Diffusion Darken’s equation

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. 

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

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. 

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... 

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... 

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. 

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 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... 

Pick s first law is useful for solving simple problems in which the concentration is known and the instantaneous flux is required, but most problems in diffusion require solving the time-dependent partial differential equation known as Pick s second law (as modified by Darken), i.e.. 

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