Bernoullian model

A portion of the PV A database is shown in Figure 3. Boxed entries indicate required input from the user. In this case, a Bernoullian model is the most likely choice. The program completes the database by calculating a chemical shift window and the coefficients from Equation 1 for each peak. 

When the polymerization mechanism is known, peak intensities in the spectra of polymers with different monomer compositions can be compared with calculated intensities based on various polymerization models [e.g., first order Bernoullian (B) or first order Markovian model]. For example, if polymerization mechanism of a vinyl monomer fits a Bernoullian model, then the fractions of triads will be given by ... 

Bovey adopts a two-state model where the preceding unit is the monomer unit immediately preceding the active chain end. The Bovey-Bernoullian model is now described at dyad levels by 4 reactions defined by the 4 probabilities of occurrence of the 4 dyads ... 

Table 13. Pentad Fractions for an Isotactic Bernoullian Model...

The resulting isotactic Bernoullian model (this is a true Bernoullian model, as the chain end effect is neglected) gives the calculated pentad distribution reported in Table 13. Each pentad contains two symmetric contributions for example, the mmmm pentad derives from the two possibilities... 

In this case the probability of insertion of a monomer with a given enantioface at site 1 is equal to the probability of insertion of a monomer with the opposite enantioface at site 2. This probability is indicated with the parameter a. The resulting expressions for pentad distribution for the syndiotactic Bernoullian model are reported in Table 15. 

Table 15. Calculated Pentad Distribution with the Syndiotactic Bernoullian Model...

Table 16. Pentad Fractions for a Syndiospecific Bernoullian Model in Presence of Site Isomerization ...

The Bernoullian model The Bernoullian model is the simplest of the statistical models used to describe copolymerisation, whereby the addition of each comonomer to a growing polymer chain is regarded as a random process. Thus, the framework for the model is that the probability of addition of a given monomer unit to a growing polymer chain is only dependent upon the mole fraction of the monomer in the feed. 

Table 2.2 Bernoullian model dyad and triad sequence abundance [4] Pa + Pb = 1...

As with the Bernoullian model, comparison between an observed and calculated sequence distribution is required to check for conformity to first-order Markov statistics. Obviously, with only two independent observations, a dyad distribution is insufficient for determining the two independent probabilities of the model. In contrast, a triad distribution provides five independent observations, so this can be used to check conformity to first-order Markov statistics. Trial values of the monomer addition probabilities can be obtained by taking appropriate combinations of the expressions shown in Table 2.3. For example, is given by... 

Corradini suggested a novel mechanism for syndiospecific chain-end control reactions in which the olefin migratory-insertion was assumed with 2,1-regiospecificity and in which the configuration of the last inserted monomer unit determines the configuration of the active sites in a subsequent rapid isomerization step. The satistical equations for this mechanism are equivalent to those for the chain-end control, Bernoullian model. 

Triad and pentad equations for the one-parameter Bernoullian model are summarized under (1-w) in Table 14. 

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