Diffusion and interdiffusion

The diffusion coefficient for a given ion in a crystal is determined, as we have seen, by the atomistic properties of the ion in the structural sites where the vacancy (or interstitial) participating in the migration process is created (see eq. 4.71). The units of diffusion (and/or self-diffusion ) are usually cm sec . Pick s first law relates the diffusion of a given ion A (Jf) to the concentration gradient along a given direction X  

If there is simultaneous diffusion of more than one component in the crystal, the flux of A in direction X depends on the individual diffusivities of all diffusing components (Darken, 1948), and the individual diffusivity coefficient in equations 4.87 and 4.88 is replaced by interdiffusion coefficient D i.e., for the simultaneous diffusion of two ions A and B, 

however, the diffusing species are electrically charged, the net flux at the interface is effectively zero, to maintain the charge neutrality. Particularly, if A and B are of the same charge, the flux of A will equal that of B  

Equation 4.90 is of particular interest when treating diffusion in nonmetallic solids, where diffusion is primarily limited to ionic species. Note that, based on equations 4.91 and 4.93, D when Xg 0. Thus, in a diluted binary mixture. 

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Studies were made of self-diffusion and interdiffusion in buried Al lOaAs/ Al GaAs/2lGaAs and AlAs/ lGaAs isotope heterostructures at 800 to 1160C. The... 

Chebotin s scientific interests were characterized by a variety of topics and covered nearly all aspects of solid electrolytes electrochemistry. He made a significant contribution to the theory of electron conductivity of ionic crystals in equilibrium with a gas phase and solved a number of important problems related to the statistical-thermodynamic description of defect formation in solid electrolytes and mixed ionic-electronic conductors. Vital results were obtained in the theory of ion transport in solid electrolytes (chemical diffusion and interdiffusion, correlation effects, thermo-EMF of ionic crystals, and others). Chebotin paid great attention to the solution of actual electrochemical problem—first of all to the theory of the double layer and issues related to the nature of the polarization at the interface of the solid electrol34e and gas electrode. 

One mechanism involves the interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancy, as represented schematically in Figure 5.3a. This mechanism is aptly termed vacancy diffusion. Of course, this process necessitates the presence of vacancies, and the extent to which vacancy diffusion can occur is a function of the number of these defects that are present significant concentrations of vacancies may exist in metals at elevated temperatures (Section 4.2). Because diffusing atoms and vacancies exchange positions, the diffusion of atoms in one direction corresponds to the motion of vacancies in the opposite direction. Both self-diffusion and interdiffusion occur by this mechanism for the latter, the impurity atoms must substitute for host atoms. 

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. 

By selecting the reference properly, the diffusion coefficients DA and DB can be made equal to each other. This value is termed the mutual diffusion (or interdiffusion) coefficient Dab- The reference frame is one across which no change in volume occurs (fixed volume) ... 

Interdiffusion, effective binary diffusion, and multicomponent diffusion may be referred to as chemical diffusion, meaning there are major chemical concentration gradients. Chemical diffusion is defined relative to self diffusion and tracer diffusion, for which there are no major chemical concentration gradients. 

Self-diffusion and tracer diffusion are described by Equation 3-10 in one dimension, and Equation 3-8 in three dimensions. For interdiffusion, because D may vary along a diffusion profile, the applicable diffusion equation is Equation 3-9 in one dimension, or Equation 3-7 in three dimensions. The descriptions of multispecies diffusion, multicomponent diffusion, and diffusion in anisotropic systems are briefly outlined below and are discussed in more detail later. 

To describe the process of the formation of such structures, it is necessary to write down the equations for a component that may be composed of several species and consider reactions among the species (Fisher and Lasaga, 1981). For example, for the case of diffusion of silver ions into a gel containing chromate ions, there are two species of Ag one is Ag , which diffuses by interdiffusion with Na, and the other is Ag2Cr04 precipitation. The diffusivity of precipitated Ag2Cr04 is negligible. Therefore,... 

Tracer diffusivities are often determined using the thin-source method. Self-diffusivities are often obtained from the diffusion couple and the sorption methods. Chemical diffusivities (including interdiffusivity, effective binary diffusivity, and multicomponent diffusivity matrix) may be obtained from the diffusion-couple, sorption, desorption, or crystal dissolution method. 

Elphick S.C., Ganguly J., and Loomis T.P. (1985) Experimental determination of cation diffusivities in aluminosilicate garnets, I experimental methods and interdiffusion data. Contrib. Mineral. Petrol. 90, 36-44. 

Methods to solve the diffusion equation for specific boundary and initial conditions are presented in Chapter 5. Many analytic solutions exist for the special case that D is uniform. This is generally not the case for interdiffusivity D (Eq. 3.25). If D does not vary rapidly with composition, it can be replaced by successive approximations of a uniform diffusivity and results in a linearization of the diffusion equation. The... 

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. 

The process is also known as chemical diffusion, interdiffusion, transport diffusion, and in the case of systems with two components, binary diffusion. 

The theoretically based interpretation of the kinetics of ion exchange that has been developed has been confirmed by experiments that have shown the interdiffusion rates in the exchanger phase to depend, as predicted, on diffusivity and selectivity factors as well as on bulk solution concentrations. For unfavorable ion-exchange isotherms these dependencies are formally similar to those observed in film diffusion [49]. 

Diffusion of Water in poly[PFSA] Membranes To describe diffusion of water through the membrane in the presence of a water activity gradient, an appropriate interdiffusion coefficient must be determined. Experimental methods used to study diffusion of water in these polymers, such as radiotracer and pulsed gradient spin-echo NMR techniques, probe intrad-iffusion coefficients, often referred to as tracer or self-diffusion coefficients, determined in the absence of a chemical potential gradient. Intra- and interdiffusion coefficients are related for the case of diffusion of a small molecule in a polymeric matrix as follows [28] ... 

Another device used to study diffusion and to measure diffusion coefficients is the Loschmidt tube illustrated in Figure 5.6. Two tubes containing fluids with different concentrations are brought together at time t = 0 and the fluids allowed to interdiffuse. After some time the tubes are separated and the compositions measured. 

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