(188aa) Mathematical Modeling of the Molecular Weight Distribution in Catalytically Degraded Polystyrene

The recycling of thermoplastic polymers is an attractive approach to the reduction of plastic waste accumulation. However, since most waste streams contain several different immiscible thermoplastics, the task requires implementing either separation protocols or compatibilization methods. The latter strategy may be achieved in several ways, one of which is the reactive processing of the mixed polymers in conditions that promote the synthesis of a graft copolymer that could act as a compatibilizer. Since polystyrene (PS) and the polyolefins (PO) are two of the majority component in household plastic waste streams, there is an interest in developing strategies for dealing with mixtures of these two types of resins. Several authors have reported the use of Friedel-Crafts alkylations to achieve the reactive compatibilization of mixtures of PS and PO [Sun and Baker, 1997; Sun et al., 1998; Díaz et al., 2002, 2007; Gao et al., 2003; Guo et al., 2007]. Unfortunately, the same reaction conditions that promote the graft copolymer synthesis also promote the degradation of PS [Díaz et al. 2009, Pukánszky et al., 1981; Karmore y Madras, 2002; Nanbu et al., 1987]. Balancing the two types of reaction requires careful tuning of the reaction conditions, a task for which a mathematical model could prove very useful.

As a preliminary step in the construction of a complete mathematical model of the graft reaction, we have studied the degradation of PS in the presence of a catalyst (AlCl3) and a cocatalyst (Styrene (S)) appropriate for a Friedel Crafts reaction. The use of the well-known method of moments allows calculation of average molecular weights. Although in principle the full molecular weight distribution (MWD) can be recovered from its moments, the method has proven to be impractical. As an alternative, we apply the transform technique of probability generating functions (pgf) [Asteasuain et al., 2002a, 2002b , 2003]. For this task we define three pgf, each one associated to the MWD expressed in number, weight or chromatographic fraction(MWDn, MWDw, MWDc). It is the inversion of these transforms what allows the calculation of the MWD.

For the system under study we first propose a kinetic mechanism and formulate the mass balances for all species involved in it. The infinitely sized set of mass balances is then transformed, producing a set of moment balances and a set of pgf balances. The finite system of moment and pgf balances must be solved simultaneously using a numerical method suited for stiff systems. The result is the numerical value of the three pgf at specific values of the transform variable z. From this information, one must invert the pgf in order to recover the MWDn, MWDw and MWDc. Since only discrete numerical values of the pgf are available, numerical inversion is the only option. Several methods are available. We have used an adaptation of the method originally proposed by Papoulis (1956) for the inversion of Laplace transforms. Details of the adaptation may be found elsewhere [Asteasuain et al. 2002b]. This particular method was selected because it provides satisfactory results with a computational effort that is modest compared to that required by other inversion methods. The result is a set of numerical values of the sought MWD. It has been previously found [Brandolin et al., 2001, Asteasuain et al., 2002b] that the best results are found when each distribution is recovered from its own pgf. For example, it is best to recover the MWDn from the number pgf.

Using the proposed model we have been able to recover the MWD for the system PS/AlCl3 and for PS/AlCl3/S. We have compared the calculated MWDc with experimental distributions obtained by GPC from samples treated in a mixer operating above 190°C. The calculated MWDc agreed well with the experimental distributions, although the predictions show some discrepancy at the low molecular weight region. This is probably due to the numerical inversion method, which is known to have more accuracy problems at the extremes of the distribution. Nevertheless, average molecular weights calculated from the recovered distributions agree within 20% with the experimental ones.

References

Asteasuain M.; Sarmoria, C.; Brandolin, A. Polymer, 43, 2513-2527, 2002a.

Asteasuain M.; Brandolin, A.; Sarmoria, C. Polymer, 43, 2529-2541, 2002b.

Asteasuain M.; Sarmoria, C.; Brandolin, A. J. Appl. Polym. Sci. 88, 1676-1685, 2003.

Brandolin, A., Asteasuain, M., Sarmoria, C., López-Rodríguez, A.R., Whiteley, K.S., del Amo Fernández, B. Polym. Eng. Sci. 41, 1156-1170, 2001.

Díaz M.F.; Barbosa, S.E.; Capiati, N.J. Polymer 48, 1058-1065, 2007.

Díaz, M.F., Barbosa, S.E., Capiati, N.J. J. Appl. Polym. Sci. 114, 3081-3086, 2009.

Díaz, M.F.; Barbosa, S.E.; Capiati, N.J. Polymer 43, 4851-4858, 2002.

Gao, Y.; Huang, H.; Yao, Z.; Shi, D.; Ke, Z.; Yin, J. J. Polym. Sci. B-Polym. Phys. 41, 1837-1849, 2003.

Guo Z.; Fang Z.; Tong L.; Xu, Z. Polym. Degrad. Stabil. 92, 545-551, 2007.

Karmore, V.; Madras, G. Ind. Eng. Chem. Res. 41, 657-660, 2002.

Nanbu, H.; Sakuma, Y.; Ishihara, Y.; Takesue, T.; Ikemura, T. Polym. Degrad. Stabil. 19, 61-76, 1987.

Papoulis, A. Quat. Appl. Math. 14, 405-414, 1956.

Pukánszky, B.; Kennedy, J.P.; Kelen, T.; Tüdõs, F. Polym. Bull. 5, 469-476, 1981.

Sun, Y.-J.; Baker, W.E. J. Appl. Polym. Sci. 65, 1385-1393, 1997.

Sun, Y.-J.; Willemse, R.J.G.; Liu, T.M.; Baker, W.E. Polymer 39, 2201-2208, 1998.