Dimensions of Random-Walk Chains

The step length does not have to be fixed to b either. Suppose the step length has some distribution, but each step follows the same distribution to yield (Ar3) = b. A typical trajectory of this type of random walk is seen in a bead-spring model. With a randomness in the orientation of Ar, and a common distribution for all of Ar Arj wiU satisfy Eq. 1.19. If Ar, and Ar, (i A j) are not correlated, the final position r yfollows the distribntion given by Eq. 1.20. 

Eignre 1.19 compares a freely jointed chain with a fixed bond length b (also called a segment length) and a bead-spring model with (Ar ) = b, both in two dimensions. Examples of a 100-step random walk are shown. The bead-spring model can have greater density flnctnations for the same Nb.  

1 End-to-End Distance and Radius of Gyration Here, we learn how to assess the dimension or the size of a polymer molecnle. We consider a linear chain consisting of N bonds of length b (Fig. 1.20). The positions of the joints are denoted by r, (i = 0, The two ends of the ith bond are at r, i and r,. It is conven- 

R is different for each configuration of the chain. Although the chain ends are not necessarily faced outward and therefore R does not always span the largest dimension of the chain, its average length is a good measure for the overall chain dimension. The root-mean-square end-to-end distance Rp (or simply end-to-end distance) of the chain is the root mean square of R  

We can regard the whole chain as roughly being contained in a sphere of diameter Rp. 

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