The kinetics of Ziegler-Natta polymerization, like almost all of the physicochemical aspects of the process, are extremely complex.
Furthermore, the most notable advances in this field have been achieved in homogeneous polymerizations. In order to apply this knowledge to heterogeneous cases (which have greater commercial value), it is generally necessary to review each piece of knowledge individually. The fundamental principles for homogeneous cases are outlined below.
These, with the necessary adjustments, can be considered in heterogeneous polymerization. The behavior observed in vitro for heterogeneous polymerizations can be summarized in Figure 3.7.
The behavior described by curve (1) generally occurs when the solid particles containing the transition metal component are relatively large. These particles consist of aggregates of very small crystals or grains, which represent the smallest elementary units that make up the physical support/catalyst particles fed into the reaction.
Figure 3.7: Observed polymerization rates
These large particles undergo a process of fragmentation, the limit of which is determined by the grains, as the compressive forces generated by the growth of molecules resulting from polymerization act on the bonds between the grains.
This is the general interpretation, typically offered by researchers from the perspective of reaction chemistry, which helps explain an increase in the “rate” of polymerization during this fragmentation stage, associated with increased accessibility to active sites at the edges of the grains as they separate from one another. Once, according to this explanation, a state of saturation is reached in the capacity for fragmentation—associated with the point at which the particles have fragmented down to their individual grains—the “rate” stabilizes.
Curve (2) corresponds to smaller particle sizes, achieved, for example, by grinding large particles to reduce their diameters. It should be noted that other factors,
Factors such as the rate at which active sites form through the in situ addition of cocatalysts and modifiers, or the presence of poisons, also affect the shape of curves (1) and (2).
Curve (3) is interpreted, for in vitro conditions, as resulting from the presence of short-lived active sites that are present at the onset of the reaction and whose activity subsequently declines.
However, other effects—such as the ability (or inability) of solid particles to dissipate reaction energy, combined with intraparticle mass transport phenomena—may be sufficient to account for these instances of local increases in the “rate” of polymerization.
In reality, this approach to explaining the curves observed in various polymerization processes is decidedly one-sided, stemming from a chemistry-centric perspective in the analysis of the situations mentioned, as well as from the in vitro nature of the observations. These phenomena will be analyzed in detail when describing, in later chapters, the REF and IRSF phenomena related to substrate/catalyst fragmentation.
Finally, curve (4) graphically represents the phenomenon of catalyst deactivation due to effects resulting from aging (loss of catalytic capacity due to internal morphological or chemical changes) or poisoning (transfer of inactive chemical species from the fluid medium) of the active site.
The termination mechanisms can be summarized in the following diagrams (the thick vertical bars represent the active sites of the transition metal):
a) Termination by spontaneous internal reaction (internal transfer to hydride),
b) by transfer from the chain to the monomer,
c) or by transfer to the group (Metal I-III)-R
d) or by reaction with active hydrogen
The relative extent of the various proposed termination mechanisms depends on the concentration of the species involved, the temperature, and, of course, the catalyst used.
For example, the termination mechanism described in (a) is virtually nonexistent in propylene polymerizations at temperatures below 80 C (which is typical in some industrial production processes).
H2 transfer is typically used to control the molecular weight, as it regenerates the active site.
It is common practice to use Langmuir-Hinschelwood-type equations locally (that is, for each active site) to quantify the steps described above. One of the methods traditionally used involves considering the reaction between two adsorbed species (Me-R and M), which compete for the same active sites on the catalyst surface.
The model presented by Odian (“Principles of Polymerization,” 1981) assumes that the two species rapidly reach adsorption equilibrium and remain in this state for the entire lifetime of the active site during polymerization. The fractions θA (for Me-R) and θM (for the monomer) are given by:
where [A] and [M] represent the concentrations of Me-R and monomer in the fluid phase adjacent to the active site, and the K values represent the equilibrium constants for adsorption.
Propagation occurs through the reaction of the monomer adsorbed at the adsorption-activated sites and the reaction of the Me-R groups on the edges of the supported transition-metal crystals. According to this scheme, the rate would be given by
where [S] is the total concentration of adsorption sites, generally expressed in moles per liter, in order to maintain the analogy with the typical Rp and kp values for homogeneous processes.
Combining the three preceding equations yields the form of the polymerization rate for the Langmuir-Hinschelwood model that has been developed:
The degree of polymerization—or, more precisely, its inverse—can be calculated by dividing the expression obtained by summing all the termination terms by the propagation term, resulting in:
An alternative model is the Rideal model, which assumes that polymerization occurs between unadsorbed monomer (directly from the liquid phase) and the active site. This mechanism is expressed as follows:
Both models, as well as others, are used to model specific cases depending on the catalytic system employed. These models, which have proven useful for unsupported catalysts, do not always accurately represent what happens with supported catalysts.
As noted at the beginning of this section, the standard procedure for tracking the polymerization rate in heterogeneous systems is generally described by first-order pseudo-kinetics as follows:
where [C] is the concentration of active sites.
Some typical values for constants and concentrations are shown in Table 3.2 for a polyethylene catalyst consisting essentially of (C2H5)2ClAl-γ-TiCl3, at 40° C.
Table 3.2: Typical Parameters for a ZN Catalyst for PE