Head-tail bias

The extent of HT bias in the polymers of substituted cycloalkenes is very dependent on (i) the location and nature of the substituent(s) and (ii) the catalyst. Monomers with a substituent at the double bond generally give strongly biased polymers with most catalysts. For example, 1-methylcyclobutene with W(=CPh2)(CO)5 gives an 85% cis polymer with HH HT TT = 1 8 1 (Katz 1976a). The presence of one or two substituents at the oc-position results in fully biased polymers with some but not all catalysts. When the substituent(s) are further removed from the double bond there is usually very little HT bias in the polymer (see, for example, Figs.l 1.9b and 11.10b). 

Some examples of monomers giving polymers with a completely regular HT stmcture are shown in Table 11.10. In most cases the polymers are either high-m or high-/ra 5 and then have exceptionally simple C NMR spectra (see Fig. 11.14). A few, however, are of intermediate cis content. 

For an a -trans polymer without HT bias, ku =k2t, and 3, = 4 so that (HT/HH), = 2. When the polymer is biased, it is more likely that Ph is dominant, as discussed above, with and k t- k, rather than Pt dominant and 

The general features of ROMP of cycloalkenes have been presented in Ch. 11. Here we shall describe the behaviour of the monocyclic alkenes and polyenes, including heterocyclic rings, taking them in order of ring size. 

Acetylenes may be regarded as the first members of the series their polymerization using olefin metathesis catalysts is described in Ch. 10. There is no recorded attempt to polymerize cyclopropene with metathesis catalysts the product would probably be cyclohexa-1,4-diene rather than polymer. 

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Head-tail Bias in Polymers of Asymmetric Monomers... 

Ring-Opening Metathesis Polymerization General Aspects 11.6.3 Head-tail bias... 

One may, by careful choice of monomer, study the potential of different catalysts to behave in a stereoselective or regioselective manner. Thus, with a symmetrical monomer such as norbomene [58], norbomadiene [59] or their 5,6 [60] or 7-substituted derivatives [61,62] we have obtained polymers with a variety of cis main chain double bond contents and distributions. In a number of the 7-substituted examples, fine structure on certain NMR resonances is observed which is attributable to tacticity effects. Conversely, one may use the unsymmetrical monomers such as 1-substituted derivatives [63] and delineate the propensity of the different catalysts to regioselectivity, which manifests itself as head-tail bias in the polymer. 

The interesting problems of ring-chain equilibria, cis/trans blockiness, head-to-tail bias, and the various methods used to determine tacticity are treated in detail in three reviews.158-160 Consequently, only some more interesting observations are discussed here. 

Figure 4. Head-to-tail bias in poly(l-methylNBE) (a),80% trans polymer made using Grubbs catalyst (b), 100% trans polymer made using RuCl, (c), 100% trans polymer made using OsClj.

The statistical test procedures that we use unfortunately are not perfect and from time to time we will be fooled by the data and draw incorrect conclusions. For example, we know that 17 heads and 3 tails can (and will) occur with 20 flips of a fair coin (the probability from Chapter 3 is 0.0011) however, that outcome would give a significant p-value and we would conclude incorrectly that the coin was not fair. Conversely we could construct a coin that was biased 60 per cent/40 per cent in favour of heads and in 20 flips see say 13 heads and 7 tails. That outcome would lead to a non-significant p-value (p = 0.224) and we would fail to pick up the bias. These two potential mistakes are termed type I and type II errors. 

The magnitude of the HT bias is clearly determined by both polar and steric effects and may also be governed to some extent by relaxation processes between propagation steps. In living systems the head species Pe (69) is generally present in higher concentration than the tail species Pj (70)307. It is likely that in most, but not all, cases of total HT... 

If you observe 2 5 heads and 75 tails more often than 50 heads and 50 tails in many sets of N = 100 coin flips, you w ould have evidence of bias. Furthermore, because there is only one sequence that is all heads, the probability of observing 100 heads in 100 flips of an unbiased coin is 9.9 x 10 , virtually zero. 

EXAMPLE 6.4 Biased coins The exponential distribution again. Let s determine a coin s bias. A coin is just a die with two sides, t = 2. Score tails fr = 1 and heads in = 2. The average score per toss a) for an unbiased coin would be 1.5. 

Flere we derive the functional form of the entropy function, S = -k Zi Pi In Pi from a Principle of Fair Apportionment. Coins and dice have intrinsic symmetries in their possible outcomes. In unbiased systems, heads is equivalent to tails, and every number on a die is equivalent to every other. The Principle of Fair Apportionment says that if there is such an intrinsic symmetry, and if there is no constraint or bias, then all outcomes will be observed with the same probability. That is, the system treats each outcome fairly in comparison with every other outcome. The probabilities will tend to be apportioned between those outcomes in the most uniform possible way, if the number of trials is large enough. Throughout a long history, the idea that every outcome is equivalent has gone by various names. In the 1700s, Bernoulli called it the Principle of Insufficient Reason in the 1920s, Keynes called it the Principle of Indifference 1. ... 

In summary, the Ukelihood L is a function returning the probability of observed outcomes (e.g. HHT), given a parameter value (i.e. pn)- We now ask ourselves how the likelihood L = 1 (1-pn) can be maximised. Mathematically this is easy calculus teUs us that dL/dpn = d/dpn [Ph(I-Ph)] = 2pn-3 Pn, which vanishes when Ph = 2/3. A plot, or a quick calculation of the second derivative, tells us that pn = 2/3 is indeed a maximum, at which point L = 4/27. The result that Ph, max l = 2/3 can be intuitively understood by stating that the coin is biased towards heads up, by a factor 2 over tail up. Indeed, with such a bias, the probability of the observed outcomes HHT, given pn = 2/3, is maximal. How does all this help understanding a key aspect behind Kriging ... 

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