Curve-fitting exponential function

A second important aspect of electrochromism is the temporal response under alternating potentials ( 0.55 V). The DG showed sharp and distinct transitions between the colored/oxidized and bleached/reduced state across the entire visible spectrum (Fig. 6.10b). This time-resolved switching behavior was analyzed in more detail at A = 630 nm (Fig. 6.10c). The DG device showed short characteristic response times of 53 ms for the bleaching step and 63 ms for the reverse process, determined by fitting exponential functions to the switching curves. These short response times which are close to video rate (24 frames per second) are enabled by the short ion diffusion distance through the only >= 5 nm thick NiO nanotube wall. 

Using SigmaPlot 5.0 (Jandel Scientific, San Rafael, CA), the sets of sedimentation equilibrium data for the paucidisperse acetylated methylated and underivatized kraft lignin fractions were successfully curve fit to functions representing sums of terms of the form expressed in Equation (20). In no case were more than four individual terms required for the fits of unprecedented accuracy that were achieved these are exemplified in Figure 6A2 and B2. The sums of the areas under the component exponential curves confirmed that the total mass of solute species observed at equilibrium never differed by more than 1% from that present before the sedimentation process began. 

Figure 12.10 Typical time traces of (a) emission intensityand (b) lifetime, measured from a single DMPBI nanocrystal, (c) Photon correlation histogram obtained from the time trace of the emission intensity (a). The lifetimes were obtained by fitting a single exponential function to the decay curves constructed for every 2000...

Figure 3.18. Time dependence of the peak position of the 1570 cm Raman band of Sj trans-stilbene in chloroform solution (filled triangle). The time dependence of the anti-Stokes/Stokes intensity ratio is also shown with open circles. The best fit of the peak position change with a single-exponential function is shown with a solid curve, while the best fit of the anti-Stokes/Stokes intensity ratio is shown with a dotted curve. The obtained lifetime for both single-exponential decay functions was 12ps. (Reprinted with permission from reference [78]. Copyright (1997) American Chemical Society.)...

Figure 4.5. Kinetic traces observed at (a) 1650cm and (b) 2096cm following 266nm photolysis (10ns, 0.4ml) of N-labeled 24 (3.1 mM) in argon-satnrated Freon-113. The dotted curves are experimental data the solid curves are the calculated best fit to a single exponential function. Reprinted with permission from Y. Wang, T. Yuzawa, H. Hamaguchi, and J. P. Toscano, J. Am. Chem. Soc., 1999,121, 2875. Copyright 1999, American Chemical Society.

Fig. 2.5.5 A study examining the conformational changes of the protein ubiquitin, showing the population ratio of the A-state to the native-state as a function of time, (a) The reaction from 0 to 120 s. (b) The reaction for the first 40 s, including curves fit to a single exponential. Reprinted with permission from Ref. [37]. Copyright (2003) American Chemical Society.

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

The data were fitted to a stretched exponential function (Eq. 4.9) setting the stretching parameter to its dielectric value. The solid lines included in Fig. 6.3 display the resulting curves. These fits lead to the Q-dependent characteristic relaxation times TKww(Q)> hich are converted to average relaxation times by Eq. 5.25 (see Fig. 6.4). 

In Figures 8 and 9 are shown the data for the dependence of the characteristic film buildup time t on Apg and U. In accord with the model, t is found to be independent of U, with only a very weak dependence on Apg indicated. This latter result could in part be a function of experimental inaccuracy. The data reduction for t introduces no assumptions beyond that needed to draw the exponential flux decline curves such as those shown in Figures 2 and 3. However, an error analysis shows that the maximum errors relative to the exponential curve fits occur at the earlier times of the experiment. This is seen in the typical error curve plotted in Figure 10. The error analysis indicates that during the early fouling stage the relatively crude experimental procedure used is not sufficiently accurate or possibly that the assumed flux decline behavior is not exponential at the early times. In any case, it follows that the accuracy of the determination of 6f is greater than that for t. 

