On the Distribution of Small Radii of Gyration of Linear Flexible Polymer Molecules

Abstract

The Fourier transform of the distribution function characterizing the three orthogonal components of the radius of gyration of linear flexible polymer molecules can be obtained from the properties of random-flight chains. In this publication we present a detailed analysis of the problem of inverting the Fourier transform of the distribution of small radii of gyration and present approximate and asymptotic expressions for the distribution. We consider random-flight chains of t statistical links. If we measure a given orghogonal component of the radius of gyration Si by the dimensionless parameter ξ=π(12)−12Si/〈Si2〉12, we find: (1) In the domain of ξ between about (1/√3) and (3/t½), the distribution of ξ is adequately approximated by Wξ(ξ)=(π32)2−1ξ−3 exp (−π2/16ξ2). The expression is independent of t and therefore, as for larger radii, independent of how we subdivide a real polymer molecule into statistical segments. We also conclude, then, that for all radii such that ξ≥(3/t12), Wξ(ξ) is independent of the detailed structural features of real polymer chains. (2) In the domain of ξ⩽(1/102t12) the distribution of ξ can be approximated by Wξ(ξ)=[2t!/(12t−1)!]ξt−1. The configurational statistics of such highly compressed coils is dependent on the number of statistical chain elements and thus on how we would choose to subdivide a real polymer molecule into such elements. We can also interpret these results to imply that the configurational statistics of such highly compressed real polymer molecules would be dependent upon the detailed structural features of the chain.