The Schulz-Flory Distribution

FIGURE 5-7 Weight fraction of x-mers for different values of p. 

FIGURE 5-8 Experimental results redrawn from the data of G. B. Taylor, JACS, 69,638 (1947). 

Back term is in the mole fraction of its respective species NX/ NX) So we can write  

which it is, then the summation term is part of a well known series that converges  

FIGURE 5-9 Derivation of the number average degree of polymerization in terms of p. 

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Reaction mechanisms and molar mass distributions The molar mass distribution of a synthetic polymer strongly depends on the polymerization mechanism, and sole knowledge of some average molar mass may be of little help if the distribution function, or at least its second moment, is not known. To illustrate this, we will discuss two prominent distribution functions, as examples the Poisson distribution and the Schulz-Flory distribution, and refer the reader to the literature [7] for a more detailed discussion. 

Note 2 In the literature, this distribution is sometimes referred to as the Flory distribution or the Schulz-Flory distribution. 

Thus, Equation 27 is in this case a possible distribution function. It is of the type of the Schulz-Flory (25) distribution function. The expressions p and alternating polymerization (chain termination). The validity of the Schulz-Flory distribution function in this example of a polymerization with reversible propagation steps is evident. This type of distribution is always present if the distribution of the chain lengths... 

Because in some cases the production of methane does not obey the Schulz-Flory distribution, kj is allowed to differ from kj. The Schulz-Flory constant is defined as a = rn/rn i, for n 3,... 

The minimum value for the rate constant k on ruthenium is calculated from the data published by Dautzenberg et al. for the on iron the data of Vannice are used. For the ruthenium case it is assumed that the methane production meets the Schulz-Flory distribution (kj = k3). The value of the rate constant of propagation, where the reaction is completely determined by this constant is shown for the different models in table II. In the first column of this table minimum k values are shown for ruthenium. The data used fpom the work of Dautzenberg and coworkers were ... 

Deviations from the Schulz-Flory distribution arc possible if secondary reactions such as cracking on acidic supports or insertion of product olefins into the growing chain occur [42]. It has been reported recently that the Schulz Flory constant a has a tendency to increase from C3 to C, [45]. This may be the reason why the values found are usually higher for methane and lower for Cj and C) j.)., as would be expected for an ideal Schulz-Flory distribution [40]. Investigations by Madon et at. on partly sulfur-poisoned iron/copper catalysts revealed a dual product distribution. This was explained by the assump tion of > 2 types of active sites for hydrocarbon chain formation, each with a slightly different value of the chain growth probability [46]. 

On zeolitic catalysis, secondary reactions have to be anticipated such as isomerization and cracking which can inlluencc the Schulz—Flory distribution. ThLs has been used by the Mobil Oil Corporation to design a dual stage process for conversion of syntltesis gas to a gasoline higl) in octane rating, fn this process, the effluent of a conventional slurry phase Fischer- Tropsch reactor is converted over acidic ZSM-5 zeolite [136). 

These catalysts seem to be able to operate at low degrees of polymerization without giving the high yields of methane predicted by the Schulz Flory distribution. Activity maintenance and long-term catalyst performance remain to be proved. However, further research in this field may lead to stable catalysts giving a highly desirable distribution of liglit a olefin products [42]. 

In a recent study, R. Pettit et at. examined the validity of tire Fischer-Tropsch carbide mechanism, the Anderson-Emmett hydroxy carbene mechanism and the Pichlcr-Schulz mediaiiism [174. In a first experiment, the Schulz Flory distribution obtained by CO/H conversion over a cobalt catalyst in the absence and in the presence of CH N] was studied. It was found that addition of CHjN resulted in a signillcant increase of the propagation rate which is in favour of the assumption of methylene as a building block, as predicted by the carbide mechanism. Furthermore, the reaction was carried out using labeled CO (90% CO and 10% CO), H2. and CHjNj in variable ratios. The number of atoms in the propenc fraction was calculated according to the three... 

Given the Schulz-Flory distribution, the mole and weight fractions of polymer with j monomer units (based on polymer) as a function of that number j are... 

Matkovskii et al. also studied ethylene oligomerization. They found that with the homogeneous TiCl —AlC HjCl catalytic system in benzene and at room temperature, the MWD of oligomers was quite similar to the Schulz-Flory distribution and its experimental average values, determined by G.P.C., agree with the theoretical ones derived by analyzing a kinetic model in an almost stationary approximation. 

