Three-component fit

Figure 5. (a) The ( A, SO,) anion symmetric streching mode of polypropylene glycol)- LiCF,SO, for 0 M ratios of 2000 1 and 6 1. Solid symbols represent experimental data after subtraction of the spectrum corre-ponding to the pure polymer. Solid curves represent a three-component fit. Broken curves represent the individual fitted components, (b) Relative Raman intensities of the fitted profiles for the ( Aj, SO,) anion mode for this system, plotted against square root of the salt concentration, solvated ions ion pairs , triple ions, (c) The molar conductivity of the same system at 293 K. Adapted from A. Ferry, P. Jacobsson, L. M. Torell, Electrnchim. Acta 1995, 40, 2369 and F. M. Gray, Solid State Ionics 1990, 40/41, 637. 

Fig. 11.6. Film thickness profiles of cavities exposed to plasmas of different precursor gases in three different positions in the vacuum vessel. The dots represent the measured profiles, the solid lines the calculated three-component fits. The profiles in each column are from the same position, the ones in a row are from the same precursor gas...

A number of authors have found that the lifetime spectra of semicrystalline polymers are best resolved into three components. In a study of PEEK (poly(ether ether ketone)), Nakanishi et al [14] found that a three component fit was best. They observed that the o-Ps lifetime (r3) did not change with an increasing amounts of crystallinity, but the o-Ps yield (I3) decreases linearly with an increasing amount of crystallininty. It was also demonstrated that I3 extrapolates to 0 at 100% crystallinity. No o-Ps lifetime was observed that might be attributed to the annihilation within the crystalline regions of the polymer. Lind et al [43] found a similar result for polypropylene where the x3 component changed very little with the amount of crystallinity, but the I3 value decreased with an increasing amount of... 

Fig. 12a—c. Three-component fit (a), residuals (b) and autocorrelation trace (c) of same synthetic decay curve as in Fig. 11. Note improvement in fittii, and that realistic values of the six parameters are recovered...

Tables I and 11 show the results of three component fits to Poly (VBuPBD) and Poly (VPPO) respectively at an excitation wavelength of 300 nm. The improvement in goodness of fit when using a three component, rather than two component model is illustrated in Figure 3 for Poly(VBuPBD) at 350 nm and 25 C.

Figure 3 Three component fit and residuals for Poly (VBuPBD) at 3 50nm and 25°C with residuals for a two component fit also shown.

Fig. 13.—Plot of InJ vs.

FIGURE 11.1 Lifetime spectrum of PDMS at 300 K (after subtraction of the source components) and the unconstrained three-component fit to the data employing the routine LifeTime (LT9.0) and allowing a distribution of lifetimes. The lines show the (convoluted) p-Ps (1), e+ (2), and o-Ps (3) decay components, the background Slt, and the fitted sum curve (1 -I- 2 -F 3 -I- Slt), The dots are the experimental data. Res shows the weighted residuals of the fit. The inset shows the spectra with added background Slt for comparison. R t) is the resolution function (FWHM = 240 ps dots). 

Figure 2b. Plots of (a) three-component fit to the same simulated three-component fluorescence decay as Figure 2a (b) auto-correlation function.

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. 

Ruckenstein and Li proposed a relatively simple surface pressure-area equation of state for phospholipid monolayers at a water-oil interface [39]. The equation accounted for the clustering of the surfactant molecules, and led to second-order phase transitions. The monolayer was described as a 2D regular solution with three components singly dispersed phospholipid molecules, clusters of these molecules, and sites occupied by water and oil molecules. The effect of clusterng on the theoretical surface pressure-area isotherm was found to be crucial for the prediction of phase transitions. The model calculations fitted surprisingly well to the data of Taylor et al. [19] in the whole range of surface areas and the temperatures (Fig. 3). The number of molecules in a cluster was taken to be 150 due to an excellent agreement with an isotherm of DSPC when this... 

The three-component models were found to fit the experimental data better than the two-component ones. Model I was found to be able to fit both LTO and cracking data very well. This model was considered the best of all models even though it is unable to calculate the HO/LO split (Hanson and Kalogerakis, 1984). 

For fluorescence decay curves of the J-aggregate LB films of [CI-MC] mixed with various matrix agents, measured with a picosecond time-resolved single photon counting system, three components of the the lifetimes fitting to exponential terms in the following equation ... 

For a three component system, the matrix T has nine elements and thus it appears that C and eventually the sum of squares are a function of nine parameters. As we will see in a moment there are actually fewer, only six, parameters to be fitted. The idea of RFA is to use the Newton-Gauss algorithm to fit this rather small number of parameters in T. 

In monoethanolamine solutions the unknown interaction parameters and equilibrium constants were determined by fitting the model to data for the three component systems CC +MEA+ O and H2S+MEA+H2O. The agreement of the fitted model with Che data was found to be good. The parameters obtained in this way were then used to predict the partial pressures of mixtures of HoS and CO2 over aqueous MEA solutions. The predictions were in good agreement with experimental data, except at the higher partial pressures. 

Monti et al. [105] also used fluorescence lifetimes to monitor the merocyanine forms of NOSH. In ethanol, they identified two components to the decay having lifetimes of 15 and 700 psec. The longer-lifetime decay is a very minor component. In the case of NOSH in ethanol, 85% of the decay was attributed to the fastest component with a lifetime of 20 psec, but to fit the decay, it was necessary to use a further two components. This is in agreement with Wilkinson et al.[64], who suggested three components to the merocyanine formation from the NOSH closed form based on picosecond time-resolved resonance Raman which they attributed to equilibration of three merocyanines trans about the (3-bond. Monti et al. further found that in acetonitrile, NOSH had one component decay with a lifetime of 20 psec. Clearly, solvent and substiments are important factors. 

The dashed lines in Figure 5 show mole fractions for the three components calculated on the basis that the ratio of k2 to ki is 0.5. The fit is reasonably good until well past the maximum in Yheg- However the growth in Y da and fall-off in Yheg at high conversions is clearly underestimated. A ratio of 0.41 as estimated above gave... 

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