Tracer diffusion concentration

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

Tracer Diffusivity Tracer diffusivity, denoted by D g is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach selr-diffusivities at the pure component limit. That is, at the limit of dilute A in B, D g D°g and... 

A simple case is the diffusion of a single type of ion in a solution containing a sufficient excess of an indifferent electrolyte (see page 116), which then occurs in the same way as in the case of a non-electrolyte. Isotope (tracer) diffusion has the same character, where a concentration gradient of the radioactive isotope of an ion, present in a much lower concentration, is formed in a solution with a much larger, constant salt concentration. 

A similar situation occurs in tracer diffusion. This type of diffusion occurs for different abundances of an isotope in a component of the electrolyte at various sites in the solution, although the overall concentration of the electrolyte is identical at all points. Since the labelled and the original ions have the same diffusion coefficient, diffusion of the individual isotopes proceeds without formation of the diffusion potential gradient, so that the diffusion can again be described by the simple form of Fick s law. 

Mechanistic Ideas. The ordinary-extraordinary transition has also been observed in solutions of dinucleosomal DNA fragments (350 bp) by Schmitz and Lu (12.). Fast and slow relaxation times have been observed as functions of polymer concentration in solutions of single-stranded poly(adenylic acid) (13 14), but these experiments were conducted at relatively high salt and are interpreted as a transition between dilute and semidilute regimes. The ordinary-extraordinary transition has also been observed in low-salt solutions of poly(L-lysine) (15). and poly(styrene sulfonate) (16,17). In poly(L-lysine), which is the best-studied case, the transition is detected only by QLS, which measures the mutual diffusion coefficient. The tracer diffusion coefficient (12), electrical conductivity (12.) / electrophoretic mobility (18.20.21) and intrinsic viscosity (22) do not show the same profound change. It appears that the transition is a manifestation of collective particle dynamics mediated by long-range forces but the mechanistic details of the phenomenon are quite obscure. 

The key physics of our model (see Eqs. (9) and (10)) is contained in the nonlocal diffusion kernels which occur after integrating over the atomic processes which produce step fluctuations. We have calculated these kernels for a variety of physically interesting cases (see Appendix C) and have related the parameters in those kernels to atomic energy barriers (see Appendix B). The model used here is close in spirit to the work of Pimpinelli et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of Einstein and co-workers and Bales and Zangwill is based on an equihbrated gas of atoms on each terrace. The concentration of this gas of atoms obeys Laplace s equation just as our probability P does. To make complete contact between the two methods however, we would need to treat the effect of a gas of atoms on the diffusion probabilities we have studied. Actually there are two effects that could be included. (1) The effect of step roughness on P(J) - we checked this numerically and foimd it to be quite small and (2) The effect of atom interactions on the terrace - This leads to the tracer diffusion problem. It is known that in the presence of interactions, Laplace s equation still holds for the calculation of P(t), but there is a concentration... 

If one component is at a trace level but with variable concentrations (e.g., from 1 to 10 ppb) and concentrations of other components are uniform, the diffusion is called tracer diffusion. An example of tracer diffusion is diffusion into a melt of uniform composition (Watson, 1991b) when the concentration of is below ppb level. Usually only for a radioactive nuclide such as or " Ca, can such low concentrations be measured accurately to obtain concentration profiles. If a radioactive nuclide diffuses into a melt that contains the element (such as Ca diffusion into a Ca-bearing melt), it is still called tracer diffusion although it may be through isotopic exchange. 

When the concentration levels are higher, such as Ni diffusion between two olivine crystals with the same compositions except for Ni content (e.g., one contains 100 ppm and the other contains 2000 ppm), it may be referred to as either tracer diffusion or chemical diffusion. Tracer diffusivity is constant across the whole profile because the only variation along the profile is the concentration of a trace element that is not expected to affect the diffusion coefficient. 

If both chemical concentration gradients and isotopic ratio gradients are present (e.g., basaltic melt with Sr/ Sr ratio of 0.705 and andesitic melt with Sr/ Sr ratio of 0.720), the homogenization of isotopic ratio is referred to as isotopic diffusion (Lesher, 1990 Van Der Laan et al., 1994), although some prefer to call it isotopic homogenization. If there are concentration gradients in both major and trace elements, the diffusion of the trace elements is referred to as trace element diffusion (Baker, 1989). Isotopic diffusion and trace element diffusion are really part of multicomponent diffusion, which is complicated to handle. Isotopic diffusion should not be confused with self-diffusion, and trace element diffusion should not be confused with tracer diffusion. 

Interdiffusion, effective binary diffusion, and multicomponent diffusion may be referred to as chemical diffusion, meaning there are major chemical concentration gradients. Chemical diffusion is defined relative to self diffusion and tracer diffusion, for which there are no major chemical concentration gradients. 

Watson (1979a) carried out tracer diffusion experiments by loading a small amount of " Ca tracer onto one surface of a cylinder. The cylinder was heated up and " Ca diffuses into the cylinder. Assume that diffusion is along the axis of the cylinder (i.e., there is no radial concentration gradient). Assume that D=10 m /s. The cylinder is 3 mm long. Calculate the diffusion profile (concentration normalized to the surface concentration) at t = 2h and t=8h. How does the concentration profile at 8 h look like when compared to that at 2 h ... 

