pubs.acs.org/Macromolecules Published on Web 06/15/2010 r 2010 American Chemical Society 5820 Macromolecules 2010, 43, 5820–5832 DOI: 10.1021/ma1005233 A Novel Stochastic Approach for the Prediction of the Exact Topological Characteristics and Rheological Properties of Highly-Branched Polymer Chains Dimitrios Meimaroglou and Costas Kiparissides* Department of Chemical Engineering, Aristotle University of Thessaloniki and Chemical Process Engineering Research Institute, P.O. Box 472, Thessaloniki, Greece 54124 Received October 16, 2009; Revised Manuscript Received May 26, 2010 ABSTRACT: A novel stochastic algorithm is described for the accurate prediction of the detailed molecular topology of highly branched polymer chains. Stochastic topological polymer chain simulations are carried out in conjunction with a polymerization kinetic MC simulator. Contrary to previous efforts on the stochastic simulation of topological features of branched polymer chains, the present approach does not apply any simplifying assumptions regarding the distributional form of “live” and “dead” polymer chain populations. The proposed stochastic approach takes explicitly into account the effects of various diffusional limitations in the termination and propagation rate constants (i.e., gel- and glass-effect) as well as the effect of branching density on the kinetics of various reactions (i.e., transfer to polymer and chain scission reactions). To demonstrate the predictive capabilities of the proposed stochastic approach, the free-radical polymerization of ethylene in an industrial high-pressure tubular reactor is investigated. It is shown that the present stochastic kinetic/topology algorithm can provide detailed information on the topological features of highly branched polymer chains (i.e, long- and short-chain branching distributions, segment seniority and priority distribu- tions, etc.). The topological information, obtained by the application of the stochastic kinetic/topology algorithm, is then used together with a 3-D molecular random-walk simulator to predict the 3-D random spatial configurations of branched polymer chains as well as some important rheological parameters (i.e., the mean radius of gyration, R g , the mean hydrodynamic radius, R h , and the average branching factor, g) of low-density polyethylene. Introduction Control of the molecular architecture of polymer chains produced in a polymerization reactor is a subject of profound interest to polymer scientists and industry as well. This originates from the well-established fact that the molecular properties (e.g., molecular weight distribution (MWD), copolymer composition distribution (CCD), long-chain branching distribution (LCBD), etc.) are directly linked with the polymer end-use properties (e.g., physical, chemical, mechanical, rheological, etc.). Hence, the elucidation of the molecular architecture of highly branched polymer chains in terms of the kinetic mechanism and polymer- ization conditions (e.g., temperature, pressure, mixing, etc.) has been the subject of a great number of theoretical and experi- mental studies. A well-known approach for the calculation of the distributed polymer molecular properties (e.g., MWD, LCBD, etc.) is the use of multivariate population balance equations (PBE). 1 In principle, dynamic PBEs can be derived to describe the time evolution of the “live” and “dead” polymer chains in a polymerization reactor in terms of a specific polymerization kinetic mechanism. In the open literature, a number of numerical methods have been proposed to reduce the resulting infinite system of differential equations into a low-order system. These include the kinetic lumping method, 2-4 the polynomial expansion method, 5 the global ortho- gonal collocation, 6 the method of moments, 7,8 the “numerical fractionation” method, 9,10 the discrete weighted Galerkin 11,12 and the orthogonal collocation on finite elements and sectional grid methods. 13 Despite their computational complexities, in a series of recent publications, 14-16 the application of orthogonal collocation on finite elements and fixed pivot methods to free-radical linear and nonlinear polymerization systems was demonstrated. An alternative approach to the above deterministic methods is the use of probabilistic tools (e.g., Monte Carlo simulations). Stochastic simulations have attracted significant attention over the last 30 years, due to their inherent capability to simulate the discrete and random nature of polymerization kinetics. The application of MC methods to complex polymerization systems has been facilitated by the dramatic increase in computer power that has gradually eliminated the main disadvantage of stochastic methods associated with their high computational requirements. It should be pointed out that stochastic simulations have an additional advantage over the commonly employed deterministic numerical methods for they can deal with multidimensional problems (e.g., three- and/or higher-dimensional problems) in a simple and efficient way, making them ideal computational tools for the calculation of the multidimensional distributed molecular properties (i.e., joint MW-LCB distribution, joint MW-CC distribution, etc.). In the past, a number of different stochastic approaches have been proposed 17,18 and applied to both homo- polymerization 19-21 and copolymerization 22-25 systems. How- ever, in most cases, a number of simplifying kinetic and chain distributional assumptions have been applied to facilitate the stochastic numerical solution. In principle, the average and distributed molecular properties of polymer chains in a polymerization system (e.g., M n , M w , MWD, CCD, etc.) can be calculated by using either deterministic or stochastic numerical methods. However, a number of mole- cular properties (i.e., the topology and distribution of short- and long-chain branches, the branching order distribution, etc.) that *Corresponding author. Tel: þ 302310 99 6211; fax: þ2310 99 6198; e-mail: cypress@alexandros.cperi.certh.gr.