Polymer Physics

Chapter 3

Conformation Statistics and Entropic Elasticity

3.1 Gaussian Distribution of End-to-End Distances

of Polymer Coils

If the internal rotation surrounding each backbone bond contains three possible
conformation states, the internal rotation of a long chain will generate an astronom-
ical amount of possible conformation states. In such a case, we could not count
them one-by-one, and thus have to make conformation statistics on the basis of a
simplified ideal chain model.
A real polymer chain can be modeled by the freely jointed chain that is consisted
of Kuhn segments. A freely jointed chain is analogous to the trajectories of random
walks. A random walk of a man who gets lost in the forest often turns back to the
starting point. Therefore for a polymer coil, one chain end exhibits a rather
stochastic location near another chain end. In mathematics, the stochastic
distributions of random events follow the central-limit theorem, i.e., the
distributions of large-enough amount of independent random events exhibit a
characteristics ofGaussian functionaround their mean value, as demonstrated in
Fig.3.1. The one-dimensional distribution of the one-end locations for a polymer
coil follows the Gaussian function around another chain end, as given by

WðxÞ¼ð

b^2

p

Þ^12 =eb

(^2) x 2
(3.1)
where
b^2 

3

2 nb^2

(3.2)

Herexis the end-to-end distance,nis the total bond number, andbis the bond
length. Since the fractions of three dimensions are independent to each other, we
obtain

W. Hu,Polymer Physics, DOI 10.1007/978-3-7091-0670-9_3,
#Springer-Verlag Wien 2013

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