Implicit propagation steps

Thus, Equation 27 is in this case a possible distribution function. It is of the type of the Schulz-Flory (25) distribution function. The expressions p and alternating polymerization (chain termination). The validity of the Schulz-Flory distribution function in this example of a polymerization with reversible propagation steps is evident. This type of distribution is always present if the distribution of the chain lengths... 

In the above derivations, the values of and have been implicitly considered as constants independent of the reaction medium and of the chain length of the polymeric chains involved in the reactions. For propagation, it is known that this is true only to a first approximation, since there is increasing evidence that the value of should be greater for the first few propagation steps, when the chain... 

However, recent experimental and theoretical work has shown that the assumptions of the implicit penultimate model are unlikely to be applicable to the majority of copol5mierization systems. We have recently published a review of this evidence (37,42), which draws on direct experimental and theoretical measures of reactivity ratios, model testing in a range of copolymerization systems, and other tests of the mechanism of the propagation step via, for example, the examination of solvent effects on reactivity ratios. These studies provide strong evidence for penultimate imit effects but, in all cases where penultimate unit effects have been measured directly, effects on radical selectivity have been shown to be significant. In other words all available evidence contradicts the assumption of the implicit penultimate model that the penultimate unit affects reactivity but not selectivity. 

When solvent effects on the propagation step occnr in free-radical copolymerization reactions, they result not only in deviations from the expected overall propagation rate, but also in deviations from the ejqiected copolymer composition and microstracture. This may be trae even in bulk copolymerization, if either of the monomers exerts a direct effect or if strong cosolvency behavior causes preferential solvation. A number of models have been proposed to describe the effect of solvents on the composition, microstmcture and propagation rate of copolymerization. In deriving each of these models, an appropriate base model for copolymerization kinetics is selected (such as the terminal model or the implicit or explicit penultimate models), and a mechanism by which the solvent influences the propagation step is assumed. The main mechanisms by which the solvent (which may be one or both of the comonomers) can affect the propagation kinetics of free-radical copolymerization reactions are as follows ... 

Note that, in loeal eoordinates. Step 2 is equivalent to integrating the equations (13). Thus, Step 2 can either be performed in loeal or in eartesian coordinates. We consider two different implicit methods for this purpose, namely, the midpoint method and the energy conserving method (6) which, in this example, coineides with the method (7) (because the V term appearing in (6) and (7) for q = qi — q2 is quadratie here). These methods are applied to the formulation in cartesian and in local coordinates and the properties of the resulting propagation maps are discussed next. 

The nonlinear part of the susceptibility was introduced into the quasi-linear finite-difference scheme via iterations, so that at any longitudinal point, the magnitude of E calculated at the previous longitudinal point was used as a zero approximation. This approach is better than the split-step method since it allows one to jointly simulate both the mode field diffraction on irregular sections of the waveguide and the self-action effect by introducing the nonlinear permittivity into the implicit finite-difference scheme which describes the propagation of the total field. 

A variety of explicit (Dufort-Frankel, Lax-Wendroff, Runge-Kutta) and implicit (approximate factorization, LU-SGS) or hybrid schemes have been employed for integration in time. Because of the complexity of the incompressible Navier-Stokes equations, stability analyses to determine critical time steps are difficult. As a general rule, the allowable time step for an explicit method is proportional to the ratio of the smallest grid size to the largest convective velocity (or the wave propagation speed for an artificial compressibility method). 

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