Freely jointed chain contour length

Hypothetical freely jointed chain with the same mean-square end-to-end distance and contour length as an actual macromolecular chain in a theta state. 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

An effective freely jointed chain of contour length L would then contain Nx = L/bx rigid finks, each of length bx, and Nx is the number of Kuhn steps in the chain (Semenov and Khokhlov 1988). Hence,... 

Problem 2.8 A real polymer chain consisting of n bonds each of length I may be usefully represented by an equivalent freely jointed chain of N links each of length b such that it will have the same end-to-end distance and the same contour length. Obtain N and b in terms of the characteristic ratio Coo of ths polymer chain. 

The simplest mathematical model of a polymer chain is the freely jointed chain. It has n links, each of length I, joined in a linear sequence with no restrictions on the angles between successive bonds. The length of the chain along its backbone is known as the contour length and is given by nl. However, for linear flexible chains, it is more usual, and more realistic, to consider the dimensions of the molecular coil in terms of the distance between the two chain ends, that is the end-to-end distance r [Fig. A2.1(a)]. 

As follows from eqn [4], the characteristic dimensions of the chain scale proportionally to the square root of the number of monomer units N irrespective of the correlations in orientation of consecutive segments. This enables an equivalent freely jointed chain of the same contour length L to be introduced but with renormalized number of the effeaive statistical segments of length I tl. 

A real chain whose conformation is governed only by local (effective) interactions along its contour can be mapped onto a freely-jointed chain by requiring that Eq. (8) is satisfied, and also that the contour length of the freely-jointed chain, Njcbjc, matches that of the real chain at full extension. 

We briefly describe three models, namely, the Gaussian Chain model. Freely Jointed Chain (FJC) model, and Worm Like Chain (WLC) model, which have been extensively used to describe the force-extension curves of biopolymers. The advantages of these models come from their simplicity and allowing one to derive analytical expressions in a simple form. In these models a polymer chain consists of N beads (monomers) of contour length L. A point in d-dimensional space represents each monomer and the distance between two consecutive monomers is R, i - R, (see O Fig. 8-1). The energy with force / (along x-direction) in Gaussian model is expressed as (Doi and Edwards 1986)... 

This rather simple result is very important and it shows to what extent a polymer chain will be coiled in a dilute solution. For example a freely-jointed chain of 10 000 segments of 2 A long will have a fully extended contour length of 20 000 A whereas the value of (r for such a chain will only be 200 A. In practice there are several factors which cause 2)1/2 jQ jjg somewhat greater than the value given for a freely-jointed chain and these are discussed below. 

Because (r ) increases with each additional restriction but remains proportional to (r ) for a freely jointed chain, we can consider a polymer molecule a freely jointed chain having n links, where is less than the number of bonds, but the length of each fink / is greater than the bond length, so that p-) is again nll and the contour length is n t. 

X 10 N and the value obtained by the simplest form, 1.45 X 10 " N m n = 918 and a = 0.31 nm for Equation 21.1). These comparisons imphed that the measurements were consistent with the theoretical predictions. The deviation between the rupture length of 260.9 nm and the fitted-contour length indicated that the polymer chain was not fully stretched at the rupture event. The reason for this was that the rupture event was a stochastic process and was dependent on many factors such as pulling speed, bond strength, and temperature. The vahdity of the freely jointed (FJC) model (dashed fine) was also checked ... 

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