Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section D / 25 / d / ii / Solution

The grafting density is the inverse area per chain:For a neutral polymer brush in a good solvent, stretching costs $F_{\mathrm{el}}/k_{B}T\sim h^2/(N{b}_0^2)$ per chain while excluded-volume interactions cost $F_{\mathrm{int}}/k_{B}T\sim v{N}^2/(h{D}^2)$. Minimization gives $h^3\sim v{N}^3{b}_0^2/D^2$. With $v\sim b_0^3$,An isolated good-solvent coil has size $\sim b_0{N}^{3/5}\simeq 502$ nm, much larger than $D=10$ nm, consistent with strong overlap. The brush height is below the 20 μm contour length. This estimate assumes good-solvent excluded volume and locally planar geometry; the source gives neither solvent quality nor particle radius, so a unique curved-brush thickness is not established. An ideal-solvent brush has different scaling.