Kinetic modeling

At the micro-scale, the polymer chain reactions take place that determine the chain microstructure which in return is linked to the final polymer properties. [72] To model the molecular processes of this length scale, a kinetic model is used. In this model, the polymerization is expressed in terms of kinetic rate constants and concentrations of reactants in order to describe the temporal course of the polymerization rates and molecular property distributions (e.g. molecular weight distributions). Since the kinetic constants needed for the model are catalyst specific, these need to be estimated using experimentally determined polymerization data, i.e. polymerization profiles for various conditions, in order to eventually simulate the polymerization behavior of the studied catalyst.

The elementary reaction steps occurring during the coordinative polymerization of olefins are very complex and not fully understood. To reduce the complexity but still being able to model

described by a set of standard reaction steps. [14, 72–74] Depending on the modeling objective and the experimental data available to estimate the numerous kinetic rate constants, a set of reaction steps is chosen which best describes the experimental observations. The general kinetic scheme (Table 2.2) for the polymerization of olefins via organometallic catalysts consists of the following basic reaction steps:

1) Catalyst activation 2) Chain initiation 3) Chain propagation 4) Chain transfer 5) Catalyst deactivation

The titanium sites of ZN catalysts are typically activated by a cocatalyst (e.g.

triethylaluminum). Other activation paths such as the activation by hydrogen, by monomer, or spontaneously are also possible. By this activation step, the potential catalyst site is converted to a vacant active site. A new polymer chain is created by the initiation step, in which one monomer molecule is added to the vacant active center forming a living polymer chain with a chain length of one. This chain can now grow by chain propagation. Here, the monomer is attached to the active site of the living chain, increasing the chain length by one monomer unit in each propagation step. The chain growth continues until a transfer reaction occurs. In the chain transfer reaction, the living polymer chain reacts with a chain transfer species. The living chain is terminated and a dead polymer chain and a vacant active site are produced. In industry, hydrogen is used to control the molecular weight since it is an effective chain transfer agent. Additionally, the transfer step may occur by other species such as monomer, cocatalyst or spontaneously (β-hydride elimination). The typical activity loss of Ziegler-Natta catalysts observed over time during the polymerization of olefins is believed to occur because of catalyst site deactivation. Both the vacant sites as well as the living polymer chains can deactivate forming a dead catalyst site or a dead polymer chain and a dead site, respectively. The deactivation step may happen spontaneously or by monomer, cocatalyst, hydrogen, or other species.

To the general kinetic scheme presented in Table 2.2 further reaction steps may be added.

Site transformation [73, 75–77] and the multi-site nature of ZN catalyst [78, 79] are often considered. For the latter case, the kinetic scheme is still valid, but would apply to each single catalyst site type. The multi-site approach is required in order to model the broad molecular weight distributions (MWD) of ZN catalysts. This can be achieved by MWD deconvolution techniques by which the number of site types is estimated. [4, 80]

Table 2.2: Overview of standard reaction steps for the kinetic modeling of coordinative polymerization. [14, 72] In this scheme, Sp, CoCat, Sa, M, H2, Pn, Dn and Sd are symbols for the potential catalyst site, cocatalyst, active catalyst site, monomer, hydrogen, living polymer of length n, dead polymer of length n and dead catalyst site, respectively.

Reaction Step Chemical Equation

A further important kinetic phenomenon observed for the polymerization of propylene is the rate enhancement by hydrogen. This can be explained by the dormant site theory (section 2.2.2) which was implemented in kinetic models in different forms. [4, 27, 36, 73, 81] The kinetic scheme can be simplified to three reaction steps (Table 2.3). [81] The dormant chain is formed by the regio-irregular 2,1-insertion of a propylene molecule into a living polymer chain. This dormant species can now be reactivated in two different ways: via hydrogen or monomer. The dormant chain can react with hydrogen to form a dead polymer chain and a hydrogenated active site (equally treated as a vacant active site). The reactivation step with hydrogen models the polymerization rate enhancement observed upon increase of the hydrogen concentration. With this kinetic step, the concentration of active sites is effectively increased, whereas the concentration of dormant chains not contributing to the polymerization rate is decreased. In a third reaction step, the reactivation of dormant chains

a living polymer chain is formed. Without this second reactivation step, the polymerization would come to a standstill in absence of hydrogen.

Table 2.3: Kinetic scheme for the rate enhancement by hydrogen based on the dormant site theory. [81] In this scheme, Pn, M, Pndorm, H2, Sa and Dn are symbols for the living polymer of length n, monomer, dormant polymer of length n, hydrogen, active catalyst site and dead polymer of length n, respectively. proceeds within the growing polymer particle. Since the (spatial) conditions within the particle such as temperature and monomer concentration directly affect the polymerization rates and therefore also the polymer properties, a special interest lies in modeling the processes of this length scale. To account for the physical processes occurring at the meso-scale, single particle models are employed connecting the micro- and macro-scale. Here, phenomena such as heat and mass transfer through the particle boundary layer and within the particle, particle growth and catalyst fragmentation are considered. Various particle models and modifications of these exist which were reviewed by Dubé et al. [82] and McKenna and Soares [83]. The two most common ones, the multigrain model (MGM) and the polymeric flow model (PFM), are explained briefly in the following.

The multigrain model was first developed by Yermakov et al. [84], Crabtree et al. [85] and Nagel et al. [86] between 1970 and 1980. Two levels of mass and heat transfer resistances are considered. The particle (macro-particle or secondary particle) is assumed to consist of an agglomerate of micro-particles (primary particles) (Figure 2.2) resembling the heterogeneous morphology of real polyolefin particles as observed by researchers such as Hock [87], Kakugo et al. [66, 67] and Noristi et al. [88]. Each micro-particle consists of a catalyst fragment covered by a layer of dead and living polymer. Monomer from the bulk phase first diffuses through the pores of the macro-particle and then through the polymer layer of the micro-particles to reach the active sites at the surface of the catalyst fragments where the polymerization occurs. The newly formed polymer accumulates there which leads to micro-particle and thus macro-particle growth. The MGM was further developed by