Chapter 4
Scaling Analysis of Real-Chain Conformations
4.1 What Is the Scaling Analysis?
We have discussed the ideal-chain model in Sect.2.2by 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 semi-
flexibility of the real polymer chains. The mean-square end-to-end distances of
chains in different models are given below.
Freely jointed chain:<Rf.j^2 >¼nb^2
Freely rotating chain:<Rf.r.^2 >¼nb^2 (1 + cosy)/(1cosy)
Hindered rotating chain:<Rh.r^2 >¼nb^2 (1 + cosy)/(1cosy)(1 +
(1
By using Gaussian statistics, the average coil sizes were also given by
<R^20 >/nb^2 (4.1)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 asthe scaling lawwiththe
scaling exponentn, as given by
Rnn (4.2)For ideal chains,n¼0.5. In this book, we use the symbol “~” to describe the
proportional relationship without further consideration of the consistency in the
units/dimensions.
A simple example can elucidate the scaling law. Suppose that we measured the
areaSof a square with a lateral lengthL, by using a small square with an areapand a
lateral lengthras the ruler, as illustrated in Fig.4.1. The resulted relationshipS¼L^2
is actually inherited from the corresponding relationship in the ruler,p¼r^2.
W. Hu,Polymer Physics, DOI 10.1007/978-3-7091-0670-9_4,
#Springer-Verlag Wien 2013
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