Phantom chain freely-jointed

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

It should be pointed out that these results also hold for the phantom freely jointed chain with fixed-length segments provided we neglect higher even-order terms in the exponent of (2.1.9) or, equivalently, higher odd-order terms in (2.1.15). 

The phantom (unperturbed) polymer chain can be represented by a hypothetical chain with n = n/C freely jointed segments each of length / = Cl If n and / are replaced by n and / in the equation for the freely jointed chains, eq. (2.95) is obtained. 

Calculation of the End-to-End Distance. The characteristic dimensions of a chain with perfectly flexible joints can be described by means of the random walk model. In this case, the N-step trajectory of a random walker—who can freely return to a site previously visited—is assimilated to a chain of length NL the only element taken into consideration is thus the bond length (L). The valence (0) and rotation (macromolecular chain is to calculate the end-to-end distance (r) connecting its two chain ends by vector analysis. 

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