Diffusion, coefficients transformation

Because the fluid is in equilibrium, any ensemble average property should not change with time. Hence, the ensemble average of (u(tf)u(t")> depends only on the relative difference of time, t — t". That is, it is a stationary process. On transforming the time variables to f and r = tr — f" (rather like the centre of diffusion coefficient transformation of Chap. 9, Sect. 2), the Green—Kubo expression for the diffusion coefficient is obtained [453, 490],... 

The study of molecular diffusion in solution by NMR methods offers insights into a range of physical molecular properties. Different mobility rates or diffusion coefficients may also be the basis for the separation of the spectra of mixtures of small molecules in solution, this procedure being referred to as diffusion-ordered spectroscopy (DOSY) [271] (Figure 5.11). In this 2D experiment, the acquired FID is transformed with respect to 2 (the acquisition time). 

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."... 

If Dh is indeed time dependent as in eq. (5) it is not obvious that C(x, t) will follow an error function expression as in eq. (3) or that >H will be thermally activated as in eq. (4). We now show that eqs. (3) and (4) still apply with a time dependent diffusion coefficient, by making a coordinate transformation (Kakalios and Jackson, 1988). The one-dimensional diffusion equation... 

When the diffusion coefficient varies with time, the fundamental transformation commonly used consists in defining a new time variable t as... 

Abbreviations D, self-diffusion coefficient ge, gradient-echo IR, inversion recovery IRFT, inversion recovery fourier transform MRS, magnetic resonance spectroscopy PD, proton density PFGSE, pulsed field gradient spin echo se, spin-echo. 

In another review, Hoffert discussed the social motivations for modeling air quality for predictive purposes and elucidated the components of a model. Meteorologic factors were summarized in terms of windfields and atmospheric stability as they are traditionally represented mathematically. The species-balance equation was discussed, and several solutions of the equation for constant-diffusion coefficient and concentrated sources were suggested. Gaussian plume and puff results were related to the problems of developing multiple-source urban-dispersion models. Numerical solutions and box models were then considered. The review concluded with a brief outline of the atmospheric chemical effects that influence the concentration of pollutants by transformation. 

The Gaussian plume foimulations, however, use closed-form solutions of the turbulent version of Equation 5-1 subject to simplifying assumptions. Although these are not treated further here, their description is included for comparative purposes. The assumptions are reflection of species off the ground (that is, zero flux at the ground), constant value of vertical diffusion coefficient, and large distance from the source compared with lateral dimensions. This Gaussian solution to Equation 5-1 is obtained under the assumption that chemical transformation source and sink terms are all zero. In some cases, an exponential decay factor is applied for reactions that obey first-order kinetics. A typical solution (with the time-decay factor) is ... 

If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

Table 3-2 Diffusion coefficients of noble gases in aqueous solutions Table 3-3 Ionic porosity of some minerals Table 4-1 Steps for phase transformations Table 4-2 Measured crystal growth rates of substances in their own melt...

Taking the time and space Fourier transforms of 2, multiplying by e iq fc and integrating over r0, we obtain the following key relationship between the diffusion coefficient and the eigenstates ... 

As for the properties themselves, there are many chemically useful autocorrelation functions. For instance, particle position or velocity autocorrelation functions can be used to determine diffusion coefficients (Ernst, Hauge, and van Leeuwen 1971), stress autocorrelation functions can be used to determine shear viscosities (Haile 1992), and dipole autocorrelation functions are related to vibrational (infrared) spectra as their reverse Fourier transforms (Berens and Wilson 1981). There are also many useful correlation functions between two different variables (Zwanzig 1965). A more detailed discussion, however, is beyond the scope of this text. 

This relationship is captured in differential-equation form as Eq. 7.60. Since the momentum and energy equations (Eqs. 7.59 and 7.62) explicitly involve r2, the radial coordinate has become a dependent variable, not an independent variable. A consequence of the Von Mises transformation is that the radial velocity v is removed as a dependent variable and the radial convective terms are eliminated, which is a bit of a simplification. However, the fact that the group of dependent variables pur2 appear within the diffusion terms is a bit of a complication. The factor pur2 plays the role of an apparent variable diffusion coefficient. ... 

In the absence of water, none of the chemical transformations described above occurs noticeably. The low diffusion coefficient of alkyl-ammonium cations between the montmorillonite layers (2) together with the strong acid character of residual water (3, 4) in this situation might provide a favorable situation which, perhaps, does not exist on other silicate surfaces with a more open structure. 

Treatment of class (c) membranes, on the other hand, presents a considerably more complicated problem. Here, S and DT in Eqs. (1) and (2) are functions of the spatial coordinates. The problem becomes much more acute if S and DT are also dependent on C 4,5). Under these conditions, transformation of Eqs. (2) into (3) is not generally possible and there are no standard methods, as in the previous cases, of fully characterizing the membrane-penetrant system 3 "5). There is usually no difficulty in determining an overall or effective solubility coefficient but the definition of useful effective diffusion coefficients is a more difficult matter, which, not surprisingly, is a major concern of current research in the field. 

---