Fast Laplace Inversion

Novel Two Dimensional NMR of Diffusion and Relaxation for Material Characterization 171 2.7.3.2 Fast Laplace Inversion - FLI... 

An important technical development of the PFG and STD experiments was introduced at the beginning of the 1990s the Diffusion Ordered Spectroscopy, that is DOSY.69 70 It provides a convenient way of displaying the molecular self-diffusion information in a bi-dimensional array, with the NMR spectrum in one dimension and the self-diffusion coefficient in the other. While the chemical-shift information is obtained by Fast Fourier Transformation (FFT) of the time domain data, the diffusion information is obtained by an Inverse Laplace Transformation (ILT) of the signal decay data. The goal of DOSY experiment is to separate species spectroscopically (not physically) present in a mixture of compounds for this reason, DOSY is also known as "NMR chromatography."... 

Whatever the excitation, the transformation of the response from the frequency to the time domain (Fig. 11.21) is done with the inverse Fourier transform, normally as the FFT (fast Fourier transform) algorithm, just as for spectra of electromagnetic radiation. Remembering that the Fourier transform is a special case of the Laplace transform with... 

Another approach presented by Yamaoka, Yano, and Tanaka, uses a fast inverse Laplace transform to generate the integrated equation data. Thus the model is described in terms of the Laplace transform equations and solved numerically. 

Yano, Y. Yamaoka, K. Tanaka, H. A non-linear least squares program, MULTI(FILT), based on fast inverse laplace transform for microcomputers. Chem. Pharm. Bull. 1989, 37, 1035-1038. 

The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected i.e., Dj, = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen s model is equivalent to Carta s [34]. 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

Considerably less work exists addressing Leveque s problem for inlet channel flow with wall reaction. Carslaw and Jaeger [58] and Petersen [68] presented solutions for plug-flow conditions, using the Laplace transform. Pancharatnam and Homsy [69] used the same technique for laminar flow. The inversion of the transformed solution is given in terms of an infinite summation with coefficients given by recurrence relations (first 24 out of 50 coefficients are tabulated). Ghez [70] considered a first-order reversible reaction with the same solution method. Moreover, asymptotic expansions in the limits of fast and slow reactions were presented. 

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