A polymer chain occupies lattice sites

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

Soft Matter at the Thermal Scale, Ch Polymer solutions and blends

Problem

A polymer chain occupies N=800N = 800 lattice sites, each of volume v0=0.15 nm3v_0 = 0.15\,\mathrm{nm^3}, dissolved in a solvent with Flory-Huggins parameter χ=0.35\chi = 0.35. ((a)) The osmotic pressure virial expansion is Πv0/kBT=ϕ/N+(12−χ)ϕ2+…\Pi v_0/k_BT = \phi/N + (\tfrac12-\chi)\phi^2 + \ldots. Writing ϕ=nNv0\phi = nNv_0 with nn the number density of chains, show that this matches the form Π=nkBT(1+B2n+…)\Pi = nk_BT(1 + B_2 n + \ldots) used for the second virial coefficient of colloidal particles, with B2=(12−χ)N2v0B_2 = (\tfrac12-\chi)N^2v_0. Evaluate B2B_2 in nm3\mathrm{nm^3}, and state whether this corresponds to net exclusion or net attraction between chains. ((b)) Using this value of χ\chi, decide whether the solvent is good or poor for this chain, and estimate the overlap volume fraction ϕ∗≈N1−3ν\phi^* \approx N^{1-3\nu} using the appropriate value of the Flory exponent ν\nu and treating each site as one step of the swollen walk, as an order-of-magnitude estimate. ((c)) Calculate the critical Flory parameter χc\chi_c and critical volume fraction ϕc\phi_c for this same chain dissolved in the same type of solvent, using χc=12(1+N−1/2)2\chi_c = \tfrac12\left(1+N^{-1/2}\right)^2 and ϕc=1/(1+N)\phi_c = 1/(1+\sqrt{N}).

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