Polymer mushroom-to-brush crossover
= Polymer mushroom-to-brush crossover
{title2=$\sigma_gR_g^2\sim1$}
End-grafted chains are approximately isolated when their lateral spacing $\sigma_g^{-1/2}$ exceeds their coil size. When the spacing becomes smaller, excluded-volume crowding favors extension normal to the substrate. The crossover criterion is a scaling estimate, not a sharp phase transition.