C3 can be obtained from PVT data of the neat polymer at temperatures and pressures where the viscosity measurements of the neat polymer are performed. 

Simha and Boyer [34] postulated that the free-volume fraction defined by Eq. (2.22) is the same for all polymers, that is,... 

This implies that the free volume fraction at the glass transition temperature is the same for all polymers and constitutes 11.3% of the total volume in the glassy state. (Many simple organic compounds have a 10% volume increase on melting, it may be pointed out.) TTiis is the largest of the theoretical values derived, but the first. Other estimates placed the free volume at about 2%. 

A relation between Tg and composition of a polymer-diluent mixture can be derived [47] in a relatively straightforward manner from the free-volume concept by postulating that the free volumes of the polymer and diluent are additive in the mixture, and that the free volume fraction has a critical value fg, which is the same for the pure polymer, the diluent and their mixtures at their respective glass temperatures. The composition of polymer-diluent mixtures is conveniently expressed in terms of the volume fractions of polymer cj)p and diluent (f>d. 

According to the free-volume theory, as we have seen earlier, the free-volume fraction / in a polymer at a temperature T above Tg can be expressed in a linear form [cf. Eq. (2.33)] ... 

---