Viscosity glass transition theory

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (see Table 1, WW) provided a rather good description of dielectric relaxation experiments in polymer melts, but they are not of considerable help in understanding molecular phenomena since they are not associated with a molecular model. In the same way, the glass transition theories account well for macroscopic properties such as viscosity, but they are based on general thermodynamic concepts as the free volume or the configurational entropy and they completely ignore the nature of molecular motions. 

Liquids lower the glass transition temperature, and according to the WLF theory, the viscosity and relaxation times are decreased. 

As discussed by Kirkpatrick [10], this slow mode is important in the theories that include mode coupling effects. Such theories have been used to quantitatively understand the anomalous long-time tails of the stress-stress correlation function and the shear-dependent viscosity [3, 30, 34], observed in computer simulations. As mentioned earlier, a theory of glass transition has also been developed based on the softening of the heat mode. 

It can be instructive to compare the two rather different mode coupling theory expressions for the viscosity one valid near the critical point and other near the liquid-glass transition point. 

In the limit of very large viscosity, such as the one observed near the glass transition temperature, it is expected that rate of isomerization will ultimately go to zero. It is shown here that in this limit the barrier crossing dynamics itself becomes irrelevant and the Grote-Hynes theory continues to give a rate close to the transition theory result. However, there is no paradox or difficulty here. The existing theories already predict an interpolation scheme that can explain the crossover to inverse viscosity dependence of the rate... 

Free Volume Theory. Free volume theory suggests that the glass transition temperature is observed for polymers when their viscosity approaches that of their liquid state. Following a derivation based on the Doolittle expression for polymer viscosity (r ) as a function of free volume (Eisenberg, 1984)... 

The free volume theory of glass transition is based on Doolittle s empirical assumption (29), which states that the viscosity, q, at T > Tg is related to the free volume fraction by the equation... 

Another attempt to understand glass transition phenomenon from the theory of viscosity was made by Eyring and co-workers (1962, 1970). In their significant structures approach, the total partition function of the liquid was expressed as a product of solid-like and gas-like contributions. Then the viscosity is obtained from the activated complex theory after ignoring the contribution from gas-like part to viscosity. The final working expression for the viscosity of a system of rigid spheres was obtained as. 

Figure 3.15 Test of the mode coupling theory power law predictions for viscosity temperature dependence for a variety of molecular and ionic liquids. Tg is the glass transition temperature determined by thermal analysis at 10 K/min scanning (After Angell, 1998).

For other systems, the relations tend to be different and variable, and there is currently no consensus about the theory. The most important point is that the decrease in viscosity with increasing temperature tends to be weaker, and in some cases much weaker, than predicted by the WLF equation, especially near the glass transition. To be sure, even then the temperature dependence is strong as compared to that for simple liquids, which tend to follow an Arrhenius type relation [t] oc exp(C/T)]. For complex mixtures, prediction of the temperature-viscosity relation from theory is currently impossible. 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

Fig. 3.53 Viscosity data from Fig. 3.51 plotted against the volume (normalized by the ambient-pressure volume). The curves are fits obtained by the free-volume theory (eqn 3.9.5), the resulting parameters being for methanol, B = 2.22 0.05 and V = 33.8 0.1 A for 1,2-propanediol, S = 6.7 0.3 and 14 = 71.9 0.5 A. Extrapolation of these fits to T = 10 Pa s gives the indicated glass-transition pressures, P. The strong curvature of the methanol data indicates the more fragile behavior of this fluid, which is reflected in the fragility parameters (calculated from equation 5 given by Cook et a .)F For methanol mp = 289 6 and for 1,2-propanediol mp = 123 5.

The liquid-glass transition has been intensively studied for many years. Despite the many papers on the subject, both experimental and theoretical, there is still no clear understanding of this transition. However a few relatively simple phenomenological theories have been developed to explain an extensive body of observations, especially those of viscosity, heat capacity, and volume. The focus of one set of these theories is on the temperature dependence of the diffusion in dense liquids. 

---