We remember that minority point defect concentrations in compounds depend on the activity of their components. This may be illustrated by the solubility of hydrogen in olivine since it depends on the oxygen potential in a way explained by the association of the dissolved protons with O" and O- as minority defects [Q. Bai, D. L. Kohlstedt (1993)]. Similarly, tracer diffusion coefficients and mobilities of Si and O are expected to depend on the activity of Si02. The value (0 lnDf/0 In aSio2)> = Si and O, should give information on the disorder type as discussed in Section 2.3. 

During self-diffusion in a pure material, whether a gas, liquid, or solid, the components diffuse in a chemically homogeneous medium. The diffusion can be measured using radioactive tracer isotopes or marker atoms that have chemistry identical to that of their stable isotope. The tracer concentration is measured and the tracer diffusivity (self-diffusivity) is inferred from the evolution of the concentration profile. 

The self-diffusion of component 1 in such a system is measured by studying the diffusion of a radioactive isotope tracer of component 1 (i.e., 1) under the condition that while there is a gradient in the tracer s concentration, c i, the sum (ci +c i) and C2 are both uniform. A possible diffusion couple is shown in Fig. 3.2. 

Figure 9.8 Isolated-boundary (Type-B) self-diffusion associated with a stationary grain boundary, (a) Grain boundary of width 6 extending downward from the free surface at y = 0. The surface feeds tracer atoms into the grain boundary and maintains the diffusant concentration at the grain boundary s intersection with the surface at the value cB(y = 0, t) = 1. Diffusant penetrates the boundary along y and simultaneously diffuses transversely into the grain interiors along x. (b) Diffusant distribution as a function of scaled transverse distance, xi, from the boundary at scaled depth, yx, from the surface. Penetration distance in grains is assumed large relative to 5.

Mass transfer in catalysis proceeds under non-equilibrium conditions with at least two molecular species (the reactant and product molecules) involved [4, 5], Under steady state conditions, the flux of the product molecules out of the catalyst particle is stoi-chiometrically equivalent (but in the opposite direction) to the flux of the entering reactant species. The process of diffusion of two different molecular species with concentration gradients opposed to each other is called counter diffusion, and if the stoichiometry is 1 1 we have equimolar counter diffusion. The situation is then similar to that considered in the case of self-or tracer diffusion, the only difference being that now two different molecular species are involved. Tracer diffusion may be considered, therefore, as equimolar counter diffusion of two identical species. 

Powerful methods for the determination of diffusion coefficients relate to the use of tracers, typically radioactive isotopes. A diffusion profile and/or time dependence of the isotope concentration near a gas/solid, liq-uid/solid, or solid/solid interface, can be analyzed using an appropriate solution of - Fick s laws for given boundary conditions [i-iii]. These methods require, however, complex analytic equipment. Also, the calculation of self-diffusion coefficients from the tracer diffusion coefficients makes it necessary to postulate the so-called correlation factors, accounting for nonrandom migration of isotope particles. The correlation factors are known for a limited number of lattices, whilst their calculation requires exact knowledge on the microscopic diffusion mechanisms. 

If the bulk process is dominating, in the electrical experiment the total conductivity (ionic and electronic) is measured, the second gives information on the tracer diffusion coefficient (D ) which is directly related to the ionic conductivity (or DQ). In the third experiment one measures the chemical diffusion coefficient (D5), which is a measure of the propagation rate of stoichiometric changes (at given chemical gradient) it is evidently a combination of ionic and electronic conductivities and concentrations.3,4,173 175... 

It is instructive to compare the three transport processes (conduction, tracer diffusion and chemical diffusion) by using chemical kinetics and for simplicity concentrating on the electron-rich electron conductor, i.e., referring to the r.h.s. of Fig. 52. The results of applying Eq. (97) are summarized in Table 5 and directly verify the conclusions. Unlike in Section VI.2. ., we now refer more precisely to bimolecular rate equations (according to Eqs. 113-115) nonetheless the pseudo-monomolecular description is still a good approximation, since only one parameter is actually varied. This is also the reason why we can use concentrations for the regular constituents in the case of chemical diffusion. In the case of tracer diffusion this is allowed because of the ideality of distribution. 

Neither the tracer diffusivity nor the self-diffusivity has much practical value except as a means to understand ordinary diffusion and as order-of-magnitude estimates of mutual diffusivities. Darken s equation [Eq. (5-230)] was derived for tracer diffusivities but is often used to relate mutual diffusivities at moderate concentrations as opposed to infinite dilution. 

Fig. 5.24. The existence of a concentration gradient for tracer iont nmHiirpc diffusion of the tracer, i.e., tracer diffusion.

In this study we have measured tracer diffusion coefficients of octadecyltrimethylammonium chloride (CigTAC) micelles in water and aqueous NaCl solutons at 35 °C. Due to its low critical micelle concentration (cmc), the smallest Ka value is as low as 0.18. We have also determined tracer diffusion coefficients for C14TAB micelles at 35 °C. 

Figures 1 and 2 show diffusion coefficients of pyrene solubilized in C igTAC and C14TAB micelles, respectively, at 35 °C. Because essentially all pyrene molecules are solubilized in the micelles, the diffusion coefficients can be interpreted as tracer diffusion coefficients of the micelles. Diffusion coefficients decrease with increasing concentration of the micelles, and increase with increasing concentration of the salts added. Diffusion coefficients at erne s, Dcmc> obtained by extrapolation, and are listed in Table 1...

The significance of cross coefficients, Laa, has been demonstrated also for intrinsic and tracer diffusion of SO2 in surface flow through carbon compacts (i). Cross coefficients must arise from direct interactions between A and A, and should be significant whenever sorbate is sufficiently concentrated for such encounters to become frequent. In the... 

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