First-order Markov statistics

In these processes, the ultimate configurational diad, that is, the last two monomer units, exerts an influence. Thus, a distinction must be made between the probabilities of forming an isotactic diad on an isotactic diad or on a syndiotactic diad. The sum of these two probabilities must equal unity, since these are the only two possibilities for addition onto a given diad  

A steady state must exist for each diad type  

The instantaneous isotactic diad mole fraction is given by 

The mole fraction of isotactic diads in the final polymer can be calculated as follows. The rate of formation for the number of isotactic diads is equal to 

From the change in the number of isotactic diads with conversion, i.e., by combining equations (16-101), (16-103), and (16-104), we obtain 

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Where the nature of the preceding dyad is important in determining the configuration of the new chiral center (Scheme 4.2), first order Markov statistics... 

Bcrnoullian statistics do not provide a satisfactory description of the tacticity. 6 This finding is supported by other work.28" 38 First order Markov statistics provide an adequate fit of the data. Possible explanations include (a) penpenultimale unit effects are important and/or (b) conformational equilibrium is slow (Section 4.2.1). At this stage, the experimental data do not allow these possibilities to be distinguished. 

It seems likely that other polymerizations will be found to depart from Bemoullian statistics as the precision of tacticity measurements improves. One study12 indicated that vinyl chloride polymerizations are also more appropriately described by first order Markov statistics. However, there has been some reassignment of signals since that time. 4 25... 

Styrene-SQ., Copolymers. I would now like to discuss two systems which illustrate the power of C-13 nmr in structural studies. The first is the styrene-SO system. As already indicated, this is of the type in which the chain composition varies with monomer feed ratio and also with temperature at a constant feed ratio (and probably with pressure as well.) The deviation of the system from simple, first-order Markov statistics, —i.e. the Lewis-Mayo copolymerization equation—, was first noted by Barb in 1952 ( ) who proposed that the mechanism involved conplex formation between the monomers. This proposal was reiterated about a decade later by Matsuda and his coworkers. Such charge transfer com-... 

From the analysis of 13C NMR spectra of polypropylenes, Doi96) found that the sequence distribution of inverted propylene units follows first-order Markov statistics. Table 4 lists the two reactivity ratios rQ and rt, for the polymerization of propylene with the soluble catalysts composed of VC14 and alkylaluminums at — 78 °C ... 

The r.h.s. of flg. 3.31 presents liquid-gas coexistence curves, of which curve I relates to the conditions of fig. 3.31a. Curve II, arises from somewhat improved lattice statistics. For curve I the chain is fully flexible, implying that each bond can bend back to coincide with the previous one. In statistical parlance it is said that the chain has no self-avoidance and obeys first-order Markov statistics. In curve II a second-order Markov approximation was used ) in which three consecutive bonds in the chain are forbidden to overlap and an energy difference of 1/kT is assigned to local sets of three that have a bend conformation. The figure demonstrates the extent of this variation T is reduced as a result of the loss of conform-... 

From the Ah-NMR analysis of poly(3,8-d2-styrene oxide) obtained using ZnEt2/H20 as initiator Figure 2), the formation of the partly crystalline fraction can be described by first order Markov statistics, while that for the amorphous fraction follows Bemoulllan statistics. Different chain propagation mechanisms are, therefore, responsible for the formation of the two different polymer fractions obtained from this particular catalyst. Consequently, the existence of two different active centers, responsible for the two polymerization mechanisms and for formation of fractions I and II, are clearly indicated. 

When these results are compared with those from the polymerization of the other monosubstituted oxiranes (9), it should be emphasized that the partly crystalline fraction of poly(styrene oxide), I, obtained in the presence of the ZnEt2/H20 catalyst is so far the only polyoxirane reported which does not follow Bernoullian statistics. Only one other case, namely that of the polymerization of phenylthiirane by a coordination catalyst system, is known until now which follows first-order Markov statistics (10). 

When the manner of addition is affected by the growing chain end, the configurations of the added units will not obey Bernoulli statistics. In the simplest case, first-order Markov statistics will operate. Addition will be characterized by two parameters because the probability of r diad generation by monomer addition to an m end unit, will not be identical with monomer addition to an r end unit, P . The probabilities of m or r diad generation by addition to m or r chain ends will be bound by the relations P = (1 — Pmr) rm 0 rr) According to first-order Markov... 

First-order Markov statistics permits two other non-bernoullian forms of addition heterotactic for P. -> 1, P 1, and stereoblock for small but finite P rm values. The smaller the value of P, or P, the larger will be the generated m or r blocks, respectively. 

