Determination of Diffusivities

The preceding analysis provides a powerful method for determining the diffusivities of species that produce an anelastic relaxation, such as the split-dumbbell interstitial point defects. A torsional pendulum can be used to find the frequency, u p, corresponding to the Debye peak. The relaxation time is then calculated using the relation r = 1/ojp, and the diffusivity is obtained from the known relationships among the relaxation time, the jump frequency, and the diffusivity. For the split-dumbbell interstitials, the relaxation time is related to the jump frequency by Eq. 8.63, and the expression for the diffusivity (i.e., D = ra2/12), is derived in Exercise 8.6. Therefore, D = a2/18r. This method has been used to determine the diffusivities of a wide variety of interstitial species, particularly at low temperatures, where the jump frequency is low but still measurable through use of a torsion pendulum. A particularly important example is the determination of the diffusivity of C in b.c.c. Fe, which is taken up in Exercise 8.22. 

Allen and E.L. Thomas. The Structure of Materials. John Wiley Sons, New York, 1999. 

Johnson. Empirical potentials and their use in calculation of energies of point-defects in metals. J. Phys. F, 3(2) 295-321, 1973. 

Schilling. Self-interstitial atoms in metals. J. Nucl. Mats., 69—70(1—2) 465—489, 1978. 

Shewmon. Diffusion in Solids. The Minerals, Metals and Materials Society, War-rendale, PA, 1989. 

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The diffusion of U and Th within a solid is, in general, very slow due to their large size and charge (Van Orman et al. 1998). Even at mantle temperatures, it is expected that a solid will not fully equilibrate with the surrounding phases (fluid, melt or other solid phases) if solid diffusion controls the equilibration. As yet, there have been no direct determinations of diffusion coefficients for any other decay chain element. 

M Southard, L Dias, K Himmelstein, V Stella. Experimental determinations of diffusion coefficients in dilute aqueous solution using the method of hydrodynamic stability. Pharm Res 8 1489-1491, 1991. 

PI Lee. Determination of diffusion coefficients by sorption from a constant, finite volume. In RW Baker, ed. Controlled Release of Bioactive Materials. New York Academic Press, 1980, pp 255-265. 

Condition 1 Direct determination of diffusivities in ZSM-5 via sorption rate measurements showed Dp/DQ > 103 and and DQ Dm. 

Application of FCS to the determination of diffusion coefficients of atoms in the gas phase has also been proposed. 

This method is commonly used to obtain the diffusion coefficient through porous membranes. The schematic diagram illustrating the best technique for application of the time-lag method for determination of diffusion transport is shown in Fig. 4. As in the test setup shown in Fig. 4 a, the soil is contained between the source and collection reservoirs. Using this technique for diffusion coefficient determination of pollutants requires that the following conditions are satisfied ... 

Fig. 4a - c. Schematic diagram illustrating the time-lag method for determination of diffusion transport of organic pollutants, as follows a column setup b pollutant concentration vs time in source and collection reservoirs during the test c 2 amount of pollutants (i.e., Qt) transported through solid particles with time after the test... 

American Welding Society, Standard Methods for Determination of Diffusible Hydrogen Content , ANSI/AWS, A4.3-93, AWS, Miami, FL, (1993). 

For experimental determination of diffusion coefficients, a large database is already available. Nonetheless, data for specific applications are often difficult to find because the data may not cover the right temperature range, mineral compositions, or fluid conditions. In geospeedometry applications, data often must be extrapolated to much lower temperatures and the accuracy of such extrapolation is difficult to assess. Because the timescale of geological processes is often in the order of Myr, and that of experiments is at most years, instrumental methods to measure very short profile are the key for the determination of diffusion coefficients that are applicable to geologic problems. 

This section describes the experimental methods and focuses on the estimation of diffusivity after the experiment. The analytical methods are not described here. Estimation of diffusivity from homogeneous reaction kinetics (e.g., Ganguly and Tazzoli, 1994) is discussed in Chapter 2 and will not be covered here. Determination of diffusion coefficients is one kind of inverse problems in diffusion. This kind of inverse problem is relatively straightforward on the basis of solutions to forward diffusion problems. The second kind of inverse problem, inferring thermal history in thermochronology and geospeedometry, is discussed in Chapter 5. 

The results above have the following applications (i) estimation of diffusive crystal dissolution distance for given crystal and melt compositions, temperature, pressure, and duration if diffusivities are known and surface concentrations can be estimated and (ii) determination of diffusivity (EBDC) and interface-melt concentrations. Those diffusivities and interface concentrations can be applied to estimate crystal dissolution rates in nature. 

Equation (4.70) is a starting point in the determination of diffusivities in liquid metal alloys, but in most real systems, experimental values are difficult to obtain to confirm theoretical expressions, and pair potentials and molecular interactions that exist in liquid alloys are not sufficiently quantified. Even semiempirical approaches do not fare well when applied to liquid alloy systems. There have been some attempts to correlate diffusivities with thermodynamic quantities such as partial molar enthalpy and free energy of solution, but their application has been limited to only a few systems. 

For the determination of diffusion coefficient of solutes (except from salts and ions) in water and dilute solutions (<10%) the Hayduk and Laudie equation is used (Lyman et al., 1990 Perry and Green, 1999) ... 

Equation (208) can be used for the determination of diffusion coefficients by writing it in the form... 

The final section (Section 5.8) introduces dynamic light scattering with a particular focus on determination of diffusion coefficients (self-diffusion as well as mutual diffusion), particle size (using the Stokes-Einstein equation for the diffusion coefficient), and size distribution. 

Dynamical study of the phase transition of the gels in spinodal regimes was described. The evolution of intensity of light scattered from the gels indicated the applicability of Cahn s linearized theory to the phase transition. Our work offers a basis for the determination of diffusion coefficient of gels in their spinodal regimes. 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

R.W. Balluffi. On the determination of diffusion coefficients in chemical diffusion. Acta Metall., 8(12) 871-873, 1960. 

As can be inferred from Fig. 4.9, A/ pyjf = 1 when both diffusion coefficients are equal (Do = Dr) (see Eq. (4.86)). When Do > Dr, A/(() )py)pk > 1, whereas when Do < Dr we find that A/(p)pyepk < 1. Hence, this ratio is sensitive to the y value and, therefore, is very useful for the determination of diffusion coefficients (in the optimum range 0.5 < rs j /DmaxT2 < 3.5, with Dmax being the largest diffusion coefficient of the redox pair). 

In this technique, the analysis of the current-time responses corresponding to the application of a sequence of potential pulses without the reestablishment of the equilibrium between them is carried out. The usefulness of this technique lies mainly in the determination of diffusion coefficients of the electroactive species when only two potentials are applied, as discussed in Sect. 4.2.2. Nevertheless, there are also other analytical applications, which are presented in this section. 

Microelectrodes also are used for the determination of diffusion coefficients. In these experiments a pulsed waveform usually is used with the current measured at a single potential (at which the process is controlled only by diffusion). Under these conditions, the current obeys the relation... 

Zuleta, M., Bjornbom, P., Lundblad, A., Nurk, G., Kasuk, H., and Lust, E. Determination of diffusion coefficients of BF4 inside carbon nanopores using the single particle microelectrode technique. J. Electroanal. Chem. 586, 2006 247-259. 

Determination of Diffuse View Factor by Contoui Integration... 

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