Topological Characteristics of Chain Molecules with Branching and Molar‐Mass Distributions

Abstract

Here, the suffix L denotes the linear chain. For the purpose of estimating branching, considerable effort has been made toward estimating the overall dimensions of isolated chain molecules by measuring the intrinsic viscosity [η] and light scattering (LS) of their dilute solutions. However, these methods are only suitable for monodispersed or fractionated polymers in the Θ state, where the effects of the excluded volume of a structural unit vanish. Furthermore, these methods are appropriate for precisely evaluating the branching degree based on conformational statistics for each branched chain molecule.Branched polymers possess a branching point distribution in addition to their wide, polydisperse molar masses. Typical longchain branched polymers, such as low-density polyethylene (LDPE), are randomly branched polydisperse polymers that are considered to be random mixtures of structural isomers. [6,7] Gel-permeation chromatography (GPC), also known as sizeexclusion chromatography (SEC), [8][9][10][11][12][13] is a popular technique used to fractionate polymers according to their volume dimension [η]M, where M is the polymer molecular weight. Thus, the chain dimensions of various branched polymers have been continuously investigated using combinations of GPC with either LS or [η] measurements. [14,15] The effects of branching degree and the type of branched structure need to be distinguished to gain structural insight into the effect of branching on polymer physical properties. The aim of the present work is to unravel these intertwined effects, where we developed a novel mathematical framework to codify the molecular architecture of any type of branching structure. In addition, we investigated the effects of molarmass distribution on the statistical and dynamic properties of unperturbed branched chains for polydisperse polymer systems. This article deals with only flexible polymers containing no loops or rings because they have numerous practical applications. We focused on the radius of gyration and the total sum of relaxation times of branched polymers with random or fixed branching points and examined the effects of molar-mass distribution on their statistical and dynamic properties.A graph-theoretical method is presented for characterizing the statistical and dynamic properties of randomly and fixed-branched chains that exhibit molecular weight distribution. The Wiener index, which is a topological index reflecting tree-shaped structures, of any branched chain estimated from its random-flight model (molecular graph) enables to express the radius of gyration and dynamic relaxation times of various chains with branching and molar-mass distributions. A homologous series of branched chains with random branching points can be obtained by comparing their Wiener indices with those of their corresponding primitive chains (where the subchains between adjacent branch or end points are reduced to single bonds). Moreover, the analogical growth of a fixed-branched structure during polymerization can be quantitatively anal...