Chain with rotation states randomly

Figure 4. Calculated s-weighted intensity function si(s) for chains with rotation states randomly distributed, the probability of trans being pt...

This is the temperature below which an amorphous rubbery polymer becomes brittle. These changes are completely reversible and depend on the molecular motion within the polymer chain. In the rubbery state of polymer melt, the chains are in rapid rotational motion (many rotations per second), but as the temperature is lowered, this movement is slowed down until it eventually stops and the polymer behaves like a frozen liquid with a completely random structure. Although the value of Tg is useful when considering polymers, any glass-forming liquid will have a similar transition, e.g. TgS for quartz 1200 °C, B2O3 250 °C, sulphur 75 °C, polyphosphoric acid — 10 °C, glycerol — 90 °C, toluene — 170°C. 

But how shall the populations and distributions of these rotation states be described The simplest model for a random chain is to ascribe to each rotation state, t, a probability, p, such that for a bond with m states... 

We have modified the conformational description of Abe et al. in curve b of Fig. 10 to allow the three rotation states for the k+2 bond to be equally weighted. While there is some improvement, the additional detail not observed in the experimental si (s) curve remains. Curves a and b are derived from a model in which the bond rotation angles are fixed at particular values. The si (s) curve marked c in Fig. 10 relates to a model in which thermal fluctuations with a standard deviation of 10 have been introduced. Their effect is minimal as observed in the study of polyethylene. If we increase the size of this "thermal" fluctuation for the k+2 bond to only 30 we start to obtain a reasonable match with the experimental curve. Curve e, marked "delocalised in Fig. 10, is obtained for a random chain in which the conformational structure is as described by Abe and Floty except that the k+2 bond is allowed to take any value between. r-lOO and -100 . This provides a most satisfactory match to the experimentally obtained si (j) curve. We note here that, of... 

The structural analysis described above and in more detail elsewhere, shows the x-ray scattering functions to be sensitive to intrachain correlations. In fact, a more "random chain model (with a delocalized rotation state for one bond) than the "random coil" chain model is required to give a satisfactory match between the experimental and model si (s) functions. A model in which the interchain correlations are minimal with no orientational correlations provides a scattering function which is in good agreement with the observed scattering. Thus there seems to be no evidence to require more local order than inherent in a dense molecular system. This is perhaps not suprising. The polyisoprene molecule has a compact cross section, almost cylindrical in nature and corresponds to the "typical molecule drawn in schematic views of the noncrystalline state. 

The preceding considerations are essentially based on the model of random-matrix ensembles proposed by Dyson and others in the 1960s. Recent works, in particular by Casati and co-workers [89], have focused on band random matrices. Such matrices naturally arise in quantum systems with subspaces coupled only to next-neighboring subspaces such as for electronic states in a chain of atoms or in the kicked rotator. In such systems, localized states are observed that present a level statistics interme-... 

Macromolecular conformations describe the positions of the atoms that occur due to rotation about the single bonds in the main chain.2 Polymer chains in solution, melt, or amorphous state exist in what is termed a random coil. The chains may take up a number of different conformations, varying with time. Figure 15.4 shows one possible conformation for a single polymer chain. In order to describe the chain, polymer scientists utilize the root mean square end-to-end distance ((r2)m), which is the average over many conformations. This end-to-end distance is a function of the bond lengths, the number of bonds, and a characteristic ratio, C, for the specific polymer. 

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