Probability parameters

The probabilities of the various dyad, triad, and other sequences that we have examined have all been described by a single probability parameter p. When we used the same kind of statistics for copolymers, we called the situation one of terminal control. We are considering similar statistics here, but the idea that the stereochemistry is controlled by the terminal unit is inappropriate. The active center of the chain end governs the chemistry of the addition, but not the stereochemistry. Neither the terminal unit nor any other repeat unit considered alone has any stereochemistry. Equations (7.62) and (7.63) merely state that an addition must be of one kind or another, but that the rates are not necessarily identical. 

It is reasonable to assume that the most probable values of the parameters have normal distributions with means equal to the values that were obtained from well test and core data analyses. These are the prior estimates. Each one of these most probable parameter values (kBj, j=l,...,p) also has a corresponding standard deviation parameter estimate. As already discussed in Chapter 8 (Section 8.5) using maximum likelihood arguments the prior information is introduced by augmenting the LS objective function to include... 

Depending on the values of the probability parameters of the branching process two paths of evolution for every population are possible either it will degenerate in some generation or it will infinitely grow. The probability of... 

It is worth emphasizing that formulas (39) and (40) may be invoked to determine the statistical characteristics of the sol-fraction provided the probability parameters are replaced in these equations by their modified values... 

A general theory of the equilibrium polycondensation of an arbitrary mixture of monomers, described by the FSSE model, has been developed [75]. Proceeding from rigorous thermodynamic considerations a branching process has been indicated which describes the chemical structure of condensation polymers and expressions have been derived which relate the probability parameters of this stochastic process to the thermodynamic parameters of the FSSE model. 

Figure 2. Experimental (—) and calculated ( ) X-ray diffraction profile for H-BOR-E, sample 10 (a) and 23 (b). The insets show a plot of the intensity disagreement factor (see Table II) vs. the fault probability parameter, p.

Figure 4. Variation of the ratio between the intensity of the reflections occurring at 2 = 14° and Z d z 15°, respectively, in the X-ray powder pattern of H-BOR-E, as a function of the fault probability parameter, p.

The variable-input norm description of the decoupled chemical bonds gives the full agreement with the chemical intuition, of r bonds in XH with changing covalent/ionic composition in accordance with the actual MO polarization and the adopted basis set representation. The more the probability parameter P deviates from the symmetrical bond (maximum covalency) value P = 1/2, due to the electronegativity difference between the central atom and hydrogen, the lower is the covalency (the higher ionicity) of this localized, diatomic bond. Therefore, in this IT description the total bond multiplicity Af = r bits is conserved for changing proportions between the overall covalency and ionicity of all chemical bonds in the system under consideration. 

A general and precise description of stereoisomerism in polymers is suggested on the basis of the repetition theory which describes the distinct patterns along a line that can be obtained from a three-dimensional motif. The probability models for describing the" stero-sequence length in various possible cases of interest in stereoregular polymers are discussed. It is shown that for describing the stereosequence structure, the simplest probability model must involve a Markov chain with four probability parameters. 

If the value of this probability parameter is a <= 1, an essentially iso tactic structure is obtained. If, on the other hand a 0, almost all the monomer units in the chain are in syndiotactic placements. If the polymer is capable of crystallization and the crystallization takes place under equilibrium conditions, then the limitation of this model is that a small melting point depression implies also a high degree of percent crystallinity. Although there are a number of systems, for example stereoregular methyl methacrylate (2, 8), in which this is true and this model is valid, this is not the case for polymers of propylene oxide from different catalysts that we discuss in another chapter (1). 

A two-parameter model represents the case where the probability of the final state of the growing chain depends upon the configuration of the last two units after a monomer unit is added. The probability parameters and some of the typical structures that can be described by this model are given in Figure 4. In this model i is the probability that a growing chain end in the + state will continue to be in + state, after a monomer unit is added a 2 is the probability that a growing chain end in the - state will result in the +... 

Stereostructures for some typical values of the four probability parameters on, (X2, as, and aA are given in Figure 6 to show that by properly choosing values of the transitional probabilities, the stereostructure of the chains can be represented with sufficient generality. 

Although several techniques have been used to characterize stereosequence distribution, we suggest that the percent crystallinity and temperature of melting measurements are more generally applicable than any other technique presently available. Bovey and co-workers (7) showed how NMR measurements can be used to determine the triad distribution in polymers such as polymethyl methacrylate in which there is sufficient difference between the NMR spectra corresponding to syndiotactic, isotactic, and heterotactic triads to allow quantitative measurements to be made. This type of measurement unfortunately is restricted to few systems and would lead to a unique description of the stereostructure of the chain only when a model involving one or two probability parameters is applicable (See Appendix I). 

Two quantities—Pi, the concentration of the isotactic triad and ai, the probability that an isotactic triad be followed by another isotactic triad—are sufficient to describe the sequence length of the isotactic units. The concentration of the various triad states can be expressed in terms of the various transition probabilities of the four-parameter model discussed previously (1). It can be shown that in the expressions for the probabilities of the various triads, the four transition probability parameters occur in such a combination that the concentration of the various triad states like Pi, etc. depend on only two independent parameters (Appendix I). 

I. Relations Between Probability (Pi) of Triad States and Transition Probability Parameters (a i). The four possible states of three successive monomer units in the polymer chain were defined previously (1) as Ei = + + E2 = +... 

Here we derive the relations between the probability of any triad of monomers in a given state in terms of the transition probabilities, and show that the four transition probability parameters occur in such a combination that the concentration of the various triad states depends only on two independent parameters. 

The initial distribution on the other hand, is the distribution of sequences in the uncrystallized amorphous polymer. This initial distribution, of course, is determined in the case of stereoregular polymers by the particular statistics and probability parameters applicable to the system. 

To demonstrate the procedures to be used to obtain probability parameters from crystallization data, let us consider the distribution which occurs if we have a single parameter model. Under these conditions the equilibrium distribution is,... 

The stereochemistry of addition to a free centre is mostly determined by interactions between the monomer and active centre during approach to the transition state. In simple cases, represented by equations (34) and (35) only the two primary components will interact, and Bernoulli statistics with a single probability parameter Pm will predominate. For Pm = 0.5, the propagation rate constants of isotactic and syndiotactic growth, kpj and k, will differ... 

FYom observations and any available prior information, Bayes theorem infers a probability distribution for the parameters of a postulated model. This posterior distribution tells all that can be inferred about the parameters on the basis of the given information. From this function the most probable parameter values can be calculated, as well as various measures of the precision of the parameter estimation. The same can be done for any quantity predicted by the model. 

What is actually the meaning of the parameters G and U with respect to probability The value U is the probability of the filling of a finite volume of an addition system by the elements of a porous system or, alternatively, the probability of encountering individual elements within a given volume. Accordingly, G is the probability of the recurrence of voids in a system. It is extremely important that, as might be expected for dimensionless probability parameters, neither G nor U depend on the nature of the elements or the absolute geometrical dimensions of the latter... 

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