(188ac) Graft Copolymers for Blend Compatibilization. Mathematical Modeling of the Grafting Process

Polystyrene (PS) and polyolefins (PO) such as polyethylene (PE) and polypropylene (PP), are major components of urban plastic waste streams. Recycling them together, avoiding previous separation of resins, is an attractive alternative in terms of time and costs. However, mixtures of PS and PO are immiscible and show poor mechanical properties. This problem may be overcome by a compatibilization that enhances the adhesion between phases, improving the mechanical properties of the material [O'Shaughnessy and Sawhney, 1996; Sundararaj and Macosko, 1995]. One possible compatibilizer is a copolymer composed of the immiscible homopolymers. To achieve the compatibilization, the copolymer may be added to the mixture or generated in situ through reactive processing [Bisio and Xanthos 1995; Barentsen and Heikens, 1974].

In this work we study the synthesis of graft copolymers of PS and PE suitable for compatibilization of PS/PE blends. In order to select appropriate operating conditions that lead to a good compatibilizer, several aspects should be taken into account. For example, it has been reported that shorter homopolymer molecules increase the amount of copolymer formed, but compatibilization has been found to be more effective when the copolymer contains long homopolymer blocks [Díaz et al., 2007]. Other important variables are the concentrations of both the catalyst and cocatalyst. Given the number of variables and their competing effects, experimental determination of the optimal grafting conditions is not straightforward. A mathematical model of the process would be a useful tool to aid in this task.

In this context, we present a mathematical model of the graft copolymerization reaction between PS and PE, achieved through a Friedel-Crafts alkylation that uses AlCl3 and styrene (S) as catalytic system. We assume that the aromatic rings present in PS, as well as the double bonds in PE, may be activated by AlCl3 alone or by the joint action of AlCl3 and S, giving as a result the graft copolymer PE-g-PS. A second route to the copolymer is the reaction of PE with benzene, a side product of chain scission, followed by reaction with PS. We assume that only one molecule of PE may graft onto a PS chain. In view of reported experimental evidence [Díaz, 2004; Nambu et al., 1987], the following side reactions involving PS have been considered: chain scission, rearrangement to produce indane structures, and chain combination. Weak and normal bonds in the PS chain give rise to scission reactions at different rates [Chiantore et al., 1981]. Propagation involving S is considered. Finally, the catalyst is assumed to deactivate by decomposition or by reaction with S.

The model considers batch, isothermal operation in a well-mixed melt. Given that the chains of PE, PS and PE-g-PS have sizes that range from one to infinity, the mass balances of the species involved lead to an infinite set of differential equations. We have resorted to transformation methods to limit its size. In this work we use two: the method of moments, and bivariate probability generating functions (pgf). The former allows calculation of several average molecular properties from the moments of the size distribution. The pgf method is appropriate for obtaining the complete molecular weight distribution (MWD) and the chemical composition distribution (CCD).

After transformation, the resulting system of 740000 differential equations and 107628 algebraic equations was solved using the software gPROMS (PSE Entreprise Ltd.). From the first three moments of the distributions the average number and weight molecular weights were calculated. The resulting pgf were inverted numerically to recover the MWD of the unreacted homopolymers and the CCD of the synthesized graft copolymer. For this purpose we used the inversion method for bivariate pgf recently proposed by Asteasuain and Brandolin (2010).

Kinetic parameters for the side reactions were estimated previously [Gianoglio Pantano et al., 2010], using experimental data corresponding to PS/AlCl3 and PS/AlCl3/S reaction systems [Díaz et al., 2007a,b, 2009]. The parameters for the graft reactions were estimated using experimental data of average molecular weights and mass of grafted PS obtained in our laboratories [Díaz et al., 2007a,b]. These data were obtained using a melt mixer at 190°C at a constant ratio of PE to PS of 80/20 w/w. Experiments were conducted at a single S concentration with a single PS, two PE resins of different molecular weights, and several concentrations of AlCl3. Several mixing speeds and times of reaction were used. Experimental data on complete MWD were not used in the parameter estimation.

The experimental measurements indicate that the mass of grafted PS increases with AlCl3 concentration. It also shows that when using PE of higher molecular weight one obtains the lowest mass of grafted PS. The proposed model is able to predict correctly these two behaviors. When comparing the calculated values of Mw of unreacted PS with those obtained experimentally, we find that although the model predicts the general trend correctly, it underestimates most values. This may probably be attributed to the particular method of separation used to isolate the unreacted PS from the mixture [Martini, 2007; Martini et al., 2006].

Both the average molecular weight and the MWD calculated for the unreacted homopolymers indicate that PE does not suffer any modification during the reactive processing. However, PS degrades as indicated by the shift of its MWD to lower molecular weights. This agrees with experimental observations [Díaz, 2004].

The calculated CCD of the copolymer for several concentrations of the catalytic system indicate that all mixtures show a similar behavior, showing a bell-shaped distribution with respect to PS and a distribution with a maximum at low molecular weights with respect to PE. This behavior is reasonable, since the model considers shorter molecules to be more reactive. In the case of PS this effect is attenuated because the rate of formation of the graft copolymer is proportional to the length of the PS homopolymer.

We analyzed calculated curves for the number MWD of one of the copolymer blocks while keeping the other one constant, for varying concentrations of catalyst. We observed that the distributions of the blocks of PE are identical, reinforcing the idea that PE only takes part of the grafting reaction. However, the number MWD of PS are sensitive to the concentration of catalyst. The distributions shift towards the lower molecular weights. This is probably due to the degradation of PS, which leaves shorter molecules available for grafting.

We have found that the CCD show signs of inaccuracy in the region of high molecular weights, probably due to numerical error propagation inherent to the numerical inversion method used [Brandolin et al., 2001; Asteasuain et al., 2003a,b].

The model developed for the graft copolymerization of PS and PE has potential for its application in an optimization tool for the process, as it is able to account for the opposing effects of competitive reactions that characterize the studied graft reaction. The use of such a tool would simplify determination of the operating conditions necessary to synthesize a sufficient amount of a graft copolymer with appropriate block lengths that would optimize the compatibilization. The model also sheds some light on the theoretical understanding of this complex process.

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