Bulk Crosslinking Copolymerization: Comparison of Different Modeling Approaches

Abstract

The predictions of four different models of crosslinking copolymerization,Kinetic Monte Carlo (KMC), statistic/kinetic Flory/Tobita (FT) model, and two kinetic approaches based on population balance equations (PBE) (solved with generating functions (GF) and numerical fractionation (MRNF), respectively), were compared. The approaches underlying more restrictive assumptions but asking for less computational effort, FT and MRNF, lead to very satisfactory predictions in terms of average properties (sol and gel fractions, degrees of polymerizations, crosslinking densities). On the other hand, fully detailed models (KMC) as well as more computationally demanding numerical solutions of the detailed PBE (GF) become necessary when the chain length distributions are required.

Transition into the gel regime for crosslinking radical polymerisation in a continuously stirred tank reactor

“…An important feature of the model expressed by (8) is that the total mass conservation is naturally satisfied in both pre-gel and gel regions,…”

“…Lazzari et al [8,9] focused on the issue of multiradicals importance employing models of three parametric dimensions. With no exceptions, the level of exactitude intended by multidimensional description was reduced in the above-mentioned studies as a consequence of further treatment by the moment approach: only average quantities are recovered but not the original multidimensional distributions that the models are based on.…”

Solution of population balance equations by logarithmic shape preserving interpolation on finite elements

“…For this reason, the differential equation set, known as population balance equations (PBE), has been used extensively since its formal introduction. [7,8] PBE have been employed to rationalize polymerization reactions [9,10,11,12,13,14,15,16,17,18,19,20,21], colloid aggregation, [22,23,3,24,25,26,27,28] and crystallization processes [29,30,31,32,33]. This short list proves the versatility of the PBE, that enable the description of virtually any type of population, focusing on some key property such as the number of monomeric units in polymer chains, or the size of crystals.…”

“…[8] Stochastic methods are indeed very interesting and may reveal structural and topological information of the assemblies, but their lengthy computational times make them unsuitable to be employed for process optimization. [18,8] Deterministic approaches overcome this hurdle and allow the quantification of kinetic rates. An overview of the available deterministic strategies to solve the PBE is discussed in the following, considering a one dimensional distribution function in its continuous formulation.…”

“…[34,35] Nevertheless, the method of moments has been proven very powerful in dealing with multidimensional and complex balances, for instance in the case of non-linear polymerizations. [11,18] An improvement of the method of moments is represented by the quadrature method of moments (QMOM) and its variations. [28,35] By solving a few more differential equations (usually 6-12), these approaches still suffer of the necessity of assuming a distribution shape when reconstructing f (x, t), but overcome the remaining issues of the method of moments.…”