Solid chemical diffusion coefficient

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. 

The above sections have focused upon homogeneous systems with a constant composition in which tracer diffusion coefficients give a close approximation to selfdiffusion coefficients. However, a diffusion coefficient can be defined for any transport of material across a solid, whether or not such limitations hold. For example, the diffusion processes taking place when a metal A is in contact with a metal B is usually characterized by the interdiffusion coefficient, which provides a measure of the total mixing that has taken place. The mixing that occurs when two chemical compounds, say oxide AO is in contact with oxide BO, is characterized by the chemical diffusion coefficient (see the Further Reading section for more information). 

Yasuda I and Hishinuma M. Electrical conductivity and chemical diffusion coefficient of Sr-doped lanthanum chromites. Solid State Ionics 1995 80 141-150. 

An example of a material (Li3Sb) with a very large Wagner factor is shown in Fig. 8.3. The effective chemical diffusion coefficient is compared with the diffusivity as a function of non-stoichiometry. These data were determined by electrochemical techniques (see Section 8.5). An increase of the diffusion coefficient is observed at about the ideal stoichiometry which corresponds to a change in the mechanism from a predominantly vacancy to interstitial mechanism. The Wagner factor W is as large as 70 000 at the ideal stoichiometry. This gives an effective diffusion coefficient which is more typical of liquids than solids. It is a common... 

Figure 5-11 illustrates the results of an oxide interdiffusion experiment. Clearly, the transport coefficients are not single valued functions of composition. From the data, one concludes that for a given composition, the chemical diffusion coefficients depend both on time and location in the sample [G. Kutsche, H. Schmalzried (1990)]. Let us analyze this interdiffusion process in the ternary solid solution Co. O-Nq. O, which contains all the elements necessary for a phenomenological treatment of chemical transport in crystals. The large oxygen ions are almost immobile and so interdiffusion occurs only in the cation sublattice of the fee crystal. When we consider the following set ( ) of structure elements... 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

Figure 5. Water uptake of Gd-doped BaCe03. The time evolution is characterized by the chemical diffusion coefficient of H20,36 Reprinted from K.-D. Kreuer, E. Schonherr, and J. Maier, Solid State Ionics 70 71 (1994) 278-284. Copyright 1994 with permission from Elsevier.

We will see that in the steady state of the blocking cells, we can extract partial conductivities, and from the transients chemical diffusion coefficients (and/or interfacial rate constants). Cell 7 combines electronic with ionic electrodes here a steady state does not occur but the cell can be used to titrate the sample, i.e., to precisely tune stoichiometry. Cell 1 is an equilibrium cell which allows the determination of total conductivity, dielectric constant or boundary parameters as a function of state parameters. In contrast to cell 1, cell 2 exhibits a chemical gradient, and can be used to e.g., derive partial conductivities. If these oxygen potentials are made of phase mixtures212 (e.g., AO, A or AB03, B203, A) and if MO is a solid electrolyte, thermodynamic formation data can be extracted for the electrode phases. 

Figure 47. The time-dependence of both the polarization and the depolarization voltage (galvanostatic mode) of a symmetrical blocking cell applied to PbO (orh.) allows the determination of the chemical diffusion coefficient (here denoted by IT).15,217 Reprinted from J. Maier, Solid State Phenom., 39/40 (1994), 35-60. Copyright 1994 with permission from Trans Tech Publications Ltd.

In this case, it is well known that the process occurs in steady state. To understand this process, one must consider it as a special case of binary diffusion, where the diffusivity of the Pd atoms is zero. Consequently, the frame of reference is the fixed coordinates of the solid Pd thin film. The interdiffusion or chemical diffusion coefficient is the diffusivity of the mobile species [20], that is, hydrogen. Then, the hydrogen flux in the Pd thin film is given by... 

With the help of Equation 5.107, as was previously done with Equation 5.86, we obtain a transport or chemical diffusion coefficient that is a result of Fick s laws. We now interpret the meaning of this coefficient if we consider diffusion in a microporous solid, as a special case of binary diffusion, where A is the mobile species and the diffusivity of the microporous framework atoms is zero, then, the frame of reference are the fixed coordinates of the porous solid consequently, we have a particular case of interdiffusion where the diffusion coefficient is simply the diffusivity of the mobile species [12,20],... 

Lawrence Stamper Darken (1909-1978) subsequently showed (Darken, 1948) how, in such a marker experiment, values for the intrinsic diffusion coefficients (e.g., Dqu and >zn) could be obtained from a measurement of the marker velocity and a single diffusion coefficient, called the interdiffusion coefficient (e.g., D = A ciiD/n + NznDca, where N are the molar fractions of species z), representative of the interdiffusion of the two species into one another. This quantity, sometimes called the mutual or chemical diffusion coefficient, is a more useful quantity than the more fundamental intrinsic diffusion coefficients from the standpoint of obtaining analytical solutions to real engineering diffusion problems. Interdiffusion, for example, is of obvious importance to the study of the chemical reaction kinetics. Indeed, studies have shown that interdiffusion is the rate-controlling step in the reaction between two solids. 

