Gibbs—DiMarzio theory

However, these chains must be packed together on the lattice. As each chain becomes more insistent on achieving a conformation close to its minimum-energy one, it has more trouble sharing the same space with neighboring chains, which are equally insistent on 

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The lattice cluster theory (LCT) for glass formation in polymers focuses on the evaluation of the system s configurational entropy Sc T). Following Gibbs-DiMarzio theory [47, 60], Sc is defined in terms of the logarithm of the microcanonical ensemble (fixed N, V, and U) density of states 0( 7),... 

We have replaced the idea of second order phase transition postulated by the Gibbs-DiMarzio theory, yet retained its successful features. Tr is treated as the thermodynamic anomaly at which the most stable hole configuration is reached under the close packing of holes and flex bonds. 

Thus, the Miller theory has trivial thermodynamics. Starting from the Gibbs-DiMarzio theory, one arrives at the Miller theory by setting Ae = U and z — 2 = 2 in Eq. (4-8) and by assuming that the conformation of one chain imposes no thermodynamic constraints which limit the possible conformations of other chains. 

In addition, since it is a lattice model, the bond fluctuation model can be used to assess the validity of the Gibbs-DiMarzio theory described in Section 4.4.1. Baschnagel et al. (1997) show that the curve of entropy versus inverse temperatures (Sc versus l/T) does not cross zero when the temperature is lowered, but instead levels out at large 1/7, in disagreement with the Gibbs-DiMarzio theory but qualitatively similar to the prediction of Miller s theory in Fig. 4-13. 

Fig. 1. T,vs. 100/Pn for sodium phosphate polymers. -OH terminated o-O Na terminated ------,.Gibbs-DiMarzio theory with Z = 8 and Z = 4, res-...

Fig. 6. The hole formation energy 2ajk), the barrier to internal rotation (/de/fi) and the glass transition temperature vs. compoafdon for NaPOjijCa(PO, counterion copolymers (2 /ft) and (Aelk) calculated from the Gibbs-DiMarzio theory at 10% intervals. T, points experimental...

Kg. 7. The hole formation energy vs. tanier to internal rotation 1 both calculated from Gibbs-DiMarzio theory. Organic Polymers O polyphosphates... 

Fig. 8. (Ae + 2tx)l2k vs. T values calculated from Gibbs-DiMarzio theory ftjr organic polymers, o pol3 hosphates...

We can demonstrate the utility of the Gibbs-DiMarzio theory by applying it to the calculation of the Tg of a random copolymer. Let us assume that the flex energy of a random copolymer composed of A and B monomer units is given by... 

Figure 5-8. Molecular weight trends in the glass transition temperature trends in cyclic and linear oligomers of poly(dimethyl siloxane), and the comparison of these data with the prediction of the Gibbs-DiMarzio theory. [Reprinted with permission from E. A. DiMarizio and C. M. Guttman, Macromolecules, 20, 1403 (1987). Copyright 1987, American Chemical Society]...

An increase in side group bulkiness generally serves to raise Tg. Ring-substituted polystyrenes show this effect and, in the case of polyacenaphthalene, the motions of the chain are so severely restricted that the Tg occurs at 264 °C. Chain microstructure is clearly important in determining chain mobility. We have already discussed the case of random copolymers on the basis of the Gibbs-DiMarzio theory and have developed equation (5-19), which shows that the Tg of a random copolymer is intermediate between the Tg s of the corresponding homopolymers. Equation (5-19) is similar in form to the empirical expression of Wood7... 

Vinyl polymers of the type CH2CHX and vinylidene polymers of the type CH2CXY are capable of existing in different stereoregular forms. These are generally referred to as isotactic, syndiotactic, and atactic forms it is here assumed that the reader is familiar with this nomenclature and the specific structures associated with it. It has been shown that steric isomerism has little effect on the Tg of vinyl polymers but profoundly affects the Tg of vinylidene polymers.12 In the latter case, the isotactic form invariably is associated with the lowest Tg values. This result has also been rationalized on the basis of the Gibbs-DiMarzio theory, but again the details are beyond the scope of this book. 

A second important intermolecular structural feature is crosslinking, which often has a strong influence on Tg. Both the ffee-volume and the Gibbs-DiMarzio theories give a result of the form ... 

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