PS674

Characteristic Ratio

The dimensionless characteristic ratio is defined as the ratio of the mean square unperturbed end-to-end distance for a real chain (or a model of a chain) to the value expected for a freely jointed chain with the same number of bonds, of the same mean square length.

C_n = <r²>₀ / nl²

By definition, the value of C_n is 1 for a freely jointed chain.
Nearly all real chains have C_n > 1.
The value of C_n shows how the real local structure and real short-range intramolecular interactions affect the mean square dimensions of the chain.
For any chain with any flexibility whatsoever, C_n approaches a limiting value as n becomes large. This limit is conveniently determined by linear extrapolation of C_n vs. 1/n to zero. C without a subscript implies the value at the asymptotic limit at large n.
For chains with stiff segments, l may be interpreted as a virtual bond that spans the rigid segments. For example, the characteristic ratio of polypeptides with planar trans amide units in the backbone is defined as
C_n = <r²>₀ / n_pl_p²
where l_p is the length (0.38 nm) of the virtual bond that extends through the peptide unit, from one C^alpha atom to the next. This single virtual bond replaces three covalent bond, C^alpha-C, C-N, and N-C^alpha. n_p is the number of virtual bonds (number of amide units).
Several examples are given in a table.

A closely related dimensionless ratio, denoted with a subscript s, can be defined using the mean square unperturbed radius of gyration instead of <r²>₀.

C_s,n = <s²>₀ / nl²

For long flexible chains,

C_n = 6 C_s,n

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July 1, 1999

Wayne L. Mattice: wlm@polymer.uakron.edu