Product Nonlinearity

One class of nonlinearity in mathematical models involves the one produced by a product of variables, such as x2, x3, xy, yz2, etc. This nonlinearity may introduce 

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Although small, this is a principal disadvantage of the simplest integral theory. The near-contact density of the products nonlinear in c is lost in the lowest-order approximation to this parameter. However, the nonzero contribution to this region is provided by a modified encounter theory outlined in Section XII. The chief merit of MET is that the argument of the Laplace transformation of n r,t) in (3.311) is shifted from 1 /td to 1/xd + ck. As a result, in the limit xD = oo we have instead of (3.313) [133] ... 

DePoy and Mason (1974) proposed a mechanism including an autocatalytic step which produces oscillations, and gave a mathematical model with product nonlinearity as in the case of the Lotka model, Lotka (1910-1, 2). 

Abstract models Rossler (those not included in Product Nonlinearity section above.)... 

Alekseev and Kol tsov (1982) analyzed a three step catalytic reaction where two intermediates (x,y) were considered. The model was given as two differential equations with product nonlinearity in x and y. 

It was established for several examples that it was possible to observe some departure from the expected proportionality between the enantiomeric excess of the catalyst and the enantiomeric excess of the product. Nonlinear effects (NLE) are categorized as a positive nonlinear effect ((-i-)-NLE) if the curve ee(product) = f(ee(catalyst)) is above the straight line characterizing the expected proportionality between ee(product) and ee(catalyst). The (-i-)-NLE has also been named asymmetric amplification [92]. A negative nonlinear effect ((-)-NLE) means that the experimental curve ee(product) =f( ee(catalyst)) lies below the straight line of the linear correlation. The departure from linearity reflects the formation of diastereomeric species (catalytically active or not) which perturb the predictions based only on mixture of enantiomeric catalysts and the... 

In the reaction kinetics context, the tenn nonlinearity refers to the dependence of the (overall) reaction rate on the concentrations of the reacting species. Quite generally, the rate of a (simple or complex) reaction can be defined in temis of the rate of change of concentration of a reactant or product species. The variation of this rate with the extent of reaction then gives a rate-extent plot. Examples are shown in figure A3.14.1. In... 

The simplest manifestation of nonlinear kinetics is the clock reaction—a reaction exliibiting an identifiable mduction period , during which the overall reaction rate (the rate of removal of reactants or production of final products) may be practically indistinguishable from zero, followed by a comparatively sharp reaction event during which reactants are converted more or less directly to the final products. A schematic evolution of the reactant, product and intenuediate species concentrations and of the reaction rate is represented in figure A3.14.2. Two typical mechanisms may operate to produce clock behaviour. 

Multichannel time-resolved spectral data are best analysed in a global fashion using nonlinear least squares algoritlims, e.g., a simplex search, to fit multiple first order processes to all wavelengtli data simultaneously. The goal in tliis case is to find tire time-dependent spectral contributions of all reactant, intennediate and final product species present. In matrix fonn tliis is A(X, t) = BC, where A is tire data matrix, rows indexed by wavelengtli and columns by time, B contains spectra as columns and C contains time-dependent concentrations of all species arranged in rows. 

Because of the extreme accuracy expected of many of these products, some include internal test weights which can be used to recaUbrate regularly and to adjust for nonlinearity. Some balances monitor changing conditions and initiate the recahbration procedure as needed. 

Cosmetics and Pharmaceuticals. The main use of hexadecanol (cetyl alcohol) is in cosmetics (qv) and pharmaceuticals (qv), where it and octadecanol (stearyl alcohol) are used extensively as emoUient additives and as bases for creams, Hpsticks, ointments, and suppositories. Octadecenol (oleyl alcohol) is also widely used (47), as are the nonlinear alcohols. The compatibiHty of heavy cut alcohols and other cosmetic materials or active dmg agents, their mildness, skin feel, and low toxicity have made them the preferred materials for these appHcations. Higher alcohols and their derivatives are used in conditioning shampoos, in other personal care products, and in ingested materials such as vitamins (qv) and sustained release tablets (see Controlled RELEASE technology). 

Modified ETEE is less dense, tougher, and stiffer and exhibits a higher tensile strength and creep resistance than PTEE, PEA, or EEP resins. It is ductile, and displays in various compositions the characteristic of a nonlinear stress—strain relationship. Typical physical properties of Tef2el products are shown in Table 1 (24,25). Properties such as elongation and flex life depend on crystallinity, which is affected by the rate of crysta11i2ation values depend on fabrication conditions and melt cooling rates. 

The cadmium chalcogenide semiconductors (qv) have found numerous appHcations ranging from rectifiers to photoconductive detectors in smoke alarms. Many Cd compounds, eg, sulfide, tungstate, selenide, teUuride, and oxide, are used as phosphors in luminescent screens and scintiUation counters. Glass colored with cadmium sulfoselenides is used as a color filter in spectroscopy and has recently attracted attention as a third-order, nonlinear optical switching material (see Nonlinear optical materials). DiaLkylcadmium compounds are polymerization catalysts for production of poly(vinyl chloride) (PVC), poly(vinyl acetate) (PVA), and poly(methyl methacrylate) (PMMA). Mixed with TiCl, they catalyze the polymerization of ethylene and propylene. 

Nonlinear dependence between ee values of chiral auxiliary compound or ligand and product of reaction with participation or formation of heterocycle 98AG(E)2922. 

PEs, as other polymers, exhibit nonlinear behavior in their viscous and elastic properties under practical processing conditions, i.e., at high-shear stresses. The MFI value is, therefore, of little importance in polymer processing as it is determined at a fixed low-shear rate and does not provide information on melt elasticity [38,39]. In order to understand the processing behavior of polymers, studies on melt viscosity are done in the high-shear rate range viz. 100-1000 s . Additionally, it is important to measure the elastic property of a polymer under similar conditions to achieve consistent product quality in terms of residual stress and/or dimensional accuracy of the processed product. 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

For a nonlinear molecule the rotational energy levels are a function of three principal moments of inertia /A, /B and /c- These are moments of inertia around three mutually orthogonal axes that have their origin (or intersection) at the center of mass of the molecule. They are oriented so that the products of inertia are zero. The relationship between the three moments of inertia, and hence the energy levels, depends upon the geometry of the molecules. 

Nonlinear least-squares programs have made time lag methods much less important. They are less accurate, for one thing. For another, the linearity of the appropriate plots, although a necessary consequence of first-order kinetics, does not constitute a proof of first-order kinetics. Certain other kinetic equations also lead to linear plots of either function. For example, Problem 2-11 presents data for a product-catalyzed reaction. The data in this case can be plotted linearly according to the Guggenheim equation, although the reaction does not follow first-order kinetics and the plot of In [A] versus time is decidedly nonlinear. 

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