Amorphous polymers free-volume theories

In addition to temperature and concentration, diffusion in polymers can be influenced by the penetrant size, polymer molecular weight, and polymer morphology factors such as crystallinity and cross-linking density. These factors render the prediction of the penetrant diffusion coefficient a rather complex task. However, in simpler systems such as non-cross-linked amorphous polymers, theories have been developed to predict the mutual diffusion coefficient with various degrees of success [12-19], Among these, the most notable are the free volume theories [12,17], In the following subsection, these free volume based theories are introduced to illustrate the principles involved. 

Therefore, although rather cumbersome, the free-volume theory permits one to prepare theoretical plots of permeability as a function of temperature, penetrant pressure, and amorphous volume fraction in the rubbery polymer. 

Amorphous polymers above Tg can be described by free volume theories [44-45] based on configurational entropy considerations. The essence of these concepts is that, above Tg, the vibrational energy of a segment is... 

Free-volume theories have been very successful in explaining concentration-dependent diffusion behavior of organic vapors in amorphous polymers, especially in cases where the penetrant is a good swelling agent for the polymer. The most significant free-volume theory is based on the works of Cohen and Turnbull (1959),... 

These relations for the oligomers agreed with the results of Sasabe et al. s work [24] on amorphous polymers, such as PVC and PVDC. The meaning of m for the oligomer was explained according to both the WLF equation and the free-volume theory developed by Cohen and Turnbull [25,26] ... 

Specifically, on the basis of free-volume theory, Doolittle (1951) has related the zero-shear viscosity (jjq) of an amorphous polymer to relative (fractional)... 

The model of free volume going back to the classical papers of Frenkel and Firing [48, 80, 144-147] has been widespread in the physics of liquid and solid states of matter. Some concepts allowing improvement in the nature of fluctuation free volume have been offered in the last 15 years [148-150]. Nevertheless, there is one more aspect of the problem, which has not been mentioned earlier. As a rule, the application of free volume theory for the description of the properties of amorphous bodies is based on a notion that the free volume characterises the structure of the indicated bodies. This postulate is due to a considerable extent to the absence of a quantitative model of the structure of the amorphous condensed state, including the structure of amorphous state polymers. Strictly speaking, one should understand that by structure we mean distribution of body elements in space [151]. It is evident that free volume microvoids cannot be structural elements and at best only mirror the structural state of the studied object. Taking the introduction of some structural elements (relaxators, see for example, [148]) into consideration has practically no influence on the structural representation of free volume. 

Physically, an elastomer is more liquid-like than solid-like, and therefore the mobility theory for a liquid seems the appropriate choice for ion-conducting polymer systems. The macroscopic viscosity of an elastomer cannot, however, be used in the Stokes equation. This is because the macroscopic viscosity is greatly enhanced by chain entanglement, which does not directly resist ion motion. However, the mobility in a liquid or elastomer can be derived according to the free-volume theory of Cohen and Turnbull, outlined below. Because it involves the glass transition temperature (Tg) as the major parameter, it is particularly applicable to amorphous polymers for which Jg is easily measured. 

The Tg or glass transition temperature is an important and measurable parameter giving information about the retrogradation behavior. Besides that, the Tg is also of great importance for the mechanical properties of the material. Beneath the glass transition temperature the material exists in an amorphous, frozen liquid structure with stiff and brittle behavior. Below its Tg the intermolecular bonds are not broken, due to the small amounts of room left for Brownian movement, as is stated in Eyring s free volume theory [6]. The specific volume increases relatively slowly with increasing temperature. When the material is heated up, at temperatures close to Tg the cohesive forces decrease drastically, the polymer expands, and the free volume increases to such an extent that there is... 

The correlation between the ionic migration and the local segmental motion of polymer chains may be subjected to closer examination. According to the free volume theory, the free volume in amorphous polymers that occurs above 7g increases in accordance with the temperature difference T — [23]. Because of the enlarged free volume above T, the polymer chain becomes locally mobile. The mode of the micro-Brownian motion is considered to be cooperative, involving several repeat units of the polymer chains. The dielectric relaxation time for the backbone motion of PPO and... 

Lipatov, Y. S. The Iso-Free-Volume State and Glass Transition in Amorphous Polymers New Development of the Theory. Vol. 26, pp. 63 —104. 

These authors were the first FGSE workers to make extensive use of the concept of free volume 42,44) and its effect on transport in polymer systems. That theory asserts that amorphous materials (liquids, polymers) above their glass transition temperature T contain unoccupied volume randomly distributed and in parcels of sufficient size to permit jumps of small molecules — and of polymer jumping segments — to take place. Since liquids have a fractional free volume fdil typically greater than that, f, of polymers, the diffusion rate both of diluent molecules and (uncrosslinked and unentangled) polymer molecules should increase with increasing diluent volume fraction vdi,. The Fujita-Doolittle expression 43) describes this effect quantitatively for the diluent diffusion ... 

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