Diffusive behavior

For times t > rR one expects g2 to settle down to its saturation value while gill continues to grow and crosses over to a diffusive behavior ... 

FIG. 14 Brownian dynamics. The arrows indicate that the particle trajectories show diffusive behavior. 

Since this is just the beginning of investigations into the diffusion behavior and intermolecular forces in ionic liquid systems, further experimental work needs to be done both with pure ionic liquids and with systems of mixtures of ionic and organic liquids. 

Iever89] Levermore, C.D. and B.M.Boghosian, Deterministic cellular automata with diffusive behavior, pages 118-129 in [mann89j. 

Criteria 1-3 are the cardinal characteristics of Fickian diffusion and disregard the functional form of D(ci). Violation of any of these is indicative of non-Fickian mechanisms. Criterion 4 can serve as a check if the D(ci) dependence is known. As mentioned, it is crucial that the sorption curve fully adhere to Fickian characteristics for a valid determination of D from the experimental data. At temperatures well above the glass transition temperature, 7 , Fickian behavior is normally observed. However, caution should be exercised when the experimental temperature is either below or slightly above 7 , where anomalous diffusion behavior often occurs. 

Figure 4 A Deborah number diagram for the polystyrene-ethylbenzene system showing the diffusion behavior as a function of weight fraction and temperature. (From Ref. 33.)...

Table I. Diffusion Behavior of Pesticides in a Soil with 10% Moisture...

Diffusion regime For times t > xd N3 /4/kBTd2, the dynamics are determined by reptation diffusion. We expect normal diffusive behavior... 

In this connection let us remark that in spite of several efforts, the relation between Lyapounov exponents, correlations decay, diffusive and transport properties is still not completely clear. For example a model has been presented (Casati Prosen, 2000) which has zero Lyapounov exponent and yet it exhibits unbounded Gaussian diffusive behavior. Since diffusive behavior is at the root of normal heat transport then the above result (Casati Prosen, 2000) constitutes a strong suggestion that normal heat conduction can take place even without the strong requirement of exponential instability. 

The tracer should not be adsorbed on or react with the surface of the vessel. Alternatively, the tracer should be chemically and physically similar to the fluid flowing, so that any adsorption (or diffusion) behavior may be replicated. 

O2 diffusion through the membrane seems to be limited by the percolation network of the diffusion path, which is not only defined by the amount of water in the membrane, but also by the different chemical structure of the membranes. It is difficult to make comparisons of gaseous diffusion behavior among polymers with different structures because polymer morphology can change drastically without appreciable changes in density, and the presence of water and the hydrogen bonds formed between polymer-water moieties also has major effects on system properties. However, some points can be made from these particular studies. 

In comparing the diffusion behavior of these two membranes at low MeOH concentrations, MeOH in BPSH 40 membranes exhibits significantly higher diffusion coefficients than those in Nafion 117. This may be the result of differences in morphology. Tapping mode AFM measurements found that, for dry membranes, the domains for BPSH 40 appeared to be 10-25 nm in diameter for Nafion 117, the domains were smaller, approximately 4-10 nm. Although one would expect restricted diffusion in both cases, it is possible that the smaller domains limit diffusion to a greater extent. 

As the concentration of MeOH increases, the divergent diffusion behavior between the two membrane types is a reflection of fhe difference in MeOH solubility and its concentration dependence within each membrane. This was verified by solvenf upfake measurements. Upon increasing MeOH concentration, Nafion 117 showed a steady increase in mass, while a sharp drop in total solution uptake was observed for BPSH 40. The lower viscosity of MeOH also affecfs fhe fluidity of the solution within the pores. The constant solvent uptake and the increased fluidity of the more concentrated MeOH solutions accounted for fhe slight increase in diffusion coefficienf of Nafion 117. For BPSH 40, increasing the MeOH concentration resulted in a decrease in MeOH diffusion. The solvent uptake measurements showed very similar behavior, indicating that the membrane excludes the solvent upon exposure to higher MeOH concentrations. 

The simple water charmel models can explain the ionomer peak and the small-angle upturn in the scattering data of fhe unoriented samples as well as of the oriented films. Interestingly, the helical structure of backbone segments is responsible for fhe sfabilify of fhe long cylindrical charmels. The self-diffusion behavior of wafer and protons in Nation is well described by the water channel model. The existence of parallel wide channels af high wafer uptake favors large hydrodynamic confributions to electro-osmotic water transport and hydraulic permeation. 

