Grafted polymer chain

A polymer chain tethered by an end to a surface. Its surface number density ${\sigma }_g$ sets a typical lateral separation $D\sim {\sigma }_g^{-1/2}$. Comparing $D$ with the isolated coil size determines whether grafted chains behave almost independently or must stretch cooperatively away from the surface.

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 $Nb$.

A dilute layer of end-tethered polymer coils whose lateral separation greatly exceeds the isolated coil size. Each coil retains approximately its isolated-solvent conformation, modified by the impenetrable tethering surface. Increasing the grafting density drives overlap and the crossover to a polymer brush.