Polymer brush

A layer of sufficiently crowded tethered polymer chains stretched away from their attachment surface. In a good solvent, overlap increases the interaction contribution to free energy; extension relieves this crowding at a cost of chain entropy. The mushroom-to-brush crossover occurs at ${\sigma }_g{R}^2\sim 1$, where $R$ is an isolated coil size. In a simple good-solvent scaling model, the brush height is $H\sim Nb(v{\sigma }_g/b{)}^{1/3}$ for effective segment excluded volume $v$ and statistical segment length $b$; if $v\sim b^3$, this becomes $H\sim Nb({\sigma }_g{b}^2{)}^{1/3}$. The scaling requires strong stretching relative to the isolated coil but extension well below the contour length $Nb$.