Bernoullian model sequence distributions

Carbon-13 nuclear magnetic resonance was used to determine the molecular structure of four copolymers of vinyl chloride and vinylidene chloride. The spectra were used to determine both monomer composition and sequence distribution. Good agreement was found between the chlorine analysis determined from wet analysis and the chlorine analysis determined by the C nmr method. The number average sequence length for vinylidene chloride measured from the spectra fit first order Markovian statistics rather than Bernoullian. The chemical shifts in these copolymers as well as their changes in areas as a function of monomer composition enable these copolymers to serve as model... 

The correctness of each model for a given copolymer system can be tested and confirmed by experimental observation. In general, when Bernoullian statistics do not describe the sequence distribution, Markovian statistics do. 

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... 

Several examples of NMR studies of copolymers that exhibit Bernoullian sequence distributions but arise from non-Bernoullian mechanisms have been reported. Komoroski and Schockcor [11], for example, have characterised a range of commercial vinyl chloride (VC)/vinylidene chloride (VDC) copolymers using carbon-13 NMR spectroscopy. Although these polymers were prepared to high conversion, the monomer feed was continuously adjusted to maintain a constant comonomer composition. Full triad sequence distributions were determined for each sample. These were then compared with distributions calculated using Bernoullian and first-order Markov statistics the better match was observed with the former. Independent studies on the variation of copolymer composition with feed composition have indicated that the VDC/VC system exhibits terminal model behaviour, with reactivity ratios = 3.2 and = 0.3 [12]. As the product of these reactivity ratios is close to unity, sequence distributions that are approximately Bernoullian are expected. 

Statistics of Copolymer Chains. A triad distribution is capable of giving even more microstructural information after comparing an experimental distribution to a calculated distribution having specifically defined chain statistics (3,6,15,16). The calculated statistical distributions can be random or some higher level of a so-called Markovian chain. These models are constructed on the premise of a conditional probability of finding any particular unit or sequence in a copolymer chain. In a perfectly random statistical model, there are no previous conditions required for one unit to follow any other particular unit. These types of statistical models are called Bernoullian or zero-order Markovian. For these statistical models, only mole fraction information is required to determine what the corresponding triad distribution would look like for a perfectly random copolymer, that is. 

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