Free volume models extended

Figure 6.17. Free-volume model extended to the multi-component system. The transition rate Vi of species i is dependent on its activation volume vf and attempt frequency Vj .

Stern, Frisch, and coworkers have extended Fujita s free-volume model to the permeation of light gases (31-33) (see Figure 3) and binary gas mixtures (34,25) (see Figure 4) through polymer membranes. The extended model was found to describe satisfactorily the dependence of permeability coefficients on pressure and temperature for a variety of light gases in polyethylene, as well the dependence on composition for several binary mixtures in the same polymer. The validity of the extended model is limited to total penetrant concentrations of up to 20-25 mol-%. 

The present contribution has shown that the creep behavior of amorphous polymers under the influence of progressing aging can be well described and predicted under any thermal prehistory applying the multiparameter model based on free volume. The only condition necessary is the knowledge of any measured equilibrium creep curve. For each material the multiparameter model with the given set of parameters allows the prediction of the behavior in volume under any complicated thermal history as well. Introducing some additional postulations, the free volume model is adapted to work at low temperatures, i.e., at temperatures below T. Next, the theory should be extended to measurements at still lower temperatures as well as to some other amorphous polymers. 

The following three multicomponent transport models have been used to explain the depression of the permeability of a component in a mixture relative to its pure component value (Fig. 21) the Petropoulos model and the competitive sorption model, both of which assume that direct competition for diflfiisive pathways within the glass is negligible, and a more general permeability model in which direct competition can occur between penetrant molecules for both sorption sites and diffusion pathways. All three of the models presented here are based upon the framework of the dual-mode model. It is worth mentioning that the site-distribution model has recently been extended to accoimt for diffusion (98) and that free volume models exist for transport in glassy polymers (99). 

The free-volume model can be extended beyond infinite dilution in the case of strongly sorbing components like CO2. As CO2 sorbs into the polymer matrix, it swells the polymer, resulting in enhanced segmental mobility and a depression in Tg and, therefore, an increase in diffusivity as indicated by Eq. (35.18) (Ismail and Loma, 2002). Chow (1980) proposed an expression relating Tg depression to diluent concentration ... 

Figure 1.5(a) shows as a typical example a computer-tomography-like atomic mono-layer representation of a bulk model for diisopropyldimethyl PEEK WC (DIDM-PEEK). In this case the oxygen-accessible free volume is obviously organized in relatively small isolated holes and the respective size distribution (cf. Figure 1.5(b)) is monomodal and extending only to hole radii of about 5 A. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

The FH model provides a first approximation for polymer solutions. As shown, both the combinatorial and the energetic terms need substantial improvement. Many authors have replaced the random van-Laar energetic term by a nonrandom local-composition term such as those of UNIQUAC, NRTL, and UNIFAC models. The combinatorial term should be extended/modified to account for the free-volume differences between solvents and polymers. 

The Avrami model was originally derived for the study of kinetics of crystallization and growth of a simple metal system, and further extended to the crystallization of polymer. Avrami assumes the nuclei develop upon cooling of polymer and the number of spherical crystals increases linearly with time at a constant growth rate in free volume. The Avratni equation is given as follow ... 

The positron annihilation lifetime spectroscopy provides the information on free volume size and their concentration in porous solids independently if they are open and closed inaccesible for odsorptives. Ortho-positronium (o-Ps) forms in free volumes and its pick-off annihilation probability depends on free volume size. o-Ps lifetimes are related to free volume size in the way described by the extended Tao-Eldrup model (ETE) [11). The intensities of respective spectrum components depend on free volume concentration. 

---