Models autocatalytic

Similar results have recently been reported by Aspnes and Heller. They proposed an autocatalytic model for photoactive systems involving metal/compound semiconductor interfaces. To explain induction times in CdS systems (.9), they suggest that hydrogen incorporated in the solid lowers the barrier to charge transfer across the interface and thereby accelerates H2 production rates. 

All these results support our kinetic interpretations of these supersaturated gelling solutions. We assume that the network growth is described by the growth of individual domains, each one ruled by the autocatalytic model (S). This system behaves like an assembly of microdomains. Sach steroid in a supersaturation state is a potential germ of microdo.main. According to distribution curves of induction times for each microdomain, the typical kinetic curves for each part A and B of the phase diagram are obtained. 

We have used an autocatalytic model originally proposed by Malkin et al. [62]. Bolgov et al. [61] found that the originally proposed autocatalytic model [62], which was valid for equal concentration of initiator and catalyst during the anionic polymerization of caprolactam, can be modified for unequal concentration of the initiator and catalyst by an autocatalytic equation of type... 

The terms in Equation 1.3 (Malkin s autocatalytic model) are described in Nomenclature. In Malkin s autocatalytic model, the concentration of the activator, [A], is defined as the concentration of the initiator times the functionality of the initiator. For a difimctional initiator [e.g., isophthaloyl-bis-caprolactam, the concentration of the activator (acyllactam) is twice the concentration of the initiator]. The term [C] is defined as the concentration of the metal ion that catalyzes the anionic polymerization of caprolactam. In a magnesium-bromide catalyzed system, the concentration of the metal ion is the same as the concentration of the caprolactam-magnesium-bromide (catalyst) because the latter is monofunctional. 

Malkin s autocatalytic model is an extension of the first-order reaction to account for the rapid rise in reaction rate with conversion. Equation 1.3 does not obey any mechanistic model because it was derived by an empirical approach of fitting the calorimetric data to the rate equation such that the deviations between the experimental data and the predicted data are minimized. The model, however, both gives a good fit to the experimental data and yields a single pre-exponential factor (also called the front factor [64]), k, activation energy, U, and autocatalytic term, b. The value of the front factor k allows a comparison of the efficiency of various initiators in the initial polymerization of caprolactam [62]. On the other hand, the value of the autocatalytic term, b, describes the intensity of the self-acceleration effect during chain growth [62]. 

In our studies, the catalyst and initiator system was comprised of caprolactam-magnesium-bromide and isophthaloyl-bis-caprolactam, respectively. We determined the optimum values of the kinetic parameters in Malkin s autocatalytic model (Eq. 1.3), which consist of k, U, and b, by regression analysis. 

MgBr+/IBT(a) Dave et al. [24] Malkin s autocatalytic model Adiabatic temperature analyzed by regression analysis 30.2 1.49 x 104 2.17... 

The role of the isothermal and pseudo-first-order reaction assumptions on the observed value of activation energy was assessed to allow comparison of our data to previous work by modifying Malkin s autocatalytic equation so that the autocatalytic term b is equal to zero. The values of the activation energy and front factor were calculated using short-time, low-conversion data. By making the autocatalytic term equal to zero, the modified Malkin autocatalytic model becomes a first-order rate reaction. Table 1.2 shows that by assuming a... 

The kinetics of anionic ring opening polymerization of caprolactam initiated by iso-phthaloyl-bis-caprolactam and catalyzed by caprolactam-magnesium-bromide satisfactorily fit Malkin s autocatalytic model below 50 percent conversion. The calculated value of the overall apparent activation energy for this system is 30.2kJ/mol versus about 70kJ/mol for Na/hexamethylene-l,6,-bis-carbamidocaprolactam as the initiator/catalyst system. 

Figure 2.3 Reaction rate versus degree of cure for an autocatalytic model—curves a and b, at temperatures Ta and Tb, respectively, have an initial reaction rate equal to zero, and curve c has an initial reaction rate different from zero...

