Alexander-de Gennes brush model ($H\sim Na(s{a}^2{)}^{(1-\nu )/(2\nu )}$)

A scaling model of overlapping end-grafted polymers assigns each chain a column of lateral width $D\sim s^{-1/2}$ at grafting density $s$. Each concentration blob contains $g\sim (D/a{)}^{1/\nu }$ statistical segments, and stacking $N/g$ blobs gives polymer brush height $H\sim ND/g$. For assumed segment volume $a^3$, the mean volume fraction is $\varphi \sim N{a}^{3}s/H$. It assumes a long many-segment-blob brush and supplies scaling powers rather than a precise chain-end distribution.