There are few studies in literature reporting pure gas permeabilities as well as separation factors of mixtures. Vuren et al. (1987) reported Knudsen diffusion behavior of pure gases for y-alumina membranes with a mean pore radius of 1.2 nm. Separation experiments with a 1 1 H2/N2 mixture showed, that the theoretical Knudsen separation factor [of 3.7, Equation 6.4)j for this mixture could be obtained (Keizer et al. 1988 see also Figure 6.2). In Figure 6.2, the effect of process parameters is also demonstrated. The separation factor is a function of the pressure ratio over the membrane, which is the ratio of the pressure on the permeate-side to that on the feed-side. For pressure ratios approaching unity, which means the pressure on both sides of the... 

For given values of double layer capacitance solution resistance Rjj and Warburg coefficient a, plots of -Z versus Z have been made for selected values of charge transfer resistance. Ret (26). It is observed that at smaller values of R t ("10 S2 cm ) relaxation due to Rct dl Warburg diffusion behavior are both clearly seen. 

Impedance measurements taken on specimens after 96 hours exposure to salt spray show a combination of R-C and Warburg diffusion behavior. This Is in agreement with the observation elsewhere (9,11). 

This solution is shown in Figure 1-9. It is a widely used solution in experiments and in modeling diffusion behavior in nature. 

In summary, the diffusion behavior of both H2O and CO2 demonstrates the importance of understanding the role of speciation in diffusion, and the very different consequences due to that role. Diffusion of a single-species component (such as Ar) usually does not depend on its own concentration (when the concentration is low), but depends on the melt composition. For a multispecies component, speciation affects the diffusion behavior. For H2O, speciation makes the diffusion behavior very complicated even at low H2O concentrations, total H2O diffusivity still depends on H2O content (because the species concentrations are not proportional), in addition to the dependence on melt composition. If species concentrations are proportional to each other and hence to the total concentration of the component, then the diffusivity is independent of the concentration of the component, as in the case of CO2 diffusion. Many multispecies components probably satisfy this condition that the concentrations of... 

Given a diffusion couple of different melts (e.g., basalt and andesite), if one wants to predict the diffusion behavior of all major components, the diffusion matrix approach is necessary. 

In principle, the diffusion matrix approach can be extended to trace elements. My assessment, however, is that in the near future diffusion matrix involving 50 diffusing components will not be possible. Hence, simple treatment will still have to be used to roughly understand the diffusion behavior of trace elements the effective binary diffusion model to handle monotonic profiles, the modified effective binary diffusion model to handle uphill diffusion, or some combination of the diffusion matrix and effective binary diffusion model. 

The diffusion behavior of components that are not the principal equilibriumdetermining component is difficult to model because of multicomponent effect. Many of them may show uphill diffusion (Zhang et al., 1989). To calculate the interface-melt composition using full thermod3mamic and kinetic treatment and to treat diffusion of all components, it is necessary to use a multicomponent diffusion matrix (Liang, 1999). The effective binary treatment is useful in the empirical estimation of the dissolution distance using interface-melt composition and melt diffusivity, but cannot deal with multicomponent effect and components that show uphill diffusion. 

Diffusive dissolution of MgO-rich olivine and diffusion profiles MgO is the principal equilibrium-determining component and its diffusion behavior is treated as effective binary. Consider the dissolution of an olivine crystal (Fo90, containing 49.5 wt% MgO) in an andesitic melt (containing 3.96 wt% MgO) at 1285°C and 550 MPa (exp 212 of Zhang et al. 1989). The density of olivine is 3198 kg/m, and that of the initial melt is 2632 kg/m. Hence, the density ratio is 1.215. To estimate the dissolution parameter a, it is necessary to know the interface melt... 

Behavior of trace element that can be treated as effective binary diffusion The above discussion is for the behavior of the principal equilibrium-determining component. For minor and trace elements, there are at least two complexities. One is the multicomponent effect, which often results in uphill diffusion. This is because the cross-terms may dominate the diffusion behavior of such components. The second complexity is that the interface-melt concentration is not fixed by thermodynamic equilibrium. For example, for zircon growth, Zr concentration in the interface-melt is roughly the equilibrium concentration (or zircon saturation concentration). However, for Pb, the concentration would not be fixed. 

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