Polymer brush

<h2 id="structure">Structure</h2>

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,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> 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,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> the average density of all monomers in a given chain is always the same up to a prefactor:

 
 
 
 ϕ
 (
 z
 ,
 ρ
 )
 =
 
 
 
 ∂
 n
 
 
 ∂
 z
 
 
 
 
 
 {\displaystyle \phi (z,\rho )={\frac {\partial n}{\partial z}}}

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

where 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
 is the altitude of the end monomer and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 the number of monomers per chain.
The averaged density profile 
 
 
 
 ϵ
 (
 ρ
 )
 
 
 {\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:

 
 
 
 ϕ
 (
 z
 )
 =
 
 ∫
 
 z
 
 
 ∞
 
 
 
 
 
 ∂
 n
 (
 z
 ,
 ρ
 )
 
 
 ∂
 z
 
 
 
 
 ϵ
 (
 ρ
 )
 
 
 
 d
 
 
 ρ
 
 
 {\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 
 
 
 
 H
 
 
 {\displaystyle H}
 
. One can show<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> that the corresponding end monomer density profile is given by:

 
 
 
 
 ϵ
 
 
 d
 r
 y
 
 
 
 (
 ρ
 ,
 H
 )
 =
 
 
 
 ρ
 
 /
 
 H
 
 
 N
 a
 
 
 1
 −
 
 ρ
 
 2
 
 
 
 /
 
 
 H
 
 2
 
 
 
 
 
 
 
 
 
 {\displaystyle \epsilon _{\rm {dry}}(\rho ,H)={\frac {\rho /H}{Na{\sqrt {1-\rho ^{2}/H^{2}}}}}}

where 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is the monomer size.
The above monomer density profile 
 
 
 
 n
 (
 z
 ,
 ρ
 )
 
 
 {\displaystyle n(z,\rho )}
 
 for one single chain minimizes the total elastic energy of the brush,

 
 
 
 U
 =
 
 ∫
 
 0
 
 
 ∞
 
 
 ϵ
 (
 ρ
 )
 
 
 
 d
 
 
 ρ
 
 
 ∫
 
 0
 
 
 N
 
 
 
 
 
 d
 
 
 n
 
 
 
 
 k
 T
 
 
 2
 N
 
 a
 
 2
 
 
 
 
 
 
 
 (
 
 
 
 ∂
 z
 (
 n
 ,
 ρ
 )
 
 
 ∂
 n
 
 
 
 )
 
 
 2
 
 
 
 
 {\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 
 
 
 
 ϵ
 (
 ρ
 )
 
 
 {\displaystyle \epsilon (\rho )}
 
, as shown in.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="from-a-dry-brush-to-any-brush">From a dry brush to any brush</h2>
As a consequence,<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> the structure of any brush can be derived from the brush density profile 
 
 
 
 ϕ
 (
 z
 )
 
 
 {\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:

 
 
 
 ϵ
 (
 ρ
 )
 =
 
 ∫
 
 ρ
 
 
 ∞
 
 
 −
 
 
 
 
 
 d
 
 
 ϕ
 (
 H
 )
 
 
 
 
 d
 
 
 H
 
 
 
 
 ϵ
 
 
 d
 r
 y
 
 
 
 (
 ρ
 ,
 H
 )
 
 
 {\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:

 
 
 
 
 
 
 F
 
 
 e
 l
 
 
 
 
 k
 T
 
 
 
 =
 
 
 
 π
 
 2
 
 
 
 24
 
 N
 
 2
 
 
 
 a
 
 5
 
 
 
 
 
 
 ∫
 
 0
 
 
 ∞
 
 
 
 {
 
 −
 
 z
 
 3
 
 
 
 
 
 
 
 d
 
 
 ϕ
 (
 z
 )
 
 
 
 
 d
 
 
 z
 
 
 
 
 }
 
 
 
