What Is the Scaling Analysis

We have discussed the ideal-chain model in Sect. 2.2 by incorporating short-range restrictions into the freely-jointed-chain model first the fixed bond angles, then the hindered internal rotation. In this way, we reached the description of semiflexibility of the real polymer chains. The mean-square end-to-end distances of chains in different models are given below. 

By using Gaussian statistics, the average coU sizes were also given by 

In summary, irrespective of the type of short-range restrictions that has been considered, we always obtain a power-law relationship between the coil size and the chain length. We refer such a power-law relationship as the scaling law with the scaling exponent v, as given by 

For ideal chains, v = 0.5. In this book, we use the symbol to describe the proportional relationship without further consideration of the crmsistency in the units/dimensions. 

A simple example can elucidate the scaling law. Suppose that we measured the area S of a square with a lateral length L, by using a small square with an area p and a lateral length r as the ruler, as illustrated in Fig. 4.1. The resulted relationship S = l is actually inherited from the corresponding relationship in the ruler, p = r.  

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