Random-Walk Model of Adsorption of a Chain-Polymer Molecule on a Long Rigid-Rod Molecule

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Abstract
A lattice model of adsorption of a flexible chain molecule on a rodlike molecule is investigated. The rodlike molecule is represented by the lattice sites on the z axis of a simple-cubic lattice; sites which are nearest neighbors to the z axis are adsorbing sites. The dimensionless adsorption energy per monomer unit is θ = ε/kT. The problem of enumerating polymer-chain configurations taking into account the increased probability of occupying adsorbing sites and the zero probability of occupying z-axis sites is formulated and solved as a random-walk problem. This model is a natural generalization of a random-walk model of adsorption on a plane solution surface. The average fraction of monomer units in adsorbing sites fR(θ) is computed in the limit in which the number of monomer units in the polymer chain approaches infinity. There is a critical value of the adsorption energy θc = ln (6/5) such that for θ<θc, fR(θ) = 0. For θ>θc, fR(θ) is an increasing function of θ with all derivatives equal to zero at θc = ln (6/5). In the analogous simple-cubic model of adsorption of a polymer-chain molecule at a plane solution surface, the same value of the critical energy has been obtained. For θln (6/5), fS(θ) is an increasing function of θ whose right-hand slope is 25 at θ = ln (6/5).