Polymer-brush volume fraction ($\varphi =N{v}_{0}s/H$)

Conservation of material in a uniform polymer brush gives its mean volume fraction as $N{v}_{0}s/H$, with material volume $v_0$ per counted segment, chain length $N$, grafting density $s$ and height $H$. Identifying $v_0$ with $a^3$ is an extra coarse-graining assumption. The Alexander-de Gennes brush model then gives $\varphi \sim (s{a}^2{)}^{(3\nu -1)/(2\nu )}$, including powers $2/3$ in a good solvent and $1/2$ in a theta solvent.