Model Flory distribution

Argonne National Laboratory, 137 aromatics, 477 Arrhenius equation, 60 ART. See advanced refining technologies ASF. See Anderson-Schultz-Flory distribution model ash... 

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). ... 

For the transformation of the macrocomposite model to a molecular composite model for the ultimate strength of the fibre the following assumptions are made (1) the rods in the macrocomposite are replaced by the parallel-oriented polymer chains or by larger entities like bundles of chains forming fibrils and (2) the function of the matrix in the composite, in particular the rod-matrix interface, is taken over by the intermolecular bonds between the chains or fibrils. In order to evaluate the effect of the chain length distribution on the ultimate strength the monodisperse distribution, the Flory distribution, the half-Gauss and the uniform distribution are considered. 

The existence of Wk related to the distribution of aggregate sizes and fulfilling the identity at Eq. (80) implies that the aggregation process leads to an equilibrium distribution. For the Flory-Stockmayer model (model 1 in Table 4) the number of assembling an acyclic k-mer from distinguishable units is [40]... 

A modified Cahn-Hilliard (CH) model [114] is used for the theoretical analysis of the impact of thermal diffusion on phase separation by taking into account an inhomogeneous temperature distribution, which couples to a concentration variation via the Soret effect. The Flory-Huggins model is used for the free energy of binary polymer-mixtures. The composition is naturally measured in terms of volume fraction 0 of a component A, which can be related to the weight fraction c by... 

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 ... 

So far two models have been employed to rationalize the solvation process the classical solution model, either the mole-fraction scale or any other concentration scale, and the Flory-Huggins model. The question is where to use which theoretical model to interpret the results of partitioning experiments, in which solute molecules distribute between two phases, a and ft. If the two phases are at equilibrium at the same temperature and the same pressure, /z = /xf. After rearrangement and applying Eq. (11-8), we can write... 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

The fractionation effect occurring in polydisperse samples is shown more explicitly in fig. 5.30. The theoretical computations (left) were performed with the SF model, assuming a Schultz-Flory distribution with M /M = 2.052 ). The experiments apply to polydisperse PS adsorbing from CCI4 on SIOj (right) ). In both cases a rather sharp fractionation occurs, with the long chains (the "winners") on the surface and the short ones (the "losers") in the solution. It... 

Figure 5.30. Adsorption fractionation of a polydisperse sample as calculated from the SF model (a) and as measured by gel permeation chromatography (b))). The calculations apply to a Schultz-Flory distribution with = 2.05 ), the experiments to the...

Non-Flory molecular weight distributions have also been attributed to the presence of several types of active sites with different probabilities for chain growth and for chain termination to olefins and paraffins (45). Two-site models have been used to explain the sharp changes in chain growth probability that occur for intermediate-size hydrocarbons on Fe-based catalysts (46,47). Many of these reports of non-Flory distributions may instead reflect ineffective dispersal of alkali promoters on Fe catalysts or inadequate mass balances and product collection protocols. Recently, we have shown that multisite models alone cannot explain the selectivity changes that occur with increasing chain size, bed residence time, and site density on Ru and Co catalysts (4,5,40,44). 

Diffiisional restrictions increase the effectiveness of olefin interception sites placed within catalyst pellets. Very high olefin hydrogenation turnover rates or site densities within pellets prevent olefin readsorption and lead to Flory distributions of lighter and more paraffinic hydrocarbons. Identical results can be obtained by introducing a double-bond isomerization function into FT catalyst pellets because internal olefins, like paraffins, are much less reactive than a-olefins in chain initiation reactions. However, light paraffins and internal olefins are not particularly useful end-products in many applications of FT synthesis. Yet, similar concepts can be used to intercept reactive olefins and convert them into more useful products (e.g., alcohols) and to shift the carbon number distribution into a more useful range. In the next section, olefin readsorption model simulations are used to explore these options in the control of FT synthesis selectivity. 

Krishna and Bell (299) described the results of their steady-state tracing experiments by the model shown in Fig. 31. The scheme is in accord with the Anderson-Schulz-Flory distribution of products, based on chain growth by the successive addition of monomers s to chain fragments C s- is different from Ci,s. ft is assumed that the probability of chain growth a is not a function of n, where... 

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. 

Concurrent with FTS mechanism studies, product distribution models were developed based on the analysis of product composition. Friedel and Anderson28 29 in the 1950s published the Anderson-Schulz-Flory (ASF) distribution model to predict the wide range of products yielded from FTS. The equation is shown as follows ... 

For instance, some nylons are produced through this mechanism. This is usually modeled under the simplifying assumption that the rate constants are independent of chain length. This assumption was proved pretty accurate, and by using the z transform it results in the Flory distribution ... 

The Flory distribution is a random distribution useful in several modes of polymerization. This distribution results from addition polymerization reactions when the only significant processes that interrupt macromolecular growth are either or both of chain transfer (to any species but the polymer) or termination by disproportionation. Likewise, this molecular weight distribution describes linear condensation polymerization when equal reactivity is assumed for all ends only when the reaction involves an equilibrium between polymerization and depolymerization. The model describes the distribution with one parameter which is the number average molecular weight. The distribution equation is ... 

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 polycondensation processes generally produce polyamides that are mixtures of polymer molecules of different molecular weights, the distribution of which usually follows a definite continuous function according to the most probable distribution model by Schulz-Flory [3]. This distribution function may, in principle, be derived from the kinetics of polymerization process, but is more readily derived from statistical considerations. In this case, the extent... 

A catalytic system formed in turbulent flows includes AC of Types III and IV (Figure 3.48). The polycentrism model of an ionic-coordinative catalytic system predicts that each active centre type is responsible for the synthesis of polymer fractions with the most probable Flory distribution of M /M = 2. Taking this into consideration, we can prove that such a narrow MWD at the initial polymerisation stage is provided by the presence of only two types of AC, formed after the hydrodynamic impact, a situation opposite to the traditional method of polymerisation. 

The results above are only valid for tetrafunctional crosslinking of monodisperse polymer. However, in many thermoreversible systems the crosslinks have functionalities that are much larger than four. Moreover, the polymers used are not monodisperse in general. In order to be able to calculate network parameters the present author [39—44] extended the Flory-Stockmayer model for polydisperse polymer which is crosslinked with f-functional crosslinks. It was possible to calculate network parameters for polymers of various molecular weight distributions (monodisperse polymer with D s M, /r3 = 1, a Schulz-Flory distribution with D = 1.5, a Flory distribution with D = 2, a cumulative... 

The polymer under consideration can be described with a Schulz-Flory distribution function with = 100 and k = 1. For the thermodynamic modeling, the quantities tm and rz must be calculated. 

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