A well-known approach for the calculation of the distributed molecular properties of polymers (e.g., MWD, LCBD, CCD, etc.), is the use of bivariate population balance equations (PBEs), (Kiparissides, 2006). In principle, based on the polymerization kinetic mechanism, one can derive dynamic PBEs to describe the time evolution of the “live” and “dead” polymer chains in a polymerization reactor. The resulting PBEs can be solved numerically using well-established numerical methods (i.e., 2-D fixed pivot, orthogonal collocation on finite elements, etc.). A major limitation in the application of these methods is the fact that they are extremely complex mathematically and they cannot be extended to more than two dimensions (i.e., they cannot be applied to trivariate or multi-dimensional problems).

To deal with the above limitation, a novel Monte Carlo (MC) algorithm was developed. The proposed algorithm can be used for the accurate prediction of the complete topological characteristics of the polyethylene polymer chains produced in high pressure tubular and autoclave LDPE polymerization reactors. Via the MC algorithm, it is possible to predict the exact size and position of every single chain branch (either short- or long-) that is attached to a given polymer chain in the polymerization mixture, thus leading to the calculation of the complete topological structure of the polymer chains. Furthermore, no prior information on the distributional form of the “live” or “dead” polymer chains is required, neither any other kind of simplifying kinetic assumptions. The algorithm considers the evolution of both “live” and “dead” polymer chain populations and proceeds in complete conjunction with the polymerization kinetic mechanism.

The basic principles governing the proposed stochastic formulation are based on the original developments of Gillespie (1977). According to the above formulation, the desired molecular properties of the polymer chains are inferred by tracking the relevant changes occurring in a sample population, containing the different molecular species (Meimaroglou et al., 2007 and 2008). The net formation rates of all the chemical reactions in the reacting system are dictated by known reaction probabilities and are directly related to the polymerization kinetic mechanism.

The proposed MC algorithm was successfully applied to the free radical polymerization of LDPE, for the prediction of a series of distributed and topological properties of the polymer chains. The validity of the mathematical model was initially verified through a direct comparison of MC predictions with available experimental measurements on the molecular weight distribution (MWD). Moreover, the MC calculated average molecular properties (i.e., Mn, Mw, etc.) were directly compared with theoretical predictions from the application of the well-established method of moments (Kiparissides et al., 2005) in order to further verify the predictive capabilities of the method.

Via the MC calculations, a number of distributed topological properties can be obtained. These properties are directly related to the number (MW-LCBD, SCBs and LCBs per molecule, SCBs and LCBs per 1000C distribution), size (SCB and LCB length distribution) and order (branching order distribution) of the polymer chain branches. Additionally, the shape of the produced polymer chains can be inferred through the dynamic prediction of the seniority-priority distributions and the mean radius of gyration. The latter can lead to a straightforward calculation of the branching index of the polymer chains, g. Finally, all the information obtained by the MC algorithm can be further employed for the prediction of the viscoelastic properties of the polymer melts.

References

Gillespie D.T.J., Phys. Chem. 1977, 81, 2340.

Kiparissides C., J. Process Contr. 2006, 16, 205.

Kiparissides C., Baltsas A., Papadopoulos S., Congalidis J.P., Richards J.R., Kelly M.B. and Ye Y., Ind. Eng. Chem. Res. 2005, 44, 2592.

Meimaroglou D. and Kiparissides C., to be submitted to J. Polym. Sci. Polym. Phys. Ed. 2008

Meimaroglou D., Krallis A., Saliakas V. and Kiparissides C, Macromolecules 2007, 40, 2224.