polymer brush

File:Tethered polymer chain.svg (at smaller length scales) to a 2D in-plane random walk and a 1D normal directed walk (at larger length scales).]]

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,{{cite journal | last1 = Milner | first1 = S. T | last2 = Witten | first2 = T. A | last3 = Cates | first3 = M. E | title = A Parabolic Density Profile for Grafted Polymers | journal = Europhysics Letters (EPL) | volume = 5 | issue = 5 | pages = 413–418 | date = 1988 | doi = 10.1209/0295-5075/5/5/006|bibcode = 1988EL......5..413M }} 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, the average density of all monomers in a given chain is always the same up to a prefactor:

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

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

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

The averaged density profile $\backslash epsilon(\backslash rho)$ 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:

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

A dry brush has a uniform monomer density up to some altitude $H$. One can show{{cite journal | last1 = Milner | first1 = S. T | last2 = Witten | first2 = T. A | last3 = Cates | first3 = M. E | title = Effects of polydispersity in the end-grafted polymer brush | journal = Macromolecules | volume = 22 | issue = 2 | pages = 853–861 | date = 1989 |bibcode = 1989MaMol..22..853M |doi = 10.1021/ma00192a057 }} that the corresponding end monomer density profile is given by:

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

where $a$ is the monomer size.

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

$U=\backslash int\_0^\backslash infty\backslash epsilon(\backslash rho)\backslash ,\{\backslash rm\; d\}\backslash rho\backslash ,\backslash int\_0^N\backslash ,\{\backslash rm\; d\}n\backslash ,\backslash frac\{kT\}\{2Na^2\}\backslash left(\backslash frac\{\backslash partial\; z(n,\backslash rho)\}\{\backslash partial\; n\}\backslash right)^2$

regardless of the end monomer density profile $\backslash epsilon(\backslash rho)$, as shown in.{{cite journal|last1=Zhulina|first1=E.B.|last2=Borisov|first2=O.V.|title=Structure and stabilizing properties of grafted polymer layers in a polymer medium|journal=Journal of Colloid and Interface Science|date=July 1991|volume=144|issue=2|pages=507–520|doi=10.1016/0021-9797(91)90416-6|bibcode=1991JCIS..144..507Z}}{{cite journal | last1 = Gay | first1 = C. | title = Wetting of a polymer brush by a chemically identical polymer melt | journal = Macromolecules | volume = 30 | issue = 19 | pages = 5939–5943 | date = 1997 |bibcode = 1997MaMol..30.5939G |doi = 10.1021/ma970107f }}

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

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

Correspondingly, the brush elastic free energy is given by:

$\backslash frac\{F\_\{\backslash rm\; el\}\}\{kT\}=\backslash frac\{\backslash pi^2\}\{24N^2a^5\}\backslash int\_0^\backslash infty\backslash left\backslash \{-z^3\backslash frac\{\{\backslash rm\; d\}\backslash phi(z)\}\{\{\backslash rm\; d\}z\}\backslash right\backslash \}\{\backslash rm\; d\}z$.

This method has been used to derive wetting properties of polymer melts on polymer brushes of the same species and to understand fine interpenetration asymmetries between copolymer lamellae{{cite journal | last1 = Leibler | first1 = L | last2 = Gay | first2 = C | last3 = Erukhimovich | first3 = I | title = Conditions for the existence of non-centrosymmetric copolymer lamellar systems | journal = Europhysics Letters (EPL) | volume = 46 | issue = 4 | pages = 549–554 | date = 1999 | doi=10.1209/epl/i1999-00277-9|bibcode = 1999EL.....46..549L }} that may yield very unusual non-centrosymmetric lamellar structures.{{cite journal | last1 = Goldacker | first1 = T | last2 = Abetz | first2 = V | last3 = Stadler | first3 = R | last4 = Erukhimovich | first4 = I | last5 = Leibler | first5 = L | title = Non-centrosymmetric superlattices in block copolymer blends | journal = Nature | volume = 398 | pages = 137 | date = 1999 |bibcode = 1999Natur.398..137G |doi = 10.1038/18191 | issue=6723}}

Polymer brushes can be used in Area-selective deposition.{{Cite journal|last1=Lundy|first1=Ross|last2=Yadav|first2=Pravind|last3=Selkirk|first3=Andrew|last4=Mullen|first4=Eleanor|last5=Ghoshal|first5=Tandra|last6=Cummins|first6=Cian|last7=Morris|first7=Michael A.|date=2019-09-17|title=Optimizing Polymer Brush Coverage To Develop Highly Coherent Sub-5 nm Oxide Films by Ion Inclusion|journal=Chemistry of Materials|volume=31|issue=22|pages=9338–9345|doi=10.1021/acs.chemmater.9b02856|issn=0897-4756}} Area-selective deposition is a promising technique for positional self-alignment of materials at a prepatterned surface.

• Dendronized polymer

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Category:Surface science

Category:Soft matter

Category:Polymer chemistry