Theoretical Treatment of Glass Transition

According to the hole theory of liquids (Eyring, 1936), molecular motion in liquids depends on the presence of holes or voids, i.e., places where there are vacancies, as illustrated in Fig. 2.22. For real materials, however. Fig. 2.22 has to be visualized in three dimensions. A similar model can also be constructed for the motion of polymer chains, the main difference being that more than one hole will now be required to be in the same locality for the movement of polymer chain segments. On this basis, the observed specific volume of a sample, v, can be described as a sum of the volume actually occupied by the polymer molecules, vq, and the free volume (empty spaces), uy, in the system [see Fig. 2.23(a)], i.e., 

The glass transition that takes place at a higher temperature can be visualized as the onset of coordinated segmental motion made possible by an increase of the free space in the polymer matrix to a size sufficient to allow this type of motion to occur. Conversely, if the temperature is decreased, the free volume will contract and eventually reach a critical value when the available space becomes insufficient -for any large scale segmental motion. The temperature at which this critical value is reached is the glass transition temperature (Tg). As the temperature decreases further below Tg, the free volume, i.e., uy in Eq. (2.13), will remain. essentially 

At temperatures above the glass transition, both the intemuclear separation between segments of neighboring chains and the number and size of holes adjust themselves continuously with changing temperature (provided sufficient time is allowed for segmental diffusion). The coefficient of expansion above Tg is therefore higher than that below Tg [see Fig. 2.23(b)]. 

Simha and Boyer (1962) postulated that the free volume at T = Tg should be defined as 

Substitution of v from Eq. (2.15) into Eq. (2.14) sXT = Tg leads to the following expression for free-volume fraction (Simha and Boyer, 1962)  

This imphes that the free volume fraetion at the glass transition temperature is the same for all polymers and constitutes 11.3% of the total volume in the glassy state. (It is interesting to note that many simple organic compounds show a 10% volume increase on melting.) This is the largest of the theoretical valnes of free volume derived, while other early estimates yielded a value of about 2%. 

Experiments to measure Tg often show that the measured value of Tg is dependent on the time allotted to the experiment and that Tg decreases as the time allotted is increased. One may therefore ask Is there an end to the decrease in Tg as the experiment is slowed How can the transition be explained on a molecular level These are the questions to which the theories of the glass transition are addressed. 

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In the following sections, the physieal meanings of hg and h are shown in the theoretical treatments of the glass transition, and for several polymers, 8p is predieted using these thermodynamic quantities. 

In chapter 1, a full theoretical treatment of the behaviour of the MTDSC signals over the glass transition region [30] has been presented. 

After observing quite a few anomalous properties of optical transitions in glasses and attributing them to the dynamics of TLS [14], the tunneling model was adopted by Reinecke [15] to explain the low-temperature line widths of optical transitions in amorphous solids using the concept of spectral diffusion. This concept had originally been developed for the description of spin resonance experiments [16] and had already been applied to the theoretical treatment of the above mentioned ultrasonic properties of glasses [17]. Soon after this step, the possibility of a connection between thermal and optical properties of amorphous solids was supported by the observation of time dependence of spectral hole widths [18]. 

Recently, alternative theoretical expressions have been developed by using classical thermodynamic treatments to describe the compositional dependence of the glass transition temperature in miscible blends and further extended also to the epoxywater systems 2S,27). The studies carried out on DGEBA epoxy resins of relatively low glass transition have shown that the plasticization induced by water sorption can be described by theoretical predictions given by ... 

Based on the considerations summarized above, it is not surprising to find that most theories of the glass transition [11-29] describe it, either explicitly or implicitly, in terms of key physical ingredients whose values strongly depend on the chain stiffness and/or the cohesive forces. These theoretical treatments invariably treat the observed value of Tg as a kinetic (rate-dependent) manifestation of an underlying thermodynamic phenomenon. However, they differ significantly in their description of the nature of this phenomenon at a fundamental level. 

There are many empirical correlations for the activation energy, as well as some theoretical treatments [19-21]. These treatments usually attempt to relate Ep to quantities such as the size of the penetrant molecule (usually expressed in terms of a penetrant diameter or the square of this diameter, and sometimes corrected for the penetrant shape if it deviates significantly from spherical), the glass transition temperature of the polymer, and whether the polymer is glassy or rubbery at the temperature of interest. The diffusivity and solubility components of the permeability are usually treated separately in such attempts, as described below. 

In the case of the glass transition, more complete experimental data are available, at least with respect to vinyl polymers. As will be discussed below, for certain types of structure a substantial effect of stereoregularity on the glass transition temperature, T, is found, and in addition a theoretical treatment is available which accounts for these effects. However, very little data is available with respect to the effect of stereoregularity on, for example, the configurational entropy S at T, on aC, the size of the heat capacity increment at T, 0 on other key thermodynamic parameters of the glassy state. ... 

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