Reaction diffusion terminal model

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

The diffusion-controlled termination model considers the termination reaction as the reaction... 

In multifunctional monomer polymerizations, the mobility of radicals through segmental diffusion falls well before their mobility through reaction diffusion at very low functional group conversions (as compared to linear polymerizations). From this point in the reaction, the termination and propagation kinetic constants are found to be related, and the termination kinetic constant as a function of conversion may actually exhibit a plateau region. Figure 6 illustrates the typical behavior of kp and k, vs conversion as predicted by a kinetic based model. 

The reaction diffusion regime was further clarified by Russell et al. [42] According to their model, the actual residual termination rate constant lie between two limiting values, a minimum, corresponding to a rigid chain, sue as polystyrene, and a maximum, corresponding to a flexible chain. It has beer found that the expression of the reaction diffusion controlled kt from Stickler e> al. [41] is the same as the minimum value proposed by Russell et al [42]. Both approaches share some common characteristics. Reaction diffusion control plays an important role in styrene homopolymerization since it is the main method of termination in later stages of the polymerization. 

Termination reactions occur between two relatively large radicals, and termination rates arc limited by the rates at which the radical ends can encounter each other. As a result, kt is a decreasing function of the dimensions of the reacting radical. The segmental diffusion coefficient and the termination rate constant increase as the polymer concentration increases from zero. This initial increase is more pronounced when the molecular weight of the polymer is high and/or when the polymerization is carried out in a medium which is a good solvent for the polymer. For similar reasons, k t is inversely proportional to the viscosity of reaction medium. A model has been proposed that accounts for these variations in k, in low-conversion radical polymerizations [15,16]. 

The propagation rate constant kp and the termination rate constant are calculated as functions of free volume, following a model developed by Anseth and Bowman." In that model, the two constants are calculated by summing different resistances for diffusion, intrinsic reaction, and reaction— diffusion (for termination). 

A model for linear polymerizations proposed by Russell el al. [11, 12] suggests the following equations for predicting the kinetic termination constant when reaction diffusion is the dominant termination mechanism ... 

Complex Photopolymerization Systems. Kinetic modeling of free-radical photopolymerizations becomes more complicated as comonomers are added to the reaction system and as different polymerization methods are used to tailor the pol5uner properties. Although free-radical reaction mechanisms still hold true, rates of propagation and termination must be reconsidered to account for variables such as differences in double bond reactivities, reaction diffusion, and chain transfer. 

Using mathematical modeling to rationalize and predict battery behavior and system performance has been widely used. Driven by commercial interests in the 1980s, the Zn/Bt2 battery was extensively simulated, and various types of models have been used to investigate the transportation of species, secondary electrode reactions, and chemical reactions in bulk electrolyte. Lee et al. [76] developed thin diffusion-layer models to assess the effects of separator and terminal resistance on current distribution and the performance of flow reactors. In this model, in... 

Buback has proposed a termination model to account for the above-mentioned course of kt with conversion, considering segmental, translational and reaction diffusion processes [121], The model is somewhat crude as compared to the subtle solvent effects discussed above and doesn t take phenomena such as solvent quality and coil dimensions into account. Nevertheless, the model has been found to accurately describe a large set of experimental data over broad range of conversions [122-125, 127, 128] and provides insights in which diffusion mechanisms are dominant at which conversions. 

Figure 5. Cartoon models of the reaction of methanol with oxygen on Cu(llO). 1 A methanol molecule arrives from the gas phase onto the surface with islands of p(2xl) CuO (the open circles represent oxygen, cross-hatched are Cu). 2,3 Methanol diffuses on the surface in a weakly bound molecular state and reacts with a terminal oxygen atom, which deprotonates the molecule in 4 to form a terminal hydroxy group and a methoxy group. Another molecule can react with this to produce water, which desorbs (5-7). Panel 8 shows decomposition of the methoxy to produce a hydrogen atom (small filled circle) and formaldehyde (large filled circle), which desorbs in panel 9. The active site lost in panel 6 is proposed to be regenerated by the diffusion of the terminal Cu atom away from the island in panel 7.

It should be taken into account that the reaction of chain propagation occurs in polymer more slowly than in the liquid phase also. The ratios of rate constants kjlkq, which are so important for inhibition (see Chapter 14), are close for polymers and model hydrocarbon compounds (see Table 19.7). The effectiveness of the inhibiting action of phenols depends not only on their reactivity, but also on the reactivity of the formed phenoxyls (see Chapter 15). Reaction 8 (In + R02 ) leads to chain termination and occurs rapidly in hydrocarbons (see Chapter 15). Since this reaction is limited by the diffusion of reactants it occurs in polymers much more slowly (see earlier). Quinolide peroxides produced in this reaction in the case of sterically hindered phenoxyls are unstable at elevated temperatures. The rate constants of their decay are described in Chapter 15. The reaction of sterically hindered phenoxyls with hydroperoxide groups occurs more slowly in the polymer matrix in comparison with hydrocarbon (see Table 19.8). 

In this paper, the kinetics and polymerization behavior of HEMA and DEGDMA initiated by a combination of DMPA (a conventional initiator) and TED (which produces DTC radicals) have been experimentally studied. Further, a free volume based kinetic model that incorporates diffusion limitations to propagation, termination by carbon-carbon radical combination and termination by carbon-DTC radical reaction has been developed to describe the polymerization behavior in these systems. In the model, all kinetic parameters except those for the carbon-DTC radical termination were experimentally determined. The agreement between the experiment and the model is very good. 

The iGLE also presents a novel approach for studying the reaction dynamics of polymers in which the chemistry is driven by a macroscopic force that is representative of the macroscopic polymerization process itself The model relies on a redefined potential of mean force depending on a coordinate R which corresponds locally to the reaction-path coordinate between an n-mer and an (n -t 1 )-mer for R = nl. The reaction is quenched not by a kinetic termination step, but through an (R(t))-dependent friction kernel which effects a turnover from energy-diffusion-limited to spatial-diffusion-Iimited dynamics. The iGLE model for polymerization has been shown to exhibit the anticipated qualitative dynamical behavior It is an activated process, it is autocatalytic, and it quenches... 

The above models describe a simplified situation of stationary fixed chain ends. On the other hand, the characteristic rearrangement times of the chain carrying functional groups are smaller than the duration of the chemical reaction. Actually, in the rubbery state the network sites are characterized by a low but finite molecular mobility, i.e. R in Eq. (20) and, hence, the effective bimolecular rate constant is a function of the relaxation time of the network sites. On the other hand, the movement of the free chain end is limited and depends on the crosslinking density 82 84). An approach to the solution of this problem has been outlined elsewhere by use of computer-assisted modelling 851 Analytical estimation of the diffusion factor contribution to the reaction rate constant of the functional groups indicates that K 1/x, where t is the characteristic diffusion time of the terminal functional groups 86. ... 

A valid kinetic model of stage 3 emulsion polymerization must account for diffusion-controlled termination and propagation reactions. Marten and Hamielec (J) have proposed such a model based on a free-volune theory and have confirmed its validity for the bulk polymerization of methyl methacrylate (7). Herein is reported an evaluation of this model for the emulsion... 

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