Variable polymerization numbers

At present it is not possible to determine which of these mechanisms or their variations most accurately represents the behavior of Ziegler-Natta catalysts. In view of the number of variables in these catalyzed polymerizations, both mechanisms may be valid, each for different specific systems. In the following example the termination step of coordination polymerizations is considered. 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

The use of a polymeric support also affords a unique opportunity to control independently the variables that may affect the chiral recognition process, which is hard to achieve with silica. For example, the type and number of reactive sites can be easily adjusted with a polymer support. We recently reported an extensive study of the... 

Figure 1 Is a flow sheet showing some significant aspects of the Iterative analysis. The first step In the program Is to Input data for about 50 physical, chemical and kinetic properties of the reactants. Each loop of this analysis Is conducted at a specified solution temperature T K. Some of the variables computed In each loop are the monomer conversion, polymer concentration, monomer and polymer volume fractions, effective polymer molecular weight, cumulative number average molecular weight, cumulative weight average molecular weight, solution viscosity, polymerization rate, ratio of polymerization rates between the current and previous steps, the total pressure and the partial pressures of the monomer, the solvent, and the nitrogen.

Polymerization Dynamics. Relationships presented were utilized for the simulation of monomer concentration, number and weight average molecular weights, and population density distributions for two experimental observations. Experimental values of these variables are in reasonable proximity of calculated values. 

The existence of three steady states, two stable and one metastable, is common for exothermic reactions in stirred tanks. Also common is the existence of only one steady state. For the styrene polymerization example, three steady states exist for a limited range of the process variables. For example, if Ti is sufficiently low, no reaction occurs, and only the lower steady state is possible. If Tin is sufficiently high, only the upper, runaway condition can be realized. The external heat transfer term, UAextiTout — Text in Equation (5.28) can also be used to vary the location and number of steady states. 

The increase in iV, and therefore in the rate as well, with initial soap concentration is thus explained. Quantitative results agree approximately with the predicted three-fifths power dependence. The prediction of an increase in polymerization rate with also has been confirmed by experiments at variable initiator concentrations.t Most important of all, the actual number of particles N calculated from Eq. (35) agrees within a factor of two with that observed. It is thus apparent that the theory of emulsion polymerization developed by Harkins and by Smith and Ewart has enjoyed spectacular success in accounting for the unique features of the emulsion polymerization process. 

In the polymerization stage, the number of events, i.e., of the polymerization starts, N, is given and the events are identified by the index n = 1... N. Start times of polymerizations are represented by continuous variables in e [0, H] in with H denoting the given scheduling horizon. As an initial condition, the first polymerization is defined to start at tn= i = t°. 

The timing and the number of the polymerization batches together with the assignments of the recipes are modeled by an integer variable N >rp. This variable denotes how many batches according to recipe rp are produced in period i. 

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