Nonlinearity sensitivity

R.I. Cukier, H.B. Levine and K.E. Shuler, Nonlinear Sensitivity Analysis of Multiparameter Model Systems, J. Comp. Phys. 26 (1978) 1-42. 

The word sensitivity is often used in describing an analytical method. Unfortunately, it is occasionally used indiscriminately and incorrectly. The most often-used definition of sensitivity is the calibration sensitivity, or the change in the response signal per unit change in analyte concentration. The calibration sensitivity is thus the slope of the calibration curve, as shown in Figure 8-14. If the calibration curve is linear, the sensitivity is constant and independent of concentration. If nonlinear,. sensitivity changes with concentration and is not a single value. 

Cukier, R. I., Levine, H. B., and Shuler, K. E. (1978) Nonlinear sensitivity analysis of multiparameter model systems, J. Comput. Phys., 26, 1-42. 

Process design and control structure evaluation and screening using nonlinear sensitivity analysis... 

Hwang, J.T. Nonlinear sensitivity analysis in chemical kinetics. Proc. Natl. Sci. Council B. ROC 6, 20-29 (1982)... 

The titration curve is the essential tool for determining the nonlinearity, sensitivity, and rangeability requirements that are important for every step from project definition through sustaining installed performance. It determines the investment required, the design details, and the expected efficiency, operability, and maintainability. Without a titration curve, you are flying blind [Ref. 1.1 and 1.2]. 

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. 

Standardizing the Method Equations 10.32 and 10.33 show that the intensity of fluorescent or phosphorescent emission is proportional to the concentration of the photoluminescent species, provided that the absorbance of radiation from the excitation source (A = ebC) is less than approximately 0.01. Quantitative methods are usually standardized using a set of external standards. Calibration curves are linear over as much as four to six orders of magnitude for fluorescence and two to four orders of magnitude for phosphorescence. Calibration curves become nonlinear for high concentrations of the photoluminescent species at which the intensity of emission is given by equation 10.31. Nonlinearity also may be observed at low concentrations due to the presence of fluorescent or phosphorescent contaminants. As discussed earlier, the quantum efficiency for emission is sensitive to temperature and sample matrix, both of which must be controlled if external standards are to be used. In addition, emission intensity depends on the molar absorptivity of the photoluminescent species, which is sensitive to the sample matrix. 

The polysdanes are normally electrical insulators, but on doping with AsF or SbF they exhibit electrical conductivity up to the levels of good semiconductors (qv) (98,124). Conductivities up to 0.5 (H-cm) have been measured. However, the doped polymers are sensitive to air and moisture thereby making them unattractive for practical use. In addition to semiconducting behavior, polysilanes exhibit photoconductivity and appear suitable for electrophotography (qv) (125—127). Polysdanes have also been found to exhibit nonlinear optical properties (94,128). 

Recently photorefractivity in photoconductive polymers has been demonstrated (92—94). The second-order nonlinearity is obtained by poling the polymer doped with a nonlinear chromophore. Such a polymer may or may not be a good photoconductor. Usually sensitizers have to be added to enhance the charge-generation efficiency. The sensitizer function of fuUerene in a photorefractive polymer has been demonstrated (93). 

Temperature detectors embedded in the motor winding give close, accurate indication of motor temperature. Both conventional resistance temperature detec tors (RTD) and special thermistors (highly temperature-sensitive nonlinear resistors) are used. With appropriate auxiliaries these devices can indicate or record motor temperature, alarm, and/or shut down the motor. 

An optical detector with appropriate electronics and readout. Photomultiplier tubes supply good sensitivity for wavelengths in the visible range, and Ge, Si, or other photodiodes can be used in the near infrared range. Multichannel detectors like CCD or photodiode arrays can reduce measurement times, and a streak camera or nonlinear optical techniques can be used to record ps or sub-ps transients. 

The relatively simple study of fluorescence and phosphorescence (based on the action of colour centres) has nowadays extended to nonlinear optical crystals, in which the refractive index is sensitive to the light intensity or (in the photorefractive variety (Agullo-Lopez 1994) also to its spatial variation) a range of crystals, the stereotype of which is lithium niobate, is now used. 

Material Relative sensitivity Output nonlinearity (% per GPa) Distortion Stress limit (GPa)... 

As the current pulse is largely dominated by the stress differences, a short duration current pulse is observed upon loading with a quiescent period during the time at constant stress. With release of pressure upon arrival of the unloading wave from the stress-free surface behind the impactor, a current pulse of opposite polarity is observed. The amplitude of the release wave current pulse provides a sensitive measure of the elastic nonlinearity of the target material at the peak pressure in question. 

Instead of the definition in Eq. (7-82), the selectivity is often written as log k,). Another way to consider a selectivity-reactivity relationship is to compare the relative effects of a series of substituents on a pair of reactions. This is what is done when Hammett plots are made for a pair of reactions and their p values are compared. The slope of an LEER is a function of the sensitivity of the process being correlated to structural or solvent changes. Thus, in a family of closely related LFERs, the one with the steepest slope is the most selective, and the one with the smallest slope is the least selective.Moreover, the intercept (or some arbitrarily selected abscissa value, usually log fco for fhe reference substituent) should be a measure of reactivity in each reaction series. Thus, a correlation should exist between the slopes (selectivity) and intercepts (reactivity) of a family of related LFERs. It has been suggested that the slopes and intercepts should be linearly related, but the conditions required for linearity are seldom met, and it is instead common to find only a rough correlation, indicative of normal selectivity-reactivity behavior. The Br nsted slopes, p, for the halogenation of a series of carbonyl compounds catalyzed by carboxylate ions show a smooth but nonlinear correlation with log... 

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