Integrated diffusion coefficient

In view of the lack of interdiffusion in the course of multiple-layer formation at the A-B interface and because of complicated mechanism of this process, calculation of integrated diffusion coefficients seems in most cases meaningless. 

The quantity is characteristic of the i-th phase and must therefore be the same in all reaction couples in which this phase occurs. Hence, the ratio of the parabolic growth-rate constants for the AB layer in the A2B-AB2, A2B—B and A-B reaction couples is 3 2 1, in full agreement with the calculations on the basis of equations (4.30), (4.31) and (4.33). That this is not an accidental coincidence can readily be verified by carrying out calculations for other chemical compounds of different stoichiometry. Similar results follow, for example, from the data by F.J.J. van Loo on the integrated diffusion coefficients in the Ti-Al system.66... 

Here D (ct ) denotes the integral diffusion coefficient for the concentration Cj00 see Eq. (8). The expression ... 

In an analysis of the diffusion equations in magnetic fields, Dumarque et al. introduced a diffusivity tensor and an effectiveness parameter related to the difference between diffusivity in the presence and in the absence of a magnetic field. In a less general, but somewhat more amenable approach to numerical estimations, Lielmezs and Musbally derived a magnetic correction factor called the arithmetic mean average fractional integral diffusion coefficient... 

The integral diffusion coefficient of iodine for diffusion from a given concentration into pure solvent has been measured at 25 °C in several solvents by Stokes, Dunlop, and Hall (Trans. Faraday Soc. 1953, 49, 886). By extrapolation they obtained values of the dif ion coefficient D at infinite dilution. These values together with viscosities of the solvent are given in table 1. 

RESULTS AND DISCUSSION, 71 Various Fickian Diffusion Curves, 72 Integral Diffusion Coefficients, 75... 

Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. Satisfactory agreement is obtained between integral diffusion coefficient from sorption and desorption rate curves and apparent diffusion coefficient from permeation rate curves (time-lag method). These rate curves were also compared to the curves predicted by Fickian-type diffusion equations. 

For sorption/desorption rate curves, a good estimation of integral diffusion coefficient can be obtained by taking the average of the diffusion coefficients calculated from first slope for the integral sorption (DJ and desorption (D ) cycle, [8]... 

In the Eq. (7.65), D a is referred as the integrated diffusion coefficient when considering the impact of pore-size distribution on diffusion. 

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

Here solute concentration C and Cp (in permeate) are expressed as mass fractions, D is the diffusion coefficient of the solute and y is the distance from the membrane. Rearranging and integrating from C - Cf when y = / the thickness of the film, to C = Cw, the concentration of solute at the membrane wall, when y=0, gives ... 

Diffusion of the fluid into the bulk. Rates of diffusion are governed by Pick s laws, which involve concentration gradient and are quantified by the diffusion coefficient D these are differential equations that can be integrated to meet many kinds of boundary conditions applying to different diffusive processes. ... 

The integrated form is x = Kt -i- C, or x = D Vt, where D is the Diffusion Coefficient. Eiquation 4.5.3. is called the Parabolic Law of Diffusion. If the growth of a phase can be fitted to this equation, then it is likely that the primary reaction mechanism involves simple diffusion. 

As reversible ion transfer reactions are diffusion controlled, the mass transport to the interface is given by Fick s second law, which may be directly integrated with the Nernst equation as a boundary condition (see, for instance. Ref. 230 232). A solution for the interfacial concentrations may be obtained, and the maximum forward peak may then be expressed as a function of the interfacial area A, of the potential scan rate v, of the bulk concentration of the ion under study Cj and of its diffusion coefficient D". This leads to the Randles Sevcik equation [233] ... 

Since the diffusion coefficient is the infinite-time integral of the velocity correlation function, we have the Einstein relation, D = kBT/Q. 

In the forthcoming sections we will consider several methods that have been used to derive different integral relaxation times for cases where both drift and diffusion coefficients do not depend on time, ranging from the considered mean transition time and to correlation times and time scales of evolution of different averages. 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

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