According to the analysis in the previous sections, the primary particle size in flame reactors is determined by the relative rates of particle collision and coalescence. For highly refractory materials, the characterislic coalescence time (12.6) depends on the solid-state diffusion coefficient, which is a very sensitive function of the temperature. The mechanisms of solid-.staie diffusion depend in a complex way on the structure of the solid. For example, a perfect cubic crystal of the substance AB consists of alternating ions A and B. Normally there are many defects in the lattice structure even in a chemically pure single crystal defect types are shown schematically in Fig. 12.8. The mechanism of diffusion in cry.stalline solids depends on the nature of the lattice defects. Three mechanisms predominate in ionic... 

The other major category of D values are termed interdiffusion or chemical diffusion coefficients. Experimentally, Ci does change significantly and D is a function of Ci. Diffusion of one or more chemical species is dependent on the opposing diffusion of another species in order to maintain a constant matrix volume and/or electrical neutrality. The diffusion in olivine of Mg in one direction and the complementary diffusion of Fe in the opposite direction represent one example. Rarely does this type of experiment employ the use of isotopically labeled species. However, in some cases isotopically-enriched H2O (T, D and/or O) has been used where the composition of the solid (melt) became significantly modified by incorporation of water into the structure. 

Since the slow solid-state diffusion of Lb in the bulk of carbon may control the rate-determining step of the intercalation process and consequently affect the power density of Li -ion batteries, the chemical diffusion coefficient of Lb (Dy ) becomes a very key kinetic parameter. Several electrochemical relaxation techniques such as... 

The chemical diffusion coefficient In solid state reactions several atoms or ions may diffuse in various directions, each with a different self-diffusion coefficient. This can develop electrochemical potentials that affect the overall diffusion coefficients. The chemical diffusion coefficient D of the atoms A and B in a compound Aj By is a weighted mean of the self-diffusion coefficients and of the diffusing atoms or ions A and B. D becomes... 

R P. Prosini, M. Lisi, D. Zanec, and M. PasqnaU [2002] Determination of the Chemical Diffusion Coefficient of Lithium in LiFeP04, Solid State Ionics 148, 45-51. 

In section 5.5.4, Pick s laws were solved for binary systems under the assumption that the chemical diffusion coefficient D is constant. This assumption is never strictly true, and is only approximately true in limiting cases. Thus, we must seek solutions for other cases as well. If we know the concentration dependence of the diffusion coefficient as well as the thermodynamics of the system, we can often reach conclusions regarding the type of disorder and the diffusion mechanism. Therefore, it is especially important in the study of point defect disorder and in the study of the elementary processes of solid state reactions that we know the concentration dependence of the diffusion coefficient. 

If we were to permit a KCl crystal to react with a RbCl crystal, or a NiO crystal with a MgO or CoO crystal, or a C0AI2O4 crystal with a MgAl204 crystal, then different situations would be observed in each case, depending upon the relative mobilities of the ionic and electronic defects. First of all, since these systems form nearly ideal solid solutions [34], the thermodynamic factor (1 H- d In y /d In in eq. (5-28) can be set equal to unity, and Fick s first law applies in its simplest form (see eq. (5-29)). Essentially, the problem in this class of reactions is to determine the chemical diffusion coefficient as a function of composition. 

If the ionic mobilities (or the component diffusion coefficients) are known, then it is possible to calculate the diffusion profile, as well as the displacement of inert markers, if we assume that local thermodynamic equilibrium is maintained and that the markers are firmly bound to the anion lattice. In order to calculate the chemical diffusion coefficient, we start once again with the flux equations of section 5.2 and eliminate the diffusion potential using the condition of electroneutrality. For an ideal solid solution we obtain the equation [13] ... 

A and B should form a complete range of solid solutions. This means that they should have the same crystal structure as well as similar molar volumes. The phenomenological transport problem here is concerned with the solution of Fick s laws for the given experimental conditions in this inhomogeneous system. The atomistic problem is concerned with the interpretation of the chemical diffusion coefficient which, for example, might have been determined by a Boltzmann-Matano analysis. It was shown in section 5.5.3 that, for the case of binary diffusion via vacancies, the chemical diffusion coefficient may be written as ... 

Mizusaki J, Fueki K (1982) Electrochemical Determinations of the Chemical Diffusion Coefficient and Nonstoichiometry in AgCl, Solid State Ionics 6 85-91... 

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