Markovian model, statistical analysis

The second-order Markovian statistics for chain growth uses four copolymerization parameters taking into account the last two monomeric units at the chain end. In order to decide which mathematical model to use, copolymerization parameters, determined from NMR spectroscopic sequence analysis, should be compared with calculated distributions using the above mentioned models. In most cases, the first-order Markovian statistics is adequate. However, Herfert, Montag and Fink [6] point out that some metallocene-based copolymers give best fit with the second-order Markovian model. 

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)... 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

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 second type of nonideal models takes into account the possible formation of donor-acceptor complexes between monomers. Essentially, along with individual entry of these latter into a polymer chain, the possibility arises for their addition to this chain as a binary complex. A theoretical analysis of copolymerization in the framework of this model revealed (Korolev and Kuchanov, 1982) that the statistics of the succession of units in macromolecules is not Markovian even at fixed monomer mixture composition in a reactor. Nevertheless, an approach based on the "labeling-erasing" procedure has been developed (Kuchanov et al., 1984), enabling the calculation of any statistical characteristics of such non-Markovian copolymers. 

The consistent kinetic analysis of the copolymerization with the simultaneous occurrence of the reactions (2.1) and (2.5) leads to the conclusion that the probabilities of the sequences of the monomer units M, and M2 in the macromolecules can not be described by a Markov chain of any finite order. Consequently, in this very case we deal with non-Markovian copolymers, the general theory for which is not yet available [6]. However, a comprehensive statistical description of the products of the complex-radical copolymerization within the framework of the Seiner-Litt model via the consideration of the certain auxiliary Markov chain was carried out [49, 59, 60]. 

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