Polymerization rate, computation

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.

Ethylene was polymerized with 2.5 pmol catalysts in toluene / CH2CI2 for 30 min. at different temperatures and 5 psig ethylene pressure. Polymerization rate was determined fitim the rate of consumption, measured by a hotwire flow meter (model 5850 D ftom Brooks Instrument Div.) connected to a personal computer throu an A/D converter. Polymaization was quenched by the addition of methanol containing HCl (5 v/v %). The polymar was washed with an excess amount of methanol and dried undw vacuum at 50 C. 

Monomer Concentration Effect. In bulk polymerizations such as those conducted in the present study, the dependence of polymerization rate on monomer concentration can be determined only on the basis of the dependence of rate on the extent of reaction. Reduction of the rate vs. time DSC traces to digital data files permits computer calculation of reaction rate as a function of monomer conversion. A computer program which yields print-out of the rate and time at given fractions of the total heat release allows computation of the order of reaction with respect to carbon-carbon double bond concentration. Assuming -80.0 cal gm l represents the... 

A digital-computer program was written to solve the equations and calculate the conversion at any time in a batch polymerization. Polymerization rates, instantaneous and cumulative molecular-weight distributions, and molecular-weight averages are also calculated at this time. 

The main objectives in modeling polymerization reactions are to compute polymerization rate and polymer properties for various reaction conditions. These two types of model outputs are not separate but they are usually very closely related. For example, an increase in reaction temperature raises polymerization rate... 

Polymerization Rate The overall polymerization rate is the only information that can be directly obtained from the online heat of reaction. In a homopolymerization reaction, it can be compnted directly, p(0=g,(0/V(- 4//) (moll s )- For (co)polymerization reactions, strictly speaking, the overall polymerization rate cannot be computed from this equation because the enthalpy of polymerization of each monomer is typically different. However, using an average (A//r) will suffice in many cases and a rough estimation can also be obtained from the equation. If a more accurate value is needed, information about the reactivity ratios must be used as it would be discussed in Section 1.2.53. 

Itis worth mentioning that there is a fair amount of information on computer simulation to achieve steady-state reaction conditions [39]. These simulations use artificial neural networks, [40] advances in computational fluid dynamics (CFD) [41], as well as combination of new experimental and modeling techniques, whence the apphcation of these techniques can lead to improved models of polymerization systems as well as the discovery of new kinetic mechanisms that control polymerization rate and properties. 

The first quantitative model, which appeared in 1971, also accounted for possible charge-transfer complex formation (45). Deviation from the terminal model for bulk polymerization was shown to be due to antepenultimate effects (46). Mote recent work with numerical computation and C-nmr spectroscopy data on SAN sequence distributions indicates that the penultimate model is the most appropriate for bulk SAN copolymerization (47,48). A kinetic model for azeotropic SAN copolymerization in toluene has been developed that successfully predicts conversion, rate, and average molecular weight for conversions up to 50% (49). 

Chains with uttdesired functionality from termination by combination or disproportionation cannot be totally avoided. Tn attempts to prepare a monofunctional polymer, any termination by combination will give rise to a difunctional impurity. Similarly, when a difunctional polymer is required, termination by disproportionation will yield a monofunctional impurity. The amount of termination by radical-radical reactions can be minimized by using the lowest practical rate of initiation (and of polymerization). Computer modeling has been used as a means of predicting the sources of chain ends during polymerization and examining their dependence on reaction conditions (Section 7.5.612 0 J The main limitations on accuracy are the precision of rate constants which characterize the polymerization. 

Peaking and Non-isothermal Polymerizations. Biesenberger a (3) have studied the theory of "thermal ignition" applied to chain addition polymerization and worked out computational and experimental cases for batch styrene polymerization with various catalysts. They define thermal ignition as the condition where the reaction temperature increases rapidly with time and the rate of increase in temperature also increases with time (concave upward curve). Their theory, computations, and experiments were for well stirred batch reactors with constant heat transfer coefficients. Their work is of interest for understanding the boundaries of stability for abnormal situations like catalyst mischarge or control malfunctions. In practice, however, the criterion for stability in low conversion... 

