Transformation between diffusion coefficients

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. 

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. 

The relationship can also be cast in terms of distributions of molecular weights for polymer systems. This transformation is achieved by relating the decay constant to its corresponding diffusion coefficient D which in turn is related to molecular weight through a Mark-Houwink like relationship. Equation 3 shows the fundamental relationship between the autocorelation function and polymer molecular weight distributions. 

Studer A, Georgiadis D, Baumgartner RW (2003) Ischemic infarct involving all arterial territories of the thalamus. Acta Neurol Scand 107 423-425 Timsit SG, Sacco RL, Mohr JP et al (1992) Early clinical differentiation of cerebral infarction from severe atherosclerotic stenosis and cardioembolism. Stroke 23 486-491 Tong DC, Adami A, Moseley ME et al (2000) Relationship between apparent diffusion coefficient and subsequent hemorrhagic transformation following acute ischemic stroke. Stroke 31 2378-2384... 

Einstein presented an equation relating the diffusion coefficient of species i (Df to the frictional coefficient in a diffusion process [ii], this can be transformed into a relationship between -> ionic mobility iq and -> diffusion coefficient D ... 

Some methods take advantage of a difference in a particular property between water and the molecule to be studied. In particular, a macromolecule usually has a shorter value for proton Tt than water and a much lower diffusion coefficient. One of the oldest methods for water signal suppression is WEFT (water elimination Fourier transform), in which an inversion recovery sequence is applied (see Fig. 2.12) with r chosen to be the time that the water signal goes through zero (Tj In 2), just as in the BIRD pulse sequence. Another method makes use of the technique described in Section 9.3 to measure diffusion coefficients. 

Undeniably, the speed vector, by its size and directional character, masks the effect of small displacements of the particle. Another difference comes from the different definition of the diffusion coefficient, which, in the case of the property transport, is attached to a concentration gradient of the property it means that there is a difference in speed between the mobile species of the medium. A second difference comes from the dimensional point of view because the property concentration is dimensional. When both equations are used in the investigation of a process, it is absolutely necessary to transform them into dimensionless forms [4.6, 4.7, 4.37, 4.44]. 

A good agreement is generally obtained between the models based on transport equations and the SDE for mass and heat molecular transport. However, as explained above, the SDE can only be applied when convective flow does not take place. This restrictive condition limits the application of SDE to the transport in a porous solid medium where there is no convective flow by a concentration gradient. The starting point for the transformation of a molecular transport equation into a SDE system is Eq. (4.108). Indeed, we can consider the absence of convective flow in a non-steady state one-directional transport, together with a diffusion coefficient depending on the concentration of the transported property... 

Interactions between oppositely charged micelles in aqueous solutions spontaneously form vesicles. The self-diffusion coefficient of water and 2H relaxation of 2H-labeled dodecyl trimethyl ammonium chloride of the dodecyl trimethyl ammonium chloride-sodium dodecyl benzenesulfonate systems show that in these mixtures there is limited growth of the micelles with changes in composition. The vesicles abruptly begin to form at a characteristic mixing ratio of the two surfactants. The transition is continuous.205 Transformation from micelle to vesicle in dodecyl trimethyl ammonium chloride-sodium perfluoro-nonanoate aqueous solution has been studied by self-diffusion coefficient measurements, and it was found that at a concentration of 35 wt% with a molar ratio of 1 1, the self-diffusion coefficient of the mixed micelles is far smaller than that of the two individual micelles.206 The characteristics of mixed surfactant... 

Coulometric titration techniques were used to measure chemical diffusion at between 700 and lOOOC. The transient current response to a potentiostatic step was transformed from the time domain to the frequency domain. The equivalent circuit which was used to fit the resultant impedance data contained an element which described the finite-length diffusion of O into the sample. Other elements which were included were the gas-phase capacitance, and the sum of the gas-phase diffusion resistance and that which was associated with the limited surface exchange kinetics of the sample. The chemical diffusion coefficient of the perovskite, Laq gSrq 2C0O3,... 

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