Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 1 / Section B / 9 / vi (second occurrence)

This is the second printed “(vi)”, concerning solvent temperature. The coil is expected to expand, so its radius of gyration increases. For the usual upper-critical-solution-temperature behavior of polystyrene in cyclohexane, heating above the stated theta condition improves solvent quality. Effective repulsion between segments favors more expanded conformations than the ideal coil.

In a Flory–Huggins solution theory description, the theta condition corresponds approximately to interaction parameter $\chi =1/2$, whereas a good solvent has a positive effective excluded volume. The asymptotic size dependence crosses toward $R_g\propto N^{\nu }$ with $\nu$ near $0.59$ for a flexible chain in three dimensions. These scaling ideas establish the expected direction, not a numerical expansion factor for a ten-degree temperature change. Its value would require measured solvent-interaction data or a calibrated crossover model.