Tracer diffusion methods

Unless a relation between / and fs is established, giadient diffusion on a polymer + solvent binary solution is of no use for measurement of fg. However, if applied to a labeled polymer dispersed very dilutely in an unlabeled polymer + solvent solution, it allows us to determine fg. The experiment may be done as follows. We prepare a pair of solutions containing an unlabeled polymer at a desired concentration, disperse in one of them a labeled species very dilutely, allow the two solutions to contact at a sharp boundary in a diffusion cell, and then measure the resulting change in the concentration distribution of the labeled species as a function of time by an optical method. The data obtained gives D of the labeled species. The value of / calculated from this D (in this case, the correction for the thermodynamic factor is not needed because the concentration of the labeled species is very low) may be equated to fg of the labeled species, because / and fg agree at infinite dilution. This idea is the basis of the tracer diffusion method for self-diffusion. 

The 1 0 tracer diffusion method was used to investigate O diffusion in reactively dc-sputtered Ir02 films. The profile measurements were performed using secondary ion... 

Nuclear magnetic resonance (NMR) spectroscopy is an especially useful molecular probe of water properties." " The self-diffusion coefficient of water Dg has been measured, using a spin-echo technique, over a considerable range of temperatures and densities. This method complements the more direct tracer diffusion method. With one interesting exception, Dg increases with increasing temperature and decreases with increasing pressure. The exception occurs for temperatures in the 275-323 K range and for pressures up to about... 

The most straightforward method of obtaining the mobility is the tracer diffusion method, whereby a small quantity of a radioactive isotope of the relevant ion is introduced, and its diffusion... 

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... 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

In the data compiled by Janz and Bansal, various methods for measuring diffusion coefficients in molten salts are mentioned. The methods may be broadly classified as electrochemical and analytical. However, some other methods have occasionally been employed. Various electrochemical methods were reviewed by Lesko. Tracer diffusion in molten salts was reviewed by Spedding in 1971, where some other methods were also mentioned. 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

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... 

Tracer diffusivities are often determined using the thin-source method. Self-diffusivities are often obtained from the diffusion couple and the sorption methods. Chemical diffusivities (including interdiffusivity, effective binary diffusivity, and multicomponent diffusivity matrix) may be obtained from the diffusion-couple, sorption, desorption, or crystal dissolution method. 

The problem of vacancy-mediated tracer diffusion in two dimensions has been investigated for a long time [40-44] and several different methods (simulation, analytical models, enumeration of trajectories, etc.) can be used to address it. The mathematics of this type of diffusion was solved first for the simplest case [41], when the diffusion of the vacancy is unbiased (all diffusion barriers are equal the tracer atom is identical to the other atoms), the lattice is two-dimensional and infinite. There is a single vacancy present that makes a nearest-neighbor move in a random direction at regular time intervals and has an infinite lifetime, as there are no traps. The solution is constructed by separating the motion of the tracer and that of the vacancy. The correlation between the moves of the tracer atom is calculated from the probability that the vacancy returns to the tracer from a direction, which is equal, perpendicular or opposite to its previous departure. The probability density distribution of the tracer atom spreads with... 

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. 

By use of the Taylor dispersion method, diffusion coefficients for pyrene solubilized in micelles of octadecyltrimethylammonium chloride (CigTAC) and tetradecyltrimethyl-ammonium bromide (C14TAB) have been measur in aqueous NaCl and NaBr solutions, respectively, at 35 °C. These values can be regarded as tracer diffusion coefficients for the micelles b use essentially all pyrene molecules are solubilized in the micelles. In the range... 

Zelsmann and co-workers [88] have reported tracer diffusion coefficients for water in Nafion membranes exposed to water vapor of controlled activity. These were determined by various techniques, including isotopic exchange across the membrane. They reported apparent self-diffiision coefficients of water much lower than those determined by Zawodzinski et al. [64], with a weaker dependence on water content, varying from 0.5 x 10 cm to 3 x 10 cm /s as the relative humidity is varied from 20 to 100%. It is likely that a different measurement method generates these large differences. In the experiments of Zelsmaim et al., water must permeate into and through the membrane from vapor phase on one side to vapor phase on the other. Since the membrane surface in contact with water vapor is extremely hydrophobic (see Table 7), there is apparently a surface barrier to water uptake from the vapor which dominates the overall rate of water transport in this type of experiment. 