The probabilities of the regiosequence pentads for commercial PVF and urea PVF are shown in Table III. For the former sample it is apparent simply by inspection that the regiosequence distribution is not Bernoullian, since Pobs(C5) and Pobs (D5) are different (2). The distributions conform to first-order Markov statistics, characterized by two reactivity ratios r0 and r 5 where r0 = k /lq, and rj — ku/k10 (kjj is the rate constant for monomer addition to terminal radical i which generates the new terminal radical j). The present pentad data is insufficient to check the validity of this model, but it is unlikely that there is any deviation, as the same model has been tested and found adequate to describe the regiosequence distribution in PVF2 (2). 

Figure 15-4. Compensation effect for various placement possibilities for first-order Markov statistics in the free radical polymerization of methyl methacrylate. For better clarity, some lines have been vertically displaced by -10.5 (for AHj/, - AH /,), 4.2 (for AH, - AH /,), 6.3 (for A H /, - A H /,), and 10.5 (for A H /, - A H /,) kJ/mol. The compensation temperature, but not the compensation enthalpy, is independent of the kind of placement occurring. (From data by H.-G. Elias and P. Goeldi.)...

The slope has the physical units of a temperature, and gives the compensation temperature To at which polymerizations in different solvents always lead to the same proportions of elementary steps, a and b. Figure 15-4 shows such a plot made for the assumption of first-order Markov statistics. 

The stereocontrol of the free radical polymerization of a monomer at a given temperature is still weakly dependent on the solvent. In this case, however, there is a linear relationship between activation enthalpy differences and the corresponding activation entropy differences for each of the possible six differences of the total four possible elementary steps of a first-order Markov statistics (Figure 20-8). These relationships are each independent of the solvent used, and, so, also, of conversion. The straight lines are parallel to each other, that is, the compensation temperature is independent of the kind of diad formation occurring. The stereocontrol for methyl methacrylate at this temperature of about 60° C is therefore independent of the solvent used. 

Figure 16-5. Compensation effect for various placement possibilities for first-order Markov statistics in the free radical polymerization of methyl methacrylate. For better clarity, some lines have been vertically displaced by -10.5 (for AHJs — 4.2 (for AhIs - Ar/ /J,...

Such relationships are known as compensation effects. The compensation temperature Tq is then the temperature for which polymerizations in various solvents will always lead to the same fraction in elementary steps a and b. Figure 16-5 shows such a plot for free radical polymerizations, assuming first-order Markov statistics. 

The stereocontrol of most free radical polymerizations appears to be governed by an end-controlled mechanism. It generally follows first-order Markov statistics with respect to diads (see also Section 16.5.2.3). The tactic-ity of the formed polymer is also influenced by the solvent used. The cause of this solvent control effect is unclear, and possibly is due at least partly to different degrees of solvation. A compensation effect (see Section 16.5.4.) exists in the relationship between the activation entropies and enthalpies for diad formation in various solvents. The compensation temperature TJj varies with monomer constitution (Table 20-11). The compensation enthalpies AAHI vary strongly according to both monomer and placement type. 

Table 1.8 Tacticity sequence probabilities expressed in terms of propagation probabilities for Bernoullian and first-order Markov statistics the parameters u and v are defined in the text...

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. 

The vast majority of copolymers described in the literature conform to the terminal model for copolymerisation and therefore exhibit sequence distributions which will in principle conform to first-order Markov statistics. Of... 

When in polymerization the addition probability of a monomer unit is independent of the type of the last unit in the growing chain (as is the case in most radical polymerizations), then the chain growth is governed by the two probabilities, Pj and P2, of the addition of monomer 1 or 2. Since these two probabilities are bound by the relation Pj +P2 = 1, the polymer statistics is determined by a single parameter p = Pi = 1 — P2. This is the so-called Bernoulli statistics. When the probability of monomer addition depends on the type of the last monomer unit in the growing chain, the system is characterized by four transition probabilities, P -, P 2-> 21> and 22> bound by the conditions f ii + i2 = l d P21 + P22 = 1, defining two independent parameters (e.g., P21 = 1 — P22 and P12 = 1 — Pn). This is the so-called first-order Markov statistics. Markov... 

The single parameter of Bernoulli statistics can be simply determined from one-unit sequences that is, from the tacticity of a stereopolymer (or the composition of a copolymer). To determine the two parameters of first-order Markov statistics, the populations of all three two-unit sequences must be known (use can be made of the necessary relations. Table 3). To verify whether the proposed order is actually valid, the populations of sequences longer by one unit must be checked by experiment (e.g., Bernoulli statistics is established by a check on the two-unit populations, etc.). 

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