First-order Markov

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... 

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. 

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-... 

If, instead, the dyad probability depends on the nature (m or r) of the preceding dyad, the distribution follows a first-order Markov process, with two independent statistical parameters and Pr , (the probability that after a m dyad a r dyad follows and vice versa, respectively). The corresponding equations are listed in Table 4, column 3. They correspond to those of a nonideal copolymerization and are reduced to the previous case when p + p = 1 ... 

More complex schemes have been proposed, such as second-order Markov chains with four independent parameters (corresponding to a copolymerization with penultimate effect, that is, an effect of the penultimate member of the growing chain), the nonsymmetric Bernoulli or Markov chains, or even non-Maikov models a few of these will be examined in a later section. Verification of these models calls for the knowledge of the distribution of sequences that become longer, the more complex the proposed mechanism. Considering only Bernoulli and Markov processes it may be said that at the dyad level all models fit the experimental data and hence none can be verified at the triad level the Bernoulli process can be verified or rejected, while all Markov processes fit at the tetrad level the validity of a first-order Markov chain can be confirmed, at the pentad level that of a second-order Maikov chain, and so on (10). 

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 ... 

Having established that a particular polymerization follows Bemoullian or first-order Markov or catalyst site control behavior tells us about the mechanism by which polymer stereochemistry is determined. The Bemoullian model describes those polymerizations in which the chain end determines stereochemistry, due to interactions between either the last two units in the chain or the last unit in the chain and the entering monomer. This corresponds to the generally accepted mechanism for polymerizations proceeding in a nonco-ordinated manner to give mostly atactic polymer—ionic polymerizations in polar solvents and free-radical polymerizations. Highly isoselective and syndioselective polymerizations follow the catalyst site control model as expected. Some syndioselective polymerizations follow Markov behavior, which is indicative of a more complex form of chain end control. 

The exact computation of P W) in this simple one-dipole model is already a very arduous task that, to my knowledge, has not yet been exactly solved. We can, however, consider a limiting case and try to elucidate the properties of the work (heat) distribution. Here we consider the limit of large ramping speed r, where the dipole executes just one transition from the down to the up orientation. A few of these paths are depicted in Fig. 13b. This is also called a first-order Markov process because it only includes transitions that occur in just one direction (from down to up). In this reduced and oversimplified description, a path is fully specified by the value of the field H at which the dipole reverses orientation. The work along one of these paths is given by... 

In Section V.B.l we have evaluated the path entropy s q) (Eq. (163)) for an individual dipole N = 1) in the approximation of a first-order Markov process. The following result has been obtained (Eq. (142)) ... 

They correspond to a first-order Markov process for the stereocontrol— i.e., a penultimate effect of the last diad on stereocontrol. 

More convenient and entirely sufficient for the present purpose is the calculation of ratio of rate constants. The calculation will be reviewed for a one-way first-order Markov process. A one-way mechanism is chosen because it is intuitively the most appropriate model for a free radical mechanism. Furthermore it has some experimental support. The assumption of a first-order Markov process does not rule out higher Markov processes. The differentiation between a first-order Markov process and higher order Markov processes is however possible experimentally in very rare cases because it involves the determination of tetrad, pentad, etc. fractions (11, 12, 13, 14). A Bemoullian process is ruled out by the analysis of the data of Table I. 

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 ... 

Table TV, the resulting calculation of VCl number average sequence length calculated using the first order Markov agrees very well with the observed sequence lengths.

A key facet of copolymerization is the possible disparity of reactivities of the monomers. Traditional procedure is to assume, at least as an approximation, that the reactivity of a growing propagating center depends only on the identity of its reactive end unit (i.e., the last monomer added), not on the composition and length of the rest of its chain [124-126] (first-order Markov or terminal model see also... 

The simple copolymer model is a first-order Markov chain in which the probability of reaction of a given monomer and a macroradical depends only on the terminal unit in the radical. This involves consideration of four propagation rate constants in binary copolymerizations, Eqs. (7-2)-(7-4). The mechanism can be extended by including a penultimate unit effect in the macroradical. This involves eight rate constants. A third-order case includes antepenultimate units and 16 rate coefficients. A true test of this model is not provided by fitting experimental and predicted copolymer compositions, since a match must be obtained sooner or later if the number of data points is not saturated by the adjustable reactivity ratios. 

To analyse bond breakage under steady loading, we take advantage of the enormous gap in time scale between the ultrafast Brownian diffusion (r 10 — 10 s) and the time frame of laboratory experiments ( 10 s to min). This means that the slowly increasing force in laboratory experiments is essentially stationary on the scale of the ultrafast kinetics. Thus, dissociation rate merely becomes a function of the instantaneous force and the distribution of rupture times can be described in the limit of large statistics by a first-order (Markov) process with time-dependent rate constants. As force rises above the thermal force scale, i.e. rj-t> k T/x, the forward transition... 

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-... 

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... 

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