Strongly nonlinear

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

Certain glass-ceramic materials also exhibit potentially useful electro-optic effects. These include glasses with microcrystaUites of Cd-sulfoselenides, which show a strong nonlinear response to an electric field (9), as well as glass-ceramics based on ferroelectric perovskite crystals such as niobates, titanates, or zkconates (10—12). Such crystals permit electric control of scattering and other optical properties. 

For quite a number of gases, Henry s law holds very well when the partial pressure of the solute is less than about 100 kPa (I atm). For partial pressures of the solute gas greater than 100 kPa, H seldom is independent of the partial pressure of the solute gas, and a given value of H can be used over only a narrow range of partial pressures. There is a strongly nonlinear variation of Heniy s-law constants with temperature as discussed by Schulze and Prausnitz [2nd. Eng. Chem. Fun-dam., 20,175 (1981)]. Consultation of this reference is recommended before considering temperature extrapolations of Henry s-law data. 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

The compressibility of polymers is strongly nonlinear at pressures of a few GPa. In order to consider the nonlinearity of the piezoelectric effect at shock pressure, it is of interest to consider the piezoelectric polarization in terms of the volume compression as shown in Fig. 5.9. The pressure-versus-volume relation for PVDF is not accurately known, but the available data certainly provide a relative measure of changes in compressibility. When considered versus volume, the piezoelectric polarization is found to to be remarkably linear. Thus, large volume compression does not appear to introduce large nonlinearities. Such a behavior will need to be considered when the theory of piezoelectricity for the heterogeneous piezoelectric polymer is developed. 

Given the complications of strongly nonlinear mechanical and electrical behaviors in a strongly coupled mode along with electrical conduction effects, it is not difficult to appreciate why the physical processes are poorly understood. 

The piezoelectric polymer investigations give new physical insight into the nature of the physical process in this class of ferroelectric polymers. The strong nonlinearities in polarization with stress are apparently more a representation of nonlinear compressibility than nonlinear electrical effects. Piezoelectric polarization appears to be linear with stress to volume compressions of tens of percent. The combination of past work on PVDF and future work on copolymers, that have quite different physical features promises to provide an unusually detailed study of such polymers under very large compression. 

Rao reported measurement of third-order optical non-linearity in the nanosecond and picosecond domains for phosphorus tetratolyl porphyrins bearing two hydroxyl groups in apical position [89]. Strong nonlinear absorption was found at both 532 nm and 600 nm. The high value of nonlinearity for nanosecond pulses is attributed to higher exited singlet and triplet states. Time resolved studies indicate an ultra-fast temporal evolution of the nonlinearity in this compound. 

Another particularly important topic in the modeling of strongly nonlinear phenomena is the occurrence of multiple fronts. For instance, in a supersonic reactive flow problem the position and speed of propagation of the shock wave and reaction front are different. 

These nitridation reactions are also characterized by very high values of activation energy. As a result, the nitridation front represents a strongly nonlinear combustion wave which may be extremely thin and corrugated. 

Numerical techniques are iterative and require considerable computer processing power. With modern desktop computers, this is usually not an issue and solutions of root uptake over days or weeks typically take a few seconds to generate. However, for some strongly nonlinear problems, such as the development of rhizosphere microbial populations (Sect. Ill), where the increase in microbial biomass may be exponential over time, processing time may become important with solutions requiring >60 min to calculate on a modern PC. 

First, we found a strong nonlinear effect on the adduct s enantiomeric excess, as indicated in Figure 1.6. The nonlinear effect strongly suggested there would be... 

A strong nonlinearity relationship was observed between ee % of chiral modifier 46 and the adduct 50. 

Kleinschmidt J, Rentsch S, Tottleben W, Wilhelmi B (1974) Measurement of strong nonlinear absorption in stilbene-chloroform solutions, explained by the superposition of two-photon absorption and one-photon absorption from the excited state. Chem Phys Lett 24 133-135... 

Blackbody radiation law proposed by J. Stefan form the basis for the radiation thermometers, with Planck s law. The radiation law is strongly nonlinear in temperature, since it depends on T4... 

Fig. 9.8. I-V curve for a Ge-NTD 31 (see Section 15.2.1.1) resistor at four different temperatures TB of the mixing chamber. Note the strong nonlinearity at low temperature.

We now turn to the quantum version of these results. In this case, the analogous cumulant expansion gives exactly the same equations for the centroids as above, while the equations for the higher cumulants are different. We can again investigate whether a trajectory limit exists. Localization holds in the weakly nonlinear case if the classical condition above is satisfied. In the case of strong nonlinearity, the inequality becomes... 

Analytical approaches applicable for small and large amplitudes (for weak and strong nonlinearity) of the oscillations in a nonlinear dynamic system subjected to the influence of a wave has been developed (Damgov, 2004 Damgov, Trenchev and Sheiretsky, 2003). 

The laser induces instantaneous vaporization of a microvolume (called a plume), and a mixture of ionized matrix and analyte molecules is released into the vacuum of the ion source. The relationship between the laser irradiance, I ascn and the number of molecules formed, Gma di, is most peculiar. There exists a threshold irradiance, peculiar to each matrix, below which ionization is not observed. Above this level, the ion production increases in a very strong, nonlinear, manner (often Gma di grows as Ilaser is raised to the eighth power). 

A study examining the relationship between ee of the catalyst and product revealed a strong nonlinear effect (Fig. 16) (78), suggestive of a complex relationship... 

As in any other mass balance model of bioprocesses, a strongly nonlinear kinetic behavior is present due to the reaction rates. These rates are given by ... 

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