Bernoullian process

A particularly interesting result in Table V is seen for the solutions containing about 66% methylene chloride. At this point a sharp break occurred in several properties (1) the syndiotactic content decreased and the isotactic content increased, (2) the yield dropped drastically, (3) the molecular weight decreased, and (4) the molecular weight distribution broadened. These changes all point to a change in the reaction mechanism, probably from one of solvated ion pairs to one of predominantly contact ion pairs at that solvent concentration. Nevertheless, the tacticities of all the polymers formed in this series were found to fit on a Bovey plot, which indicates that they were formed by a Bernoullian process and no penultimate effect was present. 

The Bernoullian process is therefore defined for a copolymer by two transition probabilities, and Pg, which reflect the mole fractions of monomers A and B within the resulting copolymer. Given the two addition probabilities, the mole fraction of any given sequence can be calculated straightforwardly. Thus, for example, the abundance of an A A dyad is given by P, whilst that of an AB dyad is equal to 2P P (the factor of two arises because the AB dyad represents both AB and BA sequences). Table 2.2 shows the set of Bernoullian expressions for the three dyad and six triad sequences in an A/B copolymer. 

Only one independent variable is necessary to describe the Bernoullian process in full since the sum of and is equal to unity. Dyad distribution data contain two independent observations which is sufficient to check for conformity to Bernoullian statistics, (Although there are three dyad types, the mole fraction of one of them is always defined by the mole fractions of the other two since the mole fraction of the total dyad is, of course, equal to one, i.e. there are only two independent observations.) However, a better test for conformity is found in a triad distribution since this contains five independent observations (there are six triads). 

It is important to note here that random copolymers which show conformity to Bernoullian statistics can result from non-Bernoullian processes under some sets of circumstances. In order to distinguish between random copolymers resulting from Bernoullian and non-Bernoullian processes, it is necessary to have access to additional information, e.g. comonomer feed information, copolymer composition versus conversion data, and so on. Examples of random copolymers resulting from non-Bernoullian processes are given in section 2.3. 

Note, in addition, that the special case = rg = 1 corresponds to a random copolymer formed by a Bernoullian process. 

As stated earlier, copolymers whose sequence distributions can be described by Bernoullian statistics (i.e. so-called random or statistical copolymers) can result either from Bernoullian or non-Bernoullian processes. Those formed by genuine Bernoullian mechanisms are relatively rare, reflecting the likeli-... 

In the polymerization of 1,5-HD, 1,2-addition and cyclization reactions can be described by Equations 1.1 through 1.3 (where r = the rate of reaction), on the condition that the 1,2-addition obeys a first-order Markovian process and the cyclization follows a Bernoullian process... 

Thus, copolymers of the same composition can have qualitatively different sequence distributions depending on the solvent in which the chemical transformation is performed. In a solvent selectively poor for modifying agent, hydrophobically-modified copolymers were found to have the sequence distribution with LRCs, whereas in a nonselective (good) solvent, the reaction always leads to the formation of random (Bernoullian) copolymers [52]. In the former case, the chemical microstructure cannot be described by any Markov process, contrary to the majority of conventional synthetic copolymers [ 10]. 

The distribution of B blocks, which are included mostly in nonadsorbed chain sections, decays exponentially and thus should obey Bernoullian statistics that correspond to a zeroth-order Markov process [10]. The average length of such blocks is close to 2, that is, the same as that of a random copolymer. In the case of A blocks, the distribution function/a (f) also decays exponentially in the initial region, which corresponds to short blocks included in the random chain sections. For longer A blocks, however, the distribution becomes significantly broader and has a local maximum at i 10 [95]. Hence, one can conclude that the distribution of A blocks strongly deviates from that known for random sequences. 

As no effect of the penultimate inserted unit is taken into account, the formation of r or m diads is a random process that follows the Bernoullian statistic. Therefore, the symmetric first-order Markovian model becomes Bernoullian when diad formation is considered. This fact explains why this model is usually called Bernoullian. 

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. 

The necessary apparatus with which to treat the fusion of stereoregular polymers is embodied in the theories already outlined. It remains to establish the sequence distribution of the structural irregularities and to question whether the crystalhne phase remains pure. The sequence distribution is a reflection of the polymerization process and the probability of the addition of a particular placement. As has been indicated, addition by either Bernoullian trial or a first-order Markovian process have been treated and integrated into the melting temperature-composition theory.(8,9,9a)... 

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