Flory-Schulz distribution, chain

This is the Flory-Schulz distribution. We discuss this distribution further after we discuss chain reactions. 

Equation 3.6, together with Equation 3.4, describes a random distribution of molecular sizes this distribution is also known as the Flory-Schulz distribution or the most probable distribution [5]. Recently, Wutz and Kricheldorf [6] proposed a model describing the frequency distribution (/ ) and formulated the weight distribution (w,) of linear chains in step-growth polymerizations considering the cyclation reaction, which is one of the most important side reactions in step-growth polymerization. 

The aforementioned expression is the geometric distribution or the Flory-Schulz distribution. The results can be illustrated by plotting the mole fraction of chain length for different values of conversion, p. 

The polymer chain seeding process aims to generate a set of polymer chains with a predetermined weight distribution. Different types of polymer weight distributions can be adopted to generate polymer chains, among which the Flory-Schulz distribution and the log-normal distribution are the most used. The Flory-Schulz distribution is based on a model of chain polymerization from Paul Flory and G. V. Schulz. Thus, the Flory-Schulz distribution is believed to be the most probable one. ... 

When determining the product selectivities, all compounds of equal carbon numbers (paraffines, olefins, isomers, and oxygen compounds) were summarized to one product fraction. The chain growth probability was determined by the Anderson-Schulz-Flory (ASF) distribution ... 

The carbon number distribution of Fischer-Tropsch products on both cobalt and iron catalysts can be clearly represented by superposition of two Anderson-Schulz-Flory (ASF) distributions characterized by two chain growth probabilities and the mass or molar fraction of products assigned to one of these distributions.7 10 In particular, this bimodal-type distribution is pronounced for iron catalysts promoted with alkali (e.g., K2C03). Comparing product distributions obtained on alkali-promoted and -unpromoted iron catalysts has shown that the distribution characterized by the lower growth probability a, is not affected by the promoter, while the growth probability a2 and the mass fraction f2 are considerably increased by addition of alkali.9 This is... 

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... 

Figure 2. Schulz-Flory product distribution dependence on the chain growth factor K.

Figure 4. Flory-Schulz chain-length distributions. Equation 3, for v= 100 (left) and v = 1 000 (right), Fy, = 1 (smaller unbroken) andFy, = 0 (taller dashed).

Overall hydrocarbon distributions on the four carbon supports are shown in Figure 11. The hydrocarbon mole fractions comply with the Anderson-Schulz-Flory (ASF) distribution. Note that hydrocarbons of carbon number up to 34 are detected for all carbon supports. However, the chain-growth probability (a) for catalyst with wood-AC support is 0.65 (at least for carbon numbers of up to 20), smaller than that of the catalysts using the other three AC supports ( 0.71). 

Zein el Deen et al. (30) studied the kinetics of the FTS on sintered oxides of iron and manganese. They observed, too, that the rate is independent of the CO partial pressure. Bub et al. (20) developed empirical expressions for the production rate of CO2 and Ci to C4 hydrocarbons on a Mn/Fe catalyst which could be used to successfully describe the conversion and selectivity in a pilot plant fixed bed reactor (2 cm ID by 80 cm length). If a catalyst like Mn/Fe gives a Schulz-Flory product distribution the hydrocarbon fraction can be calculated from the overall conversion rate and the chain growing probability a... 

The practical value of the Fischer-H opsch reaction is limited by the unfavorable Schulz-Flory distribution of hydrocarbon products that is indicative of a chain growth polymerization mechanism. In attempts to increase the yields of lower hydrocarbons such as ethylene and propylene (potentially valuable as feedstocks to replace petrochemicals), researchers have used zeolites as supports for the metals in attempts to impose a shape selectivity on the catalysis [114] or to control the performance through particle size effects. [IIS] These attempts have been partially successful, giving unusual distributions of products, such as high yields of C3 [114] or C4 hydrocarbons. [116] However, the catalysts are often unstable because the metal is oxidized or because it migrates out of the zeolite cages to form crystallites, which then give the Schulz-Flory product distribution. 

The FTS mechanism could be considered a simple polymerization reaction, the monomer being a Ci species derived from carbon monoxide. This polymerization follows an Anderson-Schulz-Flory distribution of molecular weights. This distribution gives a linear plot of the logarithm of yield of product (in moles) versus carbon number. Under the assumptions of this model, the entire product distribution is determined by one parameter, a, the probability of the addition of a carbon atom to a chain (Figure 4-7). ... 

The chain lengths of the hydrocarbons obey a statistical distribution named after Anderson, Schulz, and Flory, given by... 

The description of the product distribution for an FT reaction can be simplified and described by the use of a single parameter (a value) determined from the Anderson-Schulz-Flory (ASF) plots. The a value (also called the chain growth probability factor) is then used to describe the total product spectrum in terms of carbon number weight fractions during the FT synthesis. In the case... 

For entries 3-5 the increase in molecular weight observed can be assigned to the increase in the rate of insertion and the rate of termination remains practically the same. An increase of the rate of polymerisation with the steric bulk of the ligand is usually ascribed to the destabilisation of the alkene adduct while the energy of the transition state remains the same. As a chain transfer reaction presumably P-hydride elimination takes place or traces of water might be chain transfer agents. Chain transfer does occur, because a Schulz-Flory molecular weight distribution is found (PDI 2, see Table 12.2). Shorter chains are obtained with a polar ortho substituent (OMe, entry 2) and in methanol as the solvent, albeit that most palladium is inactive in the latter case. 

Already in the study of linear chain molecules it has become evident that the shape of the molar mass distribution and its width provide a valuable guide to the mechanism of chain formation. Best known are the most probable (or Schulz-Flory) distribution and the narrow Poisson distribution. The former is often... 

The molar mass distribution of branched materials differ most significantly from those known for Hnear chains. To make this evident the well known types of (i) Schulz-Flory, or most probable distribution, (ii) Poisson, and (iii) Schulz-Zimm distributions are reproduced. Let x denote the degree of polymerization of an x-mer. Then we have as follows. 

The production of hydrocarbons using traditional F-T catalysts is governed by chain growth (polymerization) kinetics. The theoretical equation describing the distribution of hydrocarbon products, commonly referred to as the Anderson-Schulz-Flory (ASF) equation, is... 

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