Polymer brush

Polymer molecules within a brush are stretched away from the attachment surface as a result of the fact that they repel each other (steric repulsion or osmotic pressure). More precisely,[7] they are more elongated near the attachment point and unstretched at the free end, as depicted on the drawing.

More precisely, within the approximation derived by Milner, Witten, Cates,[7] the average density of all monomers in a given chain is always the same up to a prefactor:

${\displaystyle \phi (z,\rho )={\frac {\partial n}{\partial z}}}$

${\displaystyle n(z,\rho )={\frac {2N}{\pi }}\arcsin \left({\frac {z}{\rho }}\right)}$

where ${\displaystyle \rho }$ is the altitude of the end monomer and ${\displaystyle N}$ the number of monomers per chain.

The averaged density profile ${\displaystyle \epsilon (\rho )}$ of the end monomers of all attached chains, convoluted with the above density profile for one chain, determines the density profile of the brush as a whole:

${\displaystyle \phi (z)=\int _{z}^{\infty }{\frac {\partial n(z,\rho )}{\partial z}}\,\epsilon (\rho )\,{\rm {d}}\rho }$

A dry brush has a uniform monomer density up to some altitude ${\displaystyle H}$. One can show[8] that the corresponding end monomer density profile is given by:

${\displaystyle \epsilon _{\rm {dry}}(\rho ,H)={\frac {\rho /H}{Na{\sqrt {1-\rho ^{2}/H^{2}}}}}}$

where ${\displaystyle a}$ is the monomer size.

The above monomer density profile ${\displaystyle n(z,\rho )}$ for one single chain minimizes the total elastic energy of the brush,

${\displaystyle U=\int _{0}^{\infty }\epsilon (\rho )\,{\rm {d}}\rho \,\int _{0}^{N}\,{\rm {d}}n\,{\frac {kT}{2Na^{2}}}\left({\frac {\partial z(n,\rho )}{\partial n}}\right)^{2}}$

regardless of the end monomer density profile ${\displaystyle \epsilon (\rho )}$, as shown in.[9][10]

As a consequence,[10] the structure of any brush can be derived from the brush density profile ${\displaystyle \phi (z)}$. Indeed, the free end distribution is simply a convolution of the density profile with the free end distribution of a dry brush:

${\displaystyle \epsilon (\rho )=\int _{\rho }^{\infty }-{\frac {{\rm {d}}\phi (H)}{{\rm {d}}H}}\epsilon _{\rm {dry}}(\rho ,H)}$.

Correspondingly, the brush elastic free energy is given by:

${\displaystyle {\frac {F_{\rm {el}}}{kT}}={\frac {\pi ^{2}}{24N^{2}a^{5}}}\int _{0}^{\infty }\left\{-z^{3}{\frac {{\rm {d}}\phi (z)}{{\rm {d}}z}}\right\}{\rm {d}}z}$.

This method has been used to derive wetting properties of polymer melts on polymer brushes of the same species[10] and to understand fine interpenetration asymmetries between copolymer lamellae[11] that may yield very unusual non-centrosymmetric lamellar structures.[12]

Polymer brushes can be used in Area-selective deposition.[13] Area-selective deposition is a promising technique for positional self-alignment of materials at a prepatterned surface.

• Dendronized polymer