Polymer Physics

ideal-chain modelof a single polymer chain. In order for a simple statistical
estimation, a long-enough ideal chain has been assumed, and the long-range
interactions between the structural units along the chain have been neglected.
Such an ideal polymer chain is often referred as aphantom polymer,oran
unperturbed polymer. In the following, we start with the simplest freely-jointed-
chain model, and then consider the short-range interactions along the chain. The
first short-range interactions are the fixed bond angles along the backbone atoms, as
described by the freely-rotating-chain model. The second short-range interactions
are the hindrances of internal rotation as described by the hindered-rotating-chain
model. In this way, we progressively approach the description to the semi-
flexibility of real polymers.

2.2.1 Freely Jointed Chains

The freely-jointed-chain modelconsiders only the chain connection of monomers
with no restriction on the connection angles. A common method to characterize the
semi-flexibility of polymer chains is to measure the size of a random coil consisting
of a single polymer chain. The end-to-end distance of a polymer chain is the first
quantity to characterize the coil size, which can be calculated by using the vectorR
connecting one end to the other end of the chain. Assuming the lengthbof each
bond vector contributing to the contour length of the main chain, the vector for the
end-to-end distance is the sum ofnbond vectors along the chain,

R¼b 1 þb 2 þþbn¼

Xn

i¼ 1

bi (2.1)

For a large number of polymer chains, their random-oriented end-to-end vectors
cancel each other, and their summation approaches zero. Therefore, we need to use
a scalar to avoid zero result, for example, use the square end-to-end distances. The
sum of square end-to-end distances over a large number of polymer chains
represents the characteristic size of polymer coils.

R^2 ¼

Xn

i¼ 1

bi

Xn

j¼ 1

bj¼nb^2 þ 2

XX

j>i

bibj (2.2)

Since the dot product of two bond vectors relies on the anglegijbetween them,

bibj¼b^2 cosgij (2.3)

The angles between any two bond vectors of the freely-jointed chain are
uniformly distributed between 0 and 2p, leading to a symmetric distribution of
positive and negative cosine values between 1 and1. Thus, in the summation over
a large number of such independent dot products, the positive values cancel the
negative counterparts, and eventually

2.2 Semi-Flexibility of Polymer Chains 15