Topological features of polymer chains have been used as the key controlling mechanism for the physicochemical properties of hyperbranched polymers (HPs) and, therefore, provide a significant impetus to determine their branching characteristics. Single monomer methodology (SMM) involving AB m step polymerization has been one of the routes to synthesize both compact and segmented HPs. Here, we explore Catalan and half-Catalan numbers in the context of AB m step polymerization to deduce the structural information of HPs. Our approach harnesses the concepts of combinatorics and graph theory to calculate the exact numbers of isomeric, branched and linear, structures of polymer chains. We also demonstrate that the extent of branching of a polymer chain can be measured via pathwidth and establish its bounds as a function of its length. We believe that our findings can be leveraged to design and control the architecture of HPs synthesized via SMM or any other chemistries in a straightforward manner.

Shape Memory Polymers (SMPs) are smart materials capable of transforming back to their intended shape and therefore, are used to replicate biomimetic functionalities in synthetic materials. Here, we demonstrate that polymer blending can be used as an effective technique to design SMPs. In particular, the blends consists of wax and rubber that signify soft and hard segments of the SMP, respectively. In particular, we use the standard linear viscoelastic model where the two constituents, rubber represented by Hookean spring and wax represented by Maxwellian liquid, are connected in parallel; hence, we assume that the contribution of each component in the total stress is distributed in proportion of the composition. Via simulation, we establish that wax and rubber, individually do not exhibit shape memory behavior but do so upon blending; the shape memory behavior is qualitatively validated from experiments. We also confirm that the shape memory properties of our rubber-wax blends, namely, the shape fixity and shape recovery ratios at different concentrations, is in qualitative agreement with the properties of SMPs. Our findings indicate that polymer blending can be seen as an alternative strategy to design SMPs; the properties of these polymers can be varied directly by changing the composition of the blend.

Number sequences, like Fibonacci, Fermat, Markov, Euler, Bernoulli, etc., have been popular in exemplifying a variety of scientific phenomena. Here, we explore the Catalan numbers in the context of ABm step polymerisation and develop a framework to derive its alternative closed form expression. Our approach harnesses the concepts of combinatorics and graph theory, in conjunction with kinetics of AB2 polymerisation to obtain the chain length distribution that directly gives the closed-form expression of Catalan number expressed as a bivariate distribution function. Furthermore, we validate our expression by comparing first 5000 Catalan numbers obtained from its traditional closed-form. 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.

Shape Memory Polymers (SMPs) are smart materials capable of transforming back to their intended shape and therefore, are used to replicate biomimetic functionalities in synthetic materials. Here, we demonstrate that polymer blending can be used as an effective technique to design SMPs. In particular, the blends consists of wax and rubber that signify soft and hard segments of the SMP, respectively. In particular, we use the standard linear viscoelastic model where the two constituents, rubber represented by Hookean spring and wax represented by Maxwellian liquid, are connected in parallel; hence, we assume that the contribution of each component in the total stress is distributed in proportion of the composition. Via simulation, we establish that wax and rubber, individually do not exhibit shape memory behavior but do so upon blending; the shape memory behavior is qualitatively validated from experiments. We also confirm that the shape memory properties of our rubber-wax blends, namely, the shape fixity and shape recovery ratios at different concentrations, is in qualitative agreement with the properties of SMPs. Our findings indicate that polymer blending can be seen as an alternative strategy to design SMPs; the properties of these polymers can be varied directly by changing the composition of the blend.