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.

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.