Polymer adsorption

Polymer adsorption binds a polymer near an attractive interface through contacts of some of its monomers. The binding energy competes with the loss of conformational entropy. A chain need not lie flat: noncontacting loops and tails can extend into the solvent. In a weak-adsorption blob model, the layer thickness exceeds the Kuhn length and local chain statistics remain approximately those of the free chain below that thickness.

A nonadsorbed terminal section extending from a surface-bound part of an adsorbed chain to a free chain end. It has one surface connection, whereas an adsorbed loop has two.

A nonadsorbed chain section whose two ends connect to surface-bound portions of the same adsorbed polymer. Loops extend into the solvent and contribute to the thickness, steric repulsion and conformational entropy of an adsorbed layer.

A contiguous sequence of segments making contacts with an adsorbing surface. The adsorption energy gained per contact competes with the loss of chain configurations. A chain can have several trains separated by loops rather than lying wholly flat.

Let an adsorbed Gaussian chain gain free energy $\mu >0$ per bound monomer. A pulled segment of $n$ Kuhn length steps spanning height $z$ has elastic free energy $3k_{B}T{z}^2/(2n{b}^2)$, giving $F=-(N-n)\mu +3k_{B}T{z}^2/(2n{b}^2)$. Minimizing over $n$ yields $n=(z/b)\sqrt{3k_{B}T/(2\mu )}$ and $F_{\min }=-N\mu +z\sqrt{6k_{B}T\mu }/b$. Its height derivative is the constant peeling force displayed above. The plateau lasts while $0<n<N$ and the Gaussian chain approximation remains valid. In the slab model $F_{\mathrm{ads}}=N{k}_{B}T[(b/\xi {)}^2-ϵb/\xi ]$, optimization gives $\mu =k_{B}T{ϵ}^2/4$ and $f=\sqrt{3/2}ϵk_{B}T/b$.

An adsorption blob is a subchain of size comparable to the adsorbed layer thickness $\xi$. If the free-chain size law is $R\sim b{g}^{\nu }$, it contains $g\sim (\xi /b{)}^{1/\nu }$ monomers. Confinement costs of order $k_{B}T$ per blob produce $F_{\mathrm{conf}}\sim N{k}_{B}T/g$. A simple contact fraction $b/\xi$ gives $F/(N{k}_{B}T)\sim (b/\xi {)}^{1/\nu }-ϵb/\xi$. This model predicts $\xi /b\sim {ϵ}^{-\nu /(1-\nu )}$ and binding gain per monomer of order $k_{B}T{ϵ}^{1/(1-\nu )}$. It assumes a many-blob layer and a monomer-scale attraction range; it does not replace the full surface-critical theory of polymer adsorption.