End effects in polymer adsorption: a tale of tails.

Nearly all theories for polymer adsorption neglect end effects by assuming that all the segments, whether they are close to the chain ends or somewhere in the middle of the chain, have the same spatial distribution. This assumption is correct in the bulk solution, but not in the vicinity of a surface. Only the Scheutjens-Fleer model accounts explicitly for a segment distribution which is a function of the segmental ranking number. It is shown that end effects are only negligible at such extreme dilution that the polymer molecules in the surface region do not interfere (isolated chains). In all other cases, even for adsorption from dilute solution, the polymer segments compete for surface sites and tails develop, the average length of which is mainly determined by the solution concentration. For very long chains adsorbing from solutions of finite concentration, the tail fraction even increases with chain length. Several examples are given of the distribution of segments as a function of their ranking number. Simple equations for this distribution are developed on the basis of the assumption that two eigenfunctions with nearly equal eigenvalues dominate the distribution functions. The average length and length distribution of tails under various conditions is discussed. Tails determine nearly exclusively the hydrodynamic thickness of adsorbed layers. At present, hydrodynamic measurements are most suitable to provide (indirect) information about the effect of tails. A simple recurrent equation is derived for the hydrodynamic thickness corresponding to a given segment density profile in a lattice model