The shape of the Schulz-Flory distribution and the chain length of the a-olefins are controlled by the geometric chain-growth factor K, defined as K = n(C +2)/ n(C ) (see Figure 2). For the economy of the whole process it is very important that the /f-factor can easily be adjusted by varying the catalyst composition. Usually the value is between 0.75 and 0.80. 

The Schulz-Flory distribution provides a simple means of formulating a mathematical model for the numerical values of n, m, and p, characterizing the product pattern. It is conceivable to include this ideal distribution law in the definition of an ideal FT catalyst. Actual product distributions that significantly differ from this simple pattern can then be rationalized in terms... 

The amount of in situ branching produced is a sensitive function of many variables. Catalyst and reaction parameters have a major influence on the following (a) the a-olefin generation relative to polymer formation, (b) 1-hexene generation relative to that of other linear a-olefins, (c) how sharp or flat the Schulz-Flory distribution of linear a-olefins is, and (d) how efficiently the a-olefins are incorporated as branches. 

As mentioned above, the a-olefin products that are generated in situ consist of two superimposed distributions a Schulz-Flory distribution of even-numbered linear a-olefins and a spike of 1-hexene. The amount of 1-hexene generated compared to the Schulz-Flory distribution and the flatness or sharpness of the Schulz-Flory distribution can be sensitively dependent on the catalyst, the cocatalyst, and the reaction conditions [27,238,681,682, 698-700]. 

These results again demonstrate that two independent mechanisms of olefin generation operate simultaneously. The Schulz-Flory distribution obviously arises from an ethylene growth reaction on some sites that have a strong disposition to terminate the chain growth. The 1-hexene spike probably arises from the formation of chromacycloheptane, as has been observed with other chromium catalysts [674],... 

Figure 220 is a plot of the logarithm of the mol% a-olefin produced in the presence of BEt3 against the carbon number of the olefin. In this format, the Schulz-Flory distribution stands out clearly as a straight line on which the data for all the olefins but 1-hexene fall. The 1-hexene product clearly stands out as a spike off the Schulz-Flory background. 

The data thus illustrate the simultaneous operation of two mechanisms. Various cocatalysts and other conditions can be applied to vary the 1-hexene yield relative to the yield of the a-olefins in the Schulz-Flory distribution. In some cases, it has been possible to obtain 1-hexene exclusively, along with the polymer. Likewise, the Schulz-Flory contribution can be sharp or extended, as desired. 

Most of this potency of the hydrosilanes is used to enhance the Schulz-Flory distribution of olefins. The data of Figure 219 provide one example of the olefin distribution observed when a hydrosilane-modified catalyst was used. The 1-hexene contribution is barely noticeable in comparison to the background of the chain-growth reaction. 

The shape of the Schulz-Flory distribution and the chain length of the a-al-kenes are controlled by the geometric chain-growth factor K, defined as K = n(C +2)ln C ) (Figure 1). 

With the introduction of a variety of surface-sensitive instrumental techniques and the use of transient techniques utilizing isotopic tracers, the carbide theory of chain growth was revived to represent a dominant viewpoint since 1980. Brady and Pettit showed that the decomposition of diazomethane on various transition metal catalysts and in the absence of CO and H2 primarily led to ethylene. In the presence of H2, hydrocarbons up to Cjg were formed and conformed to the Schulz-Flory distribution expected for FTS (for an example of a description of Schulz-Flory distribution, see Reference 23). It was... 

Another indicator of abiogenic hydrocarbons is that the relative concentrations of C1-C4 /i-alkanes would follow the Schulz-Flory distribution—indicative of hydrocarbons produced by the polymerization of lower homologs, in which ... 

Why do the majority of the Taranaki natural gas wells follow a hydrocarbon distribution line parallel to, but not on the Schulz-Flory distribution ... 

The Gaussian distribution is the best known distribution. It represents the error law about the arithmetic mean. Because of its frequent appearance, the Gaussian distribution is also called the normal distribution in mathematics. In contrast, a certain form of the Schulz-Flory distribution is often called the normal distribution in macromolecular science. 

An exponentially decreasing curve is obtained for A = 1 when the mole fraction is plotted against the degree of polymerization (see Figure 8-3). For this reason, and not because an exponential function appears in Equation (8-35), the Schulz-Flory distribution is called an exponential distribution. 

Consequently, the distributions become increasingly narrower for increasing degrees of coupling. The Schulz-Flory distribution can be distinguished from the logarithmic normal distributions via Equations (8-37) and (8-31). For the Schulz-Flory distribution, the following, of course, holds ... 

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