Theories of the Glass Transition

The straight line for the a relaxation process, as drawn, corresponds to an apparent activation energy of 84 kcal/mol (69). The relaxation possesses a corresponding apparent energy of activation of 35 kcal/mol. Section S.6.2.3 discusses methods of calculating these values. As a first approximation for many polymers, = 0.75 (70) at low frequencies. Bershtein et al. (71) point 

While each of these second-order transitions has a frequency dependence, the corresponding first-order melting transition for semicrystalline polymers does not. 

The basic experimental behavior of polymers near their glass transition temperatures was explored in the preceding phenomenological description. In Section 8.5, Tg was shown to decrease steadily as the time allotted to the experiment was increased. One may raise the not so hypothetical question, 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. 

The following paragraphs describe three main groups of theories of the glass transition (1, 72) free-volume theory, kinetic theory, and thermodynamic theory. Although these three theories may at first appear to be as different as the proverbial three blind men s description of an elephant, they really examine three aspects of the same phenomenon and can be successfully unified, if only in a qualitative way. 

Although Rgure 8.21 suggests small molecules, a similar model can be constructed for the motion of polymer chains, the main difference being that more than one hole may be required to be in the same locality, as cooperative motions are required (see reptation theory. Section 5.4).Thus, for a polymeric segment to move from its present position to an adjacent site, a critical void volume must first exist before the segment can jump. 

Free-volume theories of the glass transition assume that, if conformational changes of the backbone are to take place, there must be space available for molecular segments to move into. The total amount of free space per unit volume of the polymer is called the fractional free volume Vf. As the temperature is lowered from a temperature well above Jg, the volume of the polymer falls because the molecules are able to rearrange locally to reduce the free volume. When the temperature approaches Tg the molecular motions become so slow (see e.g. fig. 5.27) that the molecules cannot rearrange within the time-scale of the experiment and the volume of the material then contracts like that of a solid, with a coefficient of expansion that is generally about half that observed above Tg. 

If Fg is the fractional free volume at Tg, then ideally, above Tg, 

Assuming that the constant Q in the VFT equation (7.31) is 2.1 x 10 K and that this equation correctly describes the behaviour of a polymer with a glass-transition temperature Tg = + 52 °C and a value of r = 1 h at T,  

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In the next sections we will focus on analyzing the dynamics of supercooled liquids in more detail and discuss our findings in terms of the modecoupling theory of the glass transition, which is a liquid state theory that predicts the dynamics from the structural properties of the liquid. 

The segmental mobility can be related to the T -Tg difference using a theory of the glass transition. For example, if the reac-... 

G. Biroli and J. P. Bouchaud, Diverging length scale and upper critical dimension in the mode coupling theory of the glass transition. Europhys. Lett. 67, 21 (2004). 

As early as 1950 Fox and Fiery19 put forward the idea that the glass temperature corresponds to the iso-free-volume state. This hypothesis was developed by other authors20 21 and has won wide acceptance. It has been used with minor modifications as the basis for a number of sophisticated theories of the glass-transition... 

The Hole Theory of the Glass Transition and its Relation to Calorimetric Studies.256... 

A theoretical derivation based on the Gibbs-Di Marzio (1958) theory of the glass transition leads to the conclusion that for the series of poly(alkyl methacrylates)... 

Before we go into some details of MCT, we briefly mention that there exists another microscopic theory of the glass transition phenomenon, the replica theory [120,121], which is inspired by spin glass theory [122] and which lends some... 

The replica theory is another microscopic theory of the glass transition. Inspired by the spin glass theory,230 it lends some justification to the configurational entropy theory.216 However, details of the evolution of the dynamic susceptibility are less worked out. 

This equation can be derived from an energy theory of the glass transition. 

A rigorous theory of the glass transition of molecular liquids is not yet available. Even the best theories are either partly phenomenological or highly approximate. However, one hydrodynamic theory, loosely derived from self-consistent mode-coupling arguments... 

Experimental data are presented to show that the JG relaxation mimics the structural relaxation in its volume-pressure and entropy-temperature dependences, as well as changes in physical aging. These features indicate that the dependences of molecular mobility on volume-pressure and entropy-temperature have entered into the faster JG relaxation long before structural relaxation, suggesting that the JG relaxation must be considered in any complete theory of the glass transition. 

Ngai, K.L., Casalini, R., Capaccioli, S., Paluch, M., and Roland, C.M. (2005) Do theories of the glass transition, in which the structural relaxation time does not define the dispersion of the stmctural relaxation, need revision , J. Phys. Chem. B 109, 17356... 

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. 

Equation 6.16 is consistent in general form with previous treatments [24,119,123-126] of the effects of crosslinks on Tg, and with expectations based on statistical thermodynamic theories of the glass transition [13-17]. The new concept here is the use of Nrot as a normalization factor, to multiply the value of n. The product n-Nrot is the average number of rotational... 

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

The entropy theory of the glass transition was developed by Gibbs and DiMarzio and by Adams and Gibbs to describe polymeric systems. By mixing the polymer links with holes or missing sites on a lattice to account for thermal expansion as in a lattice gas model, they could determine the entropy of mixing and the configurational entropy of the polymer. They found a second-order transition at a temperature They then pointed... 

The theories of the glass transition should account for observed effects below and above the main glass transition. 

There are rival theories of the glass transition the Gibbs Dimarzio theory assumes that the configurational entropy of the chains approaches zero at Tg. Other researchers prefer a mode coupling theory (MCT), based on the dynamics of density fluctuations. However, it is difficult to extract a simple physical meaning from the complex equations that describe correlations between density fluctuations. Neither theory, at its current state of development, is particularly useful in understanding the properties of glassy polymers. 

For these polymers the following equation for Tg (absolute) can be deduced from the free-volume theory of the glass transition (see section 7.5.3) ... 

Wool, R. P, Twinkling fractal theory of the glass transition, /. Polym. Sci. B, 46, 2765-2778... 

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