First-order Markov model

The first-order markov model describes a polymerization where the penultimate unit is important in determining subsequent stereochemistry. Meso and racemic dyads can each react in two ways  

Meso (isotactic) and racemic (syndiotactic) triads result from Reactions 8-91 and 8-94, respectively. Reactions 8-92 and 8-93 yield the heterotactic triad. 

Four prohabilities, Pmm, Pmr, Prm, and Prr, characterize this model for propagation with the conservation relationships 

A second-order Markov model has also been described to show the effect on stereochemistry of the monomer unit behind the penultimate unit [Bovey, 1972], 

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A portion of the database for this polymer is shown in Figure 6. Literature reports that this polymer follows second-order Markov statistics ( 21 ). And, in fact, probabilities that produced simulated spectra comparable to the experimental spectrum could not be obtained with Bernoullian or first-order Markov models. Figure 7 shows the experimental and simulated spectra for these ten pentads using the second-order Markov probabilities Pil/i=0.60, Piv/i=0.35, Pvi/i=0.40, and Pvv/i=0.55 and a linewidth of 14.8 Hz. 

Syndiotactic propagation of propylene is know to be catalyzed by homogeneous vanadium catalyst (1 ). In the polypropylene samples prepared with the homogeneous catalysts, the relative population of iso-, hetero- and syndiotactic triads is in accordance with that predicted from the first order Markov model (25, 26). There is no chiral structure around the homogeneous vanadium species. The stereochemistry of the entering monomer is controlled by the chirality of the growing chain end, in contrast with the isotactic propagation. 

Table IV shows the reactivity ratios rG and r, derived from the probabilities in Table III in accord with a first-order Markov model (2), where it is assumed that the more likely propagating terminal radical structure is 1 (—CHF-) and not 0 (—CH2). This assumption is consistent with gas phase reactions of VF with mono-, di-, and trifluoromethyl radicals, which add more frequently to the CH2 carbon than to the CHF carbon (20). The reactivity ratio product is unity if Bernoullian statistics apply, and we see this is not the case for either PVF sample, although the urea PVF is more nearly Bernoullian in its regiosequence distribution. Polymerization of VF in urea at low temperature also reduces the frequency of head-to-head and tail-to-tail addition, which can be derived from the reactivity ratios according to %defect — 100(1 + ro)/(2 + r0 + r,). Our analysis of the fluorine-19 NMR spectrum shows that commercial PVF has 10.7% of these defects, which compares very well with the value of 10.6% obtained from carbon-13 NMR (13). Therefore the values of 26 to 32% reported by Wilson and Santee (21) are in error.

Table 12. Pentad Fractions for the Symmetric First-Order Markov Model...

The first-order Markov model In the first-over Markov model, the probability of monomer addition depends upon the identity of the preceding monomer unit. Hence, for an A/B copolymer, there are four possible propagation steps defined by four monomer addition probabilities ... 

Table 23 First-order Markov model dyad and triad sequence abundances...

Other models sometimes invoked include two-component models based on combinations of the models discussed above [5] and the complex participation model [6]. Examples of the use of these are given in section 2.3. The complex participation model is a modification of the first-order Markov model to take account of the formation of A-B comonomer complexes which compete with monomer during polymerisation. Thus, four propagation steps in addition to those shown for the first-order Markov model are required to describe addition of A-B and B-A complexes (i.e. it can add either way round) to the two types of growing chain and (either A or B). The monomer and comonomer complex addition probabilities are then related to the equilibrium constant for complex formation. As might be expected, this model has been applied particularly to systems that show a marked tendency towards alternation of their comonomers [7]. A probabilistic description of the complex participation model has been given by Cais et al [6]. 

Since the terminal model is the one most frequently used to describe copolymerisation, its relationship with the corresponding first-order Markov model is examined in some detail. 

These propagation steps are entirely analogous to those given for the first-order Markov model, except that addition probabilities have been replaced by rate constants. Mayo and Lewis derived the following differential equation to describe terminal model copolymerisation [8] ... 

The comonomer reactivity ratios are especially useful characteristics for a pair of monomers since a knowledge of their values allows polymer composition and microstructure to be predicted over the full range of monomer feeds. It can be shown that the monomer addition probabilities of the first-order Markov model are related to the reactivity ratios by the following expressions ... 

In a similar study by Cheng [32] on ethylene/1-butene copolymers, a two-site model was also invoked. In contrast, he found that the microstructure of ethylene/1-hexene copolymers could be described adequately using a one-site first-order Markov model [33]. Regio-irregularity in polypropylene has also been treated by Cheng using Markov statistics (see [5] and references therein). 

Free radical propagation is poorly stereocontrolled, with nearly equal proportion of meso and racemic dyads in polymerization of monosubstituted alkenes and a preference for syndiotactic placement for disubstituted monomers such as methacrylates (rr = 0.62, mm = 0.04). The sequence distrihution follows a first-order Markov model with a slight deviation from Bernoulian statistics. However, for very bulky substituents, as in polymerization of triphenylmethyl methacrylate, the preference for isotacticity was observed (mm = 0.64, rr = 0.12). Recently, complexation with Lewis acids and acidic solvents enabled to enhance stereocontrol in polymerization of vinyl esters and acrylamides, and to a smaller degree in polymerization of methacrylates (127-129). 

K. R. Sharma, First Order Markov Model Representation of Chain Seqnence Distribution of Alphamethylstyrene Acrylonitrile Prepared by Reversible Free Radical Polymerization, 229th ACS National Meeting, San Diego, CA, March 2005. 

The styrene and acrylonitrile can be copolymerized by free radical methods using a continuous stirred tank reactor (CSTR). The reactivity ratios r,2 and rj, can be taken as 0.04 and 0.41, respectively. Construct a first-order Markov model using the dyad probabilities derived in Section 11.1. 

First-order Markov model to represent chain seqnence distribution of SAN... 

Develop a first-order Markov model for the DNA sequence given in Example 11.1 in order to represent the first 60 base pairs. Calculate the transition probabilities and represent the information in the form of a suitable diagram... 

Transition Probabilities in the First-Order Markov Model to Represent DNA Sequence from Homo Sapiens... 

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