Chains in the presence of an interacting surface and different boundary conditions

Abstract

The end-to-end distribution function of a linear chain interacting with a penetrable surface with the potential uδ(z) is demonstrated to recover the case of the distribution in the presence of an impenetrable surface with different boundary conditions. The two different boundary conditions of zero probability density and of zero of the gradient of the probability density at the surface correspond to different values of u and the penetrable distribution function can thus be used to describe chains with various degrees of interactions both in the presence of penetrable or impenetrable surfaces. Density profiles of the monomeric units of the chains localized at one or both ends are described, furnishing an insight to the distortion which the interacting surface brings on the shape of a coil. The study includes an extension to the cases of ring and star macromolecules.

Spinodal Shifts of a Polymeric Blend Film on an Interacting Surface

“…The interaction parameter w is equal to half the binary cluster integral of the average potential between the units of different chains, while u a′ and u b′ are the binary cluster integrals of the mean potentials between the units of the a and b chains and the surface. The appropriate reflecting boundary conditions ensure the impenetrability of the surface describing thus an interacting impenetrable surface boundary 12 and the possibility of the chains to belong to the right half-space of volume V ) S × L. A second surface is not included at the free surface of the film, which is in touch with the air. We explain in section 2B how these halfspace statistics can describe the film behavior when the film's width is larger than the chain size.…”

Effects of Substrate Interactions in Liquid Chromatography of Star Homopolymers and Star Block Copolymers

“…The probability p ∼ exp{− u δ( z i )} of a contact, for large positive values of u , becomes small, indicating an increased repulsive character of the interactions, while negative values of u give large probabilities of contact describing attractions. Though the amount of adsorbed monomeric units depends also on the boundary conditions posed on the chains, , the region around u = 0, between repulsions and attractions, can be used to describe the situation where the chains do not feel special interactions with the surface. This arbitrariness in the values of u gives the possibility of adjustment for a proper description of experimental results.…”

“…This arbitrariness in the values of u gives the possibility of adjustment for a proper description of experimental results. By integration with respect to all positions of the units of the chain but the position of the last unit, the probability of a linear chain localized with one end at the distance z from the surface has been obtained . For reflecting boundary conditions, with zero the derivative of the probability at the surface, it is equal to with the interaction parameter U = u (6 N ) 1/2 /2 and S o = l o N 1/2 /6 1/2 the radius of gyration of the chain, written in terms of the number N of the units of the chain which is proportional to the molecular weight of the polymer and the length l o of each unit.…”

“…At large distances z where no surface effects survive, F ( z ) dies out, affecting no more P ( z ). The ideal state with u = 0 is somehow arbitrary and the amount of monomeric units adsorbed depends also on the boundary conditions posed on the chains. , Varying u all situations from the full adsorption to complete desorption can be described giving thus the possibility of the adjustment of the value of u in order to be in accord with experimental findings.…”