Calculation of Molecular Weight Distributions in Non-Linear Free-Radical Polymerization Using the Numerical Fractionation Technique

Abstract

Mathematical Modeling of non‐linear polymerization systems subject to gel formation is a challenging endeavor. At the gel point, the second and higher molecular weight moments diverge to infinity making it impossible to obtain the molecular weight distribution (MWD). The numerical fractionation (NF) technique utilizes a refinement of the method of moments to model non‐linear polymerization systems that form gel. Since the method of moments yields results in terms of average quantities, some information is lost when reconstructing the MWD using NF. As a consequence, a broad shoulder appears at the high chain length end of the MWD tail. This study demonstrates that the validity of the gamma distribution deteriorates for the broader branched polymer generations and evaluates the performance of various alternative model distributions. Proper selection of the model distribution enhances the NF‐reconstructed MWD.

Transient Behavior and Gelation of Free Radical Polymerizations in Continuous Stirred Tank Reactors

“…This is a completely different (and much simpler) system and a comparison with this work is not relevant. Other methods were applied for the simulation of long-chain branching or vinyl/divinyl copolymerization in continuous free radical polymerizations: Monte Carlo simulation, [25] but retaining the steady state, ''numerical fractionation'' technique [23,47] without transient behavior analysis and Galerkin finite-element method [26] but without quantifying the amount of gel. This has been more recently taken into account by excluding from sol the polymer molecules with some high enough degree of polymerization.…”

“…Quasi-steady state approximation, neglecting poly-radicals, and closure conditions are examples of approximating conditions that can be found in published works where numerical fractionation technique or simplified versions of the method of the moments have been used. [22] When these methods are extended to the analysis of CSTR [23] additional approximation conditions like neglecting outflow of radicals are introduced.…”

“…In order to make a comparison between the predictions of the present method with published related works of numerical fractionation technique, some simplification in the kinetic scheme (see Table 3) is inevitable: in published works where this kind of polymerization systems is analyzed by numerical fractionation [22,23] only transfer to polymer is considered as branching mechanism. Therefore, we had to suppress terminal double bonds propagation from the kinetic scheme; this does not mean that numerical fractionation could not take into account that reaction with a supplementary effort, which is not really worth doing just for the sake of a comparison of very basic prediction capabilities as it is done in this work.…”

“…For these reaction conditions, the predictions of numerical fractionation technique have been obtained following the principles presented in literature. [22,23] Saidel and Katz approximation for the third moment, [49] was used as closure condition.…”

“…Due to similar difficulties, alternative techniques for modeling non-linear free radical polymerizations have also introduced some of the aforementioned approximating conditions. Such is the case of simulations of free radical polymerizations using ''numerical fractionation'', [22,23] Mote Carlo method, [24,25] and Galerkin finite-elements method. [26,27] The problems to be faced in the modeling of this kind of systems become much tougher if gel is present and if the transient behavior of the reactor is to be described.…”