Phenomenology of the Glass Transition

The most rapid cooling rate is limited by the thermal conductivity of the organic liquid and is on the order of 1 K/s. The slowest rate is limited by the stability of the apparatus and the patience of the experimenter and is often in the range l(h K/s. The Active temperature changes by from 10 to 20 K over this range of cooling rates for organic materials near 300 K. 

Another thermal history consists of thermal equilibration at a temperature above the glass-transition range followed by rapid quench to a temperature within the glass-transition range. Isothermal annealing is then followed until equilibriinn is reached. A relaxation function is defined as  

The shape of the curves is highly nonexponential. One obvious reason is that the rate of relaxation depends on the free volume that is changing during the volume relaxation. However, attempts to explain the observed relaxation functions in terms of a single volume-dependent relaxation time have not been successful. Because equilibrium liquids can be obtained at 

In fact, the rate of relaxation is highest at the start of the volume expansion, decreases to a minimum, and then increases until the end of the experiment. This behavior can be explained in terms of a distribution of relaxation times for volume fluctuations. 

An even more dramatic maximum can be produced by isothermally annealing the sample before reheating. Much lower fictive temperatures can 

The defining property of a structural glass transition is an increase of the structural relaxation time by more than 14 orders in magnitude without the development of any long-range ordered structure.1 Both the static structure and the relaxation behavior of the static structure can be accessed by scattering experiments and they can be calculated from simulations. The collective structure factor of a polymer melt, where one sums over all scattering centers M in the system 

The thermal expansion, however, changes behavior at the glass transition, which is a phenomenon that was first analyzed in detail in a careful study by Kovacs.4 In the polymer melt, the thermal expansion coefficient is almost constant, and it is again so in the glass but with a smaller value. At the glass transition, there is therefore a break in the dependence of density on temperature that is the foremost thermophysical characteristic of the glass transition. 

The decay of the structural correlations measured by the static structure factor can be studied by dynamic scattering techniques. From the simulations, the decay of structural correlations is determined most directly by calculating the coherent intermediate scattering function, which differs from Eq. [1] by a time shift in one of the particle positions as defined in Eq. [2]  

The Fourier transform of this quantity, the dynamic structure factor S(q, ffi), is measured directly by experiment. The structural relaxation time, or a-relaxation time, of a liquid is generally defined as the time required for the intermediate coherent scattering function at the momentum transfer of the amorphous halo to decay to about 30% i.e., S( ah,xa) = 0.3. 

The temperature dependence of the a time scale exhibits a dramatic slowdown of the structural relaxation upon cooling. This temperature dependence 

As mentioned in Chapter 3, glassy relaxation processes are often associated with a fairly broad spectrum of relaxation times. A simple expression that describes this spectrum reasonably well over a wide range of time is the stretched exponential, or Kolrausch-Williams-Watts (KWW) expression (Kolrausch 1847 Williams and Watts 1970 Shlesinger and Montroll 1984)  

The KWW expression is convenient, but not sacrosanct other expressions can be used to approximate the spectrum. The Cole-Davidson spectrum is useful in the frequency domain, in particular for dielectric relaxation  

Equation (4-5) typically applies up to temperatures of Tg + 50 C or so. For higher temperatures, an Arrhenius temperature-dependence often applies for small-molecule liquids, even if they are fragile glass formers. For example. Fig. 4-6 shows a plot of 1/ logio(/oo/fp) versus temperature for propylene carbonate, where fp = Incop is the peak frequency (in 

It has been repeatedly emphasized throughout this book that the glass transition in amorphous polymers is accompanied by profound changes in their viscoelastic response. Thus the stress relaxation modulus commonly decreases 

THEORIES OF THE GLASS TRANSITION 1. Free-Volu me Theory 

In Section D of Chapter 4 the WLF equation was derived on the basis of free-volume concepts. In particular, we may write 

The fractional free volume / reaches a constant value, fg, at Tg and increases linearly above Tg with the coefficient of expansion Of. Following Chapter 4, substitution of equation (4-16) into equation (5-1) with Tg as the reference temperature yields the WLF equation where the constants C, and C2 are given by 

Knowledge of the numerical values of Cx and C2 thus leads to the parameters fg and Of through equation (5-2), if B is taken as unity. The constants C, and C2 were originally taken to be universal for all amorphous polymers with C, = 17.44 and C2 - 51.6. It was later found that C, values were indeed approximately constant for all systems but that C2 varied quite widely. This is illustrated by Table 4-1. Equation (5-2) indicates that this result means that fg is approximately constant at a value of 0.025, but that ay varies from one amorphous polymer to another. It is clear that the WLF equation predicts that Tg represents an iso-ffee-volume state. While this concept is not strictly true it is nevertheless of wide utility, as we shall see in the following discussion. 

Specific volume-temperature dependence for semicrystalline poly- 

It is worthwhile to comment on the temperature dependence of the first derivatives of volume and enthalpy. The first derivative of volume is the thermal expansion coefficient, a. 

---

As polymer liquids are cooled or compressed, the local viscosity increases. If the average local relaxation time of the system exceeds the time allowed for equilibration, the liquid follows a nonequilibrium path. This fact leads to the phenomenon of the glass transihon. The phenomenology of the glass transition is presented in Section 8.4. 

The free-volume theory gives an excellent account of many phenomena near Tg and enables much of the phenomenology of the glass transition to be understood in a qualitative and even semiquantitative way. The free volume plays the role of an order parameter, and by fitting the thermal expansivity jump at Tg, the other thermod3mamic results are approximately described. Absolute estimates of free volume are available from crystal densities and van der Waals radii, so the... 

---