Past exam of the soft condensed matter course of the University of Cambridge / 2012 / Paper 1 / 3 / c / Solution

Overlap means roughly one coil per coil volume. With segment volume $a^3$, the corresponding polymer overlap volume fraction isFor a strong good solvent, $R_g\sim a{N}^{3/5}$, and henceor about 0.063% by volume in the unit-prefactor scaling convention. The segment length cancels. A coil has no sharply defined boundary, so using $4\pi R_g^3/3$ or retaining a size amplitude changes an order-one prefactor. This calculation is a scaling estimate, not an exact volume-fraction measurement.