## 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 Kuhn1 defined a polymer chain as having NK links of length lK 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

Rrms = NK1/2 lK

Rg = (NK / 6)1/2 lK

Thus,

Rrms2 = 6 Rg2

The radius of gyration, Rg, 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 Rmax = NK lK. The ratio Rrms2/ Rmax 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 lK giving NK 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, Rrms2, increases by a factor of two,

Rrms2 / R02 = (1 - cos θ) / (1 + cos θ) ≈ 2.0

where R02 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, Nν, in the chain. Taking both effects into account, a characteristic ratio C(Nν → ∞) 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 lν:

C = R02 / (Nν lν2)

where R0 is the mean-square end-to-end distance of an unperturbed coiled polymer chain, Nν is the number of statistical skeletal units in the chain, and lν is the root-mean-square length of such a unit. This length should not be confused with the Kuhn lenght. lν 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 = (R02 / R0,r2)1/2

where R0,r2 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.

### Characteristic Ratio of Some Polymers4

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
a100% Isotactic Poly(methyl methacrylate)

##### References & Notes
1. W. Kuhn, Kolloid Zeitschrift 68, 2 (1934) and W. Kuhn, H. Kuhn Helv. Chim. Acta 26, 1394 (1934)

2. Gert Strobl, The Physics of Polymers, 3rd Edition, Heidelberg 2007

3. T. Hesse, Polymere an Phasengrenzflaechen, First Edition, Bremen 2004

4. S. Wu. Polymer International, Vol. 29, 3 (1992)

5. 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.

• ## Summary

### Free Jointed Chain

has the same spacial properties as a random flight in three-dimensions.

• The Kuhn length and the characteristic ratio are both measures of the flexibility of a polymer chain.

The lower these values are, the more ductile a polymer.

• Flexible chains have characteristic ratios close to unity, while
semiflexible and rigid polymers have very large values.

• The smaller the characteristic ratio, the more like a freely-jointed chain is the real chain.

• The ultimate limit of the characteristic ratio is two, C = 2.

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