Random-Walk Model of Chain-Polymer Adsorption at a Surface

Abstract

A random‐walk lattice model of adsorption of an isolated polymer chain at a solution surface is investigated. One‐dimensional characteristics of the monomer‐unit distribution are determined analytically in the limit of long polymer chains, neglecting the self‐excluded volume. The mean number of monomer units adsorbed in the surface layer ν(θ, N) is determined, assuming that one end of the polymer chain lies in the surface layer. The parameter N is the number of monomer units in the chain, and θ is the adsorption energy of each monomer unit in the surface layer measured in units of kT. In addition, the mean distance of the free end of the chain from the surface z(θ, N) is determined. The lattice models considered include the simple‐cubic, hexagonal‐closepacked, face‐centered‐cubic, and body‐centered‐cubic lattices. In the limit in which N→∞, both ν(θ, N) and z(θ, N) exhibit a very interesting discontinuity at a lattice‐dependent adsorption energy θ_c. For example for θ>θ_c, ν(θ, N) (which is also proportional to the average adsorption energy of a polymer chain) is proportional to N. For θ<θ_c, ν(θ, N) is proportional to a constant of order unity; and for θ=θ_c, ν(θ, N) is proportional to N^½. It is shown that the probability distribution of the end of the chain decreases exponentially with increasing distance from the surface layer for θ>θ_c. In addition, the mean number of monomer units in the kth layer from the surface is determined for N≫1 and θ>θ_c and is found to decrease exponentially with increasing k. In effect, for θ>θ_c the polymer chain exists in an adsorbed state. An improvement in the model, which includes short‐range correlation between successive steps in the random‐walk description, is also considered.