Freely rotating chain
This chain is identical with the freely jointed chain except that all of the bond angles are fixed.
The torsion angles can assume all values with equal probability.
If all of the bond angles are fixed at the same value, theta, and this value is less than 180 deg, the characteristic ratio is
Cn = (1 - cos theta)(1 + cos theta)-1 + finite correction term
The correction term, which is
2 cos theta [1 - (- cos theta)n] / [n(1 + cos theta)2]
approaches zero for long chains, due to the appearance of n in the denominator.
- Since most real bond angles are larger than 90 deg, the restriction on the bond angle in real chains will tend to increase the mean square dimensions
- The tetrahedral angle has cos theta = - 1/3. Therefore fixing all of the bonds at the tetrahedral angle produces C = 2.
If theta < 180 deg, this model predicts
<s2>0 = <r2>0 / 6
in the limit where n becomes very large :wq. (If theta = 180 deg, the chain is a rigid rod, with s2 = r2 /12.)
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December 9, 1999
Wayne L. Mattice: wlm@polymer.uakron.edu