Simple Statistical Descriptions of Long-chain Molecules

Simple Statistical Descriptions of Long-chain Molecules 

The magnitude of R in an unconstrained chain is uncorrelated with its direction, so that its value averaged over a large number of arbitrarily chosen conformations must be zero. Because thermal motion causes a real chain to change its local conformation every few picoseconds, for more familiar time scales it would therefore be perceived as a fuzzy ball with R) = 0 and a time-averaged mean square end-to-end distance given by Eq. (2) (r and im are uncorrdated so that r tm) must vanish for n m). 

In the limit of large N, the freely jointed chain model implies the probability density P(R, N) for a given R to be a Gaussian distribution [Eq. (3)], whence the frequent use of the term Gaussian chain to refer to idealized flexible linear macromolecules [4]. 

This result is a simplification in that it does not take into account long-range interactions and solvent effects, for example. It nevertheless provides a working description of conformations in solid and molten polymers, and hence forms the basis for a wide range of theoretical models in polymer physics and mechanics, some of which will be discussed in what follows. 

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