A hydrocarbon backbone is modelled as a freely rotating chain: bonds of length meet…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch The size and shape of a polymer chain

Problem

A hydrocarbon backbone is modelled as a freely rotating chain: bonds of length l=0.154 nml = 0.154\,\mathrm{nm} meet at the tetrahedral valence angle, so that successive bond vectors satisfy cos⁡θ=1/3\cos\theta = 1/3, where θ\theta is the angle between consecutive bond directions. Rotation about each bond is free, with no energetic preference between rotational states. The chain has n=4000n = 4000 bonds. ((a)) Calculate the characteristic ratio C∞C_\infty predicted by free rotation alone. ((b)) Calculate ⟨R2⟩1/2\langle R^2\rangle^{1/2} for the chain.

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