The scaling behavior of the molecular parameters of the hyperbranched polymers made by self-condensing vinyl polymerization

Abstract

The hyperbranched polymers can be made by self-condensing vinyl polymerization without gelation transition. The average molecular weights, as well as the average sizes, can reach infinite values as the reaction is quantitatively completed, and the scaling forms of the molecular parameters should exist. In the paper, based on a recursion formula, the scaling form of the number fraction distribution and the number of the n-mers are given analytically as the conversion of double bonds is near 1. The mean square radius of gyration for very large hyperbranched polymers is calculated explicitly to give a scaling exponent. Finally, a scaling relation associated with the fractal dimension and the polydispersity exponent is given clearly.

The radius of gyration for a ternary self-condensing vinyl polymerization system

“…Experimentally, the MSRG of polymers under various solvent conditions can be measured by the method of light-scattering [27,33]. Theoretically, both analytical methods and computer simulation can be employed to study the MSRG of polymers [28][29][30][34][35][36].…”

“…It is well known that, in the evaluation of MSRG of branched polymers, the Kramers theorem becomes very convenient because it can correlate the size of polymers to its purely structural property [27][28][29][30]. For a polymer without loops, the central idea of Kramers theorem for evaluating its MSRG is that through an imaginary split of the polymer, the summation over bonds can substitute for that over structural units.…”

“…Furthermore, to obtain the k-th MSRG of polymers, a simple method is adopted to give a recursion formula satisfied by the k-th and (k+1)-th MSRGs [28][29][30]. For this purpose, substituting the results of Eq.…”

“…Clearly, there are many sub-chains with various length for a branched polymers, namely, there exists a sub-chain distribution [29,30]. For a given branched polymer, the sub-chain distribution depends only on its configuration but not the solvent quality or excluded volume interaction.…”

“…Specifically, two types of partition functions are constructed from two different viewpoints and then, the equilibrium free energy, law of mass action, and the SDFs are carried out. In Section 3, a recursion formula satisfied by the k-th and (k+1)-th MSRGs is derived (to coincide with previous works such as references [28][29][30], the k-th MSRG is adopted in this paper, which in essence the abbreviated form of the k-th moment of MSRG). As a result, the first, second, and third MSRGs for two types of polymers are derived.…”