Equivalent chain
The freely jointed chain is an attractive model because of its simplicity.
It has only one structural parameter, l.
The other parameter, n, is determined by the number of copies of the structural unit in the chain.
The mean square dimensions of a real linear chain of sufficient length can be modeled by a freely jointed chain through an appropriate choice of these two parameters.
The usual procedure is to demand that the two models have
Solution of these two equations yields the following assignments for the equivalent chain:
Lequiv. chain = <r2>0 / rmax
Nequiv. chain = rmax2/ <r2>0
rmax for the real chain is usually associated with the maximum length attainable by thermally accessible adjustment of the torsion angles, maintaining the bond lengths and bond angles at their preferred values.
Illustrative Example: Polyethylene
Polyethylene has a bond angle of 112o, a bond length of 0.153 nm, and a characteristic ratio of 6.7.
The value of rmax can be associated with the length of the fully extended planar zig-zag, in which all torsion angles are in t states.
- The value of Lequiv. chain is found to be 1.23 nm, which is nearly an order of magnitude longer than the bond length.
(The step length in an equivalent chain is often much longer than the length of a covalent bond.)
- The value of Nequiv. chain is proportional to the molecular weight of the polyethylene chain.
It is 7,400 if M is 1,000,000.
N in an equivalent chain is often much smaller than the number of covalent bonds in the real chain that it represents.
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July 1, 1999
Wayne L. Mattice: wlm@polymer.uakron.edu