Concentration dependence local diffusion coefficient

FIGURE 20.3-11 Demonstration of the marked difference in concentration dependence of the true local diffusion coefficient (solid line) and the apparent diffusion coefficient (dashed line) for a hypothetical exponentially varying case such as shown in Fig. 20.3-6a. 

Unfortunately, the above equation often is used to calculate a parameter sometimes referred to as Dapp (apparent diffusivjty). Tins parameter does not have a simple physical meaning and is a complicated average of the uue local diffusion coefficient at the upstream and downstream conditions, as can be seen from the expression for 8 in Eq. (20.3-13). This parameter tends to show less dependence on the upstream pressure or concentration than the true local diffusion coefficient does when evaluated at the upstream condition. 

When the local diffusion coefficient evalneted from Eq. (20.3-6) is not too strongly concentration dependent (10-20% variation over the range of interest), ihe error associated with the use of Eq. 

We have applied FCS to the measurement of local temperature in a small area in solution under laser trapping conditions. The translational diffusion coefficient of a solute molecule is dependent on the temperature of the solution. The diffusion coefficient determined by FCS can provide the temperature in the small area. This method needs no contact of the solution and the extremely dilute concentration of dye does not disturb the sample. In addition, the FCS optical set-up allows spatial resolution less than 400 nm in a plane orthogonal to the optical axis. In the following, we will present the experimental set-up, principle of the measurement, and one of the applications of this method to the quantitative evaluation of temperature elevation accompanying optical tweezers. 

K. They noted a decay over timescales 95 and < 35 ns, respectively, which was attributed to geminate ion-pair recombination (see Fig. 33). The decay of the optical absorption is independent of the dose of radiation received and continues for about lps. Rather than displaying a dependence on time as eqn. (153), i.e. at f 3/2, the experimental results are more nearly represented by either at f 1 decay to an optical density about one tenth of the maximum or by a decay as t 1/2 to zero absorption. These effects may be the recombination of ions within a spur (or cluster of ion-pairs), which is more nearly like a homogeneous reaction. The range of electrons in propane at 100 K is 10 nm [334] and the extrapolated diffusion coefficient is 10 11 m2 s 1 [320]. The timescale of recombination is 10 ps. The locally greater concentration of ions within a spur probably leads to a faster rate of reaction and is consistent with the time-scale of the reaction observed. Baxendale et al. [395] observed the decay of the infrared optical absorption of the solvated electron in methylcyclo-hexane at 160 K. They noted that the faster decay occurring over < 50 ns was independent of dose and depended on time as t 1/2, i.e. the reaction rate decays as t 3/2, see eqn. (153). It was attributed to recombination of... 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

The analytical solutions to Fick s continuity equation represent special cases for which the diflusion coefficient, D, is constant. In practice, this condition is met only when the concentration of diffusing dopants is below a certain level ( 1 x 1019 atoms/cm3). Above this doping density, D may depend on local dopant concentration levels through electric field effects, Fermi-level effects, strain, or the presence of other dopants. For these cases, equation 1 must be integrated with a computer. The form of equation 1 is essentially the same for a wide range of nonlinear diffusion effects. Thus, the research emphasis has been on understanding the complex behavior of the diffusion coefficient, D, which can be accomplished by studying diffusion at the atomic level. 

Tn the critical region of mixtures of two or more components some physical properties such as light scattering, ultrasonic absorption, heat capacity, and viscosity show anomalous behavior. At the critical concentration of a binary system the sound absorption (13, 26), dissymmetry ratio of scattered light (2, 4-7, II, 12, 23), temperature coefficient of the viscosity (8,14,15,18), and the heat capacity (15) show a maximum at the critical temperature, whereas the diffusion coefficient (27, 28) tends to a minimum. Starting from the fluctuation theory and the basic considerations of Omstein and Zemike (25), Debye (3) made the assumption that near the critical point, the work which is necessary to establish a composition fluctuation depends not only on the average square of the amplitude but also on the average square of the local... 

In the present symposium, Dr. Raucher argued that the observed curvature in gas sorption isotherms does not arise from a site saturation mechanism such as we have described in the preceding discussion. The form of the sorption isotherm and local concentration dependent diffusion coefficient proposed by Raucher and Sefcik are given below ... 

When the diffusion coefficient depends not only on the spatial coordinate but also on the local concentration, the discretization of the diffusion equation proceeds in a similar manner, except that one has to solve at each time step not a system of linear equations, but a quite complicated set of coupled non-linear equations. 

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