Calculation of diffusion coefficients

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

The calculation of diffusion coefficient spectra using modified Twomey method (1975) are based on these premises and assumptions ... 

Key material properties for SOFC, such as the ionic conductivity as a function of temperature, are available in refs 36—39. In addition, Todd and Young ° compiled extensive data and presented estimation methods for the calculation of diffusion coefficients, thermal conductivities, and viscosities for both pure components and mixtures of a wide variety of gases commonly encountered in SOFCs. Another excellent source of transport properties for gases and mixtures involved in a SOFC is the CHEMKIN thermodynamic database. ... 

This relation has been used for the calculation of diffusion coefficients by the rate of evaporation of liquid droplets suspended in a still gas (S9, p. 17). Clearly Eq. (122) does not describe the situation properly if the diffusion produces free convection owing to changes in density. 

Oelkers, E. H., Calculation of diffusion coefficients for aqueous organic species at temperatures from 0 to 350°C , Geochimica et Cosmochimica Acta, 55, 3515-3529 (1991). 

The cyclic diluents can also be sorbed into the networks after the end-linking process [191] or they can be present during the process [193-195]. In the latter case, some are permanently trapped as will be described below, making difficult the calculation of diffusion coefficients from the extraction data. In the former case, however, D is readily calculable. For both the cyclic and linear chains, D was found to decrease with increase in Afd, and with decrease in Mc, as expected. The cyclics were found to have values of D larger than those for the linear chains, presumably because their greater compactness facilitates their transport through the network structure. 

Diffusion Model. Assuming that pore diffusion is rate limiting, a diffusion model based on Fick s second law can be utilized for calculation of diffusion coefficients from the experimental data. The model must... 

The sorption coefficient (K) in Equation (2.84) is the term linking the concentration of a component in the fluid phase with its concentration in the membrane polymer phase. Because sorption is an equilibrium term, conventional thermodynamics can be used to calculate solubilities of gases in polymers to within a factor of two or three. However, diffusion coefficients (D) are kinetic terms that reflect the effect of the surrounding environment on the molecular motion of permeating components. Calculation of diffusion coefficients in liquids and gases is possible, but calculation of diffusion coefficients in polymers is much more difficult. In the long term, the best hope for accurate predictions of diffusion in polymers is the molecular dynamics calculations described in an earlier section. However, this technique is still under development and is currently limited to calculations of the diffusion of small gas molecules in amorphous polymers the... 

Equation (11-2) offers a convenient expression for calculating the diffusion coefficient for various compounds and mixtures, but it should not be used as a substitute for experimental values of the diffusion coefficient when they are available for a particular system. References 3 and 5 to 9 present more information on calculation of diffusion coefficients. An abbreviated table of diffusion coefficients is given in Appendix A. 

In addition to elucidation of molecular structures, NMR can also extract valuable information about physicochemical parameters. Because of the omnipresence of protonated solvents in CE/CEC, mobile-phase events can be monitored with NMR. Early studies using E-NMR involved the calculation of diffusion coefficients, electrophoretic mobilities, and viscosity [27]. Stagnant mobile-phase mass transfer kinetics and diffusion effects [60] and fluid mass transfer resistance in porous media-related chromatographic stationary phases [61] have been studied with NMR spectroscopy. NMR imaging of the chromatographic process [62] and NMR microscopy of chromatographic columns [63] have also been reported. Several applications of NMR to on-line studies of CE/ and CEC/ NMR are highlighted. 

Use a Mathcad routine to implement calculation of diffusion coefficients for binary gas systems using the Wilke-Lee equation. 

Ritthichauy W Sugiyama T., Tsuji Y. (2002) Calculation of diffusion coefficient of ion in multicomponent solution for ion movement in concrete, 2002. Proceedings of Japan Concrete Institute, Tsukuba. 19-21 July 2002. Tokyo Japan Concrete Institute, p. 669-674. 

A new method for the calculation of diffusion coefficients from Q-TG data has been presented. The programmed thermodesorption of polar and nonpolar liquids from aluminium oxide and montmorillonite-Na and -La samples under quasi-isothermal conditions has been studied. The results from above methods were compared with literature data and good correlation was obtained. 

In the calculation of diffusion coefficient the Emst-Planck correction for the reduced mobility of ions should be taken into account. 

CALCULATION OF DIFFUSION COEFFICIENTS IN LIQUIDS. RAVDELAA PORAI-KOSHITS A B ZH PRIKLADKHIM... 

Nordstrom, D.K. (2012). Models, validation, and applied geochemistry Issues in science, communication, and philosophy. Applied Geochemistry, 27, 1899-1919. Oelkers, E.H. (1991). Calculation of diffusion coefficients for aqueous organic species at temperatures from 0 to 350°C. Geochimica et Cosmochimica Acta, 55, 3515-3529. Oelkers, E.H., Helgeson, H.C. (1988). Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures Aqueous tracer diffusion coefficients of ions to 1000°C and 5 kb. Geochimica et Cosmochimica Acta, 52, 63-85. 

At very long times, the solute s mean-square displacement is proportional to the time t. This proportionality is commonly used to determine the diffusion coefficients by fitting a straight line, either in an vs t plot or a log vs log t graph. At short times, however, the mean-square displacement may obey a different power law, oc t" with n < 1. In most of the calculations of diffusion coefficients by MD it has tacitly been assumed that the long-time limit has been reached in the calculation. In most cases in the literature only the diffusion coefficients have been reported and not the mean-square displacement curves upon closer inspection of the displacements, where reported [27,28,60, 65,66], one often notes that the mean-square displacement is a wobbly line that can be interpreted as straight only if the nonlinearity is attributed to statistical uncertainties for insufficient simulation lengths. 

The calculation of diffusion coefficients from equations based on some models describing the movement of matter in electrolyte solutions, in the end, a process contributing to the knowledge of their stmcture, provided we have accurate experimental data to test these equations. Thus, to understand the behavior of transport process of these aqueous systems, experimental mutual diffusion coefficients have been compared with those estimated using several equations, resulting from different models. 

Calculation of diffusion coefficient from size and shape of solute molecules and viscosity of solvent, by classical mechanics... 

These are noteworthy relations. They express neatly the way in which the diffusion coefficient (to which diffusion rates are proportional) depends on a balance between thermal energy, represented by kT, and frictional resistance to motion, represented by /. They allow the calculation of diffusion coefficients by classical hydrodynamic methods. Such values of D are important when, as often, experimental values are not available and when they are available, comparison with experiment permits tests of the assumptions of the theoretical models. 

Calculations of diffusion coefficients from hydrodynamic theory... 

An interesting report in this area concerns the kinetics of dimerization of Ag(CO)3 to Aga(CO>4 in a solid CO matrix between 30 and 37 K. Not surprisingly the rates are diffusion controlled and from calculations of diffusion coefficients at various temperatures an Arrhenius plot gave Ea= 1.86 0.3 kcal mol for the reaction. If it is assumed that Ag(CO)3 dimerizes by moving through the matrix via a CO vacancy mechanism the activation energy will be similar to that of self-diffusion in CO. A calculated value of a=2.05 kcal mol" for such a process can therefore be cited as reasonable evidence for such a mechanism. 

In liquid systems, equations exist that enable the calculation of diffusion coefficients via the knowledge of the course of dynamic viscosity. As an example. Me Manamay and Woolen stated for the diffiisivity of CO2 in some organic liquids [48] ... 

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