Topology Evolution in Polymer Modification a Ivan Kryven,* Piet D. Iedema A recent numerical method has opened new opportunities in multidimensional population balance modeling. Here, this method is applied to a full three-dimensional population balance model (PBM) describing branching topology evolution driven by chain end to backbone coupling. This process is typical for polymer modification reactions, e.g., in polyethylene, where initially linear polymer chains undergo hydro- gen abstraction, and subsequent branching or scis- sion. Topologies are distinguished by chain ends, number of branches, and number of reactive ends. The resulting time dependent trivariate distribution is utilized to extract various distributive properties of the polymer. The results exhibit excellent agreement with data from Monte Carlo simulations. 1. Introduction Modeling of polymer systems has been approached in two different ways: stochastic or Monte Carlo (MC) simulations and deterministic population balance models (PBM). The first way produces a large amount of sample molecules and requires a generalization step to evaluate their properties statistically. The second describes a polymer system quantitatively on the basis of a few of the most important properties. The MC methods are strong in predicting morphology related properties, while various different implementations are available. [1,2] In contrast, determin- istic PBMs are focusing on a small number of properties and describe a polymer system by a compact and straightfor- ward derived system of balance equations. From the viewpoint of optimizing the mass production of polymers and predicting physical properties, the deterministic methods are very attractive, as a high precision is often of high importance. Although the majority of the studies presented in literature refer to one dimensional PBMs, it is the numerical complexity one has faced when trying to solve the mathematical problem rather than a lacking demand for solutions, that have kept researchers out of investigations on multidimensional PBM. [3–5] Indeed, tackling the problem of branched topologies deterministi- cally makes unavoidable the consideration of at least two variables, chain length and amount of branches. Further- more, the amount of reactive sites/reactive ends, also forms an essential dimension, as it affects the reactivity. A lack of multidimensional PBM solvers on one hand, and attractiveness of the population palaces in a few dimen- sions on the other, has driven researchers into elaborate attempts to design hybrid methods. [3–7] The hybrid methods, which combine PBM along one dimension and a method of moments [3,4,5,7] or stochastic simulations [6,8] along the others have been introduced at a price of complicated reasoning and analytical work. However, these attempts have remained case studies of single problems and are hardly transferable to other, even similar systems. The only attempt of developing a more general hybrid method has been done for the MC-Galerkin technique by Sch€ utte and Wulkow. [6] In previous work we have taken a different course. [9] Instead of designing a hybrid method, we have invented a numerical approach capable of handling PBM in more than one dimension treating each of them equally. This permits not only an easy and transparent implementation, but also enables a better transferability to various PBM problems arising in polymer reaction engineering. The approach was successfully applied to polymerization [10] and polymer modification [9] problems in two dimensions. I. Kryven, P. D. Iedema University of Amsterdam, Science Park 904, 1098 XH, Amsterdam E-mail: i.kryven@uva.nl a Supporting Information is available from the Wiley Online Library or from the author. Full Paper ß 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/mats.201300121 7 Macromol. Theory Simul. 2014, 23, 7–14 wileyonlinelibrary.com