For ionic solids, measurement of the ionic conductivity, <7 , has long provided a method for studying their atomic diffusion [25, 209, 225, 226] (see also Chapter 3). The measurements are usually made with an alternating current (AC) bridge operating at a fixed frequency, f (typically >1 kHz), to avoid polarization effects. The early studies were restricted to measurements on single crystals, and in this case (7i and the tracer diffusion coefficient were seen to be related by the Nernst-Einstein equation [25] ... 

H concentration gradients) and the tracer diffusion coefficient D (as determined by microscopic methods at equilibrium conditions) is given by [20]... 

Attempts to obtain transport number information by various methods such as pulsed field gradient NMR [62], radio tracer diffusion [77], and potentiostat-ic polarization technique [46] have suggested that both cation and anion mobilities are important for the total ionic conductivity seen. In general, however, the nature of charge carriers in polymer electrolytes is quite complex and ion aggregates such as triple ions have been implicated in conductivity [78-79]. 

In many cases, the NMR method of measuring D is superior to tracer diffusion studies. For liquids, at least, the diffusion coefficient of the tracer does not correspond to that of the major chemical species except for very small tracer concentrations (Ahn, et al., 1972). Furthermore, the NMR techniques do not contaminate the sample, the measured value of D is not disturbed by isotope effects, and the method is not limited to the availability of suitable isotopes, although it is, of course, limited to suitable NMR nuclei and the existence of Hahn echoes. 

Figure 3 Sell-diffusion coefficients of benzene in ZSM-5 determined by the NMR tracer exchange method at 293 K ( ) and 386 K (A), plus comparison with the results of uptake measurements at 303 K (0)< 363 K ( ), and 393 K (A), transformed into self-diffusivities via Eq. (5). (From Refs. 6 and 11.)...

Many of the macroscopic techniques can be apphed to the measurement of self-diffusion by using isotopically labeled tracers. Such methods, first introduced by Barrer and Fender [97], have been widely applied in order to obtain data which should be directly comparable with microscopic self-diffusion measurements. Such comparisons are presented in several of the chapters within the present volume. 

In tracer ZLC (TZLC) [28,51,58] the experiment is similar to the standard method, but the monitored species is the deuterated form of the sorbate. This introduces an additional cost for the material and the requirement for an online mass spectrometer. The advantages are the eUmination of all possible heat effects, strict Unearity of the equiUbrium between the fluid phase and the adsorbed phase, and the possibility of measuring directly the tracer diffusivities (which shoifld be the same as the microscopically measured self-diffusivity) over a wide range of loading. To reduce the costs the carrier is prepared with a mixture of pure and deuterated hydrocarbons. It has been shown that small imbalances in the concentration of the carrier and the purge streams do not affect the desorption dynamics [58]. 

Figure 5. Tracer diffusion in cobaltous oxide as a function of oxygen pressure [and hence Co/0 ratio given by Equations 9 and 13]. The symbols (X) denote data obtained by a sectioning technique while (0) denote data by the surface decrease method. The slopes of the lines are approximately one-fourth, indicating the existence of singly ionized cation vacancies (6).

Thin-layer methods are almost exclusively used for self-diffusion and tracer diffusion coefficient measurements a very thin layer of element is deposited on the surface (mass g per unit area), and the concentration distribution after diffusion is measured, whence... 

Profile evolution techniques measitre only diffusion driven by chemical gradients (as opposed to tracer diffusion) for heterodiffirsion, but in suitable profile geometries can do so very directly with a minimitm of complicated modeling. The utility of a given method is hmited by (a) the variety of adsorbates it can monitor without major siuface perturbation, (b) the spatial resolution it can attain (including initial profile formation), and (c) the suitability of the initial profile geometry for qrrantitative analysis. 

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