Past exam of the soft condensed matter course of the University of Cambridge / 2013 / Paper 1 / 2 / c

The coil-globule transition is the contraction of an isolated flexible polymer as solvent quality worsens. A swollen chain in a good solvent has size $R\sim a{N}^{3/5}$ in the Flory approximation. At the theta temperature the effective two-segment excluded volume vanishes and $R\sim a{N}^{1/2}$. A poor solvent favors a dense polymer globule, with $R\propto N^{1/3}$ at fixed internal density.

The competition is visible in the scaling free energyFor $v>0$, elastic cost balances two-segment repulsion to give the swollen-coil law. At $v=0$, three-segment repulsion stabilizes ideal-chain scaling; for $v<0$, attraction balanced by the positive three-segment term gives $R^3\sim uN/|v|$. The dense polymer globule also has an interfacial free energy. The scaling expression explains regimes, not all fluctuation corrections or a detailed finite-chain transition curve. A finite chain crosses over smoothly; the sharp theta-point limit refers to indefinitely long chains.

A dilute-solution light-scattering experiment can follow radius of gyration while changing temperature or solvent composition. Measure at several concentrations and extrapolate toward isolated chains; check molecular-weight distribution, reversibility and the absence of growing multi-chain aggregates. dynamic light scattering measures hydrodynamic size, which is related to but distinct from radius of gyration. A falling scattering-derived size accompanied by a rising aggregate population cannot by itself establish single-chain collapse. solvent quality controls conformation through the balance of segment interactions and conformational entropy.