= Solution
<polymers at interfaces> balance favorable surface contacts against lost <conformational entropy> and interactions among neighboring chains. At an attractive solid–liquid interface, an <adsorbed polymer train> consists of consecutive bound segments. An <adsorbed polymer loop> leaves the surface and returns to it, while an <adsorbed polymer tail> extends to a free chain end. A chain can contain all three, so <adsorption> does not imply a completely flattened molecule. Surface affinity, <solvent quality>, <persistence length>, chain <length> and <concentration> control the balance and the resulting segment-density profile.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2015/paper-1-polymer-interfaces.png]
{title=Original schematic of adsorbed trains, loops and tails, and the crossover from isolated grafted mushrooms to an overlapping polymer brush}
<adsorption> gains <energy> of order $\epsilon k_BT$ per bound segment but reduces the number of accessible chain configurations. Weak <adsorption> can be described through <adsorption blobs> whose thickness exceeds the statistical segment <length>; each blob loses an <energy> of order $k_BT$ through <confinement>. Stronger attraction generally creates more contacts and a more compact layer, while loops and tails can still dominate its outer extent. A measurement of adsorbed mass alone therefore does not determine layer thickness or conformation.
End-grafted chains are different: their attachment is fixed at one end. At <grafting density> $\sigma_g$, the mean lateral spacing is $D\sim\sigma_g^{-1/2}$. At low coverage the chains behave as isolated <polymer mushrooms>. The <polymer mushroom-to-brush crossover> occurs when $\sigma_gR_G^2\sim1$. At higher coverage in a <good solvent>, crowding makes chains extend away from the surface. A free-energy balance per chain,
$$
\frac{F}{k_BT}\sim\frac{H^2}{Nb^2}+\frac{vN^2\sigma_g}{H},
$$
combines stretching <entropy> with repulsive <excluded volume> $v>0$. Minimization gives $H^3\sim vN^3b^2\sigma_g$, or
$$
\boxed{H\sim Nb(\sigma_gb^2)^{1/3}\quad\text{when }v\sim b^3.}
$$
This is a scaling result with an undetermined order-one coefficient, valid for overlapping chains stretched below their contour <length>. In a compact poor-solvent layer, a roughly fixed internal <polymer> <volume fraction> instead gives $H\propto N\sigma_g$.
Small-angle neutron scattering on end-grafted polydimethylsiloxane layers on porous silica has distinguished the good- and poor-solvent <density> exponents. The chemical system and scattering contrast matter: changing <solvent quality> should be separated from changing graft coverage, chain <molar mass> or <adsorption> strength.<Characterization of the brush regime for grafted polymer layers at the solid-liquid interface by P. Auroy and others>[Physical Review Letters 66, 719–722 (1991), abstract: actual polymer/substrate system, neutron-scattering method and the two thickness scalings]{ref}
An interface can also localize chains of a <block copolymer> when its blocks prefer different media. Such layers change <interfacial tension> and can stabilize dispersed droplets or particles through <steric repulsion>. Overlapping layers in a <good solvent> cost osmotic and conformational <free energy>; incomplete <adsorption> can instead produce <polymer bridging> between particles. Nonadsorbing chains can cause depletion attraction. Thus the presence of a <polymer> does not by itself guarantee stabilization.
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