Darken’s equation

Explain why one does not bother with Darken s equation when considering problems of diffusion of carbon in iron. 

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. 

The above equations are written for component A. Similar equations apply for component B. The concentration dependence of pore diflusivity followed Darken s equation. However, the concentration dependence of the barrier coefficient was much stronger than that expected from the analog of Darken s equation, which following other studies in this... 

The vaiue of the diffusivity is usually dependent on the amount adsorbed at lower temperatures than about 373 K, because of the large amount adsorbed [12,13]. However, when the temperature is higher than 373 K, the amount adsorbed is small, resulting in a small interaction among adsorbate molecules. Therefore, the relationship between the adsorbed amount (M ) and the equilibrium pressure (P ) was found to be linear under the conditions of pressures lower than 1.33 kPa and temperatures higher than 373 K, as typically shown in Fig.l. Hence, the diffusivities (D) are independent of the adsorbed amount and equal to the "self-diffusivities" (D ) defined by Darken s equation (Eq.[2j) [14]. 

From the viewpoint of diffusion conductance, the NG and Darken equations describe extreme, marginal cases. Darken s equation corresponds to parallel connection when the interdiffusion rate is usually determined by the more mobile component. The NG equation conforms to series (consecutive) connection at which the interdiffusion rate is mainly determined by the less mobile component - the more mobile one has to wait until slower atoms accomplish their migration. The discrepancy between equations becomes most apparent whenever the components mobility ratio is much larger or much less than one. In the general case, we may expect the correct description of interdiffusion to correspond to a certain combination of parallel and series (consecutive) connection, depending on vacancies sinks/sources effectiveness. We will see that the NG and Darken equations conform to different spatial and time scales. 

Equation (4.78) is named a Darken-type equation because it was first derived by Darken for a special situation [L. S. Darken (1948)]. 

Let us present D explicitly for the condition d//0 = 0, omitting all details of the lengthy derivation. By application of Manning s random-alloy model [A. R. Allnatt, A.B. Lidiard (1987)], and by inserting Eqns. (5.126) and (5.131) into Eqn. (5.132), for a constant oxygen potential across the diffusion zone, a Darken type equation is obtained... 

A number of other models have been used in conjunction with Equation 7 to calculate the binary phase diagrams. Among these are the quasichemical equation (35,44), a truncated Margules equation (45,46), Darken s formalism (47,48), and various forms of the chemical theory, in which associated liquid species are postulated and some assumptions are made about the physical interactions between the species (49-51). Several of these studies have considered the liquid phase properties as well as the liquidus in the parameter estimation (45,46,51). 

HARTMANN In addition to what you mentioned about chemical diffusion in silver-sulfide, we extended our measurements to a symmetrical cell with silver/silver iodide and two Pt-probes on each side of a long sample of Ag2S or Ag2Se which allowed us to establish a potential on each side and measure the EMF on each side independently from a flow of current. The relaxation of a silver concentration gradient recorded by EMF was used to measure D as a function of deviation from ideal stoichiometry. For Ag2+5S at 200 C the values of D are about 0.08 cm sec l at equilibrium with silver and 0.25 cm sec l near ideal stoichiometry. The consistency of the measurement is shown with the good agreement of the measured S values with those calculated from Darken-Wagner equation. D < is obtained from conductivity data and the thermodynamic factor calculated from the slope of the electrochemical titration curve. 

The process of solving the variation problem (Equations 12.46 and 12.47) will be performed according to the algorithm given in Equation 12.2. For the calculations, we use a set of parameters presented in Equation 12.2. To calculate the triple product sDh, we apply the expression for interdiffusion coefficient in Darken s approximations ... 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

In formulating Eqn. (5.101) and the following flux equations we tacitly assumed that they suffer no restrictions and so lead to the individual chemical diffusion coefficients >(/). If we wish to write equivalent, equations for,/(A) and/(B), and allow that v(A) = = v(B), then according to Eqn. (5.103), /(A) /(B) since Ve(A) = ]Vc(B)j. However, the conservation of lattice sites requires that j/(A) j = /(B), which contradicts the previous statement. We conclude that in addition to the coupling of the individual fluxes, defect fluxes and point defect relaxation must not only also be considered but are the key problems in the context of chemical diffusion. Let us therefore reconsider in more detail the Kirkendall effect which was introduced qualitatively in Section 5.3.1. It was already mentioned that this effect played a prominent role in understanding diffusion in crystals [A. Smigelskas, E. Kirkendall (1947) L.S. Darken (1948)]. 

The solutions to Fick s second law (Equations (9.5)-(9.7)) are based on a single diffusivity, D, whereas the Kirkendall experiments show that each species has its own diffusivity. Darken showed that Fick s second law should be written as... 

Rathbun and Babb [20] suggested that Darkens equation could be improved by raising the thermodynamic correction factor PA to a power, n, less than unity. They looked at systems exhibiting negative deviations from Raoult s law and found n = 0.3. Furthermore, for polar-nonpolar mixtures, they found n = 0.6. In a separate study, Siddiqi and Lucas [22] followed those suggestions and found an average absolute error of 3.3 percent for nonpolar-nonpolar mixtures, 11.0 percent for polar-nonpolar mixtures, and 14.6 percent for polar-polar mixtures. Siddiqi, Krahn, and Lucas (ibid.) examined a few other mixtures and... 

The diffusivity measured by the FR technique, D, is a transport diffu-sivity which has to be corrected, by using the Darken Equation (Eq. 26), to obtain the so-called corrected diffusion coefficient where the diffusion is measured at an equilibriiun pressure, Pe, which is outside the Henry s law range. This corrected diffusivity is generally taken to be the equivalent of the self-diffusion coefficient Dq ... 

A slightly modified form of expression was obtained by Ash and Barrer who used a somewhat different definition of the transport diffusivity. If the cross coefficient can be neglected (I. wO), Eq. (5.9) reduces to Eq. (5.6) with Dq = S), which is the familiar Darken equation/ originally derived for the interdiffusion of two alloys. While Eq. (5.6), being essentially a definition of Dq, is always valid it is evident that the assumption that Dq- is only true in the limiting case where In general both Z)q and are... 

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