Average Degrees of Polymerization and Molar Masses of Branched and Hyperbranched Condensation Polymers:  Recursive Probability Approach

Abstract

This article extends the "in-out" recursive probability method of Macosko and Miller to the polycondensation of mixtures of polyfunctional monomers bearing A-and/or B-groups. Explicit relationships of the theoretical average degrees of polymerization and average molar masses are established in the most general case and can be applied to any linear, hyperbranched, or network polymerizations, leading in the latter case to the value of the conversion at the gel point. These relationships reduce to those of Stockmayer in the case of reactions involving monomers bearing either A-or B-groups. The effect of the elimination of a condensation byproduct is also examined. An approximate explicit expression of the mass-average molar mass is proposed, allowing calculations with an error in the 1-3% range when water is the byproduct. Some examples are treated to illustrate the method, including AA + BB + AB, AB + B g , AB f , A f B g , AB f + B g , and more complex polymerizations. The effect of the functionality of the core molecule on the polydispersity of hyperbranched polymers is examined for various systems.

Structure and rheology of branched polyamide 6 polymers from their reaction recipe

“…Regarding the reaction modeling, the following assumptions are made: 31,32 (1) an A-group reacts only with a B-group and vice versa; (2) all reactions have equal probability; (3) no intramolecular reactions occur, i.e. no cycles are formed.…”

“…any molecular segment containing only caprolactam monomers (located between two extremities or functional groups A x B y different from A 1 B 1 ) can be described by a Flory distribution with the same average molecular weight M n,Strand , and (2) we know, a priori, that every chain extremity is of type 'A' (real or virtual). [29][30][31][32] In contrast to the actual polycondensation process and to usual Monte Carlo algorithms, the molecules are therefore not built from the different monomers (A 1 B 1 , B 2 , A 3 , . ), but directly from specic sub-chains which can be considered as building blocks of the molecules.…”

“…Some works have already been proposed in this direction for metallocene polyethylene [25][26][27] as well as for polydisperse polyamide samples. [28][29][30][31][32] In the latter case, it has been shown that classical Monte Carlo statistics lead to a good approximation of the sample, which is formed by a condensation reaction process. The objective here is to propose a simplied Monte Carlo approach, which takes advantage -while without losing accuracy -of the fact that we know a priori that every generation will be described by a Flory-type molecular weight distribution and that they will all have the same average molecular weight (whatever their seniority).…”

Synthesis and characterization of low viscosity aromatic hyperbranched polyester epoxy resin

“…According to the polycondensation between hydroxy group and carboxy group, the acid value of the HTB can be used to calculate the degree of polymerization and further reckon the number-average molecular weight. Explicit relationships about theoretical average degrees of polymerization and average molar masses are established in AB f + B g monomer polymerization case by "Recursive Probability Approach", 20 and , ,…”

Novel adhesive system based on 1,3-dimethylol-4,5-dihydroxyethyleneurea (DMDHEU) and hyperbranched polyglycerols

“…On the contrary to dendrimers that require multi-step approach involving tedious purification after each step, hyperbranched polyglycerols are synthesised in 'one-pot' synthesis which results in their substantial polydispersity (Galina and Krawczyk 2007;Fradét and Tessier 2006). For better description of molecular structures of the studied polyglycerols, their polydispersity as well as incorporation of the core molecule into the structure were investigated by means of MALDI-TOF spectrometry.…”