Figure 2.5 Degree of cure time versus time for an autocatalytic model—curves a, b, and c have temperatures Ta, Tb, and Tc, respectively, where Tc > Tb > Ta...

Han et al. [191] found that the rate of cure of a resin is greatly influenced by the presence of fibers and the type of fibers employed. The rate of reaction for resin-fiber system can be 60 percent different from that of neat resin, after a 10-min cure. A similar conclusion was presented by Mijovic and Wang [192] for graphite-epoxy composites based on TGDDM/DDS (33phr). They verified large differences (see Table 2.5) in the kinetic parameters when considering an autocatalytic model. 

Because of all these minor components (e.g., catalysts and inhibitors, added to major ones) the cure of vinyl ester resins is very complex, involving many competitive reactions. There are some new variables to account for, such as the inhibitor and initiator concentrations and induction time. Several papers [81,96,200,201] use the mechanistic approach, claiming that the phenomenological models do not explicitly include these facts, resulting in a new parameter characterization after each change in resin formulation [96]. Despite these arguments, the phenomenological approach is the most widely used and is based on an autocatalytic model which has been successfully applied to epoxy resins. Many authors [30,34,74,199,202,203] proposed the Equation 2.30 to describe the cure kinetic of unsaturated polyesters ... 

This equation includes the particular cases of an nth order reaction model with g(a) = 1, and the autocatalytic model with g(a) = 1 + la, where / is the autocatalysis intensity. In general, the form of g(a) must be determined from experimental data. 

Chapter 2 will discuss in some detail a simple isothermal autocatalytic model whereby a reactant P is converted to a product C through two intermediates A and B. The simple kinetic model can be written as... 

Fig. 2.3. Computed concentration histories for the cubic autocatalytic model with rate data from Table 2.1 except for the uncatalysed reaction rate constant feu = (a) the exponential...

Fig. 2.4. Computed concentration.histories for autocatalytic model with rate constants given exactly as in Table 2.1 (a) exponential decay of precursor (b) intermediate concentrations a(t) and 6(r), showing initial pseudo-stationary-state behaviour but subsequent development of an oscillatory period of finite duration, 1752 s < t < 3940 s.

Tracqui, P, Perault-Staub, A. M., Milhaud, G., and Staub, J. F. (1987). Theoretical study of a two-dimensional autocatalytic model for calcium dynamics at the extracellular fluid-bone interface. Bull. Math. Biol., 49, 597-613. 

The question of what happens to the system in the range of instability, and how the concentrations of A and B vary as they move away from the unstable stationary state, leads us to the study of sustained oscillatory behaviour. Before a full appreciation of the latter can be obtained, however, we must rehearse the relevant theoretical background. Fortunately the autocatalytic model is again an exemplary system with which to introduce at least the basic aspects of the Hopf bifurcation, and we will do this in the next section. 

Fig. 3.8. Representation of the onset, growth, and death of oscillations in the isothermal autocatalytic model as /z varies for reaction with the uncatalysed step included, showing emergence of the stable limit cycle at and its disappearance at n. ...

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

Some typical oscillatory records are shown in Fig. 4.6. For conditions close to the Hopf bifurcation points the excursions are almost sinusoidal, but this simple shape becomes distorted as the oscillations grow. For all cases shown in Fig. 4.6, the oscillations will last indefinitely as we have ignored the effects of reactant consumption by holding /i constant. We can use these computations to construct the full envelope of the limit cycle in /r-a-0 phase space, which will have a similar form to that shown in Fig. 2.7 for the previous autocatalytic model. As in that chapter, we can think of the time-dependent... 

For the simple autocatalytic model without decay, used as an example above, Fz does not vanish, so isola and mushroom patterns are not possible (confirming our previous results). If we take the model of 6.4, where the decay rate constant k2 is non-zero, then from eqn (6.61) we obtain... 

The behaviour of the simpler autocatalytic models in each of these three class A geometries seems to be qualitatively very similar, so we will concentrate mainly on the infinite slab, j = 0. For the single step process in eqn (9.3) the two reaction-diffusion equations, for the two species concentrations, have the form... 

