Diffusion interdiffusion coefficient

Diffusion coefficients are important for mass-transfer operations (see Chapter 8, Mass Transfer ). There are several differently defined diffusion coefficients (self-diffusion coefficient, interdiffusion [or mutual diffusion] coefficient, intradiffusion [or tracer diffusion] coefficient) this can be a source of confusion. These are delineated in standard references [15, 68, 69]. 

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

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) ... 

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). 

In a basalt-rhyolite interdiffusion experiment (Alibert and Carron, 1980), potassium concentrations CK were measured in a basalt at a given arbitrary distance y in pm between rhyolitic and basaltic liquids experimentally heated for 5000 seconds (Table 5.5 and Figure 5.4). In order to determine the diffusion coefficients, a fit of the experimental points with a polynomial is requested. Use the reduced concentration u, (the fractional deviation of the concentration at a, from the concentrations in the original liquids) given by... 

The diffusion theory states that interpenetration and entanglement of polymer chains are additionally responsible for bioadhesion. The intimate contact of the two substrates is essential for diffusion to occur, that is, the driving force for the interdiffusion is the concentration gradient across the interface. The penetration of polymer chains into the mucus network, and vice versa, is dependent on concentration gradients and diffusion coefficients. It is believed that for an effective adhesion bond the interpenetration of the polymer chain should be in the range of 0.2-0.5 pm. It is possible to estimate the penetration depth (/) by Eq. (5),... 

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,... 

Diffusion coefficient (often interdiffusivity or chemical diffusivity) daughter nuclide... 

Chemical diffusion has been treated phenomenologically in this section. Later, we shall discuss how chemical diffusion coefficients are related to the atomic mobilities of crystal components. However, by introducing the crystal lattice, we already abandon the strict thermodynamic basis of a formal treatment. This can be seen as follows. In the interdiffusion zone of a binary (A, B) crystal having a single sublattice, chemical diffusion proceeds via vacancies, V. The local site conservation condition requires that /a+/b+7v = 0- From the definition of the fluxes in the lattice (L), we have... 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

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... 

In discussing AO-BO interdiffusion, we saw that the two independent fluxes of this ternary system can lead to different chemical diffusion coefficients D. They depend upon the constraints which define the physical situation (e.g., VjuG = 0 or Vy/v = 0). The analysis of this relatively simple and fundamental situation is already rather complex. The complexity increases further if diffusion occurs between heterovalent components of compound crystals. This diffusion process is important in practice (e.g., heterovalent doping) and its treatment in the literature is not always adequate. We therefore add a brief outline of the relevant ideas for a proper evaluation of D. 

In practice, it is often feasible to reduce the multicomponent crystal in respect of its transport behavior to a quasi-binary system. Let us assume that the diffusion coefficients are DA>DB>DC, Dd, etc. The quasi-binary approach considers C, D, etc. as practically immobile, which means that A and B are interdiffusing in the im-... 

The most commonly defined diffusion coefficients are the chemical or interdiffusion D, intrinsic diffusion coefficient (D1), and tracer or selfdiffusion coefficient (D ). The relevant coefficient depends on the driving... 

There are different criterion of how to classify solid-solid interfaces. One is the sharpness of the boundary. It could be abrupt on an atomic scale as, for example, in III-IV semiconductor heterostructures prepared by molecular beam epitaxy. In contrast, interdiffusion can create broad transitions. Surface reactions can lead to the formation of a thin layer of a new compound. The interfacial structure and composition will therefore depend on temperature, diffusion coefficient, miscibility, and reactivity of the components. Another criterion is the crystallinity of the interface. The interface may be crystalline-crystalline, crystalline-amorphous, or completely amorphous. Even when both solids are crystalline, the interface may be disturbed and exhibit a high density of defects. 

Barrier metals, as opposed to alloys like AuGeNi, are employed in many thin film metallization systems to promote adhesion and prevent interdiffusion. Gold is an excellent conductor, however, it has very poor adhesion to both Si and GaAs. Gold also shortens the device lifetime when it diffuses into an active region of the device. For this reason it is used in multilayered structures such as Ta/Pt/Ta/Au (50), W/Au (50) and Cr/Au (51). SIMS, AES and RBS have all been used effectively in studying metal-metal interdiffusion, to extract diffusion coefficients, and to estimate device lifetimes. 

In view of the lack of interdiffusion in the course of multiple-layer formation at the A-B interface and because of complicated mechanism of this process, calculation of integrated diffusion coefficients seems in most cases meaningless. 

The intrinsic diffusion coefficients, Dk and DB, of a binary alloy A-B express the diffusion of the components A and B relative to the lattice planes [7], Therefore, during interdiffusion, a net flux of atoms across any lattice plane is present, where, normally, the diffusion rates of the diffusing particles A and B are different. Subsequently, this interdiffusion process provokes the shift of lattice planes with respect to a fixed axis of the sample, result which is named the Kirkendall effect [9],... 

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... 

As we have commented previously for metals, the diffusion in concentration gradients is described with the chemical diffusion coefficient, or interdiffusion coefficient. In this case, it is possible to consider that the interdiffusion and the intrinsic diffusion coefficients are equivalent, since we have only the movement of one species, that is, oxygen, by the vacancy mechanism. Subsequently, applying the Einstein relation... 

---