After phase separation, two sets of equations such as those in Table A-1 describe the polymerization but now the interphase transport terms I, must be included which couples the two sets of equations. We assume that an equilibrium partitioning of the monomers is always maintained. Under these conditions, it is possible, following some work of Kilkson (17) on a simpler interfacial nylon polymerization, to express the transfer rates I in terms of the monomer partition coefficients, and the iJolume fraction X. We assume that no interphase transport of any polymer occurs. Thus, from this coupled set of eighteen equations, we can compute the overall conversions in each phase vs. time. We can then go back to the statistical derived equations in Table 1 and predict the average values of the distribution. The overall average values are the sums of those in each phase. 

The fixed variables used in the computer simulation are shown in Table 1 along with the kinetic rate constants for the polymerization reactions. 

Gel Permeation Chromatography. The instrument used for GPC analysis was a Waters Associates Model ALC - 201 gel permeation chromatograph equipped with a R401 differential refractometer. For population density determination, polystyrene powder was dissolved in tetrahydrofuran (THF), 75 mg of polystyrene to SO ml THF. Three y -styragel columns of 10, 10, 10 A were used. Effluent flow rate was set at 2.2 ml/min. Total cumulative molar concentration and population density distribution of polymeric species were obtained from the observed chromatogram using the computer program developed by Timm and Rachow (16). 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

Experimental Materials. All the data to be presented for these illustrations was obtained from a series of polyurethane foam samples. It is not relevant for this presentation to go into too much detail regarding the exact nature of the samples. It is merely sufficient to state they were from six different formulations, prepared and physically tested for us at an industrial laboratory. After which, our laboratory compiled extensive morphological datu on these materials. The major variable in the composition of this series of foam saaqples is the aaK>unt of water added to the stoichiometric mixture. The reaction of the isocyanate with water is critical in determining the final physical properties of the bulk sample) properties that correlate with the characteristic cellular morphology. The concentration of the tin catalyst was an additional variable in the formulation, the effect of which was to influence the polymerization reaction rate. Representative data from portions of this study will illustrate our experiences of incorporating a computer with the operation of the optical microscope. 

Although the basic mechanisms are generally agreed on, the difficult part of the model development is to provide the model with the rate constants, physical properties and other model parameters needed for computation. For copolymerizations, there is only meager data available, particularly for cross-termination rate constants and Trommsdorff effects. In the development of our computer model, the considerable data available on relative homopolymerization rates of various monomers, relative propagation rates in copolymerization, and decomposition rates of many initiators were used. They were combined with various assumptions regarding Trommsdorff effects, cross termination constants and initiator efficiencies, to come up with a computer model flexible enough to treat quantitatively the polymerization processes of interest to us. 

Reaction rates are macroscopic averages of the number of microscopical molecules that pass from the reactant to the product valley in the potential hypersurface. An estimation of this rate can be obtained from the energy of the highest point in the reaction path, the transition state. This approach will however fail when the reaction proceeds without an enthalpic barrier or when there are many low frequency modes. The study of these cases will require the analysis of the trajectory of the molecule on the potential hypersurface. This idea constitutes the basis of molecular dynamics (MD) [96]. Molecular dynamics were traditionally too computationally demanding for transition metal complexes, but things seem now to be changing with the use of the Car-Parrinello (CP) method [97]. This approach has in fact been already succesfully applied to the study of the catalyzed polymerization of olefins [98]. 

From this type of analysis, one would conclude that t must be approximately 28 for a 10% reduction in protomer to cause a 95% reduction in the nucleus concentration. This is a rather startling apparent reaction order even assuming infinite cooperativity between protomers. It is recalled that Hofrichter et al. (1974) found from a similar analysis of the rate of nucleation of human hemoglobin S (HbS) at 30 C that the apparent reaction order for the nucleation of HbS aggregation was about 32. Of course, such analyses are not fully justifiable because one may not assume ideality in the solution properties of biopolymers at high concentrations, particularly at 200 mg/ml in the case of hemoglobin. The computation for the case of tubulin polymerization does, nonetheless, emphasize that nucleation would be an especially cooperative event if only tubulin, and not ring structures, played the active role in nuclei formation. 

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