Autocatalysis nonlinear

Keywords Flow trajectory Homochirality Nonlinear autocatalysis Recycling Soai reaction... 

As for theoretical research on the chiral symmetry breaking, Frank was the first to show that a linear autocatalysis with an antagonistic nonlinear chemical reaction can lead to homochirality [16]. His formulation with rate equations corresponds to the mean-field analysis of the phase transition in a nonequilibrium situation [17], and other variants have been proposed [6, 18-23]. All these analyses have been carried out only for open systems where... 

Beyond the linear autocatalyses, nonlinear effects such as a quadratic autocatalysis have been considered [6,24,31] ... 

When A is positive, as in the case of Fig. 6c, the coefficient of the cubic termB is also positive, and the velocity 4>i vanishes at three values of 0i in the range of - 1 < 0i < 1. This is possible if a strong quadratic autocatalysis > k 2 exists together with a linear recycling X > 0, or if a linear autocatalysis k and a nonlinear recycling x > 0 coexist. By following the direction indicated by the arrows for positive order parameter ends up at a definite value ... 

Fig. 7 Flow diagrams with a linear autocatalysis and nonlinear recycling (ki = /z > 0, k(1 = k2 =X = 0) similar to the Frank model, and b quadratic autocatalysis and linear recycling (k2c2 = /. > 0, fc, = ki = [ = 0). O origin, S stable fixed points, U instable fixed points...

In all the cases considered, auto catalytic processes must be present, whether linear or nonlinear. To understand the actual mechanism of autocatalysis for the Soai reaction, identification of the process at a molecular level is necessary, but is out of scope of the present review. 

Note the use of activities, as well as of an equilibrium constant based on activities. The kinetic constants for autocatalyzed and catalyzed reactions, k and k, were determined from initial reaction rates with liquid activity coefficients calculated by UNIQUAC. Near chemical equilibrium the fCT is about 6, while Kx is about 5. Table 8.7 gives activation energies and pre-exponential factors obtained by nonlinear regression. The simulation shows tbat the autocatalysis effect is neghgible below 150 °C, but it might increase to 20% at 180 °C. 

With best fidelity, the undertaken nonlinear model fitting for the stabilized samples of PE and PE-n-MMT has provided a triple-stage model scheme of successive reactions, wherein an nth-order autocatalysis reaction (Cn) was used at the first step, while a general nth-order (Fn) reaction was used for both the second and the third steps of the overall process of thermal oxidative degradation (Table 1) ... 

To make this quadratic-autocatalysis network capable of periodic behavior, the "step B— R is given a nonlinear rate equation —rB = kBRCB l(CB+K) as might result from saturation kinetics (see Section 8.3.1). To allow chaos, another step producing an otherwise inert species C is added ... 

We begin by considering the simplest form of autocatalysis, which is characterized by a rate equation with a quadratic nonlinearity ... 

The simplest possible autocatalytic reaction is the quadratic autocatalysis of Eq. [76]. We now consider the next simplest case, in which a cubic nonlinearity appears in the rate law ... 

Frontal polymerization discovered in 1972 (5) could be realized in free-radical polymerization because of its nonlinear behavior. If the top of a mixture of monomer and initiator in a tube is attached to an external heat source, die initiators are locally decomposed to generate radicals. The polymerization locally initiated is autoaccelerated by the c(xnbinatithermal autocatalysis exclusively at the top of the reaction systmn. An interface between reacted and unreacted regions, called propagating front, is thus formed. Pojman et al. extensively studied the dynamics of frontal polymerization (d-P) and its applicatim in matoials syndiesis (I -I3). 

The oscillatory chemical reactions can only be realized in some nonlinear system having feedback in the form of autocatalysis or autoinhibition [7,23-29]. 

Generally, the nonlinearity is further relates with autocatalysis, self-inhibition, or a delayed feedback loop etc., which cooperatively acted to play role in chemical oscillations [1, 3]. 

The kinetics of oscillatory chemical reaction must have nonlinearity and their rate equations be supposed to quadrant function of concentration of reactants. The kinetically control reaction steps have two values, (+ve) is associated with autocatalysis and (—ve) value is associated with auto-inhibition feedback loops [1]. 

Nonlinear phenomena in any system require some type of feedback. The most obvious source of feedback in polymerization reactions is thermal autocatalysis, often called thermal runaway in the engineering literature. The heat released by the reaction increases the rate of reaction, which increases the rate of heat release, and so on. This phenomenon can occur in almost any reaction and will be important when we consider thermal frontal polymerization. 

The chemistry for a stoichiometrically balanced reaction suggests that m = 1 and n = 2 in Eq. (2.86). For real systems, values are often close to these values but not identical. In the epoxy-amine reaction the alcohol, which may be present initially in small concentrations but is also a product of the reaction, catalyzes further reaction, resulting in autocatalysis. Since there are four unknowns ki, k2, m, and n) nonlinear regression analysis must be employed, although ki can be evaluated independently as the extrapolated reaction rate at a = 0. Autocatalytic kinetics are usually evaluated by the derivative form of the autocatalytic rate equation [Eq. (2.86)] with data coUected by isothermal method 1 measurements. Activation energy E and preexponential factor A are measured from the Arrhenius equation... 

The known mechanisms that produce oscillatory behavior have two characteristics in common. The first is autocatalysis. The product of a step must catalyze that step, as in steps 1 and 2 of the Lotka-Volterra mechanism. The second is that nonlinear differential equations occur. That is, the variables must occur with powers greater than unity or as products. A mechanism has been proposed for the BZ reaction that has 18 steps and involves 21 different chemical species. A computer simulation of the 18 simultaneous rate differential equations for the mechanism has been carried out and does produce oscillatory behavior. It also exhibits the interesting behavior that all curves in phase space corresponding to different initial states eventually approach... 

Under isothermal conditions, nonlinear models that lead to dynamic behavior typically incorporate some sort of a feedback, for example, autocatalysis, when a certain step is accelerated by a product of this step. 

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