Conformation of an Isolated Polymer Molecule at an Interface. III. Distributions of Segment Densities near the Interface and of Other Shape Parameters

Abstract

A detailed study of the conformation of an isolated polymer molecule adsorbed at a solution interface is given. It is an extension of the similar work, previously published, in which average values of the fraction θ of adsorbed segments, the number 〈r〉 of adsorbed sequences per polymer chain, and the number of segments 〈a〉 in an adsorbed sequence, 〈d〉 in a loop, and 〈d₀〉 in a terminal desorbed sequence were derived. In this work probability distributions of these shape parameters are given. In addition, we consider here the segment density distributions in the solution near the interface. The number q(z) of desorbed segments, belonging to loops, which are expected to be at distance z from the interface is derived separately from the number q₀(z) of desorbed segments belonging to terminal dangling sequences. For evaluation of these quantities, extensive use is made of the generating function method by extending the summation procedure proposed by Lifson. The contour integral appearing in Cauchy's theorem is evaluated by the steepest descent method which was shown in the previous work to give numerical values to a good approximation. The results show, among others, that the density distribution q(z) of loop segments is exponential, while the distribution q₀(z) of the segments belonging to terminal sequences is given by a difference of two exponential functions and has a maximum at an intermediate z. As the value of the segmental adsorption constant s varies past the critical value s_c, the relative numbers of desorbed segments belonging to loops and to terminal sequences vary drastically, and consequently either q(z) or q₀(z) predominates except when s is very near s_c. It is shown that the treatment can be readily extended to cases where either one or both of the polymer‐chain ends is much more strongly adsorbed than the rest, and also where the interface consists of a mixture of sites having different affinities for adsorption of polymer segments.