Bemoullian model

One component obeys the Bemoullian model the other two obey the enantiomorphic- site model. Similarly, the NMR data of fractionated copolymers can be used to demonstrate the presence of multiple components in the copolymers. An example is shown of ethylene-propylene copolymers where the NMR/fractionation data are used to show the presence of two or three catalytic sites. 

The Bemoullian model (also referred to as the zero-order Markov model) assumes that only the last monomer unit in the propagating chain end is important in determining polymer stereochemistry. Polymer stereochemistry is not affected by the penultimate unit or units... 

There are alternate criteria for testing the different models [Bovey, 1972 Chujo, 1967 Inoue et al., 1971, 1984]. The Bemoullian model requires... 

Having established that a particular polymerization follows Bemoullian or first-order Markov or catalyst site control behavior tells us about the mechanism by which polymer stereochemistry is determined. The Bemoullian model describes those polymerizations in which the chain end determines stereochemistry, due to interactions between either the last two units in the chain or the last unit in the chain and the entering monomer. This corresponds to the generally accepted mechanism for polymerizations proceeding in a nonco-ordinated manner to give mostly atactic polymer—ionic polymerizations in polar solvents and free-radical polymerizations. Highly isoselective and syndioselective polymerizations follow the catalyst site control model as expected. Some syndioselective polymerizations follow Markov behavior, which is indicative of a more complex form of chain end control. 

Figure 2.18 Test of Bemoullian model. Data for polymer 1 of Problem 2.14.

The diad and G-centered triad intensities measured for the whole polymers are reported in Table I. As expected, all three samples fitted poorly to one-conq)onent 1 order Markovian (or Bemoullian) models. The mean deviations in all three cases are larger than 1%. Instead, the data fitted reasonably well to two-component 1 order Markovian models. This finding is consistent with the latest NMR studies on alginates (16,17) which also indicated that alginates NMR data should be fitted not to one-con onent models, but to multicomponent models. 

Figure 2.15 Test of Bemoullian model with data for polymer 1 and polymer 2 of Problem 2.13.

The Bemoullian model and the first-order and second-order Markov models of the copolymerization mechanism can be tested by using the observed distribution of the triads. The results for the three models are shown in Table 7.15 for the sample with VA = 0.31%. 

It is evident that for polymer B (VA = 0.31), the Bemoullian model can be abandoned and that a significant amelioration of the results is achieved by going from first-order Markov to second-order Markov analysis. Thus the propagation statistics of polymer B is adequately described by a second-order Markov process. — G. van der Velden [24]... 

An excellent way to treat such data is to use reaction probability models.(1,2) In the NMR analysis of tacticity, it is frequently possible to distinguish whether the configuration is chain-end controlled or catalytic-site controlled during polymerization. Various statistical models have been proposed. The chain-end controlled models include Bemoullian (B), and first- and second-order Markovian (Ml and M2) statistics.(1) The simplest catalytic-site controlled model is the enantiomorphic site (E) model.(3) The relationship between the chain-end and catalytic-site controlled models and possible hybrid models have been delineated in a recent article.(4)... 

For NMR studies of polymer mixtures, the earliest approach proposed was the Coleman-Fox model.(5) This model assumes the coexistence of two interconverting Bemoullian propagating sites and was used extensively for poly(methyl methacrylate).(6-8)... 

The results for the 2-slte (B/B) model are satisfactory. The weighted sums (Table IX) indicate that there are two sites of roughly equal weights (41% w j for hexane set, and 51% W ] for ether set). The average Bemoullian reaction probabilities for the two sites are = 0.58, Pp,2 = 0.20. These results... 

For completeness, the results of the 3-state (B/B/B) model are also included in Table VIII and Table IX. As expected, the data fitted three Bemoullian polymers very well with the following Bemoullian probabilities ... 

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. 

More convenient and entirely sufficient for the present purpose is the calculation of ratio of rate constants. The calculation will be reviewed for a one-way first-order Markov process. A one-way mechanism is chosen because it is intuitively the most appropriate model for a free radical mechanism. Furthermore it has some experimental support. The assumption of a first-order Markov process does not rule out higher Markov processes. The differentiation between a first-order Markov process and higher order Markov processes is however possible experimentally in very rare cases because it involves the determination of tetrad, pentad, etc. fractions (11, 12, 13, 14). A Bemoullian process is ruled out by the analysis of the data of Table I. 

The steric pentad distributions of the polypropylene with structure (14) are in accord with the symmetric Bemoullian statistical model (Eq. 30) based on stereochemical control by the last propylene unit of a growing chain end. 

You can now test the models. So far we have only talked about Bemoullian statistics, where the stereochemistry of the addition of a monomer does not depend upon the configuration at the end of the chain. Just to make sure you understand, from NMR triad information—i.e., (mm), (mr) and (rr) data—we can readily calculate the conditional probabilities P(r/m), P(m/m), P(m/r) and P(r/r) as shown in Equations 7-32. Bemouillian statistics are followed if Equations 7-33 are obeyed. 

In the early 1940s when the polymerization theory was developed, tiie ideal, terminal, and penultimate models for fhe copolymerization were established also the possible distribution laws for the monomer sequence along the copolymer chains were defined Bemoullian, firsf- and second-order Markoffian. ... 

By determining which statistical model is followed in a polymerization, such as Bemoullian, or Markov, or other, it should be possible to understand better the mechanism of steric control. Thus the Bernoulli model describes those reactions in which the chain ends determine the steric arrangement. These are polymerizations that are carried out under conditions that yield mostly atactic polymers. The high isotactic sequences follow the enantiomorphic site model and the high syndio-tactic ones usually follow the Markov models. 

Next, the model requires the definition of the concentration and distribution (e.g., Bemoullian, Markovian, etc.) of defects. If a small concentration of defects with a random distribution is considered, defects most probably are isolated in the host polymer ID lattice. When the concentration increases, even a random distribution generates both isolated defects and a distribution of islands of various lengths (see, for instance, [90, 91]). 

Matrix-assisted laser desorption ionization MS is well represented in this area [56, 85-94]. Cox et al. [88] used MALDI-MS to characterize low molecular weight polyolefin copolymers of isobutylene and paramethylstyrene and to extract composition information from spectra. Comparison of experimental oligomer distributions to a Bemoullian statistical model revealed severe overrepresentation of oligomers with higher relative amounts of paramethylstyrene. However, good agreement between the model and experimental data was obtained by the introduction of an ionization efficiency term in the model used. The differences between average composition determined by MALDI and NMR were discussed. 

The relative concentrations of the longer configurational sequences can be calculated by using probability theory and the appropriate polymerization model. The manner in which the relative proportions of the various sequences can be calculated for the Bemoullian or terminal model of the stereochemical polymerization is shown in Table 7.6. 

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