= 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 $\sigma_gR^2\sim1$, 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 $H\sim Nb(\sigma_gb^2)^{1/3}$. The scaling requires strong stretching relative to the isolated coil but extension well below the contour length $Nb$.
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