Reaction-diffusion regime

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. 

The concept of reaction diffusion (also called residual termination) has been incorporated into a number of treatments.7 7 Reaction diffusion will occur in all conversion regimes. However at low and intermediate conversions the process is not of great significance as a diffusion mechanism. At high conversion long chains are essentially immobile and reaction diffusion becomes the dominant diffusion mechanism (when i and j are both "large" >100). The termination rate constant is determined by the value of kp and the monomer concentration. In these circumstances, the rate constant for termination k - should be independent of the chain lengths i and j and should obey an expression of the form 75... 

The possibility that adsorption reactions play an important role in the reduction of telluryl ions has been discussed in several works (Chap. 3 CdTe). By using various electrochemical techniques in stationary and non-stationary diffusion regimes, such as voltammetry, chronopotentiometry, and pulsed current electrolysis, Montiel-Santillan et al. [52] have shown that the electrochemical reduction of HTeOj in acid sulfate medium (pH 2) on solid tellurium electrodes, generated in situ at 25 °C, must be considered as a four-electron process preceded by a slow adsorption step of the telluryl ions the reduction mechanism was observed to depend on the applied potential, so that at high overpotentials the adsorption step was not significant for the overall process. 

It can be seen in the plot in Figure 11 that EA . shows a clear temperature dependence. For rising temperatures the mass transport limitation can be observed, which leads to a lowering of EAs by a factor of V2 in the pore diffusion regime down to 0, owing to the shift of the reaction from the interior of the pore system of the catalytic particle to the outer surface. In the final state, the diffusion through the boundary layer becomes the rate-limiting step of the reaction. 

Note In the strong pore diffusion regime the rate is lower but the catalyst deactivates more slowly. Actually, for the catalyst used here if we could have been free of diffusional resistances reaction rates would have been 360 times as fast as those measured. 

Our reaction A R proceeds isothermally in a packed bed of large, slowly deactivating catalyst particles and is performing well in the strong pore diffusion regime. With fresh pellets conversion is 88% however, after 250 days conversion drops to 64%. How long can we run the reactor before conversion drops to... 

Anseth et al. [146,147] have experimentally characterized the kinetic constant for a series of multifunctional methacrylate and acrylate monomers. In particular, they explored the kinetic evidence for the importance of reaction diffusion for polymerizations occurring in the high crosslinking regime. When reaction diffusion is the controlling termination mechanism, it was hypothesized that k, would be proportional to fcp[m] where [m] is the concentration of double bonds. The works of Anseth et al. [146] then characterized the proportional constant between k, and kp[m for the methacrylates and acrylates studied. 

On the basis of the Hatta number, the transformations carried out in biphasic systems can be described as slow (Ha < 0.3), intermediate (with a kinetic-diffusion regime 0.3 < Ha < 3.0), and fast (Ha > 3). These are diffusion limited and take place near the interface (within the diffusion layer). Slow transformations are under kinetic control and occur mostly in a bulk phase, so that the amount of substrate transformed in the boundary layer in negligible. When diffusion and reaction rate are of similar magnitude, the reaction takes place mostly in the diffusion layer, although extracted substrate is also present in the continuous phase, where it is transformed at a rate depending on its concentration [38, 50, 54]. 

The situation is analogous to the so-called energy diffusion regime in the Kramers picture of reactions in solution except that here the molecule acts as its own solvent [E. W. Schlag, J. Grotemeyer, and R. D. Levine, Chem. Phys. Lett. 190, 521 (1992)]. 

It might seem that the ArBs layer could grow in the A-ApBq-ArBs-AiBn-B system by the same mechanism as in the ApBq-ArBs-B and ApBq-ArBs-A Bn systems, i.e. at the expense of the phase transformation of ApBq into ArBs under the influence of reaction diffusion of the A atoms. However, this is not the case. If the growth regime of the ApBq layer in the A -ApBq-ArBs-AiBn-B system is reaction controlled with regard to component A (x < ), then there is an excess of A atoms in comparison with the... 

Undoubtedly, the most promising modehng of the cardiac dynamics is associated with the study of the spatial evolution of the cardiac electrical activity. The cardiac tissue is considered to be an excitable medium whose the electrical activity is described both in time and space by reaction-diffusion partial differential equations [519]. This kind of system is able to produce spiral waves, which are the precursors of chaotic behavior. This consideration explains the transition from normal heart rate to tachycardia, which corresponds to the appearance of spiral waves, and the fohowing transition to fibrillation, which corresponds to the chaotic regime after the breaking up of the spiral waves, Figure 11.17. The transition from the spiral waves to chaos is often characterized as electrical turbulence due to its resemblance to the equivalent hydrodynamic phenomenon. 

Growth kinetics of nanocrystals in the presence of capping agents is determined by several complex factors, and signatures of either the diffusion or the reaction-controlled regimes are... 

Variation of catalyst area. The catalytic rate is proportional to the total surface area, A, external and internal, for reactions controlled by surface kinetics. In the case of internal or pore diffusion control, the rate is proportional to A1,2 and is also a function of the catalyst shape and size [49, 53]. Under an external diffusion regime, the catalytic rate is proportional to the external surface area of the catalyst, Aex. 

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