 d
 
 
 z
 
 
 {\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<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> and to understand fine interpenetration asymmetries between copolymer lamellae<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> that may yield very unusual non-centrosymmetric <a href="/facts/Lamellar_structure/esvkPy5o">lamellar structures</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h2 id="applications">Applications</h2>
Polymer brushes can be used in Area-selective deposition.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> Area-selective deposition is a promising technique for positional self-alignment of materials at a prepatterned surface.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Dendronized_polymer/2b4uoKo9">Dendronized polymer</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Milner, S. T. (1991). "Polymer Brushes". Science. 251 (4996): 905–14. Bibcode:1991Sci...251..905M. doi:10.1126/science.251.4996.905. PMID 17847384. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Chremos, A; Douglas, JF (2018). "A comparative study of thermodynamic, conformational, and structural properties of bottlebrush with star and ring polymer melts". J. Chem. Phys. 149 (4): 044904. Bibcode:2018JChPh.149d4904C. doi:10.1063/1.5034794. PMC 11446256. PMID 30068167. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11446256" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11446256</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Halperin, A.; Tirrell, M.; Lodge, T. P. (1992). "Tethered chains in polymer microstructures". Macromolecules: Synthesis, Order and Advanced Properties. Advances in Polymer Science. Vol. 100/1. pp. 31–71. doi:10.1007/BFb0051635. ISBN 978-3-540-54490-6. <a href="978-3-540-54490-6" target="_blank">978-3-540-54490-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Chremos, A; Douglas, JF (2018). "A comparative study of thermodynamic, conformational, and structural properties of bottlebrush with star and ring polymer melts". J. Chem. Phys. 149 (4): 044904. Bibcode:2018JChPh.149d4904C. doi:10.1063/1.5034794. PMC 11446256. PMID 30068167. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11446256" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11446256</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Laradji, Mohamed; Guo, Hong; Zuckermann, Martin (1994). "Off-lattice Monte Carlo simulation of polymer brushes in good solvents". Physical Review E. 49 (4): 3199–3206. Bibcode:1994PhRvE..49.3199L. doi:10.1103/PhysRevE.49.3199. PMID 9961588. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Kaznessis, Yiannis N.; Hill, Davide A.; Maginn, Edward J. (1998). "Molecular Dynamics Simulations of Polar Polymer Brushes". Macromolecules. 31 (9): 3116–3129. Bibcode:1998MaMol..31.3116K. CiteSeerX 10.1.1.465.5479. doi:10.1021/ma9714934. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Szleifer, I; Carignano, MA (1996). Tethered Polymer Layers. Vol. XCIV. p. 165. doi:10.1002/9780470141533.ch3. ISBN 978-0-471-19143-8. {{cite book}}: |journal= ignored (help) <a href="978-0-471-19143-8" target="_blank">978-0-471-19143-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Milner, S. T; Witten, T. A; Cates, M. E (1988). "A Parabolic Density Profile for Grafted Polymers". Europhysics Letters (EPL). 5 (5): 413–418. Bibcode:1988EL......5..413M. doi:10.1209/0295-5075/5/5/006. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Milner, S. T; Witten, T. A; Cates, M. E (1988). "A Parabolic Density Profile for Grafted Polymers". Europhysics Letters (EPL). 5 (5): 413–418. Bibcode:1988EL......5..413M. doi:10.1209/0295-5075/5/5/006. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Milner, S. T; Witten, T. A; Cates, M. E (1989). "Effects of polydispersity in the end-grafted polymer brush". Macromolecules. 22 (2): 853–861. Bibcode:1989MaMol..22..853M. doi:10.1021/ma00192a057. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Zhulina, E.B.; Borisov, O.V. (July 1991). "Structure and stabilizing properties of grafted polymer layers in a polymer medium". Journal of Colloid and Interface Science. 144 (2): 507–520. Bibcode:1991JCIS..144..507Z. doi:10.1016/0021-9797(91)90416-6. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Gay, C. (1997). "Wetting of a polymer brush by a chemically identical polymer melt". Macromolecules. 30 (19): 5939–5943. Bibcode:1997MaMol..30.5939G. doi:10.1021/ma970107f. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Gay, C. (1997). "Wetting of a polymer brush by a chemically identical polymer melt". Macromolecules. 30 (19): 5939–5943. Bibcode:1997MaMol..30.5939G. doi:10.1021/ma970107f. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Gay, C. (1997). "Wetting of a polymer brush by a chemically identical polymer melt". Macromolecules. 30 (19): 5939–5943. Bibcode:1997MaMol..30.5939G. doi:10.1021/ma970107f. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Leibler, L; Gay, C; Erukhimovich, I (1999). "Conditions for the existence of non-centrosymmetric copolymer lamellar systems". Europhysics Letters (EPL). 46 (4): 549–554. Bibcode:1999EL.....46..549L. doi:10.1209/epl/i1999-00277-9. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Goldacker, T; Abetz, V; Stadler, R; Erukhimovich, I; Leibler, L (1999). "Non-centrosymmetric superlattices in block copolymer blends". Nature. 398 (6723): 137. Bibcode:1999Natur.398..137G. doi:10.1038/18191. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Lundy, Ross; Yadav, Pravind; Selkirk, Andrew; Mullen, Eleanor; Ghoshal, Tandra; Cummins, Cian; Morris, Michael A. (2019-09-17). "Optimizing Polymer Brush Coverage To Develop Highly Coherent Sub-5 nm Oxide Films by Ion Inclusion". Chemistry of Materials. 31 (22): 9338–9345. doi:10.1021/acs.chemmater.9b02856. ISSN 0897-4756. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Polymer brush open-in-new

Polymer brush