Time-delayed exponential decay

Kirkland et al. [6,167,171] dealt with the programming of S-FFF in an analysis of particle size distribution. They also used time-delayed exponential decay programming of the centrifugal force intensity, which allowed them to linearize the dependence of retention time to a logarithm of solute dimensions. Moreover, the total analysis time was shortened without sacrificing the resolution. 

Constant force field provides for highest resolution of particles in the sample with resulting highest precision. However, characterization of samples with wide size distributions is difficult and time consuming. Force field programming [89,90] removes these limitations to ensure that the entire distribution can be analyzed in a convenient time. In time delayed exponential decay the initial force field is held constant for a time equal to r and after this the force field is decayed exponentially with a time constant r. In this mode, a log-linear relationship is obtained of particle mass against retention time. This simple relationship permits a convenient calculation of the quantitative information needed for the sample. Retention time is given by ... 

The commercial Du Pont SdFFF instrument uses the time-delayed exponential decay force-field programing mentioned by Yau et. all. The centrifugal force field is changed according to... 

For fluorescent compounds and for times in die range of a tenth of a nanosecond to a hundred microseconds, two very successftd teclmiques have been used. One is die phase-shift teclmique. In this method the fluorescence is excited by light whose intensity is modulated sinusoidally at a frequency / chosen so its period is not too different from die expected lifetime. The fluorescent light is then also modulated at the same frequency but with a time delay. If the fluorescence decays exponentially, its phase is shifted by an angle A([) which is related to the mean life, i, of the excited state. The relationship is... 

In Fig. 2. R is presented for a solution of 3 M NaCl in HD0 D20 at different temperatures. All signals show an overall non-exponential decay, but are close to a single exponential for delays >3 ps. After this delay time, the signals only represent the orientational dynamics of the HDO molecules in the first solvation shell of the Cl- ion, thanks to the difference in lifetimes of the O-H- -0 and the O-H- -Cl- vibrations. At 27 °C, the orientational relaxation time constant ror of these HDO molecules is 9.6 0.6 ps. which is quite long in comparison with the value of Tor of 2.6 ps of HDO molecules in a solution of HDO in D20 [12],... 

Fig. 2. (a,b) Transient absorption on the v0h=1— 2 transition of OH/OH dimers (symbols). The spectrally integrated, anisotropy free absorption change AA is plotted as a function of the delay time between the pump centered at Ep=2950 cm 1 and the probe centered at Epr. Solid lines exponential decay with a time constant of 200 fs. Inset of Fig. (b) Fourier transform of the oscillatory component of the transient in Fig. (b) displaying an oscillation frequency of 145 cm 1. 

In Fig. 10, the transients exhibit quite different behavior from opal A to opal CT. In particular, a bi-exponential decay (Eq. 2) failed to reproduce the kinetics of opal CT. In this material, the emission is red-shifted towards 2.6 eV and the PL is strongly quenched at shorter time delays, with an unusual, non-linear kinetics in semi-log scale, indicating a complex decay channel either involving multi-exponential relaxation or exciton-exciton annihilations. Runge-Kutta integration of Eq. 5 seems to confirm the latter assumption with satisfactory reproduction of the observed decays. The lifetimes and annihilation rates are Tct = 9.3 ns, ta = 13.5 ns, 7ct o = 650 ps-1 and 7 0 = 241 ps-1, for opal CT and opal A, respectively. 

We have implemented the discrimination in the frequency domain. As is known in multifrequency phase fluorometry,17 the time-delayed fluorescence acquires a phase shift >p and a reduction in amplitude Mp upon increasing the modulation frequency m = 2irf of the sinusoidally modulated excitation. For a simple single exponential decay, this phase shift

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Figure 3.52. The kinetics of excitation quenching followed by delayed decay [solid line in (a)] in comparison to the free exponential decay with time zD [long dashed line in (a)]. The short dashed horizontal line in (a) indicates the amplitude of the delayed fluorescence yNo, whose free energy dependence is outlined in (b). The initial concentration of the excitations Nq — 0.01 M, zD = 1 us. The other parameters are D = D = 10 7 cm2/s, rc =0, Xc = 0.4 eV, c = 0.1M, A q = 10 4 M, Wi = 103 ns-1, L = 1.4 A, ct = 10A, T = 293 K. (From Ref. 189.)...

Figure 20. Integrated signals Ei and 2 for OHBA (a), ODBA (b), and HAN (c) plotted as a function of the time delay at the indicated excitation wavelength. Note the change in ordinate time scales. Signal Ei always followed the laser cross-correlation, indicating a rapid proton transfer reaction. The decay of signal 2 was fitted via single exponential decay, yielding the time constant for internal conversion of the Si keto state in each molecule. See color insert.

Franck-Condon dissociative continuum. At long times (Af = 3500 fs), a sharp photoelectron spectrum of the free NO(A, 3,v) product is seen. The 10.08 eV band shows the decay of the (NO)2 excited state. The 9.66 eV band shows both the decay of (NO)2 and the growth of free NO(A, 3,v) product. It is not possible to fit these via single exponential kinetics. However, these 2D data are fit very accurately at all photoelectron energies and all time delays simultaneously by a two-step sequential model, implying that an initial bright state (NO)2 evolves to an intermediate configuration (NO)2f, which itself subsequently decays to yield free NO(A, 3s) products [138]... 

Some numerical examples are shown in Fig. 7a. The nonexponential part of 4>v,b(t) for short times is readily seen and leads to a delayed onset of the exponential decay. The latter represents the Markovian limit (straight line in the semi-log plot). A time interval of approximately 2 x rc is required to establish the asymptotic dephasing rate 1/T2. The purely exponential case, rc = 0, is fictitious and shown in the figure only for comparison. 

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