Probabilities sequence distribution

If chains are long such that the initiation and termination reactions have a negligible effect on the average sequence distribution, then according to the terminal model, PAA, the probability that a chain ending in monomer unit MA adds another unit MA, is given by eq. 22 8... 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Applications of the method to the estimation of reactivity ratios from diad sequence data obtained by NMR indicates that sequence distribution is more informative than composition data. The analysis of the data reported by Yamashita et al. shows that the joint 95% probability region is dependent upon the error structure. Hence this information should be reported and integrated into the analysis of the data. Furthermore reporting only point estimates is generally insufficient and joint probability regions are required. 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

Such a consideration demonstrated [56] that the sequence distribution in products of arbitrary equilibrium copolycondensation can always be described by some Markov chain with the elements of the transition probability matrix ex-... 

The solution to the problem of sequence distribution as well as that of composition distribution (Eq. 69) reduces to the calculation of simple integrals. So, the probability P U of an arbitrary sequence U = MaMp in... 

Statistical copolymers are copolymers in which the sequential distribution of the monomeric units obeys known statistical laws e.g. the monomeric-unit sequence distribution may follow Markovian statistics of zeroth (Bemoullian), first, second or a higher order. Kinetically, the elementary processes leading to the formation of a statistical sequence of monomeric units do not necessarily proceed with equal a priori probability. These processes can lead to various types of sequence distribution comprising those in whieh the arrangement of monomeric units tends towards alternation, tends towards... 

The magnitude of the chemical shift differences for the different methyl carbons indicates that the presence of an asymmetric carbon in the 1,2- and i,3-units has little effect on the chemical shifts and that primarily sequence distribution affects the chemical shifts in both the 13C NMR and 11 NMR spectra. This is further emphasized by the presence of two resonances of singlet multiplicity at 32.93 and 33.115 which are attributed to the quaternary carbon of the 1,4-structure. This carbon should be unaffected by tacticity in the polymer, and so these two resonances are probably due to two different triads. 

Within the framework of the above models the problem of the calculation of the sequence distribution is solved in a quite simple way [51-53, 6]. In order to find the probability of any sequence Uk consisting of k units, it should be expressed through the sequence of Markov chain states, the probability of which is calculated usually by means of the routine procedure as a product of the few factors. The first factor 7i corresponds to the initial state Sh and each of the following factors, Vy, corresponds to the transition from the state Sj to Sj at the conditional movement along the sequence of Markov chain states. For instance, in this manner one can calculate the probability of the sequence U3 = S3[1M2M2 in the both cases of terminal model ... 

In contrast to the above mentioned models, the similar statistical description of the products of the complex-radical copolymerization occurring through the scheme (2.5) has been carried out quite recently [37, 49, 55-60]. Within the framework of this Seiner-Litt model, both copolymer composition [37,49, 55-58] and fractions of the different triads and blocks of the monomer units in the macromolecules were calculated [57]. The probability approaches which were applied in these works, are regarded as being of limited applicability in contrast to the general statistical method [49, 59, 60], By means of the latter method, the sequence distribution and composition inhomogeneity of the copolymer were completely described [49, 60] and also thorough calculations of its microstructure with the account for the tactidty were carried out [59, 60]. 

The parameters a = l/rij5 the number of which equals m(m — IX are reciprocal reactivity ratios (2.8) of binary copolymers. Markov chain theory allows one, without any trouble, to calculate at any m, all the necessary statistical characteristics of the copolymers, which are formed at given composition x of the monomer feed mixture. For instance, the instantaneous composition of the multicomponent copolymer is still determined by means of formulae (3.7) and (3.8), the sums which now contain m items. In the general case the problems of the calculation of the instantaneous values of sequence distribution and composition distribution of the Markov multicomponent copolymers were also solved [53, 6]. The availability of the simple algebraic expressions puts in question the expediency of the application of the Monte-Carlo method, which was used in the case of terpolymerization [85,99-103], for the calculations of the above statistical characteristics. Actually, the probability of any sequence MjMjWk. .. Mrl 4s of consecutive monomer units, selected randomly from a polymer chain is calculated by means of the elementary formula ... 

So far, we have used kinetics to describe the relationship of monomer feed concentration and reactivity ratios to copolymer composition. Now we will show how probability theory can be used to describe sequence distributions. Try to contain your excitement. 

We have already seen that, depending on the values of the reactivity ratios, there is a tendency to get random, alternating, blocky, etc., types of copolymers. Probability theory allows us to quantify this in terms of the frequency of occurrence of various sequences, like the triads AAA or ABA in a copolymerization of A and B monomers. The value of this information is that such sequence distributions can be measured directly by NMR spectroscopy, thus allowing a direct probe of copolymer structure and an alternative method for measuring reactivity ratios. As mentioned above, there are problems, as some spectra can be too complex and rich for easy analysis, as we will see in Chapter 7. 

Conformer sequence probabilities Radial distribution functions Scattering functions Orientation correlation functions Mechanical properties Distribution of free volume... 

A statistical analysis of the sequence distribution can be performed in terms of direct and inverted units (D and I), i.e. of units written with carbon Cl at the left or at the right, respectively. Dyads DD and II, which differ in the sense of observation, correspond to head-to-tail, ID to head-to-head and DI to tail-to-tail junctions respectively. In the same way triads of D or I units are related to longer sequences. Remember that DDD and III, IDD and IID, DDI and DII, IDI and DID cannot be distinguished from each other. An interpretation according to a first-order Markov chain requires the use of two conditional probabilities, p... 

All IR investigations of sequence distribution so far published rely on the terminal copolymerization model, which assumes that the kinetics of copolymerization are governed only by the probability that monomer units from the feed will be added to the last unit of the growing chain, and that there is only one active site present in the catalyst system, whether homogeneous or heterogeneous. As will be shown later (Section 3.4), this is only an approximation multiple active species are formed by many soluble Ziegler-Natta catalysts, so that the product of reactivity ratios determined from the normal copolymerization equation does not always exactly predict the actual sequence distribution in the copolymer. 

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