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

摘要

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 thgr; of adsorbed segments, the number lang;rrang; of adsorbed sequences per polymer chain, and the number of segments lang;arang; in an adsorbed sequence, lang;drang; in a loop, and lang;d0rang; 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 numberq(z) of desorbed segments, belonging to loops, which are expected to be at distancezfrom the interface is derived separately from the numberq0(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 distributionq(z) of loop segments is exponential, while the distributionq0(z) of the segments belonging to terminal sequences is given by a difference of two exponential functions and has a maximum at an intermediatez.As the value of the segmental adsorption constantsvaries past the critical valuesc, the relative numbers of desorbed segments belonging to loops and to terminal sequences vary drastically, and consequently eitherq(z) orq0(z) predominates except whensis very nearsc. It is shown that the treatment can be readily extended to cases where either one or both of the polymerhyphen;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.