Catalan and half-catalan numbers in hyperbranched polymers

Abstract

Number sequences, like Fibonacci, Fermat, Markov, Euler, Bernoulli, etc., have found their utlity in representing a variety of scientific phenomena in science and engineering. Here, we explore the role of Catalan and Half-Catalan numbers in the context hyperbranched polymers synthesized through step polymerization. Our approach harnesses the concepts of combinatorics and graph theory, in conjunction with the kinetics of step polymerization to obtain structural attributes of hyperbranched polymers. Specifically, we establish one-to-one correspondence between polymer chains and tree data structures and develop various metrics to calculate the exact quantities of isomorphic/non-isomorphic and branched/linear polymer chains. Further, we use the traditional kinetic models to validate our findings. In addition, we obtain the chain length distribution that directly gives the closed-form expression of Catalan number expressed as a bivariate distribution function. As an offshoot, we discuss “pathwidth”, a construct used in graph theory, as a better metrics for describing topology of polymer chains. The framework developed in this work can be extended to ABm step polymerisation and thus, facilitates topological characterisation of hyperbranched polymers (HPs) that ultimately, dictates their structure-property relationships.