Tracer diffusion compared

The advantage of the simulations compared to the experiments is that the correspondence between the tracer diffusion coefficient and the internal states of the chains can be investigated without additional assumptions. In order to perform a more complete analysis of the data one has to look at the quench-rate and chain-length dependence of the glass transition temperature for a given density [43]. A detailed discussion of these effects is far beyond the scope of this review. Here we just want to discuss a characteristic quantity which one can analyze in this context. 

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. 

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 ... 

For B = A, the tracer diffusion coefficient equals the self-diffusion coefficient, D lba= Dla- 1° Table 6-6 the self-diffusion coefficient of water and some diffusion coefficients of organic solutes in water at infinite dilution calculated with Eq. (6-31) are compared with experimental values (Reid et al., 1987). The experimental value for sucrose is from Cussler (1997). 

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. 

The Nernst-Einstein reiation can be tested by using the experimentally determined tracer-diffusion coefficients D,. to calcuiate the equivalent conductivity A and then comparing this theoreticai vaiue with the experimentally observed A. It is found that the vaiues of A caicuiated by Eq. (5.61) are distinctly greater (by 10 to 50%) than the measured values (see Table 5.27 and Fig. 5.33). Thus there are deviations from the Nernst-Einstein equation and this is strange because its deduction is phenomenological. ... 

In order for us to have flexibility in our modeling of natural water chemistry we need a way to obtain individual ion activity coefficients from mean values. To do so requires that we make an assumption, called the Macinnes convention (Macinnes 1919), which states = 7c - The convention is based on the observation that and Cr ions are of the same charge and nearly the same size, have similar electron structures (inert gas), and similar ionic mobilities. In support of this assumption, tracer diffusion coefficients, D°, of K+ and Cl" at infinite dilution are nearly equal at 19.6 and 20.3 X 10" cmVs (Lerman 1979). Also, limiting equivalent conductances, A°, of and Cl" are comparable at 73.50 and 76.35 cmV(ohm) (equiv.) at 25°C (Robinson and Stokes 1970),... 

Membrane Diffusion in Dilute Solution Environments. The measurement of ionic diffusion coefficients provides useful information about the nature of transport processes in polymer membranes. Using a radioactive tracer, diffusion of an ionic species can be measured while the membrane is in equilibrium with the external solution. This enables the determination of a selfdiffusion coefficient for a polymer phase of uniform composition with no gradients in ion or water sorption. In addition, selfdiffusion coefficients are more straightforward in their interpretation compared to those of electrolyte flux experiments, where cation and anion transport rates are coupled. 

Fig. 9 Arrhenius-plot of the parabolic rate constant measured for the growth of CoO on Co in air [91] compared with that calculated from Wagners theory and the tracer diffusion coefficient for Co in CoO [89, 90].

However, if we can assiune that the non-diffusive relaxation of the less mobile polymers is fast compared with the diffusion of the more mobile chains, we can derive a simple equation relating the mutual diffusion coefficient to the tracer diffusion coefficient and a thermodynamic factor. This equation has been applied to liquid systems, in which context it is known as the Hartley-Crank equation (Tyrell and Harris 1984), and to metals, in which context it is known as the Darken equation (Haasen 1984). The polymer version was first stated by Kramer (Kramer et al. 1984) and is often known as the fast theory . It is most compactly written in the form... 

This equation does have some convincing experimental support for the case of polymers of high relative molecular mass. Figure 4.24 shows the results of an experiment in which both tracer diffusion coefficients were measured as a function of composition (Composto et al. 1988). Equation (4.4.11) was then used to calculate the mutual diffusion coefficient, given a value of x that had been measured independently. These calculated mutual diffusion coefficients were compared with the directly measured values and good agreement was found. 

