Random-Walk Statistics The Freely Jointed Chain

2 Random-Walk Statistics The Freely Jointed Chain 

Flexible molecules, such as those in Fig. 2-7, permit rotational motions of one bond about another, so that a combinatorially huge number of configurations is accessible (Flory 1969). On length scales of tens or hundreds of such monomers, the details of the distribution of allowed bond angles average out, producing in the melt a configuration distribution equivalent to that of a random walk (see Fig. 2-8). Because of the flexibility of these molecules, even in the densely packed melt state, they remain unoriented, or isotropic, at equilibrium. 

Because of Brownian forces, each molecule in the molten state continually changes its configuration, but for long polymer molecules composed of hundreds or more monomers, the time-averaged mean-square distance R )o separating one end of the molecule from the other obeys the random-walk formula 

For a freely jointed chain. Coo = 1 but for real chains, bond angle restrictions lead to values of Coo in the range 5-10. For polystyrene, for example. Coo 9.6 at 140°C. A tabulation of Coo for various polymers can be found by looking ahead to Table 3-3, as well as to Fetters et al. (1994 see also Flory 1969). 

A quantity related to ( is the radius of gyration Rg, which is defined as the root- 

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