Figure 1-5 Determination of the order of hypothetical reactions with respect to species A. (a) The initial reaction rate method is used. The initial rate versus the initial concentration of A is plotted on a log-log diagram. The slope 2 is the order of the reaction with respect to A. The intercept is related to k. (b) The concentration evolution method is used. Because the exponential function (dashed curve) does not fit the data (points) well, the order is not 1. The solution for the second-order reaction equation (solid curve) fits the data well. Hence, the order of the reaction is 2.

Figure 14 (a) Time-dependent behavior of cation radicals in liquid -dodecane monitored at 790 nm. The dotted and the solid lines represent the experimental curve and the simulation curve, respectively. The parameters of the electron dilfusion coefficient (De) = 6.4 x 10 " cm /sec, the cation radical diffusion coefficient (D + ) = 6.0 x 10 cm /sec, the relative dielectric constant e = 2.01, the reaction radius R = 0.5 nm, and the exponential function as shown in Eq. (19) with ro = 6.6 nm were used, (b) Time-dependent distribution function obtained from fitting curve of (a), r indicates the distance between the cation radical and the electron. The solid line, dashed line, and dots represent the distribution of cation radical-electron distance at 0, 30, and 100 psec after irradiation, respectively. 

The increment in mechanical properties (tensile strength, 300% modulus, and Young s modulus) as a function of SAF is plotted in Fig. 39. In general, the higher level of SAF, which in turn indicates better exfoliation, results in high level of property enhancement. However, the level of increment with the increase in SAF is different in all three cases and follows a typical exponential growth pattern. The apparent nonlinear curve fitting of the experimental values presented in Fig. 39 is a measure of the dependence of mechanical properties on the proposed SAF function. 

The decay curve recorded at 515 nm was analyzed by the sum of two-or three-exponential functions. The fitting to the experimental data with a three-exponential function was always better that a two-exponential function. The fast decay component with a time constant of 1.1 ps (59%), corresponding to the fast rise time at 650 nm, and a very fast rise time of 70-110 fs (73%), were obtained in addition to the long decay component. From these results, it is concluded that the closed-ring form is mainly formed from the species with an absorbance maximum at 515 nm with a... 

Once the background is subtracted, the component of the spectrum due to the annihilation of ortho-positronium is usually visible (see Figure 6.5(a), curve (ii) and the fitted line (iv)). The analysis of the spectrum can now proceed, and a number of different methods have been applied to derive annihilation rates and the amplitudes of the various components. One method, introduced by Orth, Falk and Jones (1968), applies a maximum-likelihood technique to fit a double exponential function to the free-positron and ortho-positronium components (where applicable). Alternatively, the fits to the components can be made individually, if their decay rates are sufficiently well separated, by fitting to the longest component (usually ortho-positronium) first and then subtracting this from the... 

Figure 15 Representative fluorescence decay curves of single CV molecules on a PMMA film. Data accumulation time was 180 s. These curves are fitted to single exponential functions (A) for a strong fluorescent spot (1.92 ns), and (B) for a weak fluorescent spot (0.44 ns) in the bimodal histogram of Fig. 14A. (From Refs. 1, 15.)...

Figure 8.9 Time-resolved fluorescent lifetime analysis of Cy3 attached to double-stranded DNA (Iqbal et al., 2008b). Fluorescent decay curve for Cy3 attached to a 16 bp DNA duplex, showing the experimental data and the instrument response function (IRF), and the fit to three exponential functions (line). The decay curve was generated using time-correlated single-photon counting, after excitation by 200 fs pulses from a titanium sapphire laser at 4.7 MHz.

Let us address the parametrical dependence of the Hall resistance on longitudinal resistance (the parameter is temperature) for the above samples with xexponential function (R oc R xx), it was found that the exponent m varies from 0.44 to... 

---