A more extreme situation occurs when solvent diffuses into a glassy polymer. Here the tracer diffusion coefficient of the polymer is negligible compared with the tracer diffusion coefficient of the solvent. In addition the rate of non-diffusive relaxation of the polymer is also slow compared with the solvent diffusion, as well as being highly composition dependent, and it is this relaxation that is the rate-limiting step. The diffusion equation must now be modified by a term that expresses the mechanical response of the glassy polymer to the build-up of osmotic pressure caused by the ingress of the solvent (Thomas and Windle 1982). 

In order to obtain correlation factors experimentally, one may for example compare the component diffusion coefficient which has been measured in flux experiments (when a gradient of the electrochemical potential was acting upon the diffusing particles) with the tracer diffusion coefficient. One also may use the so-called isotope effect Since two isotopes of the same element in a lattice have different vibrational frequencies and accordingly different jump frequencies, their correlation effects and their diffusivities differ also. One can show that the relative change in of two isotopes is proportional to the correlation factor /. The factor of proportionality, in addition, can be evaluated independently if the isotope masses and the kinetic energies of the vibrational modes around the defect are known [5]. 

Values of the correlation coefficient may be determined by comparing the measured values of the ionic conductivity and the tracer diffusion coefficient. Thus the use of the Nernst-Einstein relation gives the following expression for the correlation coefficient ... 

The CTRW methodology described in Sect. 2.1 was used to compare the tracer diffusion coefficient Dtracer in the presence of electron trapping for different porosities and nanoparticle sizes and for trap locations on the surface or extended throughout the nanoparticles, as illustrated in Fig. 8 [37]. >tracer describes the diffusion of a single electron and is obtained from the mean-squared electron displacement using... 

An exemplary early study of tracer diffusion is shown in Figure 8.16, based on measurements by Leger, et al, who examined linear polystyrene benzene with FRS, comparing self-diffusion of 245 and 599 kDa chains with tracer diffusion... 

Kops-Werkhoven, et al. measured tracer diffusion of dilute 38 nm radius silica spheres diffusing through suspensions of 33 nm silica spheres, all in cyclo-heptane(12). The experimental method was dynamic light scattering particle motions were observed over distances large compared to their radius. They found ku =-2.7 0.3. 

The case of two-sided finite initial boundary conditions is discussed more precisely below. If we apply a diffusion source on one side of a very thin, long diffusion channel (corresponding to one-sided finite or semiinfinite boundary conditions), Eq. (6.68) applies once again (comparatively small times), provided So is replaced by double the value. Figure 6.24 shows a tracer diffusion example. 

Figure 6.23 illustrates a variety of situations . We will meet further cases, in Chapter 7, in which the fluxes and hence the gradients are kept constant at the boundary. Naturally the one-dimensional solutions are not restricted to one-dimensional systems (cf. Fig. 6.22). In a suitable pseudo one-dimensional experiment is the application of a diffusion source in the form of a thin strip (e.g. gold) onto a thin film (e.g. a thin sheet of silver) (chemical diffusion). It can also be a thin strip of the same but now radioactive material (different isotope), or the exposure of a slit-shaped opening of the otherwise sealed thin oxide film to a (radioactive or chemically modified) gas atmosphere (tracer diffusion). The analogue in D is the sandwich technique or the planar application of the diffusion source onto the surface (as already considered in Fig. 6.24). If the diffusion source is a gas phase, again sealing is necessary unless the aspect ratio is very favourable, i.e. if the extension is sufficiently small in the direction of diffusion compared with the other directions in space . Otherwise the three-dimensional solution has to be considered.

The increased (differential) capacitance is finally also largely the reason for the diminution of the diffusion coefficient, which takes place when we go from the case of chemical diffusion without trapping to chemical diffusion with trapping it is also the decisive factor when we compare defect diffusion (e.g. chemical diffusion in the case of the electron-rich electron conductor (D = Dy /o )) tracer diffusion (see Fig. 6.19). The situation is similar for the k values. As for electrical effects the capacitive components in the chemical and tracer experiment are responsible for the time behaviour. 

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