We may also briefly consider the behaviour of the simple autocatalytic model of chapters 2 and 3 under reaction-diffusion conditions. In a thermodynamically closed system this model has no multiplicity of (pseudo-) stationary states. We now consider a reaction zone surrounded by a reservoir of pure precursor P. Inside the zone, the following reactions occur ... 

We will not pursue the question of local stability of the different station-ary-state solutions in any great depth here. Qualitatively, the non-isothermal and cubic autocatalytic models have shown remarkable degrees of similarity... 

In this chapter we will concentrate on fronts and pulses, and we will illustrate these with the isothermal autocatalytic models seen previously. We start with the single cubic autocatalytic process... 

If the answer to all these questions is yes, then the autocatalytic model for the decomposition can be used with some reliability to predict the adiabatic behavior. This has to be done by specialists. 

Lente proposed a discrete-state stochastic modeling approach in which chiral amplification could be described by a quadratic autocatalytic model without considering cross-inhibition [67,68]. However, the discrepancy between the usually employed deterministic kinetic approach, which reinforces the need for cross-inhibition, and the discrete-state stochastic approach is only apparent. The discrete approach considers the repetitive reproduction of single molecules which, in the case of a chiral system, obviously are individually all enantiomerically pure. Hence, basically no amplification of the ee occurs at all during the discrete scenario. It has been indicated that deter-... 

We have presented various simple scenarios that we are aware of in relevance to the ee amplification of the Soai reaction a quadratic autocatalytic model in a monomer or homodimer system, and a linear autocatalytic model in an antagonistic heterodimer system. All these models can realize ee amplification such that the final value of the ee 0ool depends on, but is larger than, the initial value 0ol> as schematically shown in Fig. 8. The curve in the figure represents 0o - (poo for a given initial ratio qo/c, namely the ratio of the amount qo of the total chiral initiators R and S relative to that of the total reactants c. Amplification is more enhanced if the ratio of the chiral initiator qo/c is smaller. This plot also shows the possibility of increasing the final ee by repeating the reaction. 

We continue to rely extensively on the two-step (initiation - propagation or autocatalytic) model 4) to evaluate data on coking rates. Two rate constants are involved fc for the deposition of coke on a "clean" surface, i.e., with no coke around and k2 when coke is deposited adjacent to another coke deposit. The former rate constant is for an initiation step (or "non-catalytic" coking), while the latter is for the propagation step (or coking catalyzed by the presence of the coke "product") hence, typically, k2 > ki. A third parameter used in the model is M, which represents the maximum amount of coke which can be deposited on the catalyst. In terms of these three parameters, the coke level expected in a pulse reactor after the passage of R amount of reactant is given by ... 

Replication is a multiple-step reaction that usually involves sophisticated enzymic machinery. Is such a process adequately described by a straightforward linear autocatalytic model ... 

Pichaud et al. (1999) highlight the chemorheology and dielectrics of the cure of DGEBA with isophorone diamine (IPD). The kinetics are well described by an autocatalytic model and the chemorheology is well described by the Macosko model... 

Other researchers have also successfully applied the autocatalytic model [122, 134]. For fitting the conversion-time curves in this context, the conversion can be indirectly obtained from the Tg of the partly cured matrix. Georjon et al. [122] studied the evolution of Tg with cyanate conversion in the isothermal cure of un-... 

Figure 7.4 Snapshots of the spatial distribution for the autocatalytic model (7.1) in the open blinking vortex-sink flow at time intervals equal to the flow period for a supercritical Damkohler number, Da = 7.0. Note, that after a transient time a time-periodic asymptotic state is reached where the autocatalytic growth, localized on the fractal unstable manifold, is balanced by the loss of product due to the outflow from the mixing region, in this case through the point sinks.

Tracqui, P. 1993. Homoclinic tangencies in an autocatalytic model of interfacial processes at the bone surface. Physica 62D 275-89. 

---