Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2021 / Paper 2 / Section B / 22 / iv / Solution

At the theta condition, effective excluded volume approximately cancel and a freely jointed ideal chain has $⟨R^2⟩=N{b}^2$. With the stipulated segment length $b=2a$,In a good solvent repulsive excluded volume swells the coil, giving $R\sim b{N}^{\nu }$ with $\nu \simeq 0.588$ in three dimensions, or the Flory approximation $\nu =3/5$. In a poor solvent effective segment attraction favours a compact globule, whose linear size scales as $b{N}^{1/3}$ at roughly fixed internal density. The latter are scaling laws with model-dependent prefactors. A real chemical monomer need not be a statistical Kuhn length; using $2a$ as the independent segment length is part of this question's idealized model.