Mapping of continuum and lattice models for describing the adsorption of an ideal chain anchored to a planar surface

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Abstract
An ideal polymer chain anchored to a planar surface is considered by using both lattice and continuum model approaches. A general equation relating the lattice and continuum model adsorption interaction parameters is derived in a consistent way by substituting the exact continuum solution for the free chain end distribution function into the lattice model boundary condition. This equation is not mathematically exact but provides excellent results. With the use of this relation the quantitative equivalence between lattice and continuum results was demonstrated for chains of both infinite and finite length and for all three regimes corresponding to attractive, repulsive and adsorption-threshold energy of polymer-surface interaction. The obtained equations are used to discuss the distribution functions describing the tail of an anchored macromolecule and its adsorbed parts. For the tail-related properties the results are independent of the microscopic details of the polymer chain and the adsorbing surface. One interesting result obtained in the vicinity of adsorption threshold point is a bimodal tail length distribution function, which manifests chain populations with either tail or loop dominance. The properties related to the number of surface contacts contain, apart from universal scaling terms, also a nonuniversal factor depending on microscopic details of polymer-surface interaction. We derived an equation for calculating this nonuniversal factor for different lattice models and demonstrated excellent agreement between the lattice results and the continuum model.