Diffusion, chemical

The most important diffusion coefficient for chemistry and materiab science is the chemical diffusion coefficient which characterizes the diffusion kinetics of composition changes. This is formally a diffusion of neutral components, and, for ionic compounds, a charge-neutral ambipolar diffusion of at least two chemically different charged particles . A relevant example is the cheinge in stoichiometry of the oxide M2O (Fig. 6.17c) in the sense of 

Application of our general transport equation permits us to quantify the relationships. For the separate ionic and electronic flux densities we have  

Elimination of the electrical potential from Eq. (6.49) and taking account of Eq. (6.50) leads to 

In our case jo is naturally identical to the flux density of the neutral component 0 (Jo). Evidently the harmonically averaged conductivity expression in Eq. (6.51) corresponds to an effective, ambipolar conductivity Oq, expressing the fact that both ionic and electronic charge carriers are necessary and the respective resistors so-to-speak connected in series . The expression in brackets obviously represents the chemical potential gradient of the component O fio = IMy/ + 2y h ) The result is a force-flux relationship of the expected form, viz. 

The expression in brackets is — in one dimension — equivalent to — jo/( co/9x) and, on account of conservation of mass and charge, identical to —]o -/ dc yi-/dx) as well as to —je-/(5ce-/9x). It is hence precisely the chemical diffusion coefficient we have been seeking  

Let us now consider the equalization of the component concentrations in an inhomogeneous multicomponent system. We may start with Eqn. (4.33) which relates the component fluxes, jk, to the (n-1) independent forces, Vyq, of the n-compo-nent solid solution. In local equilibrium, the chemical potentials are functions of state. Hence, at any given P and T 

We now introduce the thermodynamic factors (fjm) in accordance with Eqn. (4.46) and define 

We can rewrite Vpt accordingly and obtain an analogous expression for the flux jk as Eqn. (4.33), 

In order to solve this set of (coupled) differential equations, we have to formulate the proper boundary conditions. Let us define the conditions of the simplest (onedimensional) interdiffusion experiment as follows 

With these boundary conditions, the solution can be expressed in terms of one single variable (= /]U). Let us write Eqn. (4.57) in matrix form 

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The rate of chemical diffusion in a nonfiowing medium can be predicted. This is usually done with an equation, derived from the diffusion equation, that incorporates an empirical correction parameter. These correction factors are often based on molar volume. Molecular dynamics simulations can also be used. 

Fig. 15. Light scatter and chemical diffusion lead to a loss of sharpness at (a) a photographic edge where (—) represents the image and (-) the developed...

At any plane in a Raoultian alloy system parallel to die original interface, the so-called chemical diffusion coefficient Dchem. which determines the flux of atoms at any given point, and is usually a function of the local composition so that according to Darken (1948), Dchem is given by... 

The chemical diffusion coefficient at any concentration C in the experimental diffusion profile is dren given by... 

This analysis makes possible the determination of a chemical diffusion coefficient from experimental data having made no use of a model, and which takes no account of tire atomic mechanism of diffusion, and assumes tlrat the same chemical diffusion coefficient applies to each component of the alloy. 

By way of example, Volume 26 in Group III (Crystal and Solid State Physics) is devoted to Diffusion in Solid Metals and Alloys, this volume has an editor and 14 contributors. Their task was not only to gather numerical data on such matters as self- and chemical diffusivities, pressure dependence of diffusivities, diffusion along dislocations, surface diffusion, but also to exercise their professional judgment as to the reliability of the various numerical values available. The whole volume of about 750 pages is introduced by a chapter describing diffusion mechanisms and methods of measuring diffusivities this kind of introduction is a special feature of Landolt-Bornstein . Subsequent developments in diffusion data can then be found in a specialised journal. Defect and Diffusion Forum, which is not connected with Landolt-Bdrnstein. 

M. Tammaro, J. W. Evans. Reactive removal of unstable mixed NO -I- CO adlayers Chemical diffusion and reaction front propagation. J Chem Phys 705 7795-7806, 1998. 

At this point it should be clear that this representation (Eq. (80)) is not restricted to thermal diffusion, but may equally well be formulated in a completely analogous way for the case of chemical diffusion. 

If a liquid system containing at least two components is not in thermodynamic equilibrium due to concentration inhomogenities, transport of matter occurs. This process is called mutual diffusion. Other synonyms are chemical diffusion, interdiffusion, transport diffusion, and, in the case of systems with two components, binary diffusion. 

The primary question is the rate at which the mobile guest species can be added to, or deleted from, the host microstructure. In many situations the critical problem is the transport within a particular phase under the influence of gradients in chemical composition, rather than kinetic phenomena at the electrolyte/electrode interface. In this case, the governing parameter is the chemical diffusion coefficient of the mobile species, which relates to transport in a chemical concentration gradient. 

More importantly, the chemical diffusion coefficient >chcm, instead of Dself, is the parameter that is relevant to the behavior of electrode materials. They are related by... 

It is thus much better to measure the chemical diffusion coefficient directly. Descriptions of electrochemical methods for doing this, as well as the relevant theoretical background, can be found in the literature [33, 34]. Available data on the chemical diffusion coefficient in a number of lithium alloys are included in Table 3. 

Table 3. Data on chemical diffusion in lithium alloy phases.

Figure 3. Variation of the chemical diffusion coefficient with composition in the "LiAl" phase at different temperatures [35].

Measurements were also made of the potential-composition behavior, as well as the chemical diffusion coefficient, and its composition dependence, in each of the intermediate phases in the Li-Sn system at 415 °C [39]. 

The chemical diffusion coefficient in that phase was also evaluated and found to... 

Figure 10. Composition dependence of the chemical diffusion coefficient in the Li44Sn phase at ambient temperature [43).

Table 5. Chemical diffusion data for lithium-tin phases at 25 °C.

The kinetic requirements for a successful application of this concept are readily understandable. The primary issue is the rate at which the electroactive species can reach the matrix/reactant interfaces. The critical parameter is the chemical diffusion coefficient of the electroactive species in the matrix phase. This can be determined by various techniques, as discussed above. 

This concept has also been demonstrated at ambient temperature in the case of the Li-Sn-Cd system [47,48]. The composition-de-pendences of the potentials in the two binary systems at ambient temperatures are shown in Fig. 15, and the calculated phase stability diagram for this ternary system is shown in Fig. 16. It was shown that the phase Li4 4Sn, which has fast chemical diffusion for lithium, is stable at the potentials of two of the Li-Cd reconstitution reaction plateaus, and therefore can be used as a matrix phase. 

Lithiated carbons are mostly multiphase systems. Hence, the determination of chemical diffusion coefficients for Li1 causes experimental problems because the propagation of a reaction front has to be considered. 

In the predominantly electronically conducting electrodes it is the chemical diffusion of the ions which controls the electrical current of the galvanic cell. This includes the internal electric field which is built up by the simultaneous motion of ions and electrons to establish charge neutrality [14] ... 

It should be kept in mind that all transport processes in electrolytes and electrodes have to be described in general by irreversible thermodynamics. The equations given above hold only in the case that asymmetric Onsager coefficients are negligible and the fluxes of different species are independent of each other. This should not be confused with chemical diffusion processes in which the interaction is caused by the formation of internal electric fields. Enhancements of the diffusion of ions in electrode materials by a factor of up to 70000 were observed in the case of LiiSb [15]. 

The experimental value for Agl is 1.97 FT cirT1 [16, 3], which indicates that the silver ions in Agl are mobile with nearly a thermal velocity. Considerably higher ionic transport rates are even possible in electrodes, by chemical diffusion under the influence of internal electric fields. For Ag2S at 200 °C, a chemical diffusion coefficient of 0.4cm2s, which is as high as in gases, has been measured... 

Improvement of the ionic current by fast transport in the electrodes. High electronic mobility and low electronic concentration favor fast chemical diffusion in electrodes by building up high internal electric fields [14]. This effect enhances the diffusion of ions toward and away from the solid electrolyte and allows the establishment of high current densities for the battery. 

Physically, all these prokaryotes are small, diameter about 1.0 pm and are of rigid, simple shape. They usually have little or no internal structure so that chemical diffusion is relatively rapid. Secondary compartments are rare but vesicles and vacuoles (even nuclei) are found in a very few large bacteria. We shall see that all the prokaryote cells have controlled, autocatalytic, internal metabolism, but are relatively little affected by external circumstances, except by shortage of nutrients. 

Kleinfeld, M. Wiemhofer, H.-D. 1988. Chemical diffusion coefficients and stabihty of CuInS2 and CuInSe2 from polarization measurements with point electrodes. Solid State Ionics. 28-30 1111-1115. 

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