## Freely Jointed Chain and Characteristic Ratio

The idea to model a linear polymer chain as a *freely jointed chain* which occupies space as a random coil dates back to the 1930th when Kuhn^{1} defined a polymer chain as having
*N _{K}* links of length

*l*

*with no restrictions on the angles between successive bonds. The chain and its spacial properties are the same as a random flight in three-dimensions. For example, the root-mean-square distance of the ends is given by*

_{K}*R*_{rms} = *N*_{K}^{1/2 }
*l*_{K}

And the radius of gyration

*R*_{g} = (*N*_{K }/ 6)^{1/2}
*l*_{K}

Thus,

*R*_{rms}^{2} = 6
*R _{g}*

^{2}

The radius of gyration, *R*_{g}, is the average value of the first moment of all segments of the chain with respect to the center of mass of the molecule. The length of a fully extended (rod-like)
Kuhn polymer chain is *R*_{max} = *N*_{K }
*l*_{K}. The ratio
*R _{rms}*

^{2}/

*R*

_{max}is a measure for the stiffness of a polymer chain and is called

*Kuhn length*. For example, a hypothetical freely jointed polyethylene chain has a Kuhn length of approximately 1.54 A.

The concept of a Kuhn chain is quite useful for many model predictions. However, it is an oversimplification of a real polymer chain since we replace the molecule with a hypothetical chain that behaves
like a random-flight, that is, we group *n* repeat units to a statistical segement with an average end-to-end distance of
*l*_{K}
giving *N _{K}*
statistical segments. By doing this, we loose all information of the
spacial arrangement of the repeat units.

### Freely-Rotating Chain

Real polymer chains have
fixed bond angles and the rotation about the backbone is restricted
due to steric hindrance. If we assume a fixed bond angle of *τ* = 109.5° (*θ* = 68°) between consecutive bonds (polyethylene chain), but impose no
other restrictions (no steric interaction) the mean-square distance of
the chain ends, *R*_{rms}^{2}, increases by a factor of two,

*R*_{rms}^{2}
/ *R*_{0}^{2} = (1 - cos
*θ*) / (1 + cos *θ*) ≈ 2.0

where *R*_{0}^{2} is the mean square end-to-end distance of a hypothetical chain with no restrictions
on the bond angles (Gaussian
or freely-jointed chain) and same number and length of bonds.

For real polymer chains, the rotation of bonds around the backbone is restricted due to hindered internal rotation
and due to excluded-volume effects. The excluded-volume effect simply means that two segments cannot occupy the same position in space. This effect increases with the number of
repeat units, *N*_{ν}, in the chain.
Taking both effects into account, a *characteristic ratio* *C*_{∞}(*N*_{ν} → ∞) may be introduced as a measure of the expansion of the actual end-to-end distance of the real polymer chain compared to a hypothetical ideal
chain with (statistical) bond length *l*_{ν}:

*C*_{∞} = *R*_{0}^{2} / (*N*_{ν}
*l*_{ν}^{2})

where *R*_{0} is the mean-square end-to-end distance of an unperturbed coiled polymer chain, *N*_{ν} is the number of statistical
skeletal units in the chain, and *l*_{ν}
is the root-mean-square length of such a unit. This length should not be
confused with the Kuhn lenght. *l*_{ν}
is the average end-to-end distance of a monomer unit, or in other words,
a statistical skeletal unit and in some cases a real skeletal bond length
which is an elementary rotational unit of the polymer. The Kuhn length,
on the other hand, is twice the persistence length.^{5}

Another important ratio is the *Stockmayer-Kurato ratio* *σ*.
It is measure for the stiffness of a chain, that is, for the
rotational isomerism preferences.

*σ* = (*R*_{0}^{2} / *R*_{0,r}^{2})^{1/2}

where *R*_{0,r}^{2} is the root-mean
square ene-to-end distance of the hypothetical chain with same bond
angles but with free rotation around the valence cone. σ is a measure of the effect of setric hindrance on the average chain dimension.

Compound | Experimental C_{∞} |
Predicted C_{∞} |

Polyethylene (PE) | 5.7 | 5.5 |

Polycarbonate (PC) | 2.4 | 2.7 |

Poly(vinyl chloride) (PVC) | 7.6 | 7.6 |

Poly(methyl methacrylate) (PMMA) | 7.9 | 8.1 |

i-Poly(methyl
methacrylate) (iPMMA)^{a} |
10.7 | 10.6 |

Bisphenol A Polysulfone | 2.2 | 2.2 |

Polycaprolactam (Nylon 6) | 6.2 | 6.0 |

Polystyrene (PS) | 10.8 | 10.7 |

Poly(ethylene terephthalate) (PET) | 4.10 | 3.56 |

##### References

- W. Kuhn,
*Kolloid Zeitschrift*68, 2 (1934) and W. Kuhn, H. Kuhn*Helv. Chim. Acta*26, 1394 (1934) - Gert Strobl,
*The Physics of Polymers*, 3^{rd}Edition, Heidelberg 2007 - T. Hesse,
*Polymere an Phasengrenzflaechen*, First Edition, Bremen 2004 - S. Wu.
*Polymer International*, Vol. 29, 3 (1992) *The persistence length can be qualitatively described as the length along the chain backbone that one must travel before the next statistical unit no longer correlates with the end-to-end vector. For example, the persistence length of a freely jointed chain is one segment length and that of a rod like chain is equal to the end-to-end distance*.