Analytical approximation to the Scheutjens-Fleer theory for polymer adsorption from dilute solution. 1. Trains, loops, and tails in terms of two parameters: The proximal and distal lengths

Recently an analytical self-consistent mean-field theory was proposed for homopolymer adsorption in the long-chain limit. Here we make direct contact between that formalism and the numerical lattice model of Scheutjens and Fleer. The lattice layer closest to the wall is treated in a discrete manner, whereas a continuum description is used further out. This entails a new boundary condition at the wall. Together with the self-consistency condition, this leads to two equations from which the two relevant length scales (the proximal and distal lengths) follow unambiguously. As a result, analytical solutions in closed form are obtained for both (mean-field) good solvents (from chi = 0 up to chi approximate to 0.47) and a Theta solvent (chi = 0.5). For good solvents the agreement between the full lattice calculation and the analytical model is excellent; for a Theta solvent the discrepancy can amount to about 10%. The approximations necessary to render the analytical problem tractable are carefully checked against the numerical data.