Theoretical Methods of the Size Distribution Function for the Products of Hyperbranched Polymerization

Abstract

wide application of the hyperbranched polymers. To solve this problem, people first need to understand the relationship between monomer type, reaction conditions, and product structure parameters.In fact, the theoretical development of hyperbranched polymers has been in step with the experimental work, and guides the experimental development. At first, the resin with highly free branched structure was prepared mainly based on A a , and A a +B b type approaches. [13,14] This kind of polymerization is easy to gel. Flory carried out the theoretical research on hyperbranched polymers in 1940s. [15][16][17] He first demonstrated the possibility of preparing highly branched polymers from polyfunctional monomers, and calculated the critical gel conditions and relative molecular weight distribution of three functional and four functional branched monomers by means of probability statistics. In 1952, he further predicted the possibility that AB g type monomers could form highly branched polymers via condensation polymerization and derived the molecular weight distribution function for the products obtained by the probability method. [18] From another point of view, Stockmayer thinks that polymerization is similar to that of gas liquefaction. By using statistical mechanics methods of non-ideal gases, he derived the gelation conditions and size distribution functions of products for the A a , A 2 +A a , and A 2 +A a +B 2 (a > 2) type branching reactions. [19][20][21] Like the experiment of hyperbranched polymer synthesis, a new round of theoretical research climax also originated from the work of Kim et al. in the 1980s, [7][8][9] when they first synthesized highly branched polyphenylene with AB 2 monomer, and named this kind of product hyperbranched polymer. Because of its unique structure and characteristics, it is expected that it will be widely used in many fields, which has attracted the general attention of academia and industry. Hawker et al. [22] and Kim et al. [7][8][9] proposed the definition of the degree of branching (DB) of hyperbranched polymers by comparing them with perfectly branched dendrimers, Frey et al. [23,24] derived the formula of degree of branching for AB g hyperbranched polymers by kinetic and statistical methods respectively. Müller and Yan et al. developed the kinetic theory for AB 2 polycondensation and SCVP, and derived the analytical expressions of branching degree, polymerization degree distribution function, and other molecular parameters for the products. [25,26] Then Yan and Zhou [27][28][29][30][31] extended the kinetic method to the polycondensation of more general AB g type monomers and the reaction system in the presence of the In the theory of hyperbranched polymerization, due to the difficulty of deriving isomers of conformations, the early Flory's probability method and Stockmayer's statistical mechanics method have been rarely used. In this work, a new and easy way to derive the total number of configurations is developed, which can extend the application of the two theoretical me...

Going Beyond the Carothers, Flory and Stockmayer Equation by Including Cyclization Reactions and Mobility Constraints

“…Furthermore, the rate theory has been used to predict CLDs and the onset of gelation both for batch reactors [ 41 , 42 , 43 , 44 ] and for perfectly mixed continuous flow stirred tank reactors [ 45 , 46 ]. More recently, the rate theory was applied by other groups as well [ 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 ]. Finally, Schamboeck et al [ 55 ] applied the theory of percolation [ 56 ] on a directed random graph to derive analytical expressions describing the molecular structure of branched polymers synthesized via irreversible step-growth polymerization.…”