Non-linear effect

A similar situation can occur when the chiral complex (containing a chiral bidentate for instance) can form dimeric species. These dimers may either be the catalyst or just an inactive resting state. Since free energies and rate constants for meso and racemic dimeric species may differ, non-linear behaviour of e.e. of ligand versus e.e. of product can result. 

In its most general form the polarisation of a molecule, PM, can be written as  

When a single, polarised radiation field is propagating in the material, Equation (3.80) will take the form  

When a static field, bc aligned parallel to the electric vector of the radiation field, is applied, we can replace E in Equation (3.80) by the sum of Ebc and E(jCOs(iJt-kz). Then, after expanding the squared and cubed terms, the polarisation at the radiation frequency becomes  

This may also be obtained directly from Equation (3.79), where the first term gives a contribution the second term two contributions of 

Thus the non-linear refractive index comprises a term linearly dependent on the static field, the Pockels effect, a term dependent on the square of the applied field, the Kerr effect, and a term dependent on the radiation intensity, the optical Kerr effect. 

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The current frontiers for the subject of non-equilibrium thennodynamics are rich and active. Two areas dommate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Carefiil experiments [39] confinn this theory. Non-equilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. 

Dependence of enantioselectivity on loading, (b) Non-linear effect on enantioselec-... 

We have an advantage in this situation. We know that our data contain 4 components, plus a small linear baseline, and some nonlinearities. This would lead us to expect that our data would show a rank of 5 or 6. Four dimensions are required to span the variations from the 4 different spectral components, and an additional 1 or two would be needed to span the variations due to the baselines and non-linearities. It gets a bit tricky when we realize that the baseline and non-linearity effects are rather small. Statistical indicators such as the 2-way F-test on REV s are not always able to destinguish factors that span small, but... 

Trans-polyenes H-(-HC=CH-),, -H, trans-polyenynes H-(HC=CH-C=C) -H, cumulenes H2C=(C=C) =CH2 and polyynes H-(C=C) -H have been studied (M=N-1). For eentrosymmetrie molecules, the first order hyperpolarizability p is equal to zero so that non linear effects are of second order nature. Furthermore, (the x axis goes through the middle of the C-C bonds of the polyenes, or is the intemuclear axis in the case of linear molecules) is the most important component of the second order y hyperpolarizability tensor, the other components being negligible. Both y and the mean hyperpolarizability... 

The fact that the absorption spectrum of CdS changes when electrons are produced on it may be regarded as a non-linear optical effect, i.e. a dependence of the absorption coefficient on the light intensity. Other non-linear effects, which are caused... 

This model is capable of estimating both linear and non-linear effects observed experimentally. Hence, it can also be used for optimization of the desired response with respect to the variables of the system. Two popular response surface designs are central composite designs and Box-Behnken designs. Box-Behnken designs were not employed in the experimental research described here and will therefore not be discussed further, but more information on Box-Behnken designs can be obtained from reference [15]. 

In V-I DC measurements at constant temperature, non-linear effects appear even at powers as low as 10-14W they become more evident when temperature is lowered (see Section 9.6.3 and Fig. 9.8) [19]. Such non-linearities can be interpreted by the so-called hot-electron model (HEM). 

To reduce non-linear effects, we make another assumption ... 

Gaussian PDFs are found for homogeneous inert scalar mixing in the presence of a uniform mean scalar gradient. However, for turbulent reacting flows, the composition PDF is usually far from Gaussian due to the non-linear effects of chemical reactions. 

The non-linearity effect can easily be demonstrated by the following theoretical separation of a binary mixture. Let us assume that the concentrations of A and B are the same and correspond each to half of the overall feed concentration. The feed concentration is in addition assumed to be the only parameter necessary to characterize the feed composition. The mass flow ratio in section 1 (constrained by Eq. (8)) does not depend on the feed composition. On the contrary, the upper Emit on the flow rate ratio m4 given by Eq. (10) is a function of the feed composition. Both dependencies are illustrated in Fig. 5. 

Furthermore, for centrosymmetric or non-centrosymmetric materials alike, third order non-linear effects can exist (y or /3) 0). 

Units which are used in isotopic work depend on the precision of the measurements. Generally 5 units are used for stable isotopes and correspond to permil relative deviation. It is used occasionally also for non linear effects and then they are permil (%o) deviations without reference to mass differences between the isotopes. Since the beginning of the 70s (e.g., Papanastassiou and Wasserburg 1969) thermal ionization data are often given in e units which are fractional deviation from the normal in 0.01%. With the new generation of more precise instruments, results are sometimes given in ppm (parts per million) relative to a terrestrial standard sample. 

The excess was first suggested to have a nuclear origin in stars. Almost pure O is produced in He-buming shells in massive stars, and in supemovae. On the other hand it has been shown that non-mass dependent fractionation can be produced in the laboratory by non-nuclear processes (Thiemens and Heidenreich 1983 Thiemens 1988). Similar non-linear effects have been found for O isotopes in atmospheric gases (Schueler et al. 1990 Thiemens et al. 1995). Although stellar nucleosynthesis is indeed at the origin of the O observed in the universe, the link between O isotopic anomalies in inclusions and nucleosynthesis is still under debate (Thiemens 1999 Clayton 2002). 

Silicon. Normal inclusions are spread on a mass dependent fractionation line over a few %o/amu around the solar system average. FUN inclusions display a heavy isotope enrichment from 5 to 15 %o/amu in a similar way to Mg. Non-linear effects are small and indicate an excess of Si smaller than 0.5 %o (Clayton et al. 1984). 

Figure 4. Non-linear effects for Sr, Ba, Ndand Sm in the FUN inclusions Cl andEKldl (McCulloch and Wasserburg 1978a,b Papanastassiou and Wasserburg 1978). Relative deviations from terrestrial standard ratios are plotted after normalization with the isotope pair represented with large open squares. Each isotope is labeled with its primary nucleosynthetic source. In using s-process isotopes for normalization, clear excesses in r-process nuclei are seen for Ba and Sm in EK141. Sr is normal in both inclusions except for a deficit in the p-process Sr. AsNdhas only one pure s-process isotope at mass 142, the data in EK141 have been further corrected to yield an excess in Nd identical to that of Sm as these two isotope are pure r-process nuclei expected to be produced in comparable abundances.

A detailed kinetic study of the enantioselective reduction of acetophenones, ArCOMe, to arylethanols, using a propan-2-ol-acetone couple and a chiral rhodium diamine catalyst, has been undertaken.Non-linear effects on the % ee are observed, e.g. addition of achiral ketones can both slow the reaction and raise the ee. These effects can be rationalized in terms of the difference in reactivity of diastereomeric catalytic sites. The scope for exploiting such mechanistic insights so as to maximize the enantioselectivity is discussed. 

A nice analysis of non-linear effects in Rh-chiral diamine-catalysed transfer hydrogenation has been performed that reinforces the need to consider the kinetics of all of the steps in reaction manifolds (e.g. reversible formation of diastereomeric precursors and their subsequent interaction with achiral reactants). ... 

P.N Prasad and D.J. Williams, Introduction to Non-Linear Effects in Molecules and Polymers, John Wiley, New York, 1991. 

Mixture variables, expressing the composition of the mobile phase as fi ac-tions, have the property that they add up to one (the mixture restriction). The consequence is that no intercept can be estimated when the effects of the solvents are evaluated [10,19]. Moreover interactions and quadratic effects, such as used when the independent variables are process variables, can not be estimated independently. Mathematically it is better to use blending effects only. Interpretation of these blending effects, i.e. explicitly stating what components are responsible for the non-linear effects, is not possible. 

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