Catalyst sites, statistical models

The polymer stereosequence distributions obtained by NMR analysis are often analyzed by statistical propagation models to gain insight into the propagation mechanism [Bovey, 1972, 1982 Doi, 1979a,b, 1982 Ewen, 1984 Farina, 1987 Inoue et al., 1984 Le Borgne et al., 1988 Randall, 1977 Resconi et al., 2000 Shelden et al., 1965, 1969]. Propagation models exist for both catalyst (initiator) site control (also referred to as enantiomorphic site control) and polymer chain end control. The Bemoullian and Markov models describe polymerizations where stereochemistry is determined by polymer chain end control. The catalyst site control model describes polymerizations where stereochemistry is determined by the initiator. 

Syndioselective polymerization by a Cs metallocene such as Me2C(Cp)(Flu)ZrCl2 proceeds by catalyst site control. A statistical model for syndioselective catalyst site control has been described in terms of the parameter p [Resconi et al., 2000]. Parameter p is the probability of a monomer with a given enantioface inserting at one site of the initiator p is also the probability of the monomer with the opposite enantioface inserting at the other site of the initiator. The pentad fractions are given by... 

More complex model reactions (selective butadiene hydrogenation) Apart from being accessible to surface spectroscopy, model catalysts also have the advantage that the nanoparticle morphology and surface structure can be accurately measured. This advantage allows the determination of the relative abundance of specific surface sites and calculation of surface site statistics, as shown, for example, in Table II. Knowledge of the exact number and type of available surface sites then allows calculation of more accurate (and perhaps more meaningful) turnover frequencies of catalytic reactions. 

The more complex selective 1,3-butadiene (BD) hydrogenation was also examined [56, 57]. Butadiene hydrogenation produces 1-butene, tranx-2-butene, cti-2-butene, and n-butane, with 1-butene as the desired product. Pd-Al Oj model catalysts with mean particle diameters of 2-8 nm were applied to examine size effects. The abihty to accurately determine the relative abundance of specific surface sites (such as terrace, edge, interface atoms, etc. cf. surface site statistics in Table 1, 2 of [51]) is a tremendous advantage of model catalysts. Knowledge of the exact number and type of available surface sites allows the calculation of more accurate turnover frequencies. 

Statistical modeling of pentad distributions of polypropenes prepared with these catalyst systems can be satisfactorily done using a two-site modeF - based on a mixing of a chain-end-controlled site (to model the atactic blocks) and an enantiomorphic site (for isotactic blocks). ... 

It has been subsequently shown by Inoue et al. [29] that a two site model of Ziegler-Natta polymerization of propylene [30] adequately describes the distribution of stereosequences observed in atactic PP. At one of the catalyst sites the monomer addition obeys Bernoullian statistics, and at the other site a predominance of monomer units is added in only one of the two possible configurations (R,S ord, 1). 

Costeux, S. Statistical modeling of randomly branched polymers produced by combination of several single-site catalysts Toward optimization of melt properties. AfacromoZ. (2003) 36,pp. 4168-4187... 

Statistical Models of Catalyst Sites. Pentad and higher sequence distributions from chiral homopolymers have been treated statistically by so-called two... 

If a Ziegler Natta catalyst produces both isotactic and atactic polymers simultaneously during a 1-olefin polymerization, the isotactic triad produced by the catalyst sites can be modeled statistically as (17) ... 

As an example, polybutylene is a commercially inq ortant polymer made with Ziegler-Natta catalysts. Such catalysts frequently produce more than one catalytic site, and the resulting polymer is a blend of several homopolymers differing only in propagation statistics. Previously, the two-site E/B model has been used to analyze the tacticity of this polymer.(11,12) Reasonably good fits with experimental data were observed. 

The Pti samples (182) were prepared as colloids, protected by a PVP polymer film. Layer statistics according to the NMR layer model (Eqs. 28-30) for samples with x = 0,0.2, and 0.8 are shown in Fig. 63. The metal/ polymer films were loaded into glass tubes and closed with simple stoppers. The NMR spectrum and spin lattice relaxation times of the pure platinum polymer-protected particles are practically the same as those in clean-surface oxide-supported catalysts of similar dispersion. This comparison implies that the interaction of the polymer with the surface platinums is weak and/or restricted to a small number of sites. The spectrum predicted by using the layer distribution from Fig. 63 and the Gaussians from Fig. 48 show s qualitative agreement w ith the observed spectrum for x = 0 (Fig. 64a). 

Insertion of propene into alkylzirconium model compounds derived from isospecific bis-indenyl catalysts may lead to low enantiospecificity [65]. Thus, site control alone does not lead to high stereoregularities. Molecular mechanics calculations [66] and a thorough analysis of the substituent effect on the statistical distribution of microstructure defects indicate that, firstly, the polymer chain assumes the energetically most favourable position with respect to the (asymmetric) site. According to Corradini, the indenyl ligand or the chlorine atoms of the lattice will direct the pol)rmer chain. The polymer chain occupies the "free space" near the site. In Fig. 6.21 we have schematically drawn this for the TiCls... 

The most efficient metal catalysts consist of relatively small clusters with tens or hundreds of atoms exposed to the gas phase and accessible as adsorption sites. On the other hand, a detailed description of reaction in the adsorbed layer, even in the case of methane, can include a comparable number of types of surface intermediates for instance, Mhadeshwar and Vlachos (2005) take into consideration 18 different surface species. In the latter case the authors modeled the reaction on a macroscopic surface, but anyway, certain caution is required when we apply normal kinetic equations to a reacting system where the conditions for statistical behavior are not assured. 

FIGURE 81 Statistical distribution of LCB within the MW distribution as determined from model calculations based on random incorporation. A Single-site catalyst having Mw/Mn— 2.0, B Cr/silica catalyst having MW/MN 83, assuming that sites exhibit the same relative preference for macromer incorporation as for 1-hexene. 

The polymerization reaction is a sequence of different events, such as monomer insertions, site isomerizations, and chain release reactions. The polymer chain can be seen as a permanent picture of the sequence of these events, and it is possible to use a statistical approach to study their distribution along the chain to increase our knowledge on polymerization mechanisms. As a consequence, a mathematical model of the polymerization can be built by assigning a probability at each event in our system. In the case of propene homopolymerization, this approach is (largely) used to study the mechanisms governing the stereoselectivity of the catalyst from the NMR spectrum of the polymer. In fact, the type and the relative amount of the stereosequences present in the chain are obtained from the methyl region of the spectrum and are usually determined at the pentad level (see section II.G). This distribution can be studied using insertion probabilities for propene enantiofaces, which depend on the type of stereocontrol mechanism active for the catalytic... 

Enantiomorphic Site with Chain-End Control. In the case of less stereoselective Cz-symmetric metallocene catalysts, the magnitude of chain-end control can be comparable to that of site control. In this case, obviously, the former has to be added to the model using Markovian statistics. The probability parameters are the same found for pure chain-end control p si re), i.e., the probability of insertion of a si monomer enantioface after a monomer inserted with the re face, p re si), p si si), and p re re). In this case, the metallocene chirality prevents the equiprob-ability of the si olefin insertion after a re inserted monomer (see structure on the left in Scheme 36) and re olefin insertion after a si inserted monomer (see structure on the right in Scheme 36). 

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