A polymer segment interacts with a neighbouring segment through a hard core plus an…

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 polymer segment interacts with a neighbouring segment through a hard core plus an attractive square well: the pair potential is infinite for centre-to-centre separation r<σr < \sigma, equal to −u0-u_0 for σ≤r<λwσ\sigma \le r < \lambda_{\rm w}\sigma, and zero for r≥λwσr \ge \lambda_{\rm w}\sigma, with σ=0.50 nm\sigma = 0.50\,\mathrm{nm}, λw=1.5\lambda_{\rm w} = 1.5 and well depth u0=0.50 kBTu_0 = 0.50\,k_BT. ((a)) Using the pair integral for the second virial coefficient met for colloidal particles, B2=2π∫0∞[1−e−U(r)/kBT]r2 drB_2 = 2\pi\int_0^\infty \left[1-e^{-U(r)/k_BT}\right]r^2\,dr, find B2B_2 for this segment pair. ((b)) Hence find the excluded volume parameter v=2B2v = 2B_2 and state whether this segment pair behaves, on balance, as a good-solvent (net exclusion) or poor-solvent